Disregard

To the tuning list,

Please disregard my preceding message. It would be replaced by another information.

Thanks

Mario Pizarro
piagui@...
Lima, August 07

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Here's a scale I made based on merging 12TET diatonic and my "infinity" scale
system:

nearest JI
1
*1.12* 9/8
1.25 5/4
*1.365* 15/11 or 11/8
1.495 3/2
1.674 5/3
1.88 15/8
2/1
......note *1.365* and *1.12* are just about the only note really different from
12TET diatonic and there is one "half step" instead of two (2/1 and 1.888 being
the half step).......

There is something odd...when actually using this scale 5/3 feel unbearably
FLAT as does 11/6...even in dyads like 1/1 and 15/8 OR 1/1 and 1.674/"1.68".

Call it simply cultural training...but I have a hunch something not explained
at all in JI is going on in 12TET that makes those weird ratios of 1.674 and
1.88 work much better than expected. Maybe even something to do with equal
beating, a side effect of maintaining a near circle of fifths for many tones, or
another phenomena I don't know overly much about. Any ideas as to why this
happens?

if you replace the 5/3 with 27/16 you get a higher interval and one where all of it is in a harmonic series. 45/32 iwould work for the 11 if you hear that too flat too.
but all this puts you into 5 limit.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Here's a scale I made based on merging 12TET diatonic and my "infinity" scale
> system:
>
> nearest JI
> 1
> *1.12* 9/8
> 1.25 5/4
> *1.365* 15/11 or 11/8
> 1.495 3/2
> 1.674 5/3
> 1.88 15/8
> 2/1
> ......note *1.365* and *1.12* are just about the only note really different from
> 12TET diatonic and there is one "half step" instead of two (2/1 and 1.888 being
> the half step).......
>
>
> There is something odd...when actually using this scale 5/3 feel unbearably
> FLAT as does 11/6...even in dyads like 1/1 and 15/8 OR 1/1 and 1.674/"1.68".
>
> Call it simply cultural training...but I have a hunch something not explained
> at all in JI is going on in 12TET that makes those weird ratios of 1.674 and
> 1.88 work much better than expected. Maybe even something to do with equal
> beating, a side effect of maintaining a near circle of fifths for many tones, or
> another phenomena I don't know overly much about. Any ideas as to why this
> happens?
>

Kraig>"if you replace the 5/3 with 27/16 you get a higher interval and one where
all of it is in a harmonic series. 45/32 iwould work for the 11 if you hear that
too flat too."

Ah ok...so it enables everything to "slip" into x/2^x (IE
x/2,x/4,x/8,x/16,x/32...) format fractions, thus making a straight harmonic
series?
If so...makes perfect sense...and all the brain is doing in interpreting the
7-tone scale is forming a line from the root. Come to think of it, the 1.365's
being near 11/8 begins to make sense in this light as well.

General question: do dyads high up in the harmonic series, like 27/16 or 37/32, maintain any sort of root-reinforcing property? Or does their high complexity (i.e. "lack of tunability-by-ear") preclude this? Or do they reinforce the root, but only in larger harmonic structures than dyads? If the latter, would a 32:37:48 chord have rootedness as strong as 32:40:47? Or, say, 16:20:25 vs. 16:19:24?

-Igs

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Kraig>"if you replace the 5/3 with 27/16 you get a higher interval and one where
> all of it is in a harmonic series. 45/32 iwould work for the 11 if you hear that
> too flat too."
>
> Ah ok...so it enables everything to "slip" into x/2^x (IE
> x/2,x/4,x/8,x/16,x/32...) format fractions, thus making a straight harmonic
> series?
> If so...makes perfect sense...and all the brain is doing in interpreting the
> 7-tone scale is forming a line from the root. Come to think of it, the 1.365's
> being near 11/8 begins to make sense in this light as well.
>

Dyads are hard to deal with due to the limited orientation.

I find that high numbers in equal beating chords can sound just as good and as rooted as beatless ones. It is a family though that is too big and it varies greatly.
It depends on the beat rate and the range.

I don't use any method of quantifying Consonance or Dissonance except for my ear. It seems to be the most expedient and usually assumptions are too often ripe for rude surprises.
Music has way too many variables.
While certain intervals are clear due to their acoustical properties like many lower limit context, the emotional flavor of something too can be just as clear as we see in all types of tunings used around the world.

A simple palette used to construct a scale though gives us an ease of organization and a way to allow us to stay oriented to where we are.
This is as rooted as anything i think.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> General question: do dyads high up in the harmonic series, like 27/16 or 37/32, maintain any sort of root-reinforcing property? Or does their high complexity (i.e. "lack of tunability-by-ear") preclude this? Or do they reinforce the root, but only in larger harmonic structures than dyads? If the latter, would a 32:37:48 chord have rootedness as strong as 32:40:47? Or, say, 16:20:25 vs. 16:19:24?
>
> -Igs
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> >
> > Kraig>"if you replace the 5/3 with 27/16 you get a higher interval and one where
> > all of it is in a harmonic series. 45/32 iwould work for the 11 if you hear that
> > too flat too."
> >
> > Ah ok...so it enables everything to "slip" into x/2^x (IE
> > x/2,x/4,x/8,x/16,x/32...) format fractions, thus making a straight harmonic
> > series?
> > If so...makes perfect sense...and all the brain is doing in interpreting the
> > 7-tone scale is forming a line from the root. Come to think of it, the 1.365's
> > being near 11/8 begins to make sense in this light as well.
> >
>

Kraig>"I don't use any method of quantifying Consonance or Dissonance except for
my ear. It seems to be the most expedient and usually assumptions are too often
ripe for rude surprises."

Agreed: I am starting to use that more and more as a necessary final test
because the more I play with periodicity (much as I respect its laws), the more
I realize how many exceptions there are (same issue as you have, it seems).
That...and not just testing dyads but also many possible triads and beyond.
...Plus playing the entire scale at once over 2 octaves and taking single notes
out of the mix one-by-one to see which are the "odd ones out" that don't fit the
mold so well as the others (and work specifically in tweaking those).

>"A simple palette used to construct a scale though gives us an ease of
>organization and a way to allow us to stay oriented to where we are.
This is as rooted as anything i think. "

Far as "simple pallette"...now I forget the formal name for this theory but
would appreciate it if one of you could name it for me:

It seems clear to me after lots of by-ear testing that scales with LESS highly
occurrent total interval sizes far as dyads (within about 8 cents) often sound
more level emotionally. My latest scale involves merging 12TET and the Infinity
series and

A) Hits only 5 11-limit dyads across two octaves within about 7.5 cents or less
B) Hits more like 44 dyads straight from the two versions of JI diatonic
mentioned on http://en.wikipedia.org/wiki/Five-limit_tuning (IE about 90% of the
dyads are very periodic) within about 7.5 cents or less). Those dyads include
6/5,5/4,3/2,5/3,15/8,9/8,8/5, 9/5....all our old favorites.

I am starting to think this is a key advantage of 12TET over JI diatonic and
that it explains why some people actually prefer 12TET is its 12 consistent dyad
sizes within an octave (actually this applies to any TET, though 12TET has an
advantage over many due to low number of possible dyads).

In 12TET you can say "a 6th is always 900 cents, a 7th is always 1100
cents..." with perfect clarity in 12TET and near-perfect clarity (IE within
about 7.5 cents for about 90% of dyads) in my latest scale. And (in 12TET) that
seems to really make things work EVEN though things like the major 3rd, major
6th, tritone (1.414 instead of 10/7 or 7/5) , and 7th (1.8875 instead of 15/8)
are significantly off. In other words, it seems the balance of interval sizes
almost outweighs the sourness of those intervals.

Intervalic consistency seems to matter a good bit...along with periodicity,
critical band, and other theories. Agreed?

Hi Kraig,

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:
> I don't use any method of quantifying Consonance or Dissonance except for my ear. It
> seems to be the most expedient and usually assumptions are too often ripe for rude
> surprises.
> Music has way too many variables.

I could not agree more, and in fact I'm not even remotely interested in quantifying consonance and dissonance (per se). What I AM interested in is teaching microtonality to neophytes in the most concise and useful way. I'm writing (or at least attempting) a primer on EDOs (other than 12), and I'm trying to find the simplest and most useful way to portray their harmonic properties on paper.

I am strongly of the belief that consonance and dissonance depend entirely on musical context, and thus cannot be reduced to a set of dyad-related quantities, or even a set of formulas. OTOH, things like "harmonic stability" (which is probably nothing more than "beat frequency"), "intervallic identity", and "rootedness" seem to be both quantifiable and musically useful, so I'm trying to get a better understanding of them.

Basically, I don't see much point in relating the tempered intervals in EDOs to ratios unless said ratios are actually musically-coherent. So far, I've interpreted the theory of harmonic entropy as suggesting that ratios only matter when they are simple and distant from nearby ratios of not-much-greater complexity. IOW, if an EDO has an interval close to 6/5, that's significant, but close to a 13/11 not so much, because 13/11 is VERY close to 19/16, 20/17, 25/21, etc. But I've begun to wonder if this is a whole-enough picture, since it's based almost entirely on interval complexity as opposed to otonality, which I've read is key to "rootedness". So I'm trying to figure out how important otonality actually is in determining intervallic identity and harmonic stability.

So, I need a straight answer: do the odd harmonics in the 16-64 series provide rootedness and stability, despite their high complexity? Or does rootedness drop off drastically above the 16th harmonic?

-Igs

Igs>"So, I need a straight answer: do the odd harmonics in the 16-64 series
provide rootedness and stability, despite their high complexity? Or does
rootedness drop off drastically above the 16th harmonic?"

I'll leave it up to Kraig to give a more authoritative answer, but I'll say
this much:
Igs, what you said before about a triad or 4-note chord's having several dyads
that are low-limit and only a couple that are high limit seems to make sense.

That is: the lower limit dyads often click with the brain and seem to make the
high limit dyads automatically click into place! Hence that idea seems to
explain mysteries I've seen before, like how a C E F A chord in 12TET sounds so
clear despite the nasty E-F dyad: the C-E, C-F, E-A, F-A dyads seem to "knock
it into place". IE a complex interval can act "as if it's not there" or even
"as a less complex interval" in a sea of less-complex intervals, to an extent.

Same seems to go with all those triads on your "Map of an Internal Landscape"
album with triads using the "terrible" 16/11 fifth that, oddly enough, sounded
quite resolved. The actual major and minor thirds making up that fifth though,
as I recall, were fairly low-limit...enough to apparently trick the brain into
thinking of the 16/11 as a 3/2.

>"I am strongly of the belief that consonance and dissonance depend entirely on
>musical context, and thus cannot be reduced to a set of dyad-related quantities,
>or even a set of formulas."
I don't think anything can be completely reduced to formulas...but I do think
formulas can be used to explain things a high % of the time and that musical
context can only violate them a limited (though still definitely large enough to
be significant) amount.
I'd guesstimate that if about 70% or more of the triads in a note point toward
a certain root, the chord will sound fairly resolved. And that includes if the
chord is something like C E F A IE 12:15:16:20 (which at first glance, seems to
scream "15 odd-limit, high-dissonance"), but doesn't sound high-limit at all in
real life.

i do not think there has been enough prolonged exploration of these harmonics.
I imagine it would be less but if this is a result of hearing them used in our culture or not.
we already know something about lower harmonics just from 12 et.
albeit a bit warped
Especially divorced from their lower harmonics.

it would be worth while for someone to explore and might accomplish more than speculation
say a diamond or eikosany using 6 primes from the upper range. or use viggo bruns algorithm to make a scale out of a smaller set.

While we can say certain higher ratios appoximate lower ones, if we replace the lower ones in the same context the result is going to be different.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi Kraig,
>

>
> So, I need a straight answer: do the odd harmonics in the 16-64 series provide rootedness and stability, despite their high complexity? Or does rootedness drop off drastically above the 16th harmonic?
>
> -Igs
>

I can say i am no authority on an area or range i have only touched upon. If Igs is interested in EDO specifically, i am out of my ball park completely. Possibly Gene has looked at them more than anyone

It is interesting that it takes little , or simple ratio or two , to give us our bearings. By themselves they can be put together however one wishes without a completely closed system.

This structural mutability is one of my interest in JI. One can have a scale of any number of notes that one wishes with a large enough set to choose from as a 13 limit set will do for instance .

EDOs exploit the many cyclical patterns contain within the numerical qualities of its scale and can take advantage of dual or multiple meanings.

One can do the same thing in JI also really.
at this point it becomes a choice of personal preference and the way one also chooses to think about the material as well if one wants slight fluctuations in one scale or not.
One can be as disorienting as the other depending on how one hears it and processes it.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>

>
> I'll leave it up to Kraig to give a more authoritative answer, but I'll say
> this much:
> Igs, what you said before about a triad or 4-note chord's having several dyads
> that are low-limit and only a couple that are high limit seems to make sense.
>
>

Well, after some hands-on exploration of harmonics 16-32 and odd harmonics 32-64, I can say they certainly are not beatless. I was surprised to discover how many are within a few cents of familiar tempered intervals, and how at least a few were near peaks of harmonic entropy. I think it's safe to say that being part of a harmonic series does not confer any special status in and of itself. Though I'll admit, the 19th and 27th harmonics sounded pretty good to me, whereas the 23rd and 25th did not. Evidence that my ear is thoroughly conditioned to like 12-tET? Perhaps.

-Igs

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:
>
> i do not think there has been enough prolonged exploration of these harmonics.
> I imagine it would be less but if this is a result of hearing them used in our culture or not.
> we already know something about lower harmonics just from 12 et.
> albeit a bit warped
> Especially divorced from their lower harmonics.
>
> it would be worth while for someone to explore and might accomplish more than speculation
> say a diamond or eikosany using 6 primes from the upper range. or use viggo bruns algorithm to make a scale out of a smaller set.
>
> While we can say certain higher ratios appoximate lower ones, if we replace the lower ones in the same context the result is going to be different.
>
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > Hi Kraig,
> >
>
> >
> > So, I need a straight answer: do the odd harmonics in the 16-64 series provide rootedness and stability, despite their high complexity? Or does rootedness drop off drastically above the 16th harmonic?
> >
> > -Igs
> >
>

Here is one context for the 23rd.
Erv showed me once a way to use the it in a cadence.
The idea is that you use the harmonics of F for an incomplete G7.
something along the lines of a 9-23-32 which you will notice the upper tones will give 9 as the root via difference tone 9 being G if we are looking at the harmonics of F.
It seems to reinforce the rootedness more than a simpler ratio might.

The three or four tunings i have used and learned extremely well i can say i can hear as well as 12. But i have no doubt that underneath i probably hear this matrix in relation to 12 also.
It is like a different language maybe. We learn another but are aware or how it relates to the one we were brought up with, when there are connections

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Well, after some hands-on exploration of harmonics 16-32 and odd harmonics 32-64, I can say they certainly are not beatless. I was surprised to discover how many are within a few cents of familiar tempered intervals, and how at least a few were near peaks of harmonic entropy. I think it's safe to say that being part of a harmonic series does not confer any special status in and of itself. Though I'll admit, the 19th and 27th harmonics sounded pretty good to me, whereas the 23rd and 25th did not. Evidence that my ear is thoroughly conditioned to like 12-tET? Perhaps.
>
> -Igs
>
> --- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@> wrote:
> >
> > i do not think there has been enough prolonged exploration of these harmonics.
> > I imagine it would be less but if this is a result of hearing them used in our culture or not.
> > we already know something about lower harmonics just from 12 et.
> > albeit a bit warped
> > Especially divorced from their lower harmonics.
> >
> > it would be worth while for someone to explore and might accomplish more than speculation
> > say a diamond or eikosany using 6 primes from the upper range. or use viggo bruns algorithm to make a scale out of a smaller set.
> >
> > While we can say certain higher ratios appoximate lower ones, if we replace the lower ones in the same context the result is going to be different.
> >
> >
> > --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> > >
> > > Hi Kraig,
> > >
> >
> > >
> > > So, I need a straight answer: do the odd harmonics in the 16-64 series provide rootedness and stability, despite their high complexity? Or does rootedness drop off drastically above the 16th harmonic?
> > >
> > > -Igs
> > >
> >
>

By 27 do you mean x/27 or 27/x? Either way, you'd think a lot of the
fractions would be reducible by 3/2 since 27 is reducible by dividing by 3 for
even numbered denominators IE 27/16 = 3/2 * 9/8 or 34/27 = 2/3 * 17/9.
For things like the 23rd and 25th...the only thing that can factor 25 is 5 (a
bit large and not divide-able for many numbers) and 23 is a prime...hence my
opinion that much the reason these harmonics don't sound as good is that they
can not be split into parts in the same way.

I know Carl said this sort of "splitting a fraction into its components" thing
is non-sense, but my ears often are telling me otherwise. Though if you look at
something like 17/9 you are still likely to read it as just a slightly sour
15/8...or at least I do.

________________________________
From: cityoftheasleep <igliashon@...>
To: tuning@yahoogroups.com
Sent: Mon, August 9, 2010 1:26:28 AM
Subject: [tuning] Re: High Harmonics & the Root

Well, after some hands-on exploration of harmonics 16-32 and odd harmonics
32-64, I can say they certainly are not beatless. I was surprised to discover
how many are within a few cents of familiar tempered intervals, and how at least
a few were near peaks of harmonic entropy. I think it's safe to say that being
part of a harmonic series does not confer any special status in and of itself.
Though I'll admit, the 19th and 27th harmonics sounded pretty good to me,
whereas the 23rd and 25th did not. Evidence that my ear is thoroughly
conditioned to like 12-tET? Perhaps.

-Igs

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:
>
> i do not think there has been enough prolonged exploration of these harmonics.
> I imagine it would be less but if this is a result of hearing them used in our
>culture or not.
> we already know something about lower harmonics just from 12 et.
> albeit a bit warped
> Especially divorced from their lower harmonics.
>
> it would be worth while for someone to explore and might accomplish more than
>speculation
> say a diamond or eikosany using 6 primes from the upper range. or use viggo
>bruns algorithm to make a scale out of a smaller set.
>
> While we can say certain higher ratios appoximate lower ones, if we replace the
>lower ones in the same context the result is going to be different.
>
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > Hi Kraig,
> >
>
> >
> > So, I need a straight answer: do the odd harmonics in the 16-64 series
>provide rootedness and stability, despite their high complexity? Or does
>rootedness drop off drastically above the 16th harmonic?
> >
> > -Igs
> >
>

Noting that taking a 5th from B lands you on F# AKA out-of-key for the key of C.
So what are your nearest alternatives:

A) Taking B to F gives you about 17/12 (tritone)
B) From B to G gives you around 19/12...too far from 8/5 or 11/7 to really
qualify as those IMVHO.

Note that neither of these are exactly the most desirable "standard"
intervals. So, for practical purposes, 12TET diatonic really seems to have 6
(not 7) flavors of triad available...3 major and 3 minor.

So what micro-tonal scales DO enable "all" 7 triads (triad meaning, either with
the wider interval under 8 cents from the 5th OR of either 8/5 or 11/7)? I am
starting to become interested again in 12TET diatonic as a basis for figuring
out why micro-tonality works as it does as and why JI and critical band theory
both apparently have some gaping loopholes in them in certain situations.

A) While just noodling around in it by ear, I noticed it is very tough to get a
sour chord if you omit the F from C major (IE using a six-note subset). This
has left me on a quest to find what "missing note" would best substitute for F
to make a "7 note virtually all chords possible" scale.

B) I am still fascinated with the apparent fact that, even though the semi-tone
itself is terribly dissonant (both by ear and in theoretical terms of critical
band), putting it in certain context with chords such as IE G B C E virtually
cancels out its dissonance. Makes me wonder if using one or maybe even two
half-step-like sized intervals in my own scales won't kill me. :-D

On Mon, Aug 9, 2010 at 12:56 PM, Michael <djtrancendance@...> wrote:
>
> Noting that taking a 5th from B lands you on F# AKA out-of-key for the key of C.
> So what are your nearest alternatives:
>
> A) Taking B to F gives you about 17/12 (tritone)
> B) From B to G gives you around 19/12...too far from 8/5 or 11/7 to really qualify as those IMVHO.

And yet, the superpythagorean diatonic scale in 22-equal gives minor
thirds of about 7/6... and they function identical to the 6/5 minor
thirds of 19-tet.

A conundrum!

>   Note that neither of these are exactly the most desirable "standard" intervals. So, for practical purposes, 12TET diatonic really seems to have 6 (not 7) flavors of triad available...3 major and 3 minor.

Sure, but what about chord progressions like ||: Em9 | Fmaj7/G :||?
Stuff like that is pretty common in neo-soul and R&B. You'd play E
minor (let's say aeolian) over the first chord, and G mixolydian over
the second chord. So the tonal set has EF#GABCDE in the first chord,
and EFGABCDE over the second chord, and the F/F# status switches
between the two chords.

So the question is - does one need to have a single all-encompassing
scale? The all-encompassing scale here would be meantone chromatic (or
meantone[8]), but do you really need to be playing all of those notes
at once?

chord:diatonic scale :: diatonic scale:chromatic scale, is what I'm getting at.

> I am starting to become interested again in 12TET diatonic as a basis for figuring out why micro-tonality works as it does as and why JI and critical band theory both apparently have some gaping loopholes in them in certain situations.

Likewise. :)

You should read Rothenberg's papers, Carl has them linked here:
http://www.lumma.org/music/theory/tctmo/

It might change the way you think about this stuff.

> A) While just noodling around in it by ear, I noticed it is very tough to get a sour chord if you omit the F from C major (IE using a six-note subset).  This has left me on a quest to find what "missing note" would best substitute for F to make a "7 note virtually all chords possible" scale.
>
> B) I am still fascinated with the apparent fact that, even though the semi-tone itself is terribly dissonant (both by ear and in theoretical terms of critical band), putting it in certain context with chords such as IE G B C E virtually cancels out its dissonance.  Makes me wonder if using one or maybe even two half-step-like sized intervals in my own scales won't kill me. :-D

How about Gb D Eb A Bb C F? Let it sink in for a bit. :)

-Mike

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> A) Taking B to F gives you about 17/12 (tritone)
> B) From B to G gives you around 19/12...too far from 8/5 or 11/7 to really
> qualify as those IMVHO.
>
>
> Note that neither of these are exactly the most desirable "standard"
> intervals.

You are assuming 12edo is to be used for the diatonic scale, which is far from the best choice if you want seven triads. If you choose a fifth of (56/5)^(1/6) you get an exact 7/5, which would be one option.

MikeB>"You should read Rothenberg's papers, Carl has them linked here:
http://www.lumma.org/music/theory/tctmo/"

....from the article....

>"1.1.1 The number of different intervals in a scale, normalized to the number
>of notes in it, is Rothenberg's *mean variety*.
So the idea is, the less different types of intervals available per the
number of notes, the easier for the brain to interpret?

>"1.3 Insofar as pitch memory is important in melody, the
*Miller limit* suggests scales consisting of no more than 9
octave-equivalent notes."

I have noticed this in practice and heard of it before, though not by name.
In psychology "set-theory" I've heard as well...that 9 is considered the limit.

Mathematically, though: I find it hard to get more than 7 tones without getting
a whole lot of sour intervals possible and/or much more different types of
intervals.

>"2.1.1.1 *Rothenberg stability* is supposed to
measure the ease of accomplishing such 'modal
transposition' in a given scale."
Meaning how tough a time does the brain have processing transposition?

>"3.3 Ignoring octaves (factors of 2), 3-limit just intonation can be graphed on
>a linear chain (of "fifths"), 5-limit JI on a
planar lattice, 7-limit JI on the face-centered cubic lattice..."
Ok so the number primes = the number of dimensions of the lattice IE 7-limit =
3,5,7 = 3 dimensions?

>"4.1 By forbidding the occurrence of commas in our scale, we can
delimit a finite section of the lattice known as a "block" or
"periodicity block".

4.2 We create a *pun* if we use the same name ("Eb") for both
notes in such a pair."
So I'm guessing this all goes back to tempering and "using two close notes
for one musical purpose"?

>"5.1 If we do this uniformly across the lattice, the associated
comma pumps can all be performed without causing drift in our
pitch standard."
And this is what 12TET does to circles of 5ths and 3rds?

>"This error is bounded by the
size of the comma(s) being tempered out (trivial) and the number of
concordant intervals over which it/they must vanish (complexity again)."
So I'm guessing the MORE intervals it vanishes over the less de-tuning needs
to be done per note and thus the less error?

Gene>"If you choose a fifth of (56/5)^(1/6) you get an exact 7/5, which would be
one option."
Makes sense, but wouldn't that give a terribly de-tuned octave?

What would be the best option where the period is within about 8 cents of
2/1?

I'd also say having the "alternative 5th" within 8 cents of 10/7 would be
fine as well...it does not have to be limited to being within 8 cents of only
the 7/5 interval.

I'm going to try to answer these to test my knowledge of everything -
everyone can correct me if I'm wrong.

On Mon, Aug 9, 2010 at 4:08 PM, Michael <djtrancendance@...> wrote:
>
> ....from the article....
>
> >"1.1.1 The number of different intervals in a scale, normalized to the number of notes in it, is Rothenberg's *mean variety*.
> So the idea is, the less different types of intervals available per the number of notes, the easier for the brain to interpret?

That's the theory. I don't remember how Rothenberg calculates it
exactly, but I think that means that an MOS would have a mean variety
of slightly less than 2, since all intervals will have 2 sizes except
the octave, which will have 1.

The scales used in most music around the world are MOS, but note that
there are a few exceptions, like the melodic minor and harmonic
minor/major scales. Those are "almost" MOS, but have a couple
intervals that have 3 sizes. There are also things like the blues
scale and the hexatonic scale, which aren't MOS, but are used anyway.
However, the blues scale isn't used exactly like other scales (modal
transposition doesn't occur in quite the same way it does with the
diatonic scale).

>
> >"1.3 Insofar as pitch memory is important in melody, the
> *Miller limit* suggests scales consisting of no more than 9
> octave-equivalent notes."
>
> I have noticed this in practice and heard of it before, though not by name. In psychology
> "set-theory" I've heard as well...that 9 is considered the limit.
> Mathematically, though: I find it hard to get more than 7 tones without getting a whole lot of sour intervals possible and/or much more different types of intervals.

I think the idea behind the Miller limit is the notion that the
average short term memory can accomodate 7 +/- 2 pieces of
information. I was thinking though, perhaps this limit could be
superceded though if the notes followed some kind of pattern, since
the brain would then process it via "chunking." For example, try to
remember the following 18 letters:

AOSIDH JNE KRJB ESADH

Difficult, right? Now try to remember the following 18 letters:

ABCDEF JFK ASAP LMNOP

Although they they have the same amount of letters, you'll find that
the second is way easier, since you can "chunk" the set into larger
segments that represent internally a single piece of information. This
is why mnemonic devices work so well when you're studying to try and
remember stuff. I wonder if the same principle could be used to
supercede the Miller limit as well.

> >"2.1.1.1 *Rothenberg stability* is supposed to
> measure the ease of accomplishing such 'modal
> transposition' in a given scale."
> Meaning how tough a time does the brain have processing transposition?

Meaning if you play "Happy Birthday" in G major, and then you play the
theme in E minor - although the absolute intervals have changed, the
melody is still recognizable because you still hear melodic movement
by thirds and seconds and such. One of the main focuses of
Rothenberg'g papers is to lay out a set of guidelines for deriving
scales in which that still works.

One of his key ideas, for example, is that of "propriety." A scale is
proper if the largest third is still smaller than (or at least the
same size as) the smallest fourth and so on for every interval. There
are 5 7-note proper scales in 12-tet:

C D E F G A B C (diatonic)
C D Eb F G A B C (melodic minor)
C D Eb F G Ab B C (harmonic minor)
C D E F G Ab B C (harmonic major)
C Db Eb F G A B C (locrian major, melodic phrygian, make up your own
colorful name for it)

For any of those, modal transposition is going to be easy. But for the
following 7-note scale:

C Db D Eb E F F# C

which is a bunch of half steps and then a tritone, it's going to be
very difficult, since the second from C-F# is larger than the sixth
from C-F. There will be all sorts of perceptual distortions that arise
that mangle the coherence of the melody in this case.

> >"3.3 Ignoring octaves (factors of 2), 3-limit just intonation can be graphed on a linear chain (of "fifths"), 5-limit JI on a
> planar lattice, 7-limit JI on the face-centered cubic lattice..."
> Ok so the number primes = the number of dimensions of the lattice IE 7-limit = 3,5,7 = 3 dimensions?

Yep.

> 4.2 We create a *pun* if we use the same name ("Eb") for both
> notes in such a pair."
> So I'm guessing this all goes back to tempering and "using two close notes for one musical purpose"?

Yeah, and there is apparently some debate to how puns work and what
exactly they "mean." They're interesting auditory illusions for sure -
but how do they work?

For example, when you listen to 7-tet, you still can hear elements of
"majorness" and "minorness" in it, and sometimes neutral triads just
flip back and forth between sounding "majory" and "minory." Is this
because 7-equal violates our inborn sense of harmonicity, and of how
harmonic intervals "work" in a JI sense? Or because 7-equal just
violates our existing 12-tet map, and only fits our diatonic hearing
pattern, but not the chromatic one?

> >"5.1 If we do this uniformly across the lattice, the associated
> comma pumps can all be performed without causing drift in our
> pitch standard."
> And this is what 12TET does to circles of 5ths and 3rds?

It means that chord progressions like Cmaj -> Am -> Dm7 -> G7 are
playable and will get you back to the same Cmaj without having to
throw in really detuned Marcellian wolf fifths or so. It also means
chord progressions like Dm7 -> G7 -> Cmaj work much more smoothly.

> >"This error is bounded by the
> size of the comma(s) being tempered out (trivial) and the number of
> concordant intervals over which it/they must vanish (complexity again)."
> So I'm guessing the MORE intervals it vanishes over the less de-tuning needs to be done per note and thus the less error?

Yes, but I think the idea is also that tempering can help create new
tonal systems in and of itself... and the LESS complex the comma, the
better that works. For example, if you start looking at the list of
5-limit commas, 81/80 is one of the smallest, and also not very
complex. On the other hand, for temperaments like schismatic, you
don't really get that same "effect" of merging a bunch of 3/2's and
5/4, since it takes a lot of fourths to finally hit 5/4. So for
temperaments like that, being as they're more complex, I tend to hear
them as though they were still JI, but they can make things more
efficient "under the hook" (e.g. keyboard mapping, having less notes
to deal with, etc).

-Mike

Besides 12 ET where are all these MOS scales around the world.?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

>
> The scales used in most music around the world are MOS, but note that
> there are a few exceptions, like the melodic minor and harmonic
> minor/major scales. Those are "almost" MOS, but have a couple
> intervals that have 3 sizes. There are also things like the blues
> scale and the hexatonic scale, which aren't MOS, but are used anyway.
> However, the blues scale isn't used exactly like other scales (modal
> transposition doesn't occur in quite the same way it does with the
> diatonic scale).

Actually if you look at Erv paper on MOS you will see , it is possible to have scales with more than 2 sizes.
A good example are the various pentatonics one finds in a major scale.
This is apart of Japanese history theory in fact.
It is one thing Rothenberg leaves out.
also there are intermediate steps that Viggo Bruns algorithm gets that no other system seems to get.

Really the only important point you need to make is that MOS is musically useful.
I would add these too are also and there is no reason not to take advantage of all of them.

I am not sure if we should accept Millers limit as anything more than a mild signpost.
If anyone wants to use 10 notes they really should if compelled.
You example of a gestalt grouping is a good one and i agree there is no reason to place some dogma from one study that the actual test we might have trouble with.
The history of art is filled with rules, even those based on perception, that have all fallen by the wayside. In Painting it i was said one could not do a painting with blues as the primary color, then pink. That is what the science of the day said.
Most of this research is prescientific at best.
Pseudioscience at worse.
Would music based on 9 tone rows be any better?
I doubt it.

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:
>
> Besides 12 ET where are all these MOS scales around the world.?
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> >
> > The scales used in most music around the world are MOS, but note that
> > there are a few exceptions, like the melodic minor and harmonic
> > minor/major scales. Those are "almost" MOS, but have a couple
> > intervals that have 3 sizes. There are also things like the blues
> > scale and the hexatonic scale, which aren't MOS, but are used anyway.
> > However, the blues scale isn't used exactly like other scales (modal
> > transposition doesn't occur in quite the same way it does with the
> > diatonic scale).
>

On 10 August 2010 12:54, Kraig Grady <kraiggrady@...> wrote:
> Actually if you look at Erv paper on MOS you will see , it is possible to have scales with more than 2 sizes.

By the definition we're all using, an MOS has two step sizes. I
thought that was consistent with the Letter to John Chalmers. A
generated scale is the thing like an MOS but with up to three step
sizes.

Graham

Kraig replying to Mike B>"You example of a gestalt grouping is a good one and I
agree there is no reason to place some dogma from one study that the actual
test we might have trouble with."

Through experience I've seen it is possible to go over 9 notes (and not just
have a sudden one-time change ala transposition but instead use continuous
changes). However this "hack" seems to apply a lot more to melody than chords
IE, for myself and many if not most people I know using more than 9 notes in the
chords/backing of a song results in a sense of confusion.

I'd compare Gestalt grouping to
A) Having a melodic motif that goes across over 9 keys (but still works as the
brain groups it by 'motif words' rather than "random" pitches/'letters')
...but also...
B) Transposition of a major chord in such a way that it implies a change of
root/key and adds a certain degree of shock and a sense of "switching
sections/verses" in a song.
C) On the side, I get the feeling MOS is more useful melodically than chord-wise
far as composing...and this comes from both what I have read on this list and
experience. Now scales that are not proper can, of course, be a problem for
chords along with melody, but there are obviously tons of proper scales that are
not MOS.

Kraig>"Would music based on 9 tone rows be any better?"
But isn't that like saying Gregorian chant must be remarkably more easy to
process by the brain than diatonic pop music? It seems obvious to me that
diminishing returns settle in when you try to get "too consonant or too
simple"...just as they settle in when you try to get "too dissonant" and end up
with some consonance. The Miller Limit seems a mere guideline that works most
of the time but can have notable exception, kind of like Just Intonation.

The other thing is what you've said above seems based on time, and it amazes
me just how tolerant the brain is about mixing of meters in rhythm...music
theory teaches the same old kick-snare-kick-snare or kick-hihat-kick-hihat (the
equivalent diatonic monopoly of rhythm, IMVHO)...but any good abstract drum and
bass or African poly-rhythms make it obvious the brain can easily handle much
much more timing/rhythm wise.
Anyhow, I believe history seems to make it clear that 7 tones is somewhere
around where humans typically get the best combination of interpret-ability.
You could also say 5 notes or less is "too melodically flat/boring" just as you
could even say 9 is "still not flexible enough"...but from what I've experienced
both in playing and listening, about 6-9 is about right and I wouldn't be at all
surprised if a good few people thought 5 and maybe even 10 tones were a fair
balance. In other words, the 5-9 range might not be a perfect guess, but I
suspect it's still quite close.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene>"If you choose a fifth of (56/5)^(1/6) you get an exact 7/5, which would be
> one option."
> Makes sense, but wouldn't that give a terribly de-tuned octave?

I assumed you were talking about the standard diatonic scale, which uses pure octaves and is generated by a flattened fifth.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
On the other hand, for temperaments like schismatic, you
> don't really get that same "effect" of merging a bunch of 3/2's and
> 5/4, since it takes a lot of fourths to finally hit 5/4. So for
> temperaments like that, being as they're more complex, I tend to hear
> them as though they were still JI, but they can make things more
> efficient "under the hook" (e.g. keyboard mapping, having less notes
> to deal with, etc).

There's nothing very surprising in schismatic sounding like JI, since it's a microtemperament and does in fact sound like JI. It's the least xcomplex microtemperament, and hence the first one is likely to encounter, but it's a true micro.

> So for
> temperaments like that, being as they're more complex, I tend to hear
> them as though they were still JI, but they can make things more
> efficient "under the hook" (e.g. keyboard mapping, having less notes
> to deal with, etc).

I meant "under the hood." Whoops.

-Mike

Hi Graham~ I don't know what a generated scale is.
If you look at the MOS paper written to Chalmers , he goes on and talks about binary depth. An MOS of an MOS. Since we can find cultures using these there is no reason to exclude them.They are worth while scales.
Constant Structures basically follow the template on MOS patterns of Large and small, but vary according to the limit one is using. Not that it is limited to just JI. It is unfortunate we don't have any documents where he writes about it much.
Instead we have multiple examples.

It is also limiting to refer to comma only being 'tuned out'.
The scales in question as far as number will also have the ability to choose between the two.
One has only to look at the scales in Xenharmonikon 3 to see that these intervals can indeed be used giving slight fluctuation to the scale that some might enjoy.
The point here is that when such things are figured out the knowledge can be applied in more than one way.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 10 August 2010 12:54, Kraig Grady <kraiggrady@...> wrote:
> > Actually if you look at Erv paper on MOS you will see , it is possible to have scales with more than 2 sizes.
>
> By the definition we're all using, an MOS has two step sizes. I
> thought that was consistent with the Letter to John Chalmers. A
> generated scale is the thing like an MOS but with up to three step
> sizes.
>
>
> Graham
>

I don't think a music based on 9 tone rows would be an improvement over 12.
I don't think i said anything that implied what you are saying below.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

>
>
> Kraig>"Would music based on 9 tone rows be any better?"
> But isn't that like saying Gregorian chant must be remarkably more easy to
> process by the brain than diatonic pop music? It seems obvious to me that
> diminishing returns settle in when you try to get "too consonant or too
> simple"...just as they settle in when you try to get "too dissonant" and end up
> with some consonance. The Miller Limit seems a mere guideline that works most
> of the time but can have notable exception, kind of like Just Intonation.
>
> The other thing is what you've said above seems based on time, and it amazes
> me just how tolerant the brain is about mixing of meters in rhythm...music
> theory teaches the same old kick-snare-kick-snare or kick-hihat-kick-hihat (the
> equivalent diatonic monopoly of rhythm, IMVHO)...but any good abstract drum and
> bass or African poly-rhythms make it obvious the brain can easily handle much
> much more timing/rhythm wise.
> Anyhow, I believe history seems to make it clear that 7 tones is somewhere
> around where humans typically get the best combination of interpret-ability.
> You could also say 5 notes or less is "too melodically flat/boring" just as you
> could even say 9 is "still not flexible enough"...but from what I've experienced
> both in playing and listening, about 6-9 is about right and I wouldn't be at all
> surprised if a good few people thought 5 and maybe even 10 tones were a fair
> balance. In other words, the 5-9 range might not be a perfect guess, but I
> suspect it's still quite close.
>

On 10 August 2010 22:05, Kraig Grady <kraiggrady@...> wrote:
> Hi Graham~ I don't know what a generated scale is.
> If you look at the MOS paper written to Chalmers , he goes on and talks about binary depth. An MOS of an MOS. Since we can find cultures using these there is no reason to exclude them.They are worth while scales.

A generated scale has a generating interval (Erv's term) but doesn't
close to give two steps sizes.

Yes, he talks about other things, and they are worthwhile, but he
doesn't call them MOS. If he thought they were MOS he's been keeping
it quiet for the past 35 years.

Graham

Aren't slendro and pelog generally 5 and 7-note unequal MOS's...?

-Mike

On Tue, Aug 10, 2010 at 6:44 AM, Kraig Grady <kraiggrady@...> wrote:
>
> Besides 12 ET where are all these MOS scales around the world.?

On Tue, Aug 10, 2010 at 7:54 AM, Kraig Grady <kraiggrady@...> wrote:
>
> Actually if you look at Erv paper on MOS you will see , it is possible to have scales with more than 2 sizes.
> A good example are the various pentatonics one finds in a major scale.
> This is apart of Japanese history theory in fact.
> It is one thing Rothenberg leaves out.
> also there are intermediate steps that Viggo Bruns algorithm gets that no other system seems to get.

What is the Viggo Brun algorithm?

> Really the only important point you need to make is that MOS is musically useful.
> I would add these too are also and there is no reason not to take advantage of all of them.

> I am not sure if we should accept Millers limit as anything more than a mild signpost.
> If anyone wants to use 10 notes they really should if compelled.

I agree, it's just an interesting idea...

> You example of a gestalt grouping is a good one and i agree there is no reason to place some dogma from one study that the actual test we might have trouble with.
> The history of art is filled with rules, even those based on perception, that have all fallen by the wayside. In Painting it i was said one could not do a painting with blues as the primary color, then pink. That is what the science of the day said.
> Most of this research is prescientific at best.
> Pseudioscience at worse.
> Would music based on 9 tone rows be any better?
> I doubt it.

Does research exist on these issues? I'm curious to figure out some
answer to the question of whether musical perception fundamentally
comes from some inborn JI map, or from a learned 12-tet map.

The fact that 22-equal's diatonic minor triads are 6:7:9, and that
they function almost identically to the 10:12:15 minor triads of
19-tet, suggests to me that the exact JI interpretation is not all
that important to what ends up becoming the end harmonic "function."

-Mike

Hi Mike~
If the notion that scales around the world exhibit some broad "linearity " yes
pelog is closer to one of the MOS of MOS, binary depth ones. Like the Japanese but with way more variation in pitch sizes

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Aren't slendro and pelog generally 5 and 7-note unequal MOS's...?
>
> -Mike
>
>
> On Tue, Aug 10, 2010 at 6:44 AM, Kraig Grady <kraiggrady@...> wrote:
> >
> > Besides 12 ET where are all these MOS scales around the world.?
>

Hi Graham~
I sense i am missing something here. A generated scale appears to be nothing but a generator taken out to any arbitrary point. Steinhaus mentions these, but i don't know of anyone calling them scales.
Is there an example that i am missing here?

here is also confusion of terms. In the paper he refers them to 'sub'-moments of symmetries which I would tend to refer to being apart of a MOS process. But it the distinction is useful in some way i am not objecting. They are Constant structures although there is probably a way to side track that.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 10 August 2010 22:05, Kraig Grady <kraiggrady@...> wrote:
> > Hi Graham~ I don't know what a generated scale is.
> > If you look at the MOS paper written to Chalmers , he goes on and talks about binary depth. An MOS of an MOS. Since we can find cultures using these there is no reason to exclude them.They are worth while scales.
>
> A generated scale has a generating interval (Erv's term) but doesn't
> close to give two steps sizes.
>
> Yes, he talks about other things, and they are worthwhile, but he
> doesn't call them MOS. If he thought they were MOS he's been keeping
> it quiet for the past 35 years.
>
>
> Graham
>

What exactly is an MOS of an MOS? What about the hexatonic scale, C D E F G
A C? Is that an MOS of an MOS, since it's an MOS of 7-equal?

-Mike

On Tue, Aug 10, 2010 at 7:36 PM, Kraig Grady <kraiggrady@...>wrote:

>
>
> Hi Mike~
> If the notion that scales around the world exhibit some broad "linearity "
> yes
> pelog is closer to one of the MOS of MOS, binary depth ones. Like the
> Japanese but with way more variation in pitch sizes
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Mike Battaglia
> <battaglia01@...> wrote:
> >
> > Aren't slendro and pelog generally 5 and 7-note unequal MOS's...?
> >
> > -Mike
> >
> >
> > On Tue, Aug 10, 2010 at 6:44 AM, Kraig Grady <kraiggrady@...> wrote:
> > >
> > > Besides 12 ET where are all these MOS scales around the world.?
> >
>
>
>

On Tue, Aug 10, 2010 at 1:58 PM, genewardsmith
<genewardsmith@...> wrote:
>
> There's nothing very surprising in schismatic sounding like JI, since it's a microtemperament and does in fact sound like JI. It's the least xcomplex microtemperament, and hence the first one is likely to encounter, but it's a true micro.

A microtemperament is something that is indistinguishable from JI?

-Mike

An MOS of an MOS is what in Erv's paper he calls an subMOS as i was discussing with Graham.
i was referring to the form where one has a cycle of 5 within 7.
like e f g b c with is the most common pelog form.
I don't think of Pelog being 7 equal which i see is a point of confusion.
There are some of this form but it is much rarer.
The whole indonesian tuning is very very complex that one could set up a list discussing nothing but that so this is all in the most extreme ball park.

Actually you can have an MOS of an MOS.
All ETs are MOS and you can have subsets of those which qualify. We don't normally think of an MOS with only 1 interval size.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> What exactly is an MOS of an MOS? What about the hexatonic scale, C D E F G
> A C? Is that an MOS of an MOS, since it's an MOS of 7-equal?
>
> -Mike
>
>
> On Tue, Aug 10, 2010 at 7:36 PM, Kraig Grady <kraiggrady@...>wrote:
>
> >
> >
> > Hi Mike~
> > If the notion that scales around the world exhibit some broad "linearity "
> > yes
> > pelog is closer to one of the MOS of MOS, binary depth ones. Like the
> > Japanese but with way more variation in pitch sizes
> >
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Mike Battaglia
> > <battaglia01@> wrote:
> > >
> > > Aren't slendro and pelog generally 5 and 7-note unequal MOS's...?
> > >
> > > -Mike
> > >
> > >
> > > On Tue, Aug 10, 2010 at 6:44 AM, Kraig Grady <kraiggrady@> wrote:
> > > >
> > > > Besides 12 ET where are all these MOS scales around the world.?
> > >
> >
> >
> >
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Aug 10, 2010 at 1:58 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > There's nothing very surprising in schismatic sounding like JI, since it's a microtemperament and does in fact sound like JI. It's the least xcomplex microtemperament, and hence the first one is likely to encounter, but it's a true micro.
>
> A microtemperament is something that is indistinguishable from JI?

That's what I mean by it, and I don't count anything as a micro which doesn't at least get the errors under a cent. When we were tuning things by way of comparison, Marcel noted to this surprise that when tuning his 5-limit stuff, 118 sounded like JI, whereas 53 was very slightly different. That's the difference between wafso-just and microtempering.

But not everyone is on the same page with the above definition.

So the harmonic major scale is an MOS of an MOS? How about C Db E F G Ab B C?

What about C D E F G A C, which is commonly used to produce harmonies
in gospel music?

-Mike

On Tue, Aug 10, 2010 at 9:29 PM, Kraig Grady <kraiggrady@...> wrote:
>
>
>
> An MOS of an MOS is what in Erv's paper he calls an subMOS as i was discussing with Graham.
> i was referring to the form where one has a cycle of 5 within 7.
> like e f g b c with is the most common pelog form.
> I don't think of Pelog being 7 equal which i see is a point of confusion.
> There are some of this form but it is much rarer.
> The whole indonesian tuning is very very complex that one could set up a list discussing nothing but that so this is all in the most extreme ball park.
>
> Actually you can have an MOS of an MOS.
> All ETs are MOS and you can have subsets of those which qualify. We don't normally think of an MOS with only 1 interval size.
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > What exactly is an MOS of an MOS? What about the hexatonic scale, C D E F G
> > A C? Is that an MOS of an MOS, since it's an MOS of 7-equal?
> >
> > -Mike

Kraig>"I don't think a music based on 9 tone rows would be an improvement
over 12."
Right, and I was agreeing with you on that. I was just stressing the idea of
oversimplification was not what I was promoting. In other words, I think the
Miller Limit is most likely a good guess...as in it's not gospel and impossible
to disprove for some cases, but it's not arbitrary artistic bias or
oversimplification either. Again it seems to fall within the same bounds of
things like Just Intonation and Tenney Limit: which simply work well a lot of
the time but certainly can be "broken past" in some cases.

>"The fact that 22-equal's diatonic minor triads are 6:7:9, and that
they function almost identically to the 10:12:15 minor triads of
19-tet, suggests to me that the exact JI interpretation is not all
that important to what ends up becoming the end harmonic "function.""

I'd actually say a bit of both side. In 10:12:15 the outer dyad is 15:10 = 3/2,
just like as in 9/6. Now 10:12 is 6/5 and 6:7 is 7/6 (both virtually of the
same class).
So that's two of three dyads that have relatively the same feel...and the
brain seems to be able to get enough feel by those two to "forgive" the 7:9 =
9/7 for not being a 15:12 = 5/4.

I don't think it has anything to do with 12TET vs. JI...I think it has to do
much more with things like how much "slack" you get for purity for some dyads
when you purify others. And, on the other hand, how trying for too much purity
can be like pinching pennies and give diminishing marginal return.

That and...if any of dyads in a triad or greater chord cause so much critical
band dissonance it distorts the sense of periodicity each "pure" dyad gains.
Which, to me, explains why having a 17/16-ish dyad in an inverted 7th chord is
fine but having a "triad" with two 17/16's (even though 17/16 * 17/16 = almost a
perfect 9/8) doesn't work so well and a lot of the 9/8th-ish-ness is killed.

I don't see where i was saying anything different than you but in this case of the Tenney limit i find useless and unfounded and never works.
It treats all inversions the same in all ranges and at best a giant step backwards from Helmholtz.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
>
> Kraig>"I don't think a music based on 9 tone rows would be an improvement
> over 12."
> Right, and I was agreeing with you on that. I was just stressing the idea of
> oversimplification was not what I was promoting. In other words, I think the
> Miller Limit is most likely a good guess...as in it's not gospel and impossible
> to disprove for some cases, but it's not arbitrary artistic bias or
> oversimplification either. Again it seems to fall within the same bounds of
> things like Just Intonation and Tenney Limit: which simply work well a lot of
> the time but certainly can be "broken past" in some cases.
>

i never said that these scales were MOS or that all scales were MOS.
I will say that all MOS are scales that one can recognize as such.
i thought that is what you were saying. Tetrachordal scales are a good example of scales that are not MOS. the enharmonic being another good one.
http://anaphoria.com/xen9mar.PDF
has a good explanation of these scales where the the fourths can vary in size.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> So the harmonic major scale is an MOS of an MOS? How about C Db E F G Ab B C?
>
> What about C D E F G A C, which is commonly used to produce harmonies
> in gospel music?
>
> -Mike
>
>
> On Tue, Aug 10, 2010 at 9:29 PM, Kraig Grady <kraiggrady@...> wrote:
> >
> >
> >
> > An MOS of an MOS is what in Erv's paper he calls an subMOS as i was discussing with Graham.
> > i was referring to the form where one has a cycle of 5 within 7.
> > like e f g b c with is the most common pelog form.
> > I don't think of Pelog being 7 equal which i see is a point of confusion.
> > There are some of this form but it is much rarer.
> > The whole indonesian tuning is very very complex that one could set up a list discussing nothing but that so this is all in the most extreme ball park.
> >
> > Actually you can have an MOS of an MOS.
> > All ETs are MOS and you can have subsets of those which qualify. We don't normally think of an MOS with only 1 interval size.
> >
> >
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > What exactly is an MOS of an MOS? What about the hexatonic scale, C D E F G
> > > A C? Is that an MOS of an MOS, since it's an MOS of 7-equal?
> > >
> > > -Mike
>

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:
>
> I don't see where i was saying anything different than you but
> in this case of the Tenney limit i find useless and unfounded
> and never works.
> It treats all inversions the same in all ranges

No it doesn't. -Carl

an example where an inversion is rated differently?

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@> wrote:
> >
> > I don't see where i was saying anything different than you but
> > in this case of the Tenney limit i find useless and unfounded
> > and never works.
> > It treats all inversions the same in all ranges
>
> No it doesn't. -Carl
>

yesterday at the library i ran across Rudolf Rasch article on the farey series and using as a measurement for consonance and dissonance.
Has any one made a comparison.
an interesting comment he makes at the beginning is how universally Plomp research is accepted considering that the research was extremely limited

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:
>
> i never said that these scales were MOS or that all scales were MOS.
> I will say that all MOS are scales that one can recognize as such.
> i thought that is what you were saying. Tetrachordal scales are a good example of scales that are not MOS. the enharmonic being another good one.
> http://anaphoria.com/xen9mar.PDF
> has a good explanation of these scales where the the fourths can vary in size.
>
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > So the harmonic major scale is an MOS of an MOS? How about C Db E F G Ab B C?
> >
> > What about C D E F G A C, which is commonly used to produce harmonies
> > in gospel music?
> >
> > -Mike
> >
> >
> > On Tue, Aug 10, 2010 at 9:29 PM, Kraig Grady <kraiggrady@> wrote:
> > >
> > >
> > >
> > > An MOS of an MOS is what in Erv's paper he calls an subMOS as i was discussing with Graham.
> > > i was referring to the form where one has a cycle of 5 within 7.
> > > like e f g b c with is the most common pelog form.
> > > I don't think of Pelog being 7 equal which i see is a point of confusion.
> > > There are some of this form but it is much rarer.
> > > The whole indonesian tuning is very very complex that one could set up a list discussing nothing but that so this is all in the most extreme ball park.
> > >
> > > Actually you can have an MOS of an MOS.
> > > All ETs are MOS and you can have subsets of those which qualify. We don't normally think of an MOS with only 1 interval size.
> > >
> > >
> > >
> > > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > > >
> > > > What exactly is an MOS of an MOS? What about the hexatonic scale, C D E F G
> > > > A C? Is that an MOS of an MOS, since it's an MOS of 7-equal?
> > > >
> > > > -Mike
> >
>

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:
>
> yesterday at the library i ran across Rudolf Rasch article on
> the farey series and using as a measurement for consonance and
> dissonance.
> Has any one made a comparison.

Yes. The mediant or freshman sum will produce the ratio of
least Tenney height between two ratios in lowest terms.

> an interesting comment he makes at the beginning is how
> universally Plomp research is accepted considering that the
> research was extremely limited

I agree completely!

By the way, did the Rasch article mention Joos Voss at all?

-Carl

All inversions will always be rated differently. 5/4 and 8/5
etc. -Carl

Kraig wrote:

> an example where an inversion is rated differently?
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> >
> > No it doesn't. -Carl
> >
> >
> > --- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@> wrote:
> > >
> > > I don't see where i was saying anything different than you but
> > > in this case of the Tenney limit i find useless and unfounded
> > > and never works.
> > > It treats all inversions the same in all ranges

It was in Comtemporary musical review
yes he mentioned Vos research quite extensively and I believe he proposed what i seem to catch you proposed the other day.
adding the numerator and denominator as a measurement.
With dyads it seems to work quite well.
but i think there are problems maybe with things like comparing a 7-8-11 chord with 7-9-11 where i find the later more consonant.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@> wrote:
> >
> > yesterday at the library i ran across Rudolf Rasch article on
> > the farey series and using as a measurement for consonance and
> > dissonance.
> > Has any one made a comparison.
>
> Yes. The mediant or freshman sum will produce the ratio of
> least Tenney height between two ratios in lowest terms.
>
> > an interesting comment he makes at the beginning is how
> > universally Plomp research is accepted considering that the
> > research was extremely limited
>
> I agree completely!
>
> By the way, did the Rasch article mention Joos Voss at all?
>
> -Carl
>

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:

> It was in Comtemporary musical review
> yes he mentioned Vos research quite extensively and I believe
> he proposed what i seem to catch you proposed the other day.
> adding the numerator and denominator as a measurement.

Tenney height multiplies them, which has certain advantages,
but adding them isn't terrible.

> With dyads it seems to work quite well.
> but i think there are problems maybe with things like
> comparing a 7-8-11 chord with 7-9-11 where i find the
> later more consonant.

Again we multiply, and to keep the numbers reasonable,
take the cube root (square root for dyads). In other
words, the "generalized Tenney height" of a chord is the
geometric mean of its identities.

-Carl

I wrote:

> > By the way, did the Rasch article mention Joos Voss at all?

Kraig wrote:

> It was in Comtemporary musical review
> yes he mentioned Vos research quite extensively and I believe he

I misspelled his name -- you got it right. Joos Vos.

-Carl

With say a 4 note chord then one would not octave reduce the ratios but let the spacing determine what one would multiply. The 7-9-11 unfortunately still comes out higher.
If we add we might change the formula if 9 to be 3+3 but you know i think this has to do with the them being spaced equal number apart 2 and 2.

There is always a strange implication in using these numbers that is a bit counter intuitive to me.
5 comes out looking like it is 1.6 times more dissonant, the 7 to the 1.4 times more dis. than the 5.
9 to 7 -1.2 times
11-9- 1.2 times etc.
I am not sure if this is how we experience it
--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@> wrote:
>
> > It was in Comtemporary musical review
> > yes he mentioned Vos research quite extensively and I believe
> > he proposed what i seem to catch you proposed the other day.
> > adding the numerator and denominator as a measurement.
>
> Tenney height multiplies them, which has certain advantages,
> but adding them isn't terrible.
>
> > With dyads it seems to work quite well.
> > but i think there are problems maybe with things like
> > comparing a 7-8-11 chord with 7-9-11 where i find the
> > later more consonant.
>
> Again we multiply, and to keep the numbers reasonable,
> take the cube root (square root for dyads). In other
> words, the "generalized Tenney height" of a chord is the
> geometric mean of its identities.
>
> -Carl
>

there was someone he worked with too whose name i don'y remember.
Possibly i should try to get a PDF copy?

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I wrote:
>
> > > By the way, did the Rasch article mention Joos Voss at all?
>
> Kraig wrote:
>
> > It was in Comtemporary musical review
> > yes he mentioned Vos research quite extensively and I believe he
>
> I misspelled his name -- you got it right. Joos Vos.
>
> -Carl
>

Kraig wrote:

> With say a 4 note chord then one would not octave reduce the
> ratios but let the spacing determine what one would multiply.

One never octave reduces, unless that's what one wants.
The Tenney height is NOT an octave-equivalent measure. One
always plugs in what one wants and gets the answer for that.
e.g. 8/5 = sqrt(40), 16/5 = sqrt(80), and so on.

> The 7-9-11 unfortunately still comes out higher.

7:9:11 gives cubert(693) = 8.849. Higher than what?

> There is always a strange implication in using these numbers
> that is a bit counter intuitive to me.
> 5 comes out looking like it is 1.6 times more dissonant, the
> 7 to the 1.4 times more dis. than the 5. 9 to 7 -1.2 times
> 11-9- 1.2 times etc.

Not sure what you're doing here. There's no guarantee
that one can compare a dyad to a tetrad, though taking the
appropriate root does put the numbers in the same range
regardless of the chord.

4:5 dyad is sqrt(20) = 4.47
4:5:6 triad is cubert(120) = 4.93
4:5:6:7 tetrad is 840^1/4 = 5.38
4:5:6:7:9:11 hexad is = 83160^1/6 = 6.61

but 9:11 by itself is sqrt(99) = 9.95

Seems about right since adding identities to a rooted
chord doesn't increase the dissonance much, but a bare
11:9 can be a bit wild.

-Carl

--- In tuning@yahoogroups.com, "Kraig Grady" <kraiggrady@...> wrote:
>
> there was someone he worked with too whose name i don't remember.
> Possibly i should try to get a PDF copy?

I have two Vos papers in PDF:

Subjective Acceptability of Various 12-tone Tuning Systems
and
Thresholds of Discrimination Between Pure and Tempered Intervals

but the one I really want is:
Purity ratings of tempered fifths and major thirds

which is on my library list. I would welcome anyone saving
me the trip.

-Carl

>"4:5 dyad is sqrt(20) = 4.47
4:5:6 triad is cubert(120) = 4.93
4:5:6:7 tetrad is 840^1/4 = 5.38
4:5:6:7:9:11 hexad is = 83160^1/6 = 6.61"

Is it just me or do virtually all these algorithms almost blindly assume
having a higher numerator/denominator = more dissonance?

I think that much explains why Plomp and Llevelt's theory stuck "despite
relatively little testing"...it at least attempts to fill a gaping hole in
almost any other theory: the fact higher numbered fractions are not always
worse.
Chords like 15:17:22 aren't all bad. I'd agree with something Carl said in a
link about Sethares ages ago...basing analysis on critical band helps avoid
dissonance...(even if it) can only get so and so far (without also taking
periodicity into account). But I'd also say vice-versa and that there are
several high number-fraction-containing chords which can "get away with it"
pretty well because they are well spaced so far as critical band.

What REALLY sticks out to me as an ultimate solution of sorts is (as I
understand) Igs's idea that if a chord has enough pure dyads the impure ones can
get knocked into place by them by the mind. My question to you all on the list
is what options do you think are good for explaining chords and "resolvedness"
that don't just use sums/multiplications/roots of numerators/denominators on a
"brute force periodicity analysis" basis?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> What REALLY sticks out to me as an ultimate solution of sorts is (as I
> understand) Igs's idea that if a chord has enough pure dyads the impure ones can
> get knocked into place by them by the mind. My question to you all on the list
> is what options do you think are good for explaining chords and "resolvedness"
> that don't just use sums/multiplications/roots of numerators/denominators on a
> "brute force periodicity analysis" basis?
>
Well, my little idea is more a step toward a triadic or tetradic theory. A 21/16 is pretty rough as a dyad, if it's just played as a dyad. So is a 15/11. It just so happens that 21/16 can be factored into (9/8)*(7/6), and when you play a 16:18:21 triad, I suspect your brain might hear it as its two component intervals. Also, two 7/6's come out to 49/36, which is only 3 cents from 15/11. So a 15/11 could be *almost* factored into two 7/6's, and in fact if you temper out 540/539 (I think?), they *are* the same thing. So a chord of two slightly-tempered 7/6's spans a slightly-tempered 15/11, and makes that 15/11 sound pretty consonant...because you don't really hear the 15/11, you hear the two 7/6's.

I've even gotten the very poor fifth of 16-EDO to sound nice by breaking it up into a tetrad composed of a stack of tempered 8/7's. I swear, there is something to this.

-Igs

7-9-11 is higher that 7-8-11 was the example i was having trouble with.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Kraig wrote:
>
> > With say a 4 note chord then one would not octave reduce the
> > ratios but let the spacing determine what one would multiply.
>
> One never octave reduces, unless that's what one wants.
> The Tenney height is NOT an octave-equivalent measure. One
> always plugs in what one wants and gets the answer for that.
> e.g. 8/5 = sqrt(40), 16/5 = sqrt(80), and so on.
>
> > The 7-9-11 unfortunately still comes out higher.
>
> 7:9:11 gives cubert(693) = 8.849. Higher than what?
>
> > There is always a strange implication in using these numbers
> > that is a bit counter intuitive to me.
> > 5 comes out looking like it is 1.6 times more dissonant, the
> > 7 to the 1.4 times more dis. than the 5. 9 to 7 -1.2 times
> > 11-9- 1.2 times etc.
>
> Not sure what you're doing here. There's no guarantee
> that one can compare a dyad to a tetrad, though taking the
> appropriate root does put the numbers in the same range
> regardless of the chord.
>
> 4:5 dyad is sqrt(20) = 4.47
> 4:5:6 triad is cubert(120) = 4.93
> 4:5:6:7 tetrad is 840^1/4 = 5.38
> 4:5:6:7:9:11 hexad is = 83160^1/6 = 6.61
>
> but 9:11 by itself is sqrt(99) = 9.95
>
> Seems about right since adding identities to a rooted
> chord doesn't increase the dissonance much, but a bare
> 11:9 can be a bit wild.
>
> -Carl
>

I think starting with dyads is not a good idea and think you really need to test three or more tones so i agree.
yes i think the process you are describing is valid.
I happen to have a 32 37 42 triad in my tuning which i think sounds better than the 36 you are using ( This is doubling the 21/16)

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> > What REALLY sticks out to me as an ultimate solution of sorts is (as I
> > understand) Igs's idea that if a chord has enough pure dyads the impure ones can
> > get knocked into place by them by the mind. My question to you all on the list
> > is what options do you think are good for explaining chords and "resolvedness"
> > that don't just use sums/multiplications/roots of numerators/denominators on a
> > "brute force periodicity analysis" basis?
> >
> Well, my little idea is more a step toward a triadic or tetradic theory. A 21/16 is pretty rough as a dyad, if it's just played as a dyad. So is a 15/11. It just so happens that 21/16 can be factored into (9/8)*(7/6), and when you play a 16:18:21 triad, I suspect your brain might hear it as its two component intervals. Also, two 7/6's come out to 49/36, which is only 3 cents from 15/11. So a 15/11 could be *almost* factored into two 7/6's, and in fact if you temper out 540/539 (I think?), they *are* the same thing. So a chord of two slightly-tempered 7/6's spans a slightly-tempered 15/11, and makes that 15/11 sound pretty consonant...because you don't really hear the 15/11, you hear the two 7/6's.
>
> I've even gotten the very poor fifth of 16-EDO to sound nice by breaking it up into a tetrad composed of a stack of tempered 8/7's. I swear, there is something to this.
>
> -Igs
>

Kraig wrote:

> 7-9-11 is higher that 7-8-11 was the example i was having
> trouble with.

Ah. I might go along with that. Also 7/5 vs 8/5 in the
dyadic case, which I think we discussed before. Really I
doubt our perception of sounds is one dimensional, and a
simple one-dimensional rule like Tenney height can't be
expected to capture everything. It can be useful though
and the claim I'm making is that it is the "best by test"
among such rules.

-Carl

one wishes one could just modify it in some way to knock it into place. It is hard to tell if our "cultural history" of hearing effects how we process these too.
I mean we might be biased with remembering certain intervals "acting' as dissonances so anything in that range we impose some sort of history on it first.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Kraig wrote:
>
> > 7-9-11 is higher that 7-8-11 was the example i was having
> > trouble with.
>
> Ah. I might go along with that. Also 7/5 vs 8/5 in the
> dyadic case, which I think we discussed before. Really I
> doubt our perception of sounds is one dimensional, and a
> simple one-dimensional rule like Tenney height can't be
> expected to capture everything. It can be useful though
> and the claim I'm making is that it is the "best by test"
> among such rules.
>
> -Carl
>

Copy the below text into a text editor and save it as HTML. Then load it
into your browser.
It will find the nearest JI chord given any four frequencies and/or decimal
point values (IE 4/3 = 1.333333 in decimal).

Please let me know what you think and feel free to publish and/or improve it
in any way you please. :-)

<!-- program starts below -->

<script>

function getjichord()
{

var chord = new Array(4); //[1.0, 1.3333, 1.6666666, 1.6666666]

var chord0 = document.getElementById('noteone').value * 1.0;
chord[0] = 1.0;
chord[1] = (document.getElementById('notetwo').value * 1.0) / chord0;
chord[2] = (document.getElementById('notethree').value * 1.0) / chord0;
chord[3] = (document.getElementById('notefour').value * 1.0) / chord0;

if ((isNaN(chord[1])) || (isNaN(chord[2])) || (isNaN(chord[3])))
{
alert("You must enter numeric values for all four notes in the chord. If you
want a three note chord use one of the note values twice.");
return "";
}

var a;

var w;
var x;
var y;
var z;

var minw;
var minx;
var miny;
var minz;

var one;
var two;
var three;

var canreduce;

var minerrorw = 10000000.0;
var minerrorx = 10000000.0;
var minerrory = 10000000.0;

for (w=1;w < 22; w++)
{
for (x=1;x < 22; x++)
{
for (y=1;y < 22; y++)
{

for (z=1;z < 22; z++)
{
one = Math.abs(((z*1.0)/(x*1.0)) - ((chord[2]*1.0)/(chord[0]*1.0)));
two = Math.abs(((y*1.0)/(x*1.0)) - ((chord[1]*1.0)/(chord[0]*1.0)));
three = Math.abs(((w*1.0)/(x*1.0)) -
((chord[3]*1.0)/(chord[0]*1.0)));

if ((one <= minerrorx) && (two <= minerrory) && (three <= minerrorw))
{
minw = w;
minx = x;
miny = y;
minz = z;
minerrorw = three;
minerrorx = one;
minerrory = two;
}

} //z

} //y
} //x
} //w

canreduce = true
while (canreduce)
{

canreduce = false;

if (((minx % 2) == 0) &&
((miny % 2) == 0) &&
((minz % 2) == 0) &&
((minw % 2) == 0))
{
minw = minw / 2;
minx = minx / 2;
miny = miny / 2;
minz = minz / 2;
canreduce = true;
}

if (((minx % 3) == 0) &&
((miny % 3) == 0) &&
((minz % 3) == 0) &&
((minw % 3) == 0))
{
minw = minw / 2;
minx = minx / 2;
miny = miny / 2;
minz = minz / 2;
canreduce = true;
}

}

document.getElementById('result').innerHTML = 'The nearest Ji chord is ' + (minx
+ " " + miny + " " + minz + " " + minw);

} //end getjichord

</script>

<body>
<table>
<tr><td>Enter frequency of first note in chord</td><td><input
id="noteone"/></td></tr>
<tr><td>Enter frequency of second note in chord</td><td><input
id="notetwo"/></td></tr>
<tr><td>Enter frequency of third note in chord</td><td><input
id="notethree"/></td></tr>
<tr><td>Enter frequency of fourth note in chord</td><td><input
id="notefour"/></td></tr>
</table>

<input type="button" value="calculate JI chord"
onclick="javascript:getjichord();">

<br/><br/>
<span id="result"></span>

</body>

</html>

Igs>"So a 15/11 could be *almost* factored into two 7/6's...a chord of two
slightly-tempered 7/6's spans a slightly-tempered 15/11, and makes that 15/11
sound pretty consonant...because you don't really hear the 15/11, you hear the
two 7/6's. "

But that's the thing, looking at that it seems obvious to me 2 out of the
three possible dyads are good and the brain simply manages to align the bad
third dyad based on the other good two. For the record, I agree you need more
than just a dyad or two to make this work...but rather something with 3+ dyads
available (which, indeed, means a triadic or larger chord).
Question for you...how would this work on 4-note chords? The are six possible
dyads (right?)...what the required ratio of good-to-bad dyads needed in that
case...in your opinion and/or can you give an example?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Copy the below text into a text editor and save it as HTML. Then load it
> into your browser.
> It will find the nearest JI chord given any four frequencies and/or decimal
> point values (IE 4/3 = 1.333333 in decimal).
>

Hi Michael,
there exist already an HTML-code for that in the web, see:
http://superspace.epfl.ch/approximator/
"
It based on so-called Stern-Brocot tree algorithm which was discovered independently by the German mathematician Moriz Stern (1858) and by the French clock producer Achille Brocot (1860).
"
here the well working HTML-code from the site:

<script language="JavaScript" type="text/JavaScript">

function recalc_onclick(ctl) {

document.formc.pA1out.value=document.formc.pA1in.value;

var x = document.formc.pA1in.value;
var eps = document.formc.pA1in2.value
var n = x;
var nfin = x;
var a = Math.floor(n);
var p0 = 1;
var q0 = 0;
var p1 = a;
var q1 = 1;
var p2 = p1;
var q2 = q1;

while(x - a && Math.abs(n - p2 / q2) > eps)
{
x = 1 / (x - a);
a = Math.floor(x);
p2 = a * p1 + p0;
q2 = a * q1 + q0;
p0 = p1;
q0 = q1;
p1 = p2;
q1 = q2;
}
err=nfin-p2/q2;
document.formc.pA1out.value=q2;
document.formc.pA1out2.value=p2;
document.formc.pA1out3.value=err;

}
</script>

have a lot of fun with that
bye
Andy

Andy>"Hi Michael,
there exist already an HTML-code for that in the web, see:
http://superspace.epfl.ch/approximator/
"
It based on so-called Stern-Brocot tree algorithm which was discovered
independently by the German mathematician Moriz Stern (1858) and by the French
clock producer Achille Brocot (1860)."

It is pretty cool and I already use something like that regularly on
http://www.mindspring.com/~alanh/fracs.html
...which gives not one but several fractions with various levels of error to the
original value.

But my program does not find the fractional representation of a single value
but for an entire 4-note chord.

For example if you put in the following to my program
1
1.33333
1.5
1.833333

...it will return...

6 8 9 11

And for
1
1.341
1.66666

....it will return....
3 4 5

Oddly enough, may program does not convert decimals to fractions...it simply
tries to minimize the error between fractions and possible whole numbered
chords. Call it "brute force"...but it seems to work quite well. Even chords
like 12:15:16:19 can be found with it.
BTW, I did find a small error with my code: this part that says

if (((minx % 3) == 0) &&
((miny % 3) == 0) &&
((minz % 3) == 0) &&
((minw % 3) == 0))
{
minw = minw / 2;
minx = minx / 2;
miny = miny / 2;
minz = minz / 2;
canreduce = true;
}

...should be changed to....

if (((minx % 3) == 0) &&
((miny % 3) == 0) &&
((minz % 3) == 0) &&
((minw % 3) == 0))
{
minw = minw / 3;
minx = minx / 3;
miny = miny / 3;
minz = minz / 3;
canreduce = true;
}

if (((minx % 5) == 0) &&
((miny % 5) == 0) &&
((minz % 5) == 0) &&
((minw % 5) == 0))
{
minw = minw / 5;
minx = minx / 5;
miny = miny / 5;
minz = minz / 5;
canreduce = true;
}

if (((minx % 7) == 0) &&
((miny % 7) == 0) &&
((minz % 7) == 0) &&
((minw % 7) == 0))
{
minw = minw / 7;
minx = minx / 7;
miny = miny / 7;
minz = minz / 7;
canreduce = true;
}

...in order to reduce the fractions in the chord properly in most cases.

http://lumma.org/tuning/erlich/2000.08.EntropyMinimizer.txt

I found this particularly fascinating as it seems to say
A) In some cases tempered chords have less harmonic entropy than even much lower
odd-limit pure ones in some cases.
B) What appears to be a huge tip off the Igs's theory, that having enough
consonant dyads can allow a dissonant one not to interfere with the chord
Here's a "screaming example" of this from the link

>"The next most concordant tetrad was a surprise -- a very modern "augmented
>octave" chord,
0 388 886 1274¢
or 12:15:20:25 or 1/1:5/4:5/3:25/12. It contains two 4:5s, two 3:5s, and
one 3:4s, all concordant enough to counteract the great discordance of
the 12:25."

...and isn't that much the same mysterious phenomenon we hear in the oddly
consonant 12:15:16:20 "C E F A" chord?

In short, correct me if you see a loophole in this argument...but I think
it's pretty clear that having a large percentage of dyads within a chord be
close to low-limit fractions

is ultimately more important than making the entire chord "low-limit".

In the article the first 4-note chord that wasn't just a triad with a note an
octave higher was 9:12:15:20. Yes, 15-odd-limit!
And also high on the list was 24:30:40:45. Yep, (apparently) 45-limit. Then
if you look closer you see 5/3. 5/4, 13/8, 3/2, 4/3, and 9/8.
If that doesn't raise some doubts as to how odd-limit "always works"...I really
wonder how zombie-fied the world of JI has become.

1
1.12
1.1947
1.338
1.5
1.674
1.792
2

Just dyads in this scale (ones all possible dyads between 1/1 and 4/1 in this
scale are within 8 cents of) include

1.125 (9/8)
1.2 (6/5)
1.25 (5/4)
1.3333 (4/3)
1.42857 (10/7)
1.5 (3/2)
1.6 (8/5)
1.6666 (5/3)
1.8 (9/5)
1.875 (15/8)

Firstly, I'll admit this scale is essentially a "strategically tempered version
of JI diatonic" with rather similar ratios from the root tone.
The good news is, so far as I see it, it virtually eliminates diatonic JI's
issue of favoring some roots and/or triads over others...in this scale virtually
all chords from all roots are equally strong (and nearly so good as pure ones).
Plus, if used in an "adaptive JI" type program it should eliminate commatic
shift in many cases.

Do any of you know of any other 7-tone scales which would be competitive
with this scale so far as having all possible dyads from all possible root tones
within 8 cents of the above-listed low-limit ratios (or other equally low-limit
ones)?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> http://lumma.org/tuning/erlich/2000.08.EntropyMinimizer.txt
>
> I found this particularly fascinating as it seems to say
> A) In some cases tempered chords have less harmonic entropy than even much lower
> odd-limit pure ones in some cases.

It's all very interesting, but where is the evidence the harmonic entropy calculations for chords actually produce a good model for chord consonance? With the dyads, we can use our ears on a relatively small number of candidates, but how are we to evaluate the rating of tetrads?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> Firstly, I'll admit this scale is essentially a "strategically tempered version
> of JI diatonic" with rather similar ratios from the root tone.

I would not call it that; it seems to me it's an irregularly tempered diatonic scale, with flat fifths offset by a couple of pure fifths. But it's also close to the following 7-limit JI scale: [28/25, 448/375, 75/56, 3/2, 375/224, 224/125, 2].

Michael/Me>"http://lumma.org/tuning/erlich/2000.08.EntropyMinimizer.txt"

Gene>"It's all very interesting, but where is the evidence the harmonic entropy
calculations for chords actually produce a good model for chord consonance?
With the dyads, we can use our ears on a relatively small number of candidates,
but how are we to evaluate the rating of tetrads?"

Ultimately...one surefire way is to test a whole bunch of chords on a whole
bunch of people and look for patterns between that and how various "systems"
(harmonic entropy, periodicity, critical band dissonance...) rate those chords
(assuming strongly harmonic timbre IE a guitar sound).

But short of that...I think an easy test is to play various tetrads blindly
to ourselves (IE without seeing what they are) on several occasions and rating
them each time, then taking the average. I don't think evaluating larger chords
by ear has to be much harder with larger-than-dyad combinations...I, for one,
can hear significant differences between "nearby" versions of chords the same
way I can hear them with dyadic intervals. The only problem I've found is when
you squeeze over 5 or so tones per octave tones start to "blur" together, making
it harder to recognize differences...so I think a good analysis rule should
involve chords with up to 7 or so tones played/held at once with no more than 5
tones per octave.

I'm not saying harmonic entropy is a perfect idea...but that, at least for
me, it seems to have a higher accuracy with 4+ note chords than the process of
rating chords by odd-limit does...plus it can rate tempered intervals directly
(apparently).

The ultimate "theory", IMVHO, would take periodicity and critical band and
harmonic entropy all into consideration and have a formula that gives a certain
weight to each. But the only way to do this, I'm guessing, is a whole lot of
testing and then a whole lot of research into correlating how much of each
system's weighting should be applied to add together a curve that pretty much
matches the curve of listener responses.

Gene>"But it's also close to the following 7-limit JI scale: [28/25, 448/375,
75/56, 3/2, 375/224, 224/125, 2]."
Interesting, so what's the formal name for said scale?

Gene wrote:

> Michael wrote:
>
> > http://lumma.org/tuning/erlich/2000.08.EntropyMinimizer.txt
> >
> > I found this particularly fascinating as it seems to say
> > A) In some cases tempered chords have less harmonic entropy
> > than even much lower odd-limit pure ones in some cases.
>
> It's all very interesting, but where is the evidence the
> harmonic entropy calculations for chords actually produce a
> good model for chord consonance? With the dyads, we can use
> our ears on a relatively small number of candidates, but how
> are we to evaluate the rating of tetrads?

These entropy calculations were dyadic, and therefore should
NOT be expected to be a good model of chord consonance.

The best study of chord consonance I know of, either inside
or outside academia, was the "Tuning Lab" tetrads survey of
Erlich & Pehrson, done before you joined. The fact that it
is the best I know of gives an idea of the sad state of the
inquiry. I've never seen the complete results and don't
even know how many participated. But I do have these notes:

me intervals g.m. d.e.
-----------------------------
1 388 702 970 1 22-23
2 318 816 1020 2 20-21
3 498 702 886 3 7-8
4 386 702 1088 6 1
5 268 702 970 11 6
6 316 702 1018 8 2
7 184 498 886 7 3-4
8 498 886 1384 9 5
9 204 702 1088 5 9-10
10 202 702 974 4 25-26
11 268 582 970 30 22-23
12 272 772 974 24 25-26
13 388 886 1274 13 24
14 318 818 1320 15 32-33
15 384 588 1086 23 18-19
-----------------------------

So I only completed half the survey for some reason -
there were 30 chords total. g.m. is Tenney height.
d.e. looks suspiciously like dyadic entropy but I'm
not sure!

-Carl

Carl>"These entropy calculations were dyadic, and therefore should
NOT be expected to be a good model of chord consonance."

Well is there any suggested model that allows analysis of a chord "all at
once" instead of as a weighing of results for dyadic section besides just
"rating purely by ear"?

The only system I have heard of which truly seems to do that is odd-limit
and, as I've said before, harmonic entropy's results seems to be, on the
average, closer to the ratings of what I hear by ear.

>"The best study of chord consonance I know of, either inside or outside
>academia, was the "Tuning Lab" tetrads survey of Erlich & Pehrson"

..which seems to give, as the best tetrachord...
>"(0) 388 702 970"
So that's about 1/1 5/4 3/2 7/4 (AKA 4:5:6:7), right?

Which I agree on as being virtually the most resolved sounding. BTW...if d.e.
means an equivalent of (dyadic) harmonic entropy I have no clue why something
this simple would have such a high DE...if so that truly is a strike to the
harmonic entropy theory.

It also gives nearly the same rating to:

>"(0) 204 702 1088"
1/1 9/8 3/2 13/8 AKA 8:9:12:13

>"(0)202 702 974"
1/1 9/8 3/2 9/5
...Whereas I would say the second one is substantially less resolved sounding.
The odd thing is that virtually all (and not just most) of the dyads here sound
good individually, but not nearly so good as a whole. I'd at least in part
chalk it up to the fact the 13/8 has a wider critical band dissonance from 3/2
than the 9/5 does. In fact replacing 11/6 instead of 9/5 has a similar
"increased resolved-ness" effect despite forming a very high limit (and no where
near low-limit approximation) dyad.

>"So I only completed half the survey for some reason -
there were 30 chords total. g.m. is Tenney height.
d.e. looks suspiciously like dyadic entropy but I'm
not sure!"
One thing is for sure...(guessing they surveyed all chords just by
ear)...if DE is harmonic entropy and TE is tenney height...either this theory is
way off or the other theories have very little in common with a good answer.
From checking out a few of these chords I'm still leaning toward Harmonic
Entropy as the best theory for tetrachords...but think virtually all the
theories have a big gaping hole in their apparent ignorance of critical band and
critical band has a big gaping ignorance of periodicity. Argh! :-D

1) Here's an odd and IMVHO critical band supporting test

Try playing the chord
3:4:6 (very low limit)

Now try playing 15:22:30 1/1 22/15 2/1 (much higher limit, a "22/15" chord)

My ears actually relate to the second as more resolved sounding...but what about
yours?
I think it's pretty clear (likely to many if not most people) that 3:4:6 is
not a clear leader in resolved-ness at best and clearly the worse chord at worst
far as resolved-ness.

2) Another odd example. Try the chord 8:9:10. Now try 36:40:45 AKA 1/1 10/9
5/4. Oddly enough, they sound about the same to me far as resolved-ness with
the 8:9:10 sounding only slightly better...the 8:9:10 doesn't seem to have a
clear lead.
Oddly enough, this seems to suggest periodicity and (root tone) critical band
can both have little effect...but the harmonic entropy graph and Sethares'
culmulative critical band dissonance graph seem to make more sense in this case
as they show between 12/11 and 9/8 have about the same "entropy level".

Note comparing 8:9:10 and 36:40:45 is like comparing major to minor...it's an
interval reversal.
The only difference is the intervals are so close that the brain doesn't seem
to care much (IE there's no clear advantage here as there is in resolved-ness of
a diatonic major vs. minor triad).
Which seems to suggest equally spaced intervals may be ideal for clustered
chords...move the 11/10 or 9/8 (or anything in between) dyad formed by the
second note in either chord either further up or down and the chord sounds less
and less resolved. This seems to imply critical band dissonance is kicking in
as well...and perhaps over-riding the effect of periodicity for such clustered
intervals.

3) Now try 3:5:10 AKA 1/1 5/3 10/3 vs. 1/1 11/6 10/3 AKA 6:11:20. By this
point things are so spread out that periodicity seems to dominate and the 3:5:10
has a clear advantage.

4) But trying even 5:7:9 vs. 30:42:55 AKA 1/1 7/5 11/6...the latter still
sounds more resolved...though at 1/1 7/5 21/11 the lack of periodicity finally
seems to make the 5:7:9 sound substantially clearer.

5) Now for something in-between, try the mildly spread out chord of 12:15:20 AKA
1/1 5/4 5/3 vs. 1/1 11/9 5/3 AKA 9:11:15. Here the resolvedness actually seems
about the same...and I'm guessing the not-so-dramatic overall periodicity
advantage of the second chord as a whole works in its favor while the
not-so-dramatic advantage of the first chord's simpler dyads and slight critical
band advantage (15/12 vs. 11/9) work in the first chord's favor.

-------------------------------

So it seems to be a pattern (so far, at least)
1) Second or smaller intervals are perhaps mostly effected by critical band over
anything else. You have to screw up periodicity pretty miserably for the
mistakes to be heard in clustered chords.

2) Intervals up to about 5/4 (with another bump at about 4/3) are effected
significantly by the critical band, but periodicity begins to become more
important as you move further out.
In the case of the 22/15 chord, the periodicity is "not so good", but the
large distance advantage so far as critical band seems to manage to make up for
it.

3) In chords with large intervals (IE those over 3/2), lack of periodicity can
really begin to make things sound tense...but it still takes a lot of lack of
periodicity (think having half or more of the dyads as having a 'nearest JI
fraction' denominator of 11 or higher) to really send things astray.

4) Of course, added overall "odd limit" periodicity or dyadic periodicity or
critical band distance almost never hurts...but it seems to become a problem
when you throw everything in one direction (dyadic periodicity, overall
periodicity, or critical band) and forget about the other two you can find loads
of examples where following a theory "perfectly" actually gives you a more
"sour" chord.

Thoughts? Suggestions?

Michael wrote:

>> These entropy calculations were dyadic, and therefore should
>> NOT be expected to be a good model of chord consonance.
>
> Well is there any suggested model that allows analysis of a
> chord "all at once" instead of as a weighing of results for
> dyadic section besides just "rating purely by ear"?

I wonder how many times I've answered this. Did you have
trouble paying attention in school?

> The only system I have heard of which truly seems to do
> that is odd-limit

Nope, odd-limit is another dyadic concept.

> It also gives nearly the same rating to:
>
> >"(0) 204 702 1088"
> 1/1 9/8 3/2 13/8 AKA 8:9:12:13

You mean 15/8.

> >"(0)202 702 974"
> 1/1 9/8 3/2 9/5

974 cents is an approximate 7/4.

-Carl

Carl>> Well is there any suggested model that allows analysis of a
>> chord "all at once" instead of as a weighing of results for
>> dyadic section besides just "rating purely by ear"?
>"I wonder how many times I've answered this. Did you have
trouble paying attention in school?"
And I wonder how many times you post completely unproductive accusations and
don't seem to question if maybe you by odd chance don't always explain things in
perfect clarity (or completeness).
Here is a basic synopsis some of the theories (as I understand them):

1) Critical Band (Sethares/Plomp and Llevelt): dyadic (all of Sethares formulas
are based on comparing dyadic roughness and then taking accumulated sums)

2A) Periodicity of complete chords IE odd-limit of chords: 5:6:7:8 has an
odd-limit of 7...you need to see the whole chord to get this, which is why I say
it's not dyadic.
2B) Odd-limit per dyad: taking each dyad individually IE 8/7, 7/6, 6/5,
7/5...and taking a guess at what the 'entropy' is based on the average amount of
entropy of all the dyads. Obviously dyadic.

3) By Tenney Height the chord 5:6:7:8 would be the 4th root of 5*6*7*8. I'd say
it's debatable whether this is dyadic of not, since each additional note in the
chord means one additional multiplication is needed for the calculation. It
rather seems to assume if one number in the chord is huge, it always makes the
chord sound much worse, even if all the other numbers and dyads used in the
chord are very simple.

4) Harmonic Entropy of chords...likes Sethares, is based on a curve and sum of
dyads, only it rates dyads based an ambiguity rather than beating.

5) The Erlich survey you gave me of chords...gives no explanation (at least so
far as you said it) for the order of consonance...this leading me to believe it
is done by ear unless noted otherwise. You seem to have claimed it is not
dyadic, but not really explained why.

Ok, wise guy....so if you're such the productive critic, what's missing?

>> >"(0) 204 702 1088"
>> 1/1 9/8 3/2 13/8 AKA 8:9:12:13
>You mean 15/8.
Right, my bad. Should be 8:9:12:15.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> These entropy calculations were dyadic, and therefore should
> NOT be expected to be a good model of chord consonance.

Why not? Is there evidence that chord concordance derives from any other property than the aggregate concordance of its component dyads? If so, I'd like to see it. Only because aggregate dyadic concordance has (thus far) been the most accurate and helpful means I've found of finding good chords in weird tunings. When I work with EDOs where the fifth is too impure to serve as a stable concordance in its own right, I look at whichever few intervals *are* close enough to a simple ratio, and then I put them together. So far, I've had nothing but success; the chords I find in this way always seem to be the "nicest" chords in each system.

I mean, it just seems illogical to me to imagine a chord made up of discordant dyads could sound concordant when they're all played together, or that a chord made up of concordant dyads could sound anything but concordant when played together. So what other measurable property to a chord could there be, which determines its concordance?

(Note that I'm not trying to make a point here, I'm just asking a question. I don't presume that I have an adequate understanding of the subject to make points about it.)

-Igs

Igs wrote:

> > These entropy calculations were dyadic, and therefore should
> > NOT be expected to be a good model of chord consonance.
>
> Why not? Is there evidence that chord concordance derives
> from any other property than the aggregate concordance of
> its component dyads? [snip]
> I mean, it just seems illogical to me to imagine a chord made
> up of discordant dyads could sound concordant when they're all
> played together, or that a chord made up of concordant dyads
> could sound anything but concordant when played together.

Compare a utonal hexad with a otonal one.

> So what other measurable property to a chord could there be,
> which determines its concordance?

Its fit to a harmonic series?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Compare a utonal hexad with a otonal one.

Interesting! I played a 4:5:6:7:8:9, and then a 1/(4:5:6:7:8:9) and the difference was pronounced. I'm not entirely sure I'd call it a difference in consonance, but the difference in mood was striking. Having interval complexity decrease as pitch increases seems to make for a less-stable sound than having interval complexity increase as pitch increases. This must also be why I've always found supermajor chords to be so much less stable than subminor.

> > So what other measurable property to a chord could there be,
> > which determines its concordance?
>
> Its fit to a harmonic series?

I'm not entirely sure I'd agree with that. I'd think a chord like 1/1-5/4-3/2-7/4-9/4-8/3 would be more concordant than a 4:5:6:7:9:11 chord, but the latter one fits the harmonic series better. Even a 1/1-6/5-3/2 is bound to be more concordant than a 6:7:9. There's gotta be more factors going on there.

Carl>"Compare a utonal hexad with a otonal one."
Wouldn't that be the old major vs. minor triad argument?

I'll agree o-tonality sounds better than u-tonality...but how do you justify
that as being a symptom of being related to the (o-tonal) harmonic series and
not just of general progression from larger to smaller intervals as one
progresses to higher frequencies?

After all, the critical band itself progressed toward smaller intervals at
higher frequencies as well and has nothing derived from the harmonic series.

Igs>"So what other measurable property to a chord could there be,
> which determines its concordance?"

Carl>"Its fit to a harmonic series?"

Well here we go again... Carl I hear what you are saying, I just don't agree
with it...and yes I can understand something without agreeing with it. Sure,
something very low in the harmonic series IE 3:4:5 or 6:7:8 is almost always
going to work.

But try 8:12:13 vs. 60:75:88. Note 60:75:88 has a 5/4 dyad from 76/60 and a
not-so-far from 7/6 dyad of 88/75...but perhaps more importantly a huge critical
band advantage in having a closest dyad of about 7/6 vs. 13/12!
Now if that doesn't at least bring into question if not outright debunk your
supposed theory of fit to harmonic series providing a no-fail determination of
consonance...Lord knows how stuck you are on promoting it.

>"Having interval complexity decrease as pitch increases seems to make for a
>less-stable sound than having interval complexity increase as pitch increases.
>This must also be why I've always found supermajor chords to be so much less
>stable than subminor."

Not going to argue with that...I also see major chords as having a more stable
"mood" (along with more stable consonance). However still what bothers me is
who says this is an artifact of the harmonic series when critical band also gets
smaller with increasing frequencies.

For example look at the fairly non-JI-compliant chord of

#1) 1 1.2625 1.4834

Now reverse the dyads to form

#2) 1 1.175 1.4834

Notice that although neither of these tie closely to low parts of the harmonic
series, the second one (IE the one going from smallest to largest dyads) sounds
more tense than the first...just like minor to major. My point is the whole
largest to smallest dyad pattern within chords certainly does not appear to be
exclusive to the harmonic series.

Igs wrote:

> Having interval complexity decrease as pitch increases seems
> to make for a less-stable sound than having interval complexity
> increase as pitch increases. This must also be why I've always
> found supermajor chords to be so much less stable than subminor.

That's a generalization of two experiences -- are you sure
it's the right one?

Subminor is 6:7:9. Supermajor is 14:18:21. So that's also
consistent with my harmonic series explanation.

> > > So what other measurable property to a chord could there
> > > be, which determines its concordance?
> >
> > Its fit to a harmonic series?
>
> I'm not entirely sure I'd agree with that. I'd think a
> chord like 1/1-5/4-3/2-7/4-9/4-8/3 would be more concordant
> than a 4:5:6:7:9:11 chord, but the latter one fits the
> harmonic series better.

You'd think it would, but did you try it? The 32/21
is a doosey.

> Even a 1/1-6/5-3/2 is bound to be more concordant than a
> 6:7:9. There's gotta be more factors going on there.

10:12:15 does have greater Tenney height than 6:7:9. I'm
not sure it's more consonant. In the case of dyads we also
found that when the geometric mean of the Tenney height was
greater than 9, it stopped predicting consonance well.

-Carl

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> But try 8:12:13 vs. 60:75:88. Note 60:75:88 has a 5/4 dyad from 76/60 and a
> not-so-far from 7/6 dyad of 88/75...but perhaps more importantly a huge critical
> band advantage in having a closest dyad of about 7/6 vs. 13/12!
> Now if that doesn't at least bring into question if not outright debunk your
> supposed theory of fit to harmonic series providing a no-fail determination of
> consonance...Lord knows how stuck you are on promoting it.
>

Actually, I don't think it *does* debunk the theory. I thought about just such an example a lot last night, and I think because of the fact 60:75:88 is actually *closer* to 4:5:6 than 8:12:13, the theory holds. IOW, it's not enough just to look at the harmonics and say "this one is higher harmonics, so it's going to sound more discordant". Higher harmonics can approximate lower harmonics, so it's actually the middle harmonics that are going to be the most discordant, since they're the maximum distance from the lowest harmonics. This is definitely borne out by harmonic entropy, where maxima occur pretty close to 11/8 and 13/8.

Also, consider a chord like 16:20:25 vs. 4:5:6. Since major thirds are more consonant than minor according to pretty much every theory, why isn't two major thirds more consonant than a major plus a minor? Or why isn't a 9:12:16 more consonant than a 4:5:6, since the fourth is more consonant than the major third (also according to pretty much every theory). These examples show that having greater pitch-distance between intervals is not the key, and also that being composed of simple dyads is also not the key.

What I think might debunk this theory, at least as a theory of *consonance*, is Bohlen-Pierce music. 3:5:7 implies a harmonic series very strongly, but I'll be damned if I'd call it "consonant". Also, I think a lot of us here might actually prefer 4:5:6 over 3:4:5, due to our cultural bias for the former voicing.

The problem with theories like this, IMHO, is that they conflate "stability" (or "concordance", if you prefer) with "consonance". Two very different phenomena. Intervals like 11/6 or 7/5 are more stable than intervals like 13/11 or 27/16, but I'd be lying through my teeth if I told you I thought the first two sound "nicer" or "more resolved" or "calmer" or what have you. A triadic theory of consonance could probably be binary: if a triad is close enough to 4:5:6 or 1/(4:5:6) for the ear to "round it", it's consonant. If not? It's dissonant. Why? Because of our cultural custom. We are accustomed most strongly to major and minor triads, so we will prefer things that sound like them over things that don't (unless we're into that "avant garde" stuff, in which case we're not representative of the general population anyway and our opinion shouldn't be taken in to account). Our preference will run from stable&familiar, to unstable&familiar, to stable&unfamiliar, to unstable&unfamiliar. I simply can't imagine someone rating a 12-tET major chord as sounding "less resolved" than a pure 5:6:7 chord. The latter is essentially a diminished triad, which most people find the tempered version of to be the strongest dissonance available. A Just dissonance is still a dissonance.

This is why, as a guitarist, I find the term "stability" more useful. Stability tells me nothing more (or less) than how much fuzz gain I can apply to a chord before it turns into mush. To me, it is more important to know that there are stable intervals somewhere in the tuning than to know whether the tuning approximates a major triad or not. The first question I always ask about a new tuning system: what can I use as a power-chord? In 16-EDO, for instance, I find the 900-cent ~27/16 nicer-sounding than the 1050-cent ~11/6, but the 1050-cent interval works as a power-chord while the 900-cent interval doesn't. This is why I like 16-EDO better than 14 or 15 or 17; all of those do a better job of approximating major and minor triads, but neither has as many stable intervals as 16.

But I digress...the point is, Carl's harmonic series theory probably works as well as it does because 4:5:6=a major triad, and thus the harmonic series is consistent with our culturally-infused preferences.

>"Actually, I don't think it *does* debunk the theory. I thought about just such
>an example a lot last night, and I think because of the fact 60:75:88 is
>actually *closer* to 4:5:6 than 8:12:13, the theory holds."

How so? Sure, the first dyad is the same (5/4), but 88/75 is a "gaping"
43.82931999 or so cents from 6/5 and 6/4 (3/2) is a huge 37 or so cents from
88/60 IE 22/15.

>"IOW, it's not enough just to look at the harmonics and say "this one is higher
>harmonics, so it's going to sound more discordant".
Of course...but it seems obvious to me (look at the above example) that these
are higher harmonics with pretty lousy approximations to lower harmonics. There
no easy "temperament relation" I can see that can be exploited.

>"Higher harmonics can approximate lower harmonics"
Right but, to nearly 40 cents? IMVHO if we could forgive such a huge error
virtually everything could be "easily summarize-able in lower harmonics" to the
point there wouldn't be any higher harmonic that wasn't instantly
summarize-able.

>"so it's actually the middle harmonics that are going to be the most
discordant"
So wait, 22/15 and 88/75 aren't middle harmonics? To me they seem dead smack
in the centers between 10/7 and 3/2 (22/15)...and 88/75 between 7/6 and 6/5?

So yes, I agree with your points about rounding higher to lower harmonics,
o-tonality vs. u-tonality and such, but don't see how they in the vaguest apply
to my above example...which still seems neither harmonically o-tonal/u-tonal or
to relate closely to lower harmonics for the most part.

If you really want to prove you point, it would be great to see numeric
examples...and in similar detail to those I gave.

>"60:75:88 is actually *closer* to 4:5:6 than 8:12:13, the theory holds."

My overall point has nothing to do with what harmonic series closer is closer
to what other one, but rather saying that NON-harmonic series chords can share
similar properties to harmonic series ones.

Specifically, my point is that although o-tonality obviously works in many
cases for consonance, the favoring of the brain of chords which place
larger-spaced ratios at lower frequencies is not exclusive to chords near
patterns from the harmonic series (IE 4:5:6, 5:8:9, etc.).

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Actually, I don't think it *does* debunk the theory. I thought about just such
> >an example a lot last night, and I think because of the fact 60:75:88 is
> >actually *closer* to 4:5:6 than 8:12:13, the theory holds."
>
> How so? Sure, the first dyad is the same (5/4), but 88/75 is a "gaping"
> 43.82931999 or so cents from 6/5 and 6/4 (3/2) is a huge 37 or so cents from
> 88/60 IE 22/15.

Well, obviously it's quite a bit off, but it's a whole hell of a lot closer to 4:5:6 than is the 8:12:13 you used to compare it with. And if you play that chord, I *dare* you to tell me you don't hear it as a less-stable/more-vibrating version of a major triad. Can you honestly say that if someone played you blind a series of intervals within 50 cents sharp or flat of 3/2, of which one of them was 22/15, you could reliably single out the 22/15 as well as you could single out 3/2? I just can't imagine that 22/15 has a unique psychoacoustic identity that is totally unrelated to a 3/2.

> >"IOW, it's not enough just to look at the harmonics and say "this one is higher
> >harmonics, so it's going to sound more discordant".
> Of course...but it seems obvious to me (look at the above example) that these
> are higher harmonics with pretty lousy approximations to lower harmonics. There
> no easy "temperament relation" I can see that can be exploited.

Lousy, yes. But still approximations. I.e. 22/15 is still more an approximation of a fifth than it is an approximation of any other "tunable-by-ear" interval. If you had to round it to the nearest simple ratio, it would be 3/2.

> >"Higher harmonics can approximate lower harmonics"
> Right but, to nearly 40 cents? IMVHO if we could forgive such a huge error
> virtually everything could be "easily summarize-able in lower harmonics" to the
> point there wouldn't be any higher harmonic that wasn't instantly
> summarize-able.

Actually, that's kind of the whole point of harmonic entropy, that this is possible, except for intervals at the local maxima, where multiple higher harmonics/odd-limit ratios coincide in the same pitch-range, and all are too distant from nearby simple ratios to be interpreted either way.

> >"so it's actually the middle harmonics that are going to be the most
> discordant"
> So wait, 22/15 and 88/75 aren't middle harmonics? To me they seem dead smack
> in the centers between 10/7 and 3/2 (22/15)...and 88/75 between 7/6 and 6/5?

Not exactly. 13/9 is the classical mediant between 3/2 and 10/7, and the noble mediant is closer to 16/11. 22/15 is just on the 3/2 side of 16/11, so it falls into 3/2's field of attraction. Between 7/6 and 6/5, the classical mediant is 13/11, and that's pretty close to the noble mediant as well. 88/75 is distinctly on the 7/6 side. So no, these are not middle harmonics.

> So yes, I agree with your points about rounding higher to lower harmonics,
> o-tonality vs. u-tonality and such, but don't see how they in the vaguest apply
> to my above example...which still seems neither harmonically o-tonal/u-tonal or
> to relate closely to lower harmonics for the most part.

Again, your chord is basically a major third plus a sharp subminor third, which translates to a major triad (4:5:6) with a somewhat-flattened fifth. It is still distinctly identifiable as a major triad, despite the mistuning. The 8:12:13 triad is a perfect fifth with an added neutral sixth, which is not remotely like a major triad (4:5:6). Hence, the more complex triad is more "consonant", because it is closer to a major triad. It is not as consonant as a 4:5:6, but it is still more consonant than an 8:12:13.

> >"60:75:88 is actually *closer* to 4:5:6 than 8:12:13, the theory holds."
>
> My overall point has nothing to do with what harmonic series closer is closer
> to what other one, but rather saying that NON-harmonic series chords can share
> similar properties to harmonic series ones.

Every chord can be a harmonic series chord. The harmonic series is infinite, and no matter what the chord, you can find an arbitrarily-good approximation of it somewhere in the series.

> Specifically, my point is that although o-tonality obviously works in many
> cases for consonance, the favoring of the brain of chords which place
> larger-spaced ratios at lower frequencies is not exclusive to chords near
> patterns from the harmonic series (IE 4:5:6, 5:8:9, etc.).

Your example failed to show this, and in fact only supported the point that you were trying to disprove (i.e. that a badly-tuned 4:5:6 sounds more consonant to our ears than a well-tuned 8:12:13).

>"Your example failed to show this, and in fact only supported the point that you
>were trying to disprove (i.e. that a badly-tuned 4:5:6 sounds more consonant to
>our ears than a well-tuned 8:12:13)."

Firstly, I'm favoring the 60:75:88, not the 8:12:13. Read my original
message.....

I'm saying I believe 60:75:88 sounds better despite being high in the
harmonic series and not "temper-able" to anything low in the series. You seem
to think the "fact" 60:75:88 comes across to you as a 4:5:6, despite my never
bringing it up as a point in my topic, is utterly relevant to the 60:75:88's
sounding better.
I am trying to prove that having close fit to the harmonic series (close
meaning...within 12 or so cents) is not always a good predictor of consonance.
And I am using 60:75:88 because 2 of its 3 dyads are nowhere near within 12
cents of anything low in the harmonic series...as an example of a "weird chord
that works surprisingly well". You seem to say the 60:75:88 is somehow
equivalent of "really" low in the harmonic series despite these far-off dyads.

Argh...I'm digging up my original message...
>>>>>>>>>>>original message>>>>>>>>>>>>>>>>"
Carl>"Its fit to a harmonic series?"
Well here we go again... Carl I hear what you are saying, I just don't agree
with it...and yes I can understand something without agreeing with it. Sure,
something very low in the harmonic series IE 3:4:5 or 6:7:8 is almost always
going to work.
But try 8:12:13 vs. 60:75:88. Note 60:75:88 has a 5/4 dyad from 75/60 and a
not-so-far from 7/6 dyad of 88/75...but perhaps more importantly a huge critical
band advantage in having a closest dyad of about 7/6 vs. 13/12!"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>"
...yes, I said advantage as in....I'm favoring the 60:75:88!

Note I mention my points of difference in terms of critical band and dyads
formed AND make my point relating to things low vs. high in the harmonic series,
and not specific to a certain chord.

But, ok, you appear to be taking what I take as a long shot anyhow and saying
my argument is really about 4:5:6 and "low in the harmonic series" has to mean
4:5:6 and then assuming things like 35+ cent deviations from dyads in 4:5:6
still point strongly to 4:5:6.

Now you replied:
>"Well, obviously it's quite a bit off, but it's a whole hell of a lot closer to
>4:5:6 than is the 8:12:13 you used to compare it with. And if you play that
>chord, I *dare* you to tell me you don't hear it as a less-stable/more-vibrating
>version of a major triad."

I'm not talking about tonal "character" here, I'm talking about stability.
An on-topic comparison would be get a listener to tell me they find 8:12:13 more
stable/resolved sounding than 60:75:88.

>"Actually, that's kind of the whole point of harmonic entropy, that this is
>possible, except for intervals at the local maxima, where multiple higher
>harmonics/odd-limit ratios coincide in the same pitch-range, and all are too
>distant from nearby simple ratios to be interpreted either way."
Right that's the basic idea. But again look at my original message. I noted
two things that I think cause more sourness in the 8:12:13 chord. Critical band
and the amount of periodicity of its dyads. Nothing about being pro or anti
harmonic entropy (harmonic entropy is not equal to periodicity).

But if I get what you are saying, you are saying that since dyads in
60:75:88 are not "that far" from the humps in the harmonic entropy graph IE
http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937...that the
brain not only hears it as a 4:5:6, but ALSO hear it as being as resolved as a
4:5:6. Now 88/75 = a bit over 7/6...near a relative "low point" on the curve
and vaguely near 6/5. 75/60, I agree, is a 5/4 (as I admitted before) and
obviously acts as one (IE from a 4:5:6 chord). This leads me toward the 88/60
aka 22/15. It is pretty close to the high entropy point south of 3:2 thus
indicating a "sour note" on his graph.
Thus harmonic entropy analysis seems to give a sweet dyad, a not so sweet
dyad, and a screaming in pain sour dyad. :-D So even if your argument is that
"the reason 60:75:88 works has to do with how well it works via harmonic entropy
which thus 'makes it a 4:5:6' " I still am left with many doubts.

>"the noble mediant is closer to 16/11. 22/15 is just on the 3/2 side of 16/11,
>so it falls into 3/2's field of attraction."
But the Harmonic Entropy curve doesn't seem to have a sudden jump, but a
continuous, well, curve. Thus, yes, 22/15 is rated better than 16/11 on that
curve...I'm just saying that it is not rated as better by much...the rate of
"better-ness acceleration" is not that high at that point...at least from what I
can read of the graph.

>"Lousy, yes. But still approximations. I.e. 22/15 is still more an approximation
>of a fifth than it is an approximation of any other "tunable-by-ear" interval.
>If you had to round it to the nearest simple ratio, it would be 3/2."
Exactly, it's a "lousy" 5th by Harmonic Entropy (HE) standards. And my
argument here is about resolved-ness: I'm taking the points the HE and
periodicity cause the differences in my example and you seem to be introducing a
point that you think HE (instead) explains it. But I'm saying, "even" HE
doesn't appear to explain it and 22/15 comes across as "not much more resolved
than the 16/11 mediant"...regardless of if the brain can still scrape itself to
recognize "yeah, it's a type of fifth".

>"Hence, the more complex triad is more "consonant", because it is closer to a
>major triad. It is not as consonant as a 4:5:6, but it is still more consonant
>than an 8:12:13."
Well ok (finally?) we're back to talking (or at least trying to talk about)
consonance rather than degrees of ambiguity. Could the rest of you weigh in to
this comparison, by ears?

Okay, let me try to clean this up for you.

We both agree that the 60:75:88 is more consonant than the 8:12:13. However, you are arguing that this fact disproves Carl's harmonic series model, whereas I am arguing that it supports it. My argument is that despite the fact that 60:75:88 is too far off to sound "just like" 4:5:6, it is close enough to sound like a "very out of tune" 4:5:6. 8:12:13 does not sound *anything* like a 4:5:6 at all, not even remotely. 8:12 is about 315 cents sharp of 4:5, and 12:13 is about, what, 175 cents flat of 5:6. What I am arguing, thus, is that Carl's harmonic series model suggests that the closer a chord is to 4:5:6, the more consonant it will be. Carl never said this, but I think he would have to in order to make sense. If his theory didn't state this, then it would be quite susceptible to the objection you are raising. I never said 60:75:88 is as resolved-sounding or as consonant as 4:5:6, just that it's *closer* to 4:5:6 than the 8:12:13. Since it's closer to 4:5:6, it "fits" the lower harmonics "better" than the 8:12:13—a perfect example of higher harmonics being closer to low harmonics than the "middle" harmonics. 60:88 is closer to a fifth than is 8:13. 60:75 is closer to a major third than is 8:12. Get it?

-Igs

http://tonalsoft.com/enc/h/harmonic-entropy.aspx

As I understand it, harmonic entropy is the degree to which there is ambiguity
between nearby intervals. So something like 11/10 has high entropy as it can be
fairly easily replaced by 12/11 or 10/9 and something like 3/2 has low entropy
and isn't easily replaced or mis-interpreted as another ratio.

So a side question becomes...how far can you go from a chord BEFORE the brain
stops being able to round it well to the nearest low part of the harmonic
series.
Is 22/15 still clearly decipherable as an 5th, or is it ambiguous "noise"
according to harmonic entropy? How about 18/11? Or 10/7? How about other tones
more than 13 cents from the nearest local low points in the harmonic entropy
graph.

I bring this up as Igs and I were having a discussion on another thread and I
realized that the idea of something having to be less than 10 cents or so to be
considered "tempered"/"easily interpretable" rather than "comma-tic/rough" and
the idea of harmonic entropy in some cases appear to say completely opposite
things!

What is your take?

I wrote:

> did you try it?

That wasn't a rhetorical question. -Carl

>"We both agree that the 60:75:88 is more consonant than the 8:12:13. However,
>you are arguing that this fact disproves Carl's harmonic series model, whereas
>I am arguing that it supports it"

Actually I'm saying his model IS valid in general, but has exception IE
doesn't cover everything, especially cases involving a lot of difference in both
critical band (88/75 beating far less than 13/12) and having generally low-limit
dyads (88/75 being not-so-far from 7/6 dyad and 75/60 being 5/4) despite being
high-limit as a full chord.

>"it is close enough to sound like a "very out of tune" 4:5:6. 8:12:13 does not
>sound *anything* like a 4:5:6 at all, not even remotely"
What to say. I agree with you on that, but that's not the topic. The topic
is comparing how resolved the chords sound, not "how much they sound like
low-limit chords". The thing I keep seeing with your argument that bugs me...is
you seem to be saying the brain can hear anything vaguely near a low-limit chord
like 4:5:6 or 5:6:7 as that chord and that even larger-than-commatic dyadic
differences in 2 of 3 dyads in a chord can be thrown aside. ...Now if such were
true wouldn't a chord like 1/1 6/5 11/8 supposedly be much more consonant than a
16:18:21 simply because the former is "closer to a 5:6:7...which is lower down
the series than a 16:18:21"?

>"Since it's closer to 4:5:6, it "fits" the lower harmonics "better" than the
>8:12:13—a perfect example of higher harmonics being closer to low harmonics
>than the "middle" harmonics. 60:88 is closer to a fifth than is 8:13. 60:75
>is closer to a major third than is 8:12. Get it?"

Yes...you clearly appear to be saying that what matters most is NOT how
accurate a chord is (IE how closely its dyads approach pure, the critical band
dissonance of the dyads)...but rather which low-limit chord it is nearest...even
if it's a downright lousy estimation of that chord. Better to have a Porsche
with misfiring spark plugs than a finely-tuned Volkswagen (even if Porsche is
based on Volkswagen)...in other words.

To make it clear again...I think it works most of the time (say 85-90%)...but
there are enough counter-examples to make me wonder if there is a hack. Yet
another example is the C E F A chord...which doesn't reduce to ANYTHING low in
the harmonic series, yet sounds just as good as several low-limit chords, at
least to my ears. The point? I figure if we find a hack we can create scales
with plenty of sweet chords without, say, just building them from de-tuning
notes in stacked triads...thus resulting in more possibilities for scales.

I feel funny because before you seems to be promoting the idea that 2 of 3
dyads in a chord allows the third dyad to be off (seems to open a lot of
possibilities)...but now you seem to be saying nothing creative can be done with
it unless than overall structure rounds some low-in-the-harmonic-series chord
(tear tear). :-D This is sad because it seems to say the only good way to go is
by simply enabling the same old 5-limit (and maybe occasionally 7-limit) type
chords/moods in different ways.

>"What I am arguing, thus, is that Carl's harmonic series model suggests that
>the closer a chord is to 4:5:6, the more consonant it will be"

Carl...perhaps you will have to clarify this.

(Carl) when you say "order in the harmonic series"...how far can a chord be
away from a low-harmonic-series chord to still count as "low in the harmonic
series"? Can it have notes 12+ cents away? Does it need to have at two of the
dyads to be within a certain error, or three, or even just one? Are you talking
about your ideal in terms of harmonic entropy or not and how so?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> What to say. I agree with you on that, but that's not the topic. The topic
> is comparing how resolved the chords sound, not "how much they sound like
> low-limit chords".

I am not a proponent of the "harmonic series as a model of consonance" theory, just so we can get that straight. I was only pointing out that your objection wasn't the right objection to make to it, since the theory can address and incorporate it. But FWIW, if the theory is correct, then "how resolved a chord sounds" is basically the same as "how much it sounds like a low-limit chord".

> The thing I keep seeing with your argument that bugs me...is
> you seem to be saying the brain can hear anything vaguely near a low-limit chord
> like 4:5:6 or 5:6:7 as that chord and that even larger-than-commatic dyadic
> differences in 2 of 3 dyads in a chord can be thrown aside. ...Now if such were
> true wouldn't a chord like 1/1 6/5 11/8 supposedly be much more consonant than a
> 16:18:21 simply because the former is "closer to a 5:6:7...which is lower down
> the series than a 16:18:21"?

Also FWIW, I think 5:6:7 is an excellent example of a gaping failure of the theory to predict consonance, because 5:6:7 is, at least to my ears, a "beatless dissonance". It does not sound restful or resolved to me at all, though it is very stable. Even a pure 5:6:7 sounds less resolved than a 16:18:21 to me. Though the 16:18:21 is far less stable, and doesn't sound very "clean" in low registers with harmonic timbres. The 5:6:7 is OTOH very clean-sounding.

> Yes...you clearly appear to be saying that what matters most is NOT how
> accurate a chord is (IE how closely its dyads approach pure, the critical band
> dissonance of the dyads)...but rather which low-limit chord it is nearest...even
> if it's a downright lousy estimation of that chord. Better to have a Porsche
> with misfiring spark plugs than a finely-tuned Volkswagen (even if Porsche is
> based on Volkswagen)...in other words.

Not which "matters most". What matters most is, and always has been, the aesthetic preferences of listeners and composers, and that simply cannot be predicted or formalized with 100% accuracy (maybe 80% or 90% on a good day). Really, what I think is going on is that 4:5:6 is the most stable representation of a major chord, which is the most familiar chord in our culture. It is, in a sense, our ideal of "what sounds good", and I think that due to cultural conditioning, people will prefer something that sounds familiar--even if out-of-tune--to something that sounds very unfamiliar, even if it's very in-tune.

> To make it clear again...I think it works most of the time (say 85-90%)...but
> there are enough counter-examples to make me wonder if there is a hack. Yet
> another example is the C E F A chord...which doesn't reduce to ANYTHING low in
> the harmonic series, yet sounds just as good as several low-limit chords, at
> least to my ears. The point? I figure if we find a hack we can create scales
> with plenty of sweet chords without, say, just building them from de-tuning
> notes in stacked triads...thus resulting in more possibilities for scales.

Beyond a certain level of "chord complexity", preferences simply become too unpredictable, and this is the pitfall of conflating "sounding nice" with "sounding in-tune". There's no "hack" for aesthetic preferences. In many cultures, people can sing neutral thirds and neutral seconds reliably, at least melodically or against a drone, and probably think those intervals sound quite nice. Here in the West, those intervals are so unfamiliar that most people consider them dissonances, or at best "alien" intervals that they have no way of interpreting. Humanistically-speaking, I think you can get away at the very least with any scale, so long as its intervals are spread somewhat-evenly across an interval somewhat close to an octave; with such a scale, someone somewhere will find the music pleasant. I guarantee it.

> I feel funny because before you seems to be promoting the idea that 2 of 3
> dyads in a chord allows the third dyad to be off (seems to open a lot of
> possibilities)...but now you seem to be saying nothing creative can be done with
> it unless than overall structure rounds some low-in-the-harmonic-series chord
> (tear tear). :-D This is sad because it seems to say the only good way to go is
> by simply enabling the same old 5-limit (and maybe occasionally 7-limit) type
> chords/moods in different ways.

Like I said, the only thing I was arguing was that your objection to Carl's theory wasn't valid. I believe Carl's theory is pretty valid for our culture, but not universally-valid for humankind and not necessarily the most helpful theory for dealing with tunings where a near-4:5:6 isn't an option.

However, Carl has quite neatly shown that putting simple dyads together is not by any means always effective. Lots of simple dyads, when combined, make nasty chords, as I noted with chords like 12:15:25 (a stack of 5/4's), or 9:12:16 (a stack of 4/3's), or even 5:6:7. So while I used to think that if a dyad could be split into two simpler dyads (the way 21/16 splits into 9/8 and 7/6), that dyad could be rendered consonant if played as a triad of its simpler components, I see now that that theory fails in more places than it succeeds.

That said, I *still* don't understand why I find 16:18:21 to be so dang nice-sounding. Even if I'm hearing it as a 9:10:12, I still don't know why I'd like that better than 5:6:7, but I undeniably do. If Carl's theory can explain that, I guess I'm sold.

> >"What I am arguing, thus, is that Carl's harmonic series model suggests that
> >the closer a chord is to 4:5:6, the more consonant it will be"
>
> Carl...perhaps you will have to clarify this.

Indeed, I hope he does.

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Igs wrote:
>
> > Having interval complexity decrease as pitch increases seems
> > to make for a less-stable sound than having interval complexity
> > increase as pitch increases. This must also be why I've always
> > found supermajor chords to be so much less stable than subminor.
>
> That's a generalization of two experiences -- are you sure
> it's the right one?
>
> Subminor is 6:7:9. Supermajor is 14:18:21. So that's also
> consistent with my harmonic series explanation.

Indeed.

> > > > So what other measurable property to a chord could there
> > > > be, which determines its concordance?
> > >
> > > Its fit to a harmonic series?
> >
> > I'm not entirely sure I'd agree with that. I'd think a
> > chord like 1/1-5/4-3/2-7/4-9/4-8/3 would be more concordant
> > than a 4:5:6:7:9:11 chord, but the latter one fits the
> > harmonic series better.
>
> You'd think it would, but did you try it? The 32/21
> is a doosey.

And so it is! I did indeed try it, and you are quite correct.

However, I've given this more thought. Backing off the on the degree "polyphony" (in the MIDI/synth sense, not the compositional sense) from hexads to triads, of which the latter are generally more intelligible from a consonance standpoint, can you explain why I hear 5:6:7 as being more dissonant than 16:18:21? Or why I find 4:5:6 more consonant than 3:4:5? These, I can assure you, I have tried, and my preference is undeniably for the higher-harmonic chords, even if they are somewhat "less stable". The more chords I examine, the more I come to be convinced that "stability"/"concordance" and "consonance" are very different and not necessarily related. What do you say to this?

-Igs

> > Even a 1/1-6/5-3/2 is bound to be more concordant than a
> > 6:7:9. There's gotta be more factors going on there.
>
> 10:12:15 does have greater Tenney height than 6:7:9. I'm
> not sure it's more consonant. In the case of dyads we also
> found that when the geometric mean of the Tenney height was
> greater than 9, it stopped predicting consonance well.
>
> -Carl
>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> I'm saying I believe 60:75:88 sounds better despite being high in the
> harmonic series and not "temper-able" to anything low in the series. You seem
> to think the "fact" 60:75:88 comes across to you as a 4:5:6, despite my never
> bringing it up as a point in my topic, is utterly relevant to the 60:75:88's
> sounding better.

22/15 is sometimes considered to be a blown fifth, so Carl is not alone in taking it to be a fifth of sorts.

Hi Igs,

I haven't been following the miles of stuff Michael and you
have been writing, but none of what I've sampled makes the
least bit of sense to me.

>I believe Carl's theory is pretty valid for our culture, but
>not universally-valid for humankind

Why do believe this? Tried

http://jn.physiology.org/cgi/content/abstract/89/3/1603

or

http://www.mmk.ei.tum.de/persons/ter/top/virtualp.html

?

>and not necessarily the most helpful theory for dealing with
>tunings where a near-4:5:6 isn't an option.

I don't know where you got the 4:5:6 thing.

>can you explain why I hear 5:6:7 as being more dissonant
>than 16:18:21?

You do? In what registers? That's a baffling report to me.
Try again maybe?

>The more chords I examine, the more I come to be convinced that >"stability"/"concordance" and "consonance" are very different
>and not necessarily related. What do you say to this?

The situation is comparable to that of IQ. There may be
many different kinds of intelligence, but there's also a
single principle component which correlates them. And
there may be no intuitive name for that component -- it may
not be a "kind of intelligence" itself. Psychologists
call it "g".

It's the same with consonance. Unless you can define and
measure "stability" vs. "concordance" vs. "consonance",
you're probably not going to get anywhere talking about them.
The psych way to do it is to quiz people using all the
different terms you can think of and then do a PCA on the
results. In lieu of that, people have just ignored any
distinctions, as that has a reasonable chance of getting the
same result.

>Or why I find 4:5:6 more consonant than 3:4:5?

This is widely reported, and here's where I contradict what
I just said. It's been suggested that one of the "kinds of
consonance" should be called "rootedness" (Paul Erlich
coined it). Chords get a rootedness boost if their virtual
fundamental is an octave duplicate of their lowest note.

I definitely find 4:5:6 more consonant than 3:4:5, but the
difference is pretty slight. If I start inverting the
chord across the keyboard, the difference disappears.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> I definitely find 4:5:6 more consonant than 3:4:5, but the
> difference is pretty slight. If I start inverting the
> chord across the keyboard, the difference disappears.

Of course this has long been a factor in common practice music, where 3:4:5 is regarded as less stable than 4:5:6.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Igs,
>
> I haven't been following the miles of stuff Michael and you
> have been writing, but none of what I've sampled makes the
> least bit of sense to me.
>
> >I believe Carl's theory is pretty valid for our culture, but
> >not universally-valid for humankind
>
> Why do believe this? Tried
>
> http://jn.physiology.org/cgi/content/abstract/89/3/1603
>
> or
>
> http://www.mmk.ei.tum.de/persons/ter/top/virtualp.html

I'll look into them! But I know there is a ton of world music out there where pitches are commonly performed quite off from what should be considered "consonant", and yet people find it pleasing as heck to listen to.

> >can you explain why I hear 5:6:7 as being more dissonant
> >than 16:18:21?
>
> You do? In what registers? That's a baffling report to me.
> Try again maybe?

No dice, Carl. 5:6:7 is more stable, but I hear it as basically a diminished chord: angry, tense, and dark. 16:18:21 is kind of more like some sort of suspended chord, floaty, ethereal, and peaceful, and while it does beat noticeably, it does so in a peaceful way that I hear as quite resolved. I like it best at middle C or higher. Trust me, I've given myself a lot of time with this one, it's not my ears playing tricks on me.

> >The more chords I examine, the more I come to be convinced that >"stability"/"concordance" and "consonance" are very different
> >and not necessarily related. What do you say to this?
>
> The situation is comparable to that of IQ. There may be
> many different kinds of intelligence, but there's also a
> single principle component which correlates them. And
> there may be no intuitive name for that component -- it may
> not be a "kind of intelligence" itself. Psychologists
> call it "g".
>
> It's the same with consonance. Unless you can define and
> measure "stability" vs. "concordance" vs. "consonance",
> you're probably not going to get anywhere talking about them.
> The psych way to do it is to quiz people using all the
> different terms you can think of and then do a PCA on the
> results. In lieu of that, people have just ignored any
> distinctions, as that has a reasonable chance of getting the
> same result.

I thought "consonant" just meant "pleasing to listen to" and "stability/concordance" meant "degree of beatlessness". One is measurable pretty much only statistically, using surveys, and the other is measurable mathematically. It's long been assumed (at least by some folks here) that they're interchangeable, but I know I'm not alone in thinking they're not. I mean, if people hated beating, modulation effects would not be so popular!

> >Or why I find 4:5:6 more consonant than 3:4:5?
>
> This is widely reported, and here's where I contradict what
> I just said. It's been suggested that one of the "kinds of
> consonance" should be called "rootedness" (Paul Erlich
> coined it). Chords get a rootedness boost if their virtual
> fundamental is an octave duplicate of their lowest note.
>
> I definitely find 4:5:6 more consonant than 3:4:5, but the
> difference is pretty slight. If I start inverting the
> chord across the keyboard, the difference disappears.

Aha! Well, it's good to know I'm not going crazy then. So you admit, there may be another component to it beyond "fit" to a harmonic series? Perhaps 16:18:21 is more rooted than 5:6:7, and that's why I like it better? How do you calculate that?

-Igs

Igs wrote:

> I'll look into them! But I know there is a ton of world music
> out there where pitches are commonly performed quite off from
> what should be considered "consonant", and yet people find it
> pleasing as heck to listen to.

This has to do with consonance... how exactly?

> > >can you explain why I hear 5:6:7 as being more dissonant
> > >than 16:18:21?
> >
> > You do? In what registers? That's a baffling report to me.
> > Try again maybe?
>
> No dice, Carl. 5:6:7 is more stable,
[snip]
> but I hear it as
> basically a diminished chord: angry, tense, and dark.

That's all fine and well, but I thought we were talking
about consonance. "Stable" and "tense" sound like they
may be in the right ballpark. The fact that you're
thinking in terms of "diminished" and "suspended" is not
encouraging - that may be cultural conditioning.

> I thought "consonant" just meant "pleasing to listen to"

It doesn't *mean* anything. It's a perception, or part
of a perception. It can be described in various ways.
It's kind of like "green". When we say "green" I'm fairly
confident we're talking about the same abstract thing even
though there's no way to know for certain. If we try to
nail down "green" we get into trouble - as you'll know well
from your philosophy background.

>and "stability/concordance" meant "degree of beatlessness"

To use tuning list vernacular (almost nobody does, even on
the tuning list), consonance actually refers to musical
tension, related to ALL the cues that music can convey,
whereas concordance relates to what I've been talking about
in this thread so far. In psychoacoustics, the latter is
called "sensory consonance" or "consonance" for short and
that's what I've been doing in this thread. There we don't
want anything to do with melody, rhythm, cultural
conditioning, artistic intent, mood imagery, etc. We just
want the instantaneous sensation of "restfulness" and "purity".

Beating is actually not the main object of sensory consonance
research -- that is roughness, which is related to beating.
But there is also the "tonalness" of Terhardt and the
"rootedness" of Erlich. Those are "kinds of consonance"
that any single rule like Tenney height lumps together.

>It's long been assumed (at least by some folks here) that
>they're interchangeable,

?

> So you admit, there may be another component to it beyond
> "fit" to a harmonic series?

??? When did I ever say anything like that?

> Perhaps 16:18:21 is more rooted than 5:6:7, and that's
> why I like it better? How do you calculate that?

You need to find the virtual fundamental of the chord.
Not all chords have a single dominant VF. 16:18:21 is
usually approximating 1/1 9/8 4/3 and it probably doesn't.
The higher you voice it the more likely it is to evoke
16:18:21 proper, in which case it gets VF 16 (or 4 or 1,
etc). 5:6:7 has VF 4 in most cases and therefore takes
a 'rootedness hit' if you believe in such things.

-Carl

Igs, you said

>"But FWIW, if the theory is correct, then "how resolved a chord sounds" is
>basically the same as "how much it sounds like a low-limit chord". "
...but also said
"I am not a proponent of the "harmonic series as a model of consonance" theory,
just so we can get that straight."

So let me get this right...you are saying you don't believe chords have to
comply with the series directly, but that they do need to be close enough for
the brain to snap it to the series (which to you, apparently...close enough can
be well over 20 cents off for 2 of 3 notes in a triad, for example)?
What can I say, I agree with that idea in general, but not with such a
generous slack for error. Something more like 12-14 cents error, ok...but
20+...not to my ears. To me that's almost like saying a (shrunk) 9:11:13 chord
and a wider 4:5:6 chord are comparable in resolved-ness...pretty big jump IMVHO.

ME> The thing I keep seeing with your argument that bugs me...is
> you seem to be saying the brain can hear anything vaguely near a low-limit
>chord
>
> like 4:5:6 or 5:6:7 as that chord and that even larger-than-commatic dyadic
> differences in 2 of 3 dyads in a chord can be thrown aside. ...Now if such
>were
>
> true wouldn't a chord like 1/1 6/5 11/8 supposedly be much more consonant than
>a
>
> 16:18:21 simply because the former is "closer to a 5:6:7...which is lower down

> the series than a 16:18:21"?

Igs>"Also FWIW, I think 5:6:7 is an excellent example of a gaping failure of the
theory to predict consonance, because 5:6:7 is, at least to my ears, a
"beat-less dissonance"....Even a pure 5:6:7 sounds less resolved than a 16:18:21
to me. Though the 16:18:21 is far less stable, and doesn't sound very "clean" in
low registers with harmonic timbres. The 5:6:7 is OTOH very clean-sounding."

I have noticed this as well, especially for things like chords starting at the
5th harmonic. Some parts of the series, no matter how periodic and
"major"/"o-tonal" mathematically, just seem to have a sour mood.

>"Not which "matters most". What matters most is, and always has been, the
>aesthetic preferences of listeners and composers, and that simply cannot be
>predicted or formalized with 100% accuracy (maybe 80% or 90% on a good day)."
I'd be happy with a steady 80%...and have yet to find a well-established
theory that will give me that. I've found having 70% of the chord's dyads sound
good individually usually means a good chord more than anything else (sometimes
this means the dyads have good critical band, other times periodicity, other
times neither AKA the 5:6:7 example failing).

>"There's no "hack" for aesthetic preferences. In many cultures, people can sing
>neutral thirds and neutral seconds reliably, at least melodically or against a
>drone, and probably think those intervals sound quite nice."
I don't think there is a foolproof hack...but I am suspicious something could
be close enough to convince people neutral seconds and thirds and such can be
used in chords and not, say, for the type of monophonic structures often in
ethnic music. One fleeting example of this: Jacky Ligon's work for the "Crack
My Pitch Up" album; people who usually think micro-tonal music is wrist-slitting
music think it's oddly relaxing. And he uses neutral seconds extensively in
chords and harmony.

>"Humanistically-speaking, I think you can get away at the very least with any
>scale, so long as its intervals are spread somewhat-evenly across an interval
>somewhat close to an octave; with such a scale, someone somewhere will find the
>music pleasant. I guarantee it."
So you are saying ET scales have an instant advantage? I think the problem is
people have trouble making or finding scales that are "multi-tonal" rather than
"a-tonal". Meaning that people think
A) TET scale like 12TET are "tonal" because certain chords can be depended on
the sound resolved regardless of transposition and certain notes are stressed
(namely tonic and dominant as 'relaxed' chord roots).
B) People think anything not TET must drop virtually all ability to have certain
dependable intervals across the scale so they call it "a-tonal" as in "lacking
any dominant tones" rather than "multi-tonal". The idea that you can have, say,
two different types of second of which you get different ones by switching the
root (IE in Jacky Ligon's scales) and still have the ability to use them in
similarly strong and confident ways.

IMVHO, the term a-tonal should be scrapped...it seems to give the impression
that odd chords and unequal temperaments can never point to a reliable sense of
resolved-ness.

>"However, Carl has quite neatly shown that putting simple dyads together is not
>by any means always effective. Lots of simple dyads, when combined, make nasty
>chords, as I noted with chords like 12:15:25 (a stack of 5/4's), or 9:12:16 (a
>stack of 4/3's), or even 5:6:7."
Well 5:6:7 fails under both theories. But you're right (to my ears) about
Carl's point with the stack of 5/4's and 4/3rd's. Then again I've found that's
almost universally a bad idea to stack the same interval on top of itself
regardless of the theory...it's basically a recipe for commas. Also when you
think about it...you are pointing to the same harmonic series with equal
strength in two different places...how on earth would you expect you brain to
put importance on one over the other clearly/easily?

>"So while I used to think that if a dyad could be split into two simpler dyads
>(the way 21/16 splits into 9/8 and 7/6), that dyad could be rendered consonant
>if played as a triad of its simpler components, I see now that that theory fails
>in more places than it succeeds."
Hmm...here's a challenge: find an example of a sour chord with sweet dyads
that

A) Doesn't use stacked interval (as Carl's do)
B) Doesn't use any dyads with bizarrely close critical band (IE 13/12 or closer)
C) Isn't 5:6:7 :-D

>"That said, I *still* don't understand why I find 16:18:21 to be so dang
>nice-sounding. Even if I'm hearing it as a 9:10:12, I still don't know why I'd
>like that better than 5:6:7, but I undeniably do. If Carl's theory can explain
>that, I guess I'm sold."

Call me pessimistic but, I'm betting Carl's "theory" can't because, so far,
I've seen it as very much pointing to "anything low enough in 'odd limit' is
automatically golden".

I find the 8th and 16th harmonic very odd. 15/8 even by itself, for example,
is relatively high limit but sounds rather resolved to my ear and more strong
than, say 11/6 or 9/5 despite being more "shaky/beating". I'd say it's a "mood
thing"...but it would be great if someone could explain it mathematically.
The other thing obvious to me is 16:18:21 has 9/8 and 7/6 in it (yep, the
classic 2 out of 3 solid dyads with little critical band dissonance). Try
16:19:21...I swear it's even better because the brain seems to like progression
in chord from lower/larger to higher/smaller intervals...no clue why but I
haven't found any counter-examples for that one no matter how close/far a chord
is from the beginning of the harmonic series.

Igs>"I thought "consonant" just meant "pleasing to listen to" and
"stability/concordance" meant "degree of beat-lessness". One is measurable
pretty much only statistically, using surveys, and the other is measurable
mathematically. It's long been assumed (at least by some folks here) that
they're interchangeable, but I know I'm not alone in thinking they're not."

I don't consider beatless-ness "instant consonance". Nor do I consider
rooted-ness that. The one common thread I see is too much error in any one of
these things almost always results in a sense of tension.
Got two 17/16 intervals beating in your chord?...it will very likely sound a
bit like a skipping record. Got a whole bunch of not-so-periodic 11-limit dyads
in your chord or consecutive intervals IE 4/3 * 4/3?...your brain is likely
going to be tripping over itself trying to find a solid root tone. The trick,
I'm thinking, is moderate use of such "errors" actually adds pleasant variety to
music, simply don't let any one of these things get too out of control or it
will start to sound random; as if the musician has no goal.

I just call it "resolved-ness" for that reason...because
consonance/concordance/beating/etc. all fall under the umbrella of helping
determine how much sense of tension is in a chord. Making and resolving
tension, to me, becomes the end goal...because, in the end, controlling tension,
riding the edge of a cliff without "going too random" and falling off, is a huge
part of what makes music exciting.

>"I mean, if people hated beating, modulation effects would not be so popular!"
True...I guess you could say the difference is "does the beating sound
controlled or not"? Sure some people are good at making LFO's roll with the
rhythm of music and add flow and texture, but putting an amplitude modulator on
random settings or ridiculously deep settings (often like the result of
excessive beating in chords) sounds like a broken record (often literally),
trust me...

>"The higher you voice it the more likely it is to evoke
16:18:21 proper, in which case it gets VF 16 (or 4 or 1,
etc). 5:6:7 has VF 4 in most cases and therefore takes
a 'rootedness hit' if you believe in such things."

So wait...
A) "5:6:7 makes the brain site 4 as the VF" (why that and not 5)?
B) "16:18:21 proper, in which case it gets VF 16 (or 4 or 1,
etc)."
So B) has no obvious defined root and thus jumps to being perceived as having
root 1 and thus having a root lower and more stable in the series?

I'm still trying to figure out how/why 5:6:7 takes a "rootedness hit"....

On Mon, Aug 16, 2010 at 9:56 PM, Carl Lumma <carl@...> wrote:
>
> Why do believe this? Tried
>
> http://jn.physiology.org/cgi/content/abstract/89/3/1603

Wow. So in the fulltext, there's this snippet:

"This unit had 2 frequency peaks near 9.5 kHz (CF1) and 19 kHz (CF2),
respectively (indicated by arrows). Note that CF1 and CF2 are
harmonically related. B, a: 3-D FRA of a multipeaked unit with 3
frequency peaks. b: dot raster showing that this unit was responsive
to 3 frequency regions near 7, 10.5,and 21 kHz, respectively, at 60 dB
SPL. C, a: 3-D FRA of a multipeaked unit that was responsive to 3
frequency regions near 4, 8,and 16 kHz, respectively, at 20 dB SPL"

Some of these frequencies are pretty random. There are neurons that
only respond to 7, 10.5, and 21 kHz? What about every other multiple
of 3.5 kHz? I only have time to skim this right now but this seems
really interesting.

> >can you explain why I hear 5:6:7 as being more dissonant
> >than 16:18:21?
>
> You do? In what registers? That's a baffling report to me.
> Try again maybe?

Perhaps also that 5:6:7 can be easily envisioned as a diminshed chord
within the diatonic "map," and 16:18:21 sounds like some kind of
quasi-suspended C-D-F type thing. From a diatonic standpoint, C-D-F is
more functionally consonant and relaxing and pleasant sounding than
C-Eb-Gb is, at least to my ears.

However, if you "flip your brain" around to not hear it in a diatonic
sense, but rather just in terms of the timbres produced - 5:6:7 to me
certainly sounds more harmonic, resonant, consonant, periodic,
whatever etc. In that regard.

> It's the same with consonance. Unless you can define and
> measure "stability" vs. "concordance" vs. "consonance",
> you're probably not going to get anywhere talking about them.
> The psych way to do it is to quiz people using all the
> different terms you can think of and then do a PCA on the
> results. In lieu of that, people have just ignored any
> distinctions, as that has a reasonable chance of getting the
> same result.

> This is widely reported, and here's where I contradict what
> I just said. It's been suggested that one of the "kinds of
> consonance" should be called "rootedness" (Paul Erlich
> coined it). Chords get a rootedness boost if their virtual
> fundamental is an octave duplicate of their lowest note.
>
> I definitely find 4:5:6 more consonant than 3:4:5, but the
> difference is pretty slight. If I start inverting the
> chord across the keyboard, the difference disappears.

This might also be explained by the fact that there's often a note in
the bass as well which mimics the lowest note of the triad. Perhaps
we're culturally conditioned to imagine this note too, sort of
quasi-putting it in there mentally. Take the following two chords, for
instance:

2:4:5:6
2:3:4:5

In this context, to my ears, 2:3:4:5 is way more consonant than
2:4:5:6. But now let's double the bottom note instead of just doubling
the C:

2:4:5:6
3:6:8:10

Now the 2:4:5:6 is more consonant.

-Mike

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> So wait...
> A) "5:6:7 makes the brain site 4 as the VF" (why that and not 5)?

The GCD is 1. The VFs presumably exist in octave-equivalent
land.

> B) "16:18:21 proper, in which case it gets VF 16 (or 4 or 1,
> etc)."
> So B) has no obvious defined root and thus jumps to being
> perceived as having root 1 and thus having a root lower and more
> stable in the series?

The chord may actually be 16:18:21 in higher registers.
Then its musical root (16) corresponds to its VF and it
is "rooted".

> I'm still trying to figure out how/why 5:6:7 takes a
> "rootedness hit"....

4 != 5

-Carl

Mike wrote:

> > Why do believe this? Tried
> >
> > http://jn.physiology.org/cgi/content/abstract/89/3/1603
>
> Wow. So in the fulltext, there's this snippet:
>
> "This unit had 2 frequency peaks near 9.5 kHz (CF1) and
> 19 kHz (CF2), respectively (indicated by arrows). Note that
> CF1 and CF2 are harmonically related. B, a: 3-D FRA of a
> multipeaked unit with 3 frequency peaks. b: dot raster
> showing that this unit was responsive to 3 frequency regions
> near 7, 10.5,and 21 kHz, respectively, at 60 dB SPL. C,
> a: 3-D FRA of a multipeaked unit that was responsive to
> 3 frequency regions near 4, 8,and 16 kHz, respectively,
> at 20 dB SPL"
>
> Some of these frequencies are pretty random.

They're all ratios of 2 or 3.

> There are neurons that only respond to 7, 10.5, and 21 kHz?
> What about every other multiple of 3.5 kHz?

Ah. There are millions of neurons doing this. The way
neurons work, they can only respond to absolute
frequency(ies), but neurons are cheap, so the brain just
throws them at the problem.

> However, if you "flip your brain" around to not hear it in a
> diatonic sense, but rather just in terms of the timbres
> produced - 5:6:7 to me certainly sounds more harmonic,
> resonant, consonant, periodic, whatever etc. In that regard.

Indeed, and I bet the vast majority of listeners would agree.

> Take the following two chords, for instance:
>
> 2:4:5:6
> 2:3:4:5
>
> In this context, to my ears, 2:3:4:5 is way more consonant than
> 2:4:5:6.

Ditto.

> But now let's double the bottom note instead of just
> doubling the C:
>
> 2:4:5:6
> 3:6:8:10
>
> Now the 2:4:5:6 is more consonant.

Ditto.

-Carl

I wrote:

> Ah. There are millions of neurons doing this. The way
> neurons work, they can only respond to absolute
> frequency(ies), but neurons are cheap, so the brain just
> throws them at the problem.

Also note, in humans, vowels are defined by *absolute*
frequencies. See for instance:
http://cnx.org/content/m15459/latest/#img-voweltable

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> That's all fine and well, but I thought we were talking
> about consonance. "Stable" and "tense" sound like they
> may be in the right ballpark. The fact that you're
> thinking in terms of "diminished" and "suspended" is not
> encouraging - that may be cultural conditioning.
>
> > I thought "consonant" just meant "pleasing to listen to"
>
> It doesn't *mean* anything. It's a perception, or part
> of a perception. It can be described in various ways.
> It's kind of like "green". When we say "green" I'm fairly
> confident we're talking about the same abstract thing even
> though there's no way to know for certain. If we try to
> nail down "green" we get into trouble - as you'll know well
> from your philosophy background.

Whoa, man. No need to get all metaphysical. "Consonant" is a word, and words have meanings. Or at least definitions. As far as I understand the use of the term "consonant", it describes a sound that is capable of producing a specific brain-state (i.e. pleasure). "Green" is not a good analogy to "consonant". "Green" is a good analogy for "major third" (or, more properly, "C"), perhaps, but the proper visual analog of "consonant" is something like "pretty". We can measure "green" as stimulation of specific nerves by specific frequencies of electromagnetic radiation, with appropriate neurological sequelae in the visual cortex. We can safely predict all light within the range of "green" will stimulate the optic nerves uniformly in all healthy individuals, and that there will be at least some uniformity in the neurological sequelae. Different cultures divide the color spectrum differently, of course, so at some points the sequelae will diverge depending on culture and individual associations with the initial stimuli. But to a point, there is uniformity. "Pretty" is something that we cannot reduce in such a way; the stimuli capable of producing the brain-state associated with "pretty" are too non-uniform to be modeled accurately and completely. If being "green" was all it took for something to be "pretty", that'd be one thing; but it clearly isn't.

> >and "stability/concordance" meant "degree of beatlessness"
>
> To use tuning list vernacular (almost nobody does, even on
> the tuning list), consonance actually refers to musical
> tension, related to ALL the cues that music can convey,
> whereas concordance relates to what I've been talking about
> in this thread so far. In psychoacoustics, the latter is
> called "sensory consonance" or "consonance" for short and
> that's what I've been doing in this thread.
>
So in tuning list vernacular, "consonance" is related to "all the cues music can convey", but in psychoacoustics, it means the same as "concordance" in tuning list vernacular (which takes into account only certain selected cues), and you have been using the psychoacoustics definition instead of the tuning list vernacular definition. Yeah, that's a little confusing, seeing as how this is the tuning list.

> >It's long been assumed (at least by some folks here) that
> >they're interchangeable,
>
> ?

Don't give me that question-mark! You *just* described how psychoacousticians use "consonance" to mean the same thing as "concordance", and that you have also been conflating them, in spite of the fact that here on the tuning list, they are considered distinct (at least by some). You *just* gave me a definition of consonance that is different from that of concordance, which means there is basis for a semantic distinction. Are you retracting that in the same breath, saying consonance *shouldn't* be (or isn't) distinct from concordance?

> There we don't
> want anything to do with melody, rhythm, cultural
> conditioning, artistic intent, mood imagery, etc. We just
> want the instantaneous sensation of "restfulness" and "purity".

Oh, of course. Because, you know, music has nothing to do with cultural conditioning, artistic intent, mood imagery, etc. If you want to study and understand music, of course it makes sense to leave out all the things that distinguish "music" from "sound". Psychoacoustics is to music what color psychology is to art. Knowing the effect the color green has on human mood won't tell you why (or whether) a Van Gogh painting is beautiful, nor will it tell you how to paint like him. It'll tell you what effect staring at a green wall has on human mood, nothing more. It's valuable in its own right, but not necessarily in any aesthetic sense. But if color psychologists started running around saying "green is a prettier color than red because it's more relaxing to the mood", I think we'd all know that to be nonsense. This is why it is important not to conflate measurable objective properties--like concordance--with subjective properties like consonance (aka "sonic prettiness") or musical usefulness. It leads to making claims that cannot be tested or substantiated meaningfully.

> > So you admit, there may be another component to it beyond
> > "fit" to a harmonic series?
>
> ??? When did I ever say anything like that?
>
When you mentioned "rootedness", specifically as being distinct from "fitting a harmonic series".

-Igs

On Tue, Aug 17, 2010 at 3:58 AM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > That's all fine and well, but I thought we were talking
> > about consonance. "Stable" and "tense" sound like they
> > may be in the right ballpark. The fact that you're
> > thinking in terms of "diminished" and "suspended" is not
> > encouraging - that may be cultural conditioning.
> >
> > > I thought "consonant" just meant "pleasing to listen to"
> >
> > It doesn't *mean* anything. It's a perception, or part
> > of a perception. It can be described in various ways.
> > It's kind of like "green". When we say "green" I'm fairly
> > confident we're talking about the same abstract thing even
> > though there's no way to know for certain. If we try to
> > nail down "green" we get into trouble - as you'll know well
> > from your philosophy background.
>
> Whoa, man. No need to get all metaphysical. "Consonant" is a word, and words have meanings. Or at least definitions. As far as I understand the use of the term "consonant", it describes a sound that is capable of producing a specific brain-state (i.e. pleasure). "Green" is not a good analogy to "consonant". "Green" is a good analogy for "major third" (or, more properly, "C"), perhaps, but the proper visual analog of "consonant" is something like "pretty".

The strongest argument for there being multiple types of "consonance" is
what you mentioned above, and what I responded to - 5:6:7 vs 16:18:21.

5:6:7 is a rough C-Eb-Gb, which is a very "dissonant" chord. 16:18:21
is a mistuned C-D-F, and C-D-F is much less dissonant. However, if you
stop thinking in terms of what the "chords" produced are and listen to
the actual timbres produced, 5:6:7 "fuses" into a single note better
than 16:18:21.

But when we compare 5:6:7 and 16:18:21, it's almost like comparing a
pure 6/5 and a sharp 5/4. Which is more consonant - the pure interval
of higher complexity and the mistuned interval of lower complexity?
Then you have to factor in that when placed within a diatonic
template, the other notes make 5/4 perceived as a "happier" interval
and 6/5 as a "sadder" or "darker" interval, and we're often used to
imagining these other notes getting involved. Which is more consonant?

I think that a maj7 chord is very pretty and stable and much more so
than a dom7 chord. So I think that 8:10:12:15 is much prettier
sounding than 4:5:6:7. But, 4:5:6:7 is definitely more resonant and
periodic and concordant and whatever you want to call it. So which is
more consonant in the end?

-Mike

MikeB>"I think that a maj7 chord is very pretty and stable and much more so
than a dom7 chord. So I think that 8:10:12:15 is much prettier
sounding than 4:5:6:7. But, 4:5:6:7 is definitely more resonant and
periodic and concordant and whatever you want to call it. So which is
more consonant in the end?"

IMVHO, whichever one produces the greatest sense of relaxation. It seems to
be people on here often seem to care more if a chord points strongly toward a
root tone than if it invokes a positive sense of emotion. And while something
that points toward a root tone can serve as a guard to prevent and very sour
emotions...I hesitate to think "virtual pitch/tonality" is an instant recipe for
more agreeable emotions in music.
A side note, same goes with critical band dissonance. Avoiding it seems to
guarantee something won't be annoyingly sour, but it does not make a chord
generally guaranteed as "sweet" either. Hence we seem to have a lot of theories
that do a great job of avoiding quite sour tones (say less than 6 out of a
rating of 10)...but surprisingly few (or none?) that study what emotions on top
of avoiding these pitfalls make the chords "sweet".

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I think that a maj7 chord is very pretty and stable and much more so
> than a dom7 chord. So I think that 8:10:12:15 is much prettier
> sounding than 4:5:6:7. But, 4:5:6:7 is definitely more resonant and
> periodic and concordant and whatever you want to call it. So which is
> more consonant in the end?

We can't use your personal views on "pretty" as the basis for a definition. We could use resonant and periodic and concordant and whatever, as these are audible.

Igs wrote:

> Whoa, man. No need to get all metaphysical.

[followed by a heavy paragraph of metaphysics]

> So in tuning list vernacular, "consonance" is related to "all
> the cues music can convey", but in psychoacoustics, it means
> the same as "concordance" in tuning list vernacular

Right. What could be clearer! :-\

> and you have been using the psychoacoustics definition instead
> of the tuning list vernacular definition. Yeah, that's a little
> confusing, seeing as how this is the tuning list.

Sorry.

> > >It's long been assumed (at least by some folks here) that
> > >they're interchangeable,
> >
> > ?
>
> Don't give me that question-mark! You *just* described how
> psychoacousticians use "consonance" to mean the same thing as
> "concordance", and that you have also been conflating them,

I haven't been conflating them, I've been following your
usage. If you were using the musical definition of consonance
to talk about Tenney height and chords in isolation, you
were conflating them.

The question mark is asking you who you think has "long
assumed ... they're interchangeable".

> Oh, of course. Because, you know, music has nothing to do
> with cultural conditioning, artistic intent, mood imagery, etc.
> If you want to study and understand music, of course it makes
> sense to leave out all the things that distinguish "music" from
> "sound".

I keep trying to tell people, sensory consonance has almost
nothing to do with music!

> Psychoacoustics is to music what color psychology is to art.

Color moods are much weaker than sensory consonance, and color
psychology is essentially pseudoscience. But sure, the analogy
is (kinda) valid.

> > > So you admit, there may be another component to it beyond
> > > "fit" to a harmonic series?
> >
> > ??? When did I ever say anything like that?
> >
> When you mentioned "rootedness", specifically as being distinct
> from "fitting a harmonic series".

No, when did I ever say "fit to the harmonic series" was the
only component!

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > Whoa, man. No need to get all metaphysical.
>
> [followed by a heavy paragraph of metaphysics]

Nothing in that paragraph was metaphysics. I talked about defining words in terms of measurable objective components, invoking at most physics, psychology, and semantics.

> I haven't been conflating them, I've been following your
> usage. If you were using the musical definition of consonance
> to talk about Tenney height and chords in isolation, you
> were conflating them.

I don't know what Tenney height is, and I don't think taking chords in isolation necessarily conflates concordance and consonance. Though FWIW, I suppose I haven't been using the "tuning list vernacular" definition, either, and I haven't been as consistent in my word usage as I should be. I suppose the trouble is that my idea of what "concordance" means may not be complete or accurate, and I've been using it too carelessly. I just don't know the right terms to use to say that a 5:6:7 sounds stable/beatless but not resolved/pleasant/restful, whereas a 12-tET major triad (or a 16:18:21) sounds resolved/pleasant/restful but not stable/beatless. I used "concordance" to describe the property that 5:6:7 possesses but 16:18:21 (or a 12-tET major triad) doesn't, and "consonance" to refer to what 5:6:7 doesn't possess but 16:18:21 (or a 12-tET major triad) does. Evidently this is not the right usage.

I've also been assuming too much about what you mean by "fit to a harmonic series", since you haven't done much to explain what that means and I've tried to make it make sense to myself using my own reasoning (which may have led to errors).

> The question mark is asking you who you think has "long
> assumed ... they're interchangeable".

Anybody who maintains the psychoacoustic usage. I dunno, maybe it's not as popular here as I assumed it? But didn't you say that the "tuning list vernacular" definition isn't actually popular here on the tuning list?

> > > > So you admit, there may be another component to it beyond
> > > > "fit" to a harmonic series?
> > >
> > > ??? When did I ever say anything like that?
> > >
> > When you mentioned "rootedness", specifically as being distinct
> > from "fitting a harmonic series".
>
> No, when did I ever say "fit to the harmonic series" was the
> only component!

Sorry, my bad. I assumed too much. How many components do you feel there are?

-Igs

I mean this in the best way possible.

From where I sit I propose a re-set of the conversation before a death
spiral ensues.

I personally like the clarity with which Caleb described what he is
after. Perhaps that is a good tact to take.

Igs - I know you are after stuff important to you - my view, with
all due respect, is that it is getting off track and bogged down in
details.

Thus I suggest a restatement of the initial position / question.

Again, I mean this is the best possible way - we all want to learn or
share knowledge.

Chris

On Tue, Aug 17, 2010 at 4:34 PM, cityoftheasleep
<igliashon@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > > Whoa, man. No need to get all metaphysical.
> >
> > [followed by a heavy paragraph of metaphysics]
>
> Nothing in that paragraph was metaphysics. I talked about defining words in terms of measurable objective components, invoking at most physics, psychology, and semantics.
>

[switching to tuning list vernacular for the rest of
this thread...]

Igs wrote:

> I don't know what Tenney height is,

Well that sucks, since we've just finished a detailed
discussion of it. And three weeks ago, I explained it in
painstaking detail for dyads with graphs and everything.
Does the number 70 ring a bell?

> I just don't know the right terms to use to say that a 5:6:7
> sounds stable/beatless but not resolved/pleasant/restful,
> whereas a 12-tET major triad (or a 16:18:21) sounds
> resolved/pleasant/restful but not stable/beatless.

This description makes sense to me. But which chord is
more concordant? Mike and I say 5:6:7, and I think most
people would agree, especially when the root is below A440.

> I used "concordance" to describe the property that 5:6:7
> possesses but 16:18:21 (or a 12-tET major triad) doesn't, and
> "consonance" to refer to what 5:6:7 doesn't possess but
> 16:18:21 (or a 12-tET major triad) does. Evidently this is
> not the right usage.

In tuning list vernacular this sounds right to me. Except
that bare chords don't have consonance unless you assume
they'll be used in a particular genre of music. However
maybe low rootedness -- a kind of concordance -- explains
why diminished chords were chosen as dissonances in Western
music. If so, you might say that 5:7:9 is more tonal, less
rough, and less rooted than 16:18:21. That might capture
your observations without assuming any particular genre
of music. All speculation of course.

> I've also been assuming too much about what you mean by
> "fit to a harmonic series", since you haven't done much to
> explain what that means and I've tried to make it make
> sense to myself using my own reasoning (which may have
> led to errors).

I think I asked you the question. In reply, you went off
to assassinate strawmen with Michael.

Tenney height measures fit to the harmonic series. So does
harmonic entropy. They are both quantitative, and you can
pick either one and say it's what I meant. Then nobody can
ever argue with us about what we mean. They'll have to
argue that we're wrong, and that's a lot more fun.

>> The question mark is asking you who you think has "long
>> assumed ... they're interchangeable".
>
> Anybody who maintains the psychoacoustic usage.

Who precisely? And you said especially on this list. Who?

> But didn't you say that the "tuning list vernacular"
> definition isn't actually popular here on the tuning list?

Because this is only the 298,472th time people have talked
past each other on this mailing list, there have been efforts
to standardize terminology. A mailing list about an esoteric
subject needs more specific terminology than the average
boffin on the street. However, people also don't pay
attention to such efforts, or reject them, and then we're
arguing about what the standard should be (a clear case
of Parkinson's law
http://en.wikipedia.org/wiki/Parkinson%27s_Law_of_Triviality
) and so on. So these efforts tend to fail. I used to
observe the concordance/consonance one religiously. But
generally folks who have questions were coming from
somewhere else and it bounced off them, and when I started
in defining them their eyes glazed over, and eventually I
found I was the only one using them (and monz, bless him).
So I went back to trying to ape whatever the person was
already comfortable with.

>> No, when did I ever say "fit to the harmonic series" was the
>> only component!
>
> Sorry, my bad. I assumed too much. How many components do
> you feel there are?

I have no idea! But I did mention three that have been
floating around.

-Carl

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:

> Thus I suggest a restatement of the initial position / question.

And I suggest that if people want to compare chords, it's best to provide midi file examples.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> In tuning list vernacular this sounds right to me. Except
> that bare chords don't have consonance unless you assume
> they'll be used in a particular genre of music. However
> maybe low rootedness -- a kind of concordance -- explains
> why diminished chords were chosen as dissonances in Western
> music.

Note also that 4/3 has sometimes been classed as a dissonance, whereas 5/4 always seems to be seen as consonant.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> Well that sucks, since we've just finished a detailed
> discussion of it. And three weeks ago, I explained it in
> painstaking detail for dyads with graphs and everything.
> Does the number 70 ring a bell?

Oh! *facepalm* I guess I didn't put together that "Tenney height" = n*d. Cool.

> > I just don't know the right terms to use to say that a 5:6:7
> > sounds stable/beatless but not resolved/pleasant/restful,
> > whereas a 12-tET major triad (or a 16:18:21) sounds
> > resolved/pleasant/restful but not stable/beatless.
>
> This description makes sense to me. But which chord is
> more concordant? Mike and I say 5:6:7, and I think most
> people would agree, especially when the root is below A440.
>
> > I used "concordance" to describe the property that 5:6:7
> > possesses but 16:18:21 (or a 12-tET major triad) doesn't, and
> > "consonance" to refer to what 5:6:7 doesn't possess but
> > 16:18:21 (or a 12-tET major triad) does. Evidently this is
> > not the right usage.
>
> In tuning list vernacular this sounds right to me. Except
> that bare chords don't have consonance unless you assume
> they'll be used in a particular genre of music.

Right. What I think would be helpful is a term other than consonance or concordance that could describe this property, since "consonance" is relative to genre of music (I'm glad I'm allowed to say that!), but this property I'm experiencing isn't.

> However
> maybe low rootedness -- a kind of concordance -- explains
> why diminished chords were chosen as dissonances in Western
> music. If so, you might say that 5:7:9 is more tonal, less
> rough, and less rooted than 16:18:21. That might capture
> your observations without assuming any particular genre
> of music. All speculation of course.

Hmm...is "rootedness" the term I'm looking for? I'd be interested in how rootedness is calculated. Would suspended chords, like 8:9:12 or 6:8:9, also be more rooted than 5:6:7? Because I hear them as being more relaxed/resolved as well.

> I think I asked you the question. In reply, you went off
> to assassinate strawmen with Michael.

Heh, Michael sure does have an unending supply of straw men for me to practice my aim on.

> Tenney height measures fit to the harmonic series. So does
> harmonic entropy. They are both quantitative, and you can
> pick either one and say it's what I meant. Then nobody can
> ever argue with us about what we mean. They'll have to
> argue that we're wrong, and that's a lot more fun.

But aren't both of these primarily dyadic measures, and the whole original point was that dyadic measures fail in predicting chord concordance?

> Who precisely? And you said especially on this list. Who?

Oh, gosh. I haven't exactly kept a record of every time some has used "consonance" to mean "concordance".

> So I went back to trying to ape whatever the person was
> already comfortable with.

Well, at least you tried. I imagine it must be very frustrating to you having to re-explain all these established concepts to neophytes like myself every year or so.

> > Sorry, my bad. I assumed too much. How many components do
> > you feel there are?
>
> I have no idea! But I did mention three that have been
> floating around.

So you mean the psychoacousticians don't have it all figured out?

Thank goodness.

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Note also that 4/3 has sometimes been classed as a dissonance,
> whereas 5/4 always seems to be seen as consonant.

Maybe Margo can correct or qualify this, but in the medieval
period, I believe 5/4 was considered dissonant.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > Note also that 4/3 has sometimes been classed as a dissonance,
> > whereas 5/4 always seems to be seen as consonant.
>
> Maybe Margo can correct or qualify this, but in the medieval
> period, I believe 5/4 was considered dissonant.

I think it was 81/64 which was considered dissonant.

Igs wrote:

> Oh! *facepalm* I guess I didn't put together that "Tenney
> height" = n*d. Cool.

Right. If you're doing larger chords, it's technically
"generalized Tenney height". If you take the square, cube
root etc, to be politically correct, call it something else,
like maybe "geomean". I call all of them Tenney height
because I'm a rebel.

> > > I used "concordance" to describe the property that 5:6:7
> > > possesses but 16:18:21 (or a 12-tET major triad) doesn't,
> > > and "consonance" to refer to what 5:6:7 doesn't possess
> > > but 16:18:21 (or a 12-tET major triad) does. Evidently
> > > this is not the right usage.
[snip]
> What I think would be helpful is a term other than
> consonance or concordance that could describe this property,
> since "consonance" is relative to genre of music (I'm glad
> I'm allowed to say that!), but this property I'm
> experiencing isn't.

It's hard to know if it isn't, because indoctrination with
the Western musical language starts in the womb, and there
self experimentation is fraught with bias. But ok, let's say
for the sake of argument it isn't conditioned in this case.

> Hmm...is "rootedness" the term I'm looking for? I'd be
> interested in how rootedness is calculated. Would suspended
> chords, like 8:9:12 or 6:8:9, also be more rooted than 5:6:7?
> Because I hear them as being more relaxed/resolved as well.

Rootedness was sheer speculation. It never got to the point
of calculation. Of the kinds of concordance, only roughness
has quantitative formulas to distinguish it. Everything
else is more like IQ.

What I've been suggesting is that chords with a single
dominant virtual fundamental are slightly more concordant
than a general concordance rule would indicate IFF the
lowest note in the chord is in the same pitch class as
(music theory jargon for "is an octave duplicate of") that
virtual fundamental. You can ask Paul if that accurately
captures his idea.

There are algorithms (e.g. Terhardt's) for calculating a
chord's VF. I don't have much experience with them.
Harmonic entropy and Tenney height will be low if a chord
has a single dominant VF. Provided you get through all
that, I don't know how much of a boost to grant.

> > Tenney height measures fit to the harmonic series. So does
> > harmonic entropy. They are both quantitative, and you can
> > pick either one and say it's what I meant. Then nobody can
> > ever argue with us about what we mean. They'll have to
> > argue that we're wrong, and that's a lot more fun.
>
> But aren't both of these primarily dyadic measures, and the
> whole original point was that dyadic measures fail in
> predicting chord concordance?

Tenney height works for any size chord as long as the
geomean doesn't exceed 10 or so. See my plentiful recent
examples. It's fundamentally NOT a dyadic measure.

Harmonic entropy is also fundamentally NOT a dyadic measure,
though to date only the dyadic version has been computed.
In lieu of real higher-adic entropy values, Paul used average
dyadic entropy of chords and scales in the thread Michael
originally pointed to. That was an expedient only, but the
results were still interesting.

> Oh, gosh. I haven't exactly kept a record of every time
> some has used "consonance" to mean "concordance".

Well I was just doing that, but I wasn't conflating the
two concepts, I was trying to distinguish them!

Generally you should know what you're talking about if
you're going to accuse published scientists, or mailing
list colleagues who may be reading, of doing something
wrong.

> So you mean the psychoacousticians don't have it all
> figured out?
>
> Thank goodness.

Yes but also it's not terribly important. The single
parameter does a good enough job for musical purposes; at
least at the present time, there are bigger fish to fry.

-Carl

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > >
> > > Note also that 4/3 has sometimes been classed as a dissonance,
> > > whereas 5/4 always seems to be seen as consonant.
> >
> > Maybe Margo can correct or qualify this, but in the medieval
> > period, I believe 5/4 was considered dissonant.
>
> I think it was 81/64 which was considered dissonant.

Though in early Renaissance music I hear a marked tendency to end on chords without thirds, which seems to disappear in the later Renaissance. Whether that reflects a view that they are still somehow dissonant, even though the sweetness of the otonal thirds was admired, I don't know.

Gene wrote:

> > > Note also that 4/3 has sometimes been classed as a dissonance,
> > > whereas 5/4 always seems to be seen as consonant.
> >
> > Maybe Margo can correct or qualify this, but in the medieval
> > period, I believe 5/4 was considered dissonant.
>
> I think it was 81/64 which was considered dissonant.

Yes ok, but that doesn't mean 5/4 was considered consonant.
If it had been, they probably wouldn't have tuned 3rds 81/64.

-Carl

On Tue, Aug 17, 2010 at 1:53 AM, Carl Lumma <carl@...> wrote:
>
> I wrote:
>
> > Ah. There are millions of neurons doing this. The way
> > neurons work, they can only respond to absolute
> > frequency(ies), but neurons are cheap, so the brain just
> > throws them at the problem.
>
> Also note, in humans, vowels are defined by *absolute*
> frequencies. See for instance:
> http://cnx.org/content/m15459/latest/#img-voweltable
>
> -Carl

Interesting. I studied speech and formants at school for a bit, but
didn't realize that the same neural machinery was involved. I would
presume that absolute pitch is somehow involved here as well.

Also interesting that people can identify formant frequencies so
easily - frequencies which themselves are often not even present in
the sound - but that tonal memory turns out to be such a problem.

-Mike

Carl Lumma wrote:
> I wrote:
> >> Ah. There are millions of neurons doing this. The way
>> neurons work, they can only respond to absolute
>> frequency(ies), but neurons are cheap, so the brain just
>> throws them at the problem.
> > Also note, in humans, vowels are defined by *absolute*
> frequencies. See for instance:
> http://cnx.org/content/m15459/latest/#img-voweltable
> > -Carl

Actually that's not exactly true, as there are differences in formant frequencies between male and female voices, adults and children, etc. But roughly speaking, there are three significant numbers: the frequency of the first (lowest) formant, the frequency of the second formant (or the frequency difference between the first and second formants), and the lowering of the third formant.

On Tue, Aug 17, 2010 at 12:17 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"I think that a maj7 chord is very pretty and stable and much more so
> than a dom7 chord. So I think that 8:10:12:15 is much prettier
> sounding than 4:5:6:7. But, 4:5:6:7 is definitely more resonant and
> periodic and concordant and whatever you want to call it. So which is
> more consonant in the end?"
>
>    IMVHO, whichever one produces the greatest sense of relaxation.  It seems to be people on here often seem to care more if a chord points strongly toward a root tone than if it invokes a positive sense of emotion.  And while something that points toward a root tone can serve as a guard to prevent and very sour emotions...I hesitate to think "virtual pitch/tonality" is an instant recipe for more agreeable emotions in music.

Rather than work with relaxation, though, I'm more concerned with
"clarity of feeling," and there is a very clear feeling produced by
some very "unrelaxing" chords. Either way, though, that just isn't
what harmonic entropy is supposed to measure. It's supposed to measure
how easily things fuse into a single note. I don't think that figuring
out why certain chords are "relaxing" or "happy" or "sad" is an
insurmountable task - but it will require figuring out how "cultural
conditioning" works and how learned elements interact with musical
perception. I'm reading Rothenberg for the second time now and I
really think he's onto something with what he's got. But I don't think
he has any explanations for why Cmaj -> Dm7 -> Em7 -> Fmaj7 is such a
happy and relaxing chord progression though.

>    A side note, same goes with critical band dissonance.  Avoiding it seems to guarantee something won't be annoyingly sour, but it does not make a chord generally guaranteed as "sweet" either.  Hence we seem to have a lot of theories that do a great job of avoiding quite sour tones (say less than 6 out of a rating of 10)...but surprisingly few (or none?) that study what emotions on top of avoiding these pitfalls make the chords "sweet".

I'm so down with crunchy cluster chords that I just don't understand
and I've never understood and I will never understand. As Claude
Debussy once put it, "B D E# F# G# A# E#"

-Mike

Igs> I've also been assuming too much about what you mean by
> "fit to a harmonic series", since you haven't done much to
> explain what that means and I've tried to make it make
> sense to myself using my own reasoning (which may have
> led to errors).

Carl>"I think I asked you the question. In reply, you went off
to assassinate strawmen with Michael."
And this is productive how, Carl? There's nothing useful in attacking
others over un-named issues. Particularly if it looks like the attack is done
simply because someone doesn't agree with you and you seem to assume just
because they don't agree means they must have been too dumb to understand you.
I, for one, think Igs is right when he says "since you haven't done much to
explain what that means"...meaning that a lot of the fault rests in you're not
explaining things clearly (I had the same impression he did).

So please...explain...what you favored theories of consonance are and how you
rate them in terms of importance...

>"you might say that 5:7:9 is more tonal, less rough, and less rooted than
>16:18:21."
Agreed there. But I'll tell you this much, even though I listened to your
discussion on how 5:7:9 points to 4 (a root other than 5) and this hurts its
rooted-ness.
I'd be interested to hear more examples of how rootedness "works" because I have
yet to see the pattern that defines it clearly.

On the side...your saying something akin to "16:18:21 more consonant than
5:7:9...you must be crazy" to Igs before seems again to hint that you're heavily
biased toward periodicity/tonality over all other theories (say you had good
periodicity/tonality in a but bad rooted-ness, critical band dissonance, badly
symmetrical beating, etc.). Such statements do seem to sound like tonality is
your pet hypothesis, whether you mean them to or not. Just saying maybe it
wouldn't hurt to avoid extreme statements like that which confuse people like
Igs and I...not just to you, Carl, but anyone on the list.

Carl>"I used to observe the concordance/consonance one religiously..."
I still do. Concordance as related to calculate-able psycho-acoustics (IE
critical band, periodicity, harmonic entropy, Tenney height, etc.) and
consonance as the overall sense of relaxation/pleasantness (though typically
referring to having a strong sense of rooted-ness/together-ness).

Personally since various types of concordance also fit under the definition of
consonance...I typically forgive confusion between the two by others and call
the overall goal "resolved-ness", reflecting a sense of peace in the notes not
biased toward any one theory (IE not the way consonance is often specific to
having a root). So, to me, if by odd chance a chord sounds relaxed...even if it
breaks every concordance theory known to man, it can still be strong on
"resolved-ness".
So what makes resolved-ness more than just an opinion? A significant
percentage of people showing a chord/dyad/etc. sounds more relaxed to them.
15/8 in particular surprises me this way as it sounds much more resolved to me
than any of the major concordance theories say it "should" sound, as does the C
E F A chord.

>"As Claude Debussy once put it, "B D E# F# G# A# E#"
I'll say this much...stacking near 9/8ths doesn't bother me a bit. Give me a
C D E G A B or any inversion and it sounds fine to my ears. It's just getting
more clustered than that...that makes it really start to bite IMVHO.

> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > > >
> > > > Note also that 4/3 has sometimes been classed as a dissonance,
> > > > whereas 5/4 always seems to be seen as consonant.
> > >
> > > Maybe Margo can correct or qualify this, but in the medieval
> > > period, I believe 5/4 was considered dissonant.
> >
> > I think it was 81/64 which was considered dissonant.

(reply by Carl)
> Yes, ok, but that doesn't mean that 5/4 was considered consonant.
> If it had been, they probably wouldn't have tuned 3rds 81/64.

Hi, Gene and Carl.

Really there are two levels on which to answer this question, at least
as far as Continental European music of the 13th-14th centuries is generally concerned.

First, there's a simply categorical statement: _thirds are unstable_.
Sooner or later, either a major or minor third is going to be followed
by some stable sonority, often a full 2:3:4 trine or a 3:2 fifth, for
example. This holds regardless of the fine points of tuning, just as
a closing minor seventh would be out of place in Bach whether tuned
9/5, 16/9, or 7/4.

If we want to use typical 13th-14th century language, we can call
thirds "imperfect concords" -- although some Italian authors refer
to them at times as "tolerable dissonances." Note that categories
of this kind can also apply, especially in a 13th-century framework
or a 14th-century one admitting lots of 13th-century liberties
(e.g. Guillaume de Machaut) to things like 4:6:9, 9:12:16, 6:8:9,
or 8:9:12 -- pleasing, and with a degree of blend, but definitely
unstable.

From personal experience, I can tell you that having some very
well-meaning and skillful singers read a piece in 13th-century
style from site and accidentally conclude a phrase with a major
third and fifth above the lowest note where a stable fifth was
intended sounds "wrong" -- whether this sonority was tuned
4:5:6, 0-400-700 cents, 64:81:96, or whatever (this was
a capella, so lots of things were possible).

Carl, I much agree that tuning major thirds at 81/64 -- or often
in my favorite neo-medieval temperaments somewhere around 14/11
or 33/26, in a way a "modern" variation on Pythagorean -- fits
nicely with the stylistic assumption that thirds are active
and unstable intervals.

However, Gene, there is the interesting question of 5/4 in this
kind of medieval Continental European setting, up to around 1400
or so (you make an interesting observation on the following era
which I'll address below).

One view is that it, or the almost identical Pythagorean
diminished fourth at a schisma narrower (8192/6561), or 384.36
cents, is in the same general category as other augmented or
diminished intervals. This is the view of Johannes Boen in 1357,
who considers such a diminished fourth, a comma narrow of a
ditone at 81/64, as a "consonance by situation" (_consonantia
per accidens_) if supported by a regular major third below,
e.g. E-G#-C. The same rule applies to the augmented fourth with
a minor third below, e.g. D-F-B. (Boen notes that the perfect
fourth, likewise, becomes more concordant when supported by
an appropriate interval below, here the fifth, e.g. D-A-D.
While 2:3:4 can thus continue to be seen as an array of the
three simplest and most concordant non-unisonal dyads not
exceeding an octave, the fourth has a more problematic and
debated status than in the 13th century, when it ranks
with but after the fifth.)

Curiously, one meets a similar kind of principle in Renaissance
music where a diminished fourth might be around 32:25 (precisely
so in 1/4-comma meantone): it is acceptable in certain
circumstances, but most typically occurs between two upper voices
both forming usual concords with the lowest voice.

Around 1400 or so, we do have a shift, as written sharps are
often tuned on keyboards as flats, so that schismatic thirds
become the norm in certain positions, and contrast with usual
Pythagorean thirds between diatonic steps. And by sometime
around 1450, experimenting with such systems leads to meantone
temperament making approximate 5/4 thirds the general norm.

And if we think of the early to middle of the 15th century
as the early Renaissance, this transition ties in directly
with your following comment, Gene.

> Though in early Renaissance music I hear a marked tendency to
> end on chords without thirds, which seems to disappear in the
> later Renaissance. Whether that reflects a view that they
> are still somehow dissonant, even though the
> sweetness of the otonal thirds was admired, I don't know.

This is an interesting question, and I might nuance my language
a bit to describe the situation in the young Dufay's epoch around
1420-1440, the situation around 1500, and the situation in the
later 16th century (say Lasso or Palestrina).

In the early and middle 15th century, while sonorities with thirds
(and sixths) become more and pervasive, there's still the
expectation that we'll eventually cadence on a trine or fifth.
This expectation, which in many ways is coupled with directed
progressions and organizational schemes not too different from
"classic" ones of the 14th century (which I still find myself
using in the 21st century), feels very basic to me in the
appeal of many of Dufay's early chansons, fauxbourdon settings,
and some Mass settings and motets.

As things move on, and likely we are moving into the early
meantone era of keyboard tuning in more and more places,
sonorities with thirds are reaching a status of "almost
stable." In lots of the music of Busnois and Ockeghem,
for example, I'd still expect to hear a final sonority
of 2:3:4 (or, with the wider spacing that becomes common,
often 1:2:3:4 or the like), or maybe even a simple octave
(common in some three-voice writing).

Around 1500, say in a popular four-voice style like the
Cancionero de Palacio in Spain, I'd expect to hear
a third-plus-fifth-or-sixth type of sonority (as Zarlino
puts it later, in 1558) just about anywhere within a
phrase, but with a final cadence on 2:3:4 or the like
still very common and delightful -- yes, in keyboard
performance, in 1/4-comma meantone or the like where
those concluding fifths are indeed "bouncy."

In this epoch, we also encounter a minor third plus
a fifth as a closing sonority, for example in
Josquin, which in meantone would approximate
10:12:15. One clue reinforcing the conclusion
that raising the minor third to major as an
implicit performance practice is _not_ necessarily
meant is the advocacy of the Ab-C# meantone tuning
by Ramos (1482, whose 5-limit JI monochord is
distinct from his discussion of a practical
keyobard, which Mark Lindley persuasively
analyzes as a meantone instrument). If a
closing third were meant to be major by
unwritten alteration -- as will soon become
true for the most part -- then G# would
hardly be omissible for closes in the
Third and Fourth Modes (E Phrygian with
an authentic or plagal range). Yet Ramos
considers it optional, and of lower
priority than Ab, while Arnold Schlick
(1511) describes an irregular temperament
where Ab is acceptable but G# marginal,
meant for an ornamental E-G# leading to
a cadence on A, for example, rather than
a fully harmonious E-G# as a closing
major third.

By 1523, the date of Pietro Aron's
first edition of the _Toscanello in
Musica_, the "otonal" preference
of which you speak does seem in
place, at least in Aron's milieu:
altering a minor third to major
is seen as enhancing the beauty
of the music. And in his treatise
on the modes of 1525, Aron shows
the alteration of closing thirds
from minor to major, a typical
alteration of the 16th and early
17th centuries from this point on.
Such a situation fits nicely with
Eb-G# meantone, which supplies
that E-G#-B sonority for Phrygian.

However, there still seems a
feeling, which can sometimes
hold to a degree even in the
late 16th century when Zarlino's
_harmonia perfetta_ based on
the division of the 3:2 fifth
into 4:5:6 (or better yet with
a voicing "in the order of
the sonorous numbers" like
2:3:4:5, with the major tenth
preferred to the simple third),
that a close on a simple unison
or octave in two-voice writing,
or on 2:3:4 or 1:2:3:4 or the
like in multi-voice writing,
is somehow in a sense more
"conclusive" or "formal."
This may occur in liturgical
music, for example, but also
in secular pieces such as
certain madrigals.

By the 1550's, we have
statements from Vicentino
(1555) and Zarlino (1558)
in favor of "richness of
harmony" or "perfect
harmony" including a
third plus a fifth or
sixth -- or their octave
extensions -- wherever
possible. And many final
closes follow this
axiom. However, the taste
at times for an especially
conclusive final sonority
makes itself felt.

(In a Gothic setting, the
same ideal of "richness"
or "perfection" applies,
but with 2:3:4 rather
than 4:5:6 as the
"threefold perfection
of harmony." They are
the harmonic or otonal
divisions of the octave
and fifth respectively.)

Thus we move from the
active and unstable thirds
of the Gothic Era to the
mixture of regular Pythagorean
and schismatic thirds around
the early 15th century, the
adoption of meantone in the
middle to later portion of
the century, and the
acceptance of thirds as
conclusive by the early
16th century, which seems
to lead to the "otonal"
preference expressed by
some observations and
examples of Aron in the
1520's, and in a more
developed way by Vicentino
and Zarlino.

And this isn't a complete
account, because we have
accounts of English music
in the 13th and early 14th
centuries, as well as some
surviving pieces including
the famous _Sumer is icumen
in_, which do point both
to a treatment of thirds as
"the best concords," upon
which one may end as well as
begin, and to vocal tunings
"mollifying" these thirds
by performing them at or
near 5/4 or 6/5 (as
Christopher Page puts it).

Nor have I gotten into
14th-century variations also
discussed and advocated by
Page where cadential major
thirds and sixths tend to
be considerably _wider_ than
Pythagorean, something that
may apply not only to
Marchettus of Padua and his
early 14th-century Italian
style, but to lots of French
music including that of Machaut
(Page's view and also mine).

Anyway, Gene, your observations
seem perceptive to me as far
as Renaissance trends; and Carl,
I much agree with your point
that tuning a major third at
81/64 (or 14/11 or 33/26) may
itself signal a style where
such thirds are regarded as
"_imperfect_ concords" with
lots of stress on the first
word of that expression.

Best,

Margo

Wow, that's a keeper. Thanks Margo! The point that the
interval of a 3rd was considered unstable regardless of
tuning, is well-taken (Mike B. might want to look into
that as well).

-Carl

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Hi, Gene and Carl.
>
> Really there are two levels on which to answer this question,
> at least as far as Continental European music of the 13th-14th
> centuries is generally concerned.
>
> First, there's a simply categorical statement: _thirds are
> unstable_. Sooner or later, either a major or minor third is
> going to be followed by some stable sonority, often a full
> 2:3:4 trine or a 3:2 fifth, for example. This holds regardless
> of the fine points of tuning, just as a closing minor seventh
> would be out of place in Bach whether tuned 9/5, 16/9, or 7/4.
>
> If we want to use typical 13th-14th century language, we can call
> thirds "imperfect concords" -- although some Italian authors refer
> to them at times as "tolerable dissonances." Note that categories
> of this kind can also apply, especially in a 13th-century framework
> or a 14th-century one admitting lots of 13th-century liberties
> (e.g. Guillaume de Machaut) to things like 4:6:9, 9:12:16, 6:8:9,
> or 8:9:12 -- pleasing, and with a degree of blend, but definitely
> unstable.
>
> From personal experience, I can tell you that having some very
> well-meaning and skillful singers read a piece in 13th-century
> style from site and accidentally conclude a phrase with a major
> third and fifth above the lowest note where a stable fifth was
> intended sounds "wrong" -- whether this sonority was tuned
> 4:5:6, 0-400-700 cents, 64:81:96, or whatever (this was
> a capella, so lots of things were possible).
>
> Carl, I much agree that tuning major thirds at 81/64 -- or often
> in my favorite neo-medieval temperaments somewhere around 14/11
> or 33/26, in a way a "modern" variation on Pythagorean -- fits
> nicely with the stylistic assumption that thirds are active
> and unstable intervals.
>
> However, Gene, there is the interesting question of 5/4 in this
> kind of medieval Continental European setting, up to around 1400
> or so (you make an interesting observation on the following era
> which I'll address below).
>
> One view is that it, or the almost identical Pythagorean
> diminished fourth at a schisma narrower (8192/6561), or 384.36
> cents, is in the same general category as other augmented or
> diminished intervals. This is the view of Johannes Boen in 1357,
> who considers such a diminished fourth, a comma narrow of a
> ditone at 81/64, as a "consonance by situation" (_consonantia
> per accidens_) if supported by a regular major third below,
> e.g. E-G#-C. The same rule applies to the augmented fourth with
> a minor third below, e.g. D-F-B. (Boen notes that the perfect
> fourth, likewise, becomes more concordant when supported by
> an appropriate interval below, here the fifth, e.g. D-A-D.
> While 2:3:4 can thus continue to be seen as an array of the
> three simplest and most concordant non-unisonal dyads not
> exceeding an octave, the fourth has a more problematic and
> debated status than in the 13th century, when it ranks
> with but after the fifth.)
>
> Curiously, one meets a similar kind of principle in Renaissance
> music where a diminished fourth might be around 32:25 (precisely
> so in 1/4-comma meantone): it is acceptable in certain
> circumstances, but most typically occurs between two upper voices
> both forming usual concords with the lowest voice.
>
> Around 1400 or so, we do have a shift, as written sharps are
> often tuned on keyboards as flats, so that schismatic thirds
> become the norm in certain positions, and contrast with usual
> Pythagorean thirds between diatonic steps. And by sometime
> around 1450, experimenting with such systems leads to meantone
> temperament making approximate 5/4 thirds the general norm.
>
> And if we think of the early to middle of the 15th century
> as the early Renaissance, this transition ties in directly
> with your following comment, Gene.
[snip]
> This is an interesting question, and I might nuance my language
> a bit to describe the situation in the young Dufay's epoch around
> 1420-1440, the situation around 1500, and the situation in the
> later 16th century (say Lasso or Palestrina).
>
> In the early and middle 15th century, while sonorities with thirds
> (and sixths) become more and pervasive, there's still the
> expectation that we'll eventually cadence on a trine or fifth.
> This expectation, which in many ways is coupled with directed
> progressions and organizational schemes not too different from
> "classic" ones of the 14th century (which I still find myself
> using in the 21st century), feels very basic to me in the
> appeal of many of Dufay's early chansons, fauxbourdon settings,
> and some Mass settings and motets.
>
> As things move on, and likely we are moving into the early
> meantone era of keyboard tuning in more and more places,
> sonorities with thirds are reaching a status of "almost
> stable." In lots of the music of Busnois and Ockeghem,
> for example, I'd still expect to hear a final sonority
> of 2:3:4 (or, with the wider spacing that becomes common,
> often 1:2:3:4 or the like), or maybe even a simple octave
> (common in some three-voice writing).
>
> Around 1500, say in a popular four-voice style like the
> Cancionero de Palacio in Spain, I'd expect to hear
> a third-plus-fifth-or-sixth type of sonority (as Zarlino
> puts it later, in 1558) just about anywhere within a
> phrase, but with a final cadence on 2:3:4 or the like
> still very common and delightful -- yes, in keyboard
> performance, in 1/4-comma meantone or the like where
> those concluding fifths are indeed "bouncy."
>
> In this epoch, we also encounter a minor third plus
> a fifth as a closing sonority, for example in
> Josquin, which in meantone would approximate
> 10:12:15. One clue reinforcing the conclusion
> that raising the minor third to major as an
> implicit performance practice is _not_ necessarily
> meant is the advocacy of the Ab-C# meantone tuning
> by Ramos (1482, whose 5-limit JI monochord is
> distinct from his discussion of a practical
> keyobard, which Mark Lindley persuasively
> analyzes as a meantone instrument). If a
> closing third were meant to be major by
> unwritten alteration -- as will soon become
> true for the most part -- then G# would
> hardly be omissible for closes in the
> Third and Fourth Modes (E Phrygian with
> an authentic or plagal range). Yet Ramos
> considers it optional, and of lower
> priority than Ab, while Arnold Schlick
> (1511) describes an irregular temperament
> where Ab is acceptable but G# marginal,
> meant for an ornamental E-G# leading to
> a cadence on A, for example, rather than
> a fully harmonious E-G# as a closing
> major third.
>
> By 1523, the date of Pietro Aron's
> first edition of the _Toscanello in
> Musica_, the "otonal" preference
> of which you speak does seem in
> place, at least in Aron's milieu:
> altering a minor third to major
> is seen as enhancing the beauty
> of the music. And in his treatise
> on the modes of 1525, Aron shows
> the alteration of closing thirds
> from minor to major, a typical
> alteration of the 16th and early
> 17th centuries from this point on.
> Such a situation fits nicely with
> Eb-G# meantone, which supplies
> that E-G#-B sonority for Phrygian.
>
> However, there still seems a
> feeling, which can sometimes
> hold to a degree even in the
> late 16th century when Zarlino's
> _harmonia perfetta_ based on
> the division of the 3:2 fifth
> into 4:5:6 (or better yet with
> a voicing "in the order of
> the sonorous numbers" like
> 2:3:4:5, with the major tenth
> preferred to the simple third),
> that a close on a simple unison
> or octave in two-voice writing,
> or on 2:3:4 or 1:2:3:4 or the
> like in multi-voice writing,
> is somehow in a sense more
> "conclusive" or "formal."
> This may occur in liturgical
> music, for example, but also
> in secular pieces such as
> certain madrigals.
>
> By the 1550's, we have
> statements from Vicentino
> (1555) and Zarlino (1558)
> in favor of "richness of
> harmony" or "perfect
> harmony" including a
> third plus a fifth or
> sixth -- or their octave
> extensions -- wherever
> possible. And many final
> closes follow this
> axiom. However, the taste
> at times for an especially
> conclusive final sonority
> makes itself felt.
>
> (In a Gothic setting, the
> same ideal of "richness"
> or "perfection" applies,
> but with 2:3:4 rather
> than 4:5:6 as the
> "threefold perfection
> of harmony." They are
> the harmonic or otonal
> divisions of the octave
> and fifth respectively.)
>
> Thus we move from the
> active and unstable thirds
> of the Gothic Era to the
> mixture of regular Pythagorean
> and schismatic thirds around
> the early 15th century, the
> adoption of meantone in the
> middle to later portion of
> the century, and the
> acceptance of thirds as
> conclusive by the early
> 16th century, which seems
> to lead to the "otonal"
> preference expressed by
> some observations and
> examples of Aron in the
> 1520's, and in a more
> developed way by Vicentino
> and Zarlino.
>
> And this isn't a complete
> account, because we have
> accounts of English music
> in the 13th and early 14th
> centuries, as well as some
> surviving pieces including
> the famous _Sumer is icumen
> in_, which do point both
> to a treatment of thirds as
> "the best concords," upon
> which one may end as well as
> begin, and to vocal tunings
> "mollifying" these thirds
> by performing them at or
> near 5/4 or 6/5 (as
> Christopher Page puts it).
>
> Nor have I gotten into
> 14th-century variations also
> discussed and advocated by
> Page where cadential major
> thirds and sixths tend to
> be considerably _wider_ than
> Pythagorean, something that
> may apply not only to
> Marchettus of Padua and his
> early 14th-century Italian
> style, but to lots of French
> music including that of Machaut
> (Page's view and also mine).
>
> Anyway, Gene, your observations
> seem perceptive to me as far
> as Renaissance trends; and Carl,
> I much agree with your point
> that tuning a major third at
> 81/64 (or 14/11 or 33/26) may
> itself signal a style where
> such thirds are regarded as
> "_imperfect_ concords" with
> lots of stress on the first
> word of that expression.
>
> Best,
>
> Margo
>

On Tue, Aug 17, 2010 at 12:39 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> We can't use your personal views on "pretty" as the basis for a definition. We could use resonant and periodic and concordant and whatever, as these are audible.

That was the point of my message. You misunderstood what I was saying.
But, since you bring it up, I do think that it can be studied exactly
why certain chords "function" as they do when placed within a diatonic
scale, even if cultural elements have to be identified and studied as
well. Music semiology is apparently an entire field of study that I
know nothing about (and you guys probably know more than me).

-Mike

Margo,

Thanks for that response. I found it very enlightening in the context
of this discussion. The fact that thirds were considered unstable -
regardless of their tuning - suggests to me that there is some sort of
psychoacoustic and cognitive processing going on with the scale that
is completely independent of periodicity and such. The fact also that
your singers, alive in the 21st century, could also hear the major
third as being "wrong" when placed within the context of a 13th
century composition suggests also that it's not entirely cultural.
Perhaps it has something to do with how ones cognitive facilities end
up adapting, by the end of the composition, to process the information
being presented. The order or way in which information is presented,
as well as the depth of the information being presented, would likely
be very significant here. ("Information" in this context would mean
intervals, scale degrees, chords, harmonic movement, etc.)

And, similar to the rest of the human organism, whatever adaptations
that prove useful in processing the incoming information would
persist, thus leading to the cultural differences and expectations in
processing music that we are all so familiar with. Now if I could only
adapt to Miracle[10]...

But seriously, has anyone studied the above in much depth? Seems like
Rothenberg has laid the groundwork for it all, although I don't know
if he has any explanation for why a third might be consonant in one
setting and dissonant in another, assuming the same map is being used.

-Mike

On Wed, Aug 18, 2010 at 2:35 AM, Margo Schulter <mschulter@...> wrote:
>
//snip
> Anyway, Gene, your observations
> seem perceptive to me as far
> as Renaissance trends; and Carl,
> I much agree with your point
> that tuning a major third at
> 81/64 (or 14/11 or 33/26) may
> itself signal a style where
> such thirds are regarded as
> "_imperfect_ concords" with
> lots of stress on the first
> word of that expression.
>
> Best,
>
> Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

If a
> closing third were meant to be major by
> unwritten alteration -- as will soon become
> true for the most part -- then G# would
> hardly be omissible for closes in the
> Third and Fourth Modes (E Phrygian with
> an authentic or plagal range).

Renaissance composers were positively addicted to minor triads, so I wonder why this unwritten alteration rule came about. Perhaps minor triads were regarded as just as important as major triads, but not stable enough to close on?

> By 1523, the date of Pietro Aron's
> first edition of the _Toscanello in
> Musica_, the "otonal" preference
> of which you speak does seem in
> place, at least in Aron's milieu:
> altering a minor third to major
> is seen as enhancing the beauty
> of the music.

I was referring to something I recall reading concerning what was said about the music; in terms of practice it seems to me Renaissance composers were very apt to use a lot of minor triads even in the later Renaissance, and more so earlier, where I suspect they may have been more common than major ones, as a minor feel to the music is so common. I wonder what a census of chords in the music of various composers would show?

> (In a Gothic setting, the
> same ideal of "richness"
> or "perfection" applies,
> but with 2:3:4 rather
> than 4:5:6 as the
> "threefold perfection
> of harmony."

Was that related to theology by any chance?

Hello, Gene.

> Renaissance composers were positively addicted to minor triads, so I
> wonder why this unwritten alteration rule came about. Perhaps minor
> triads were regarded as just as important as major triads, but not
> stable enough to close on?

Indeed the ideal of the era as expressed by Zarlino (1558), and which
I share when improvising or composing in this kind of style, is an
alternation of the "harmonic" or "natural" division (4:5:6) and the
"arithmetic" or "artificial" division (10:12:15, or in terms of string
ratios, 6:5:4), leading to sense of fluidity and balance.

In choosing terminology, without hopefully failing to address your
question in the terms it poses, I would emphasize that we are not
dealing with "major/minor" in the sense of late 17th-19th century
tonality. Thus many restrictions of later music don't apply, while, as
Doug Leedy has eloquently written as a lover of meantone, the sheer
sonority of saturated 5-limit combinations is at center stage,
often together with some of the most sensitive sophisticated
contrapuntal writing of any age.

Certainly there _is_, especially starting around the 1520's, a
recognition of the contrast or polarity between Zarlino's "natural"
arrangements of a fifth, major third, and minor third, or an outer
major or minor sixth plus a major or minor third and a perfect fourth
(e.g. C-E-G, E-G-C, and also G-C-E in Zarlino's view despite the
traditionally more cautious Renaissance treatment of the fourth), and
the "artificial" arrangements (e.g. D-F-A, F-A-D, and A-D-F, the last
of which Zarlino considers least consnnant). Note that while a theory
of inversion is generally deemed to have been introduced in the early
17th century (e.g. Johannes Lippius in his germinal works of 1610 and
1612), Zarlino's theory entails the result that octave transposition
of voices, in later terms "inversion," will not alter the basic nature
of a "natural" division (e.g. 4:5:6, 5:6:8, or 3:4:5) or an
"artificial" one (e.g. 10:12:15, 12:15:20, 15:20:24).

Here I'm using frequency rather than string ratios, but this is
actually in line with Zarlino's concept of the "sonorous numbers,"
since he takes the _senario_ or first six natural numbers, and
determines from them the natural order 1:2:3:4:5:6, a construction
thus synonymous with the 17th-century and later concept of the
harmonic series.

As you say, while the "harmonic" or "natural" division with the major
third below and the minor third above is regarded as more conclusive
by around the 1520's, and definitely so by the era of Vicentino and
Zarlino, the ideal within a piece is of variety and balance.

Zarlino's general guideline is very apt: if you write a harmonic or
natural division, tend to follow it with an arithmetic or artificial
one, and vice versa.

This fits both with the nature of the twelve modes, which often
encourage such an alternation, and also with the patterns of
counterpoint and polyphony which promote this fluidity also.

For lots of 16th-century and early 17th-century music, I find that
while something may sound to me quite routine, people listening from a
later major/minor perspective may note "the mixture of major and
minor" or remark on some "modulation" where I might perceive a usual
internal cadence or the like.

Here I use the term "modal" freely, because the structure of the modes
as textbook "scales" is only one factor, with routine inflections and
especially the rules of vertical progressions playing an equally
important role in shaping the polyphonic texture and procedures.

In part, the textbook forms of the modes do give a hint that a mixture
of natural and artificial divisions (or in later terms major and minor
triads) will be very common, in contrast to the often more one-sided
qualities of later major/mionr tonality. Thus Dorian has a major sixth
above the final, and Mixolydian a minor seventh; the often fluid
nature of these degrees serves both melodic variety and the kind of
harmonic alternation recommended by Zarlino.

Additionally, while cadences often involve motion of the bass by a
fifth or a fourth (a common point with later tonality), motion within
a phrase by seconds or thirds is also very common, with Tomas de
Santa Maria (1557, published in 1565) offering as part of his treatise
on improvising or composing for polyphonic instruments such as
keyboards and lute a kind of guide to 16th-century harmony in four
parts which highlights some of these patterns the bass may follow.
These characteristic bass motions often promote Zarlino's ideal of
alternating -- early and often -- between natural and artificial
divisions as part of the routine and beguiling color of the music.

Thus one side of the reality which you are picking up is that pieces
in modes with a major third above the final (Ionian, Mixolydian, and
Lydian) have a lot more arithmetic divisions or minor triads than a
later piece in major.

The other side of the coin is that pieces in modes with a minor third
above the final (Dorian, Aeolian, Phrygian) may have a lot more
harmonic divisions or major triads than a later piece in minor, and
here I'm not speaking only of the close where a major third was deemed
more conclusive.

> By 1523, the date of Pietro Aron's
> first edition of the _Toscanello in
> Musica_, the "otonal" preference
> of which you speak does seem in
> place, at least in Aron's milieu:
> altering a minor third to major
> is seen as enhancing the beauty
> of the music.

> I was referring to something I recall reading concerning what was
> said about the music; in terms of practice it seems to me
> Renaissance composers were very apt to use a lot of minor triads
> even in the later Renaissance, and more so earlier, where I suspect
> they may have been more common than major ones, as a minor feel to
> the music is so common. I wonder what a census of chords in the
> music of various composers would show?

The idea of a census might indeed be interesting! I can offer a few
guesses as to what such a census might show, meant as an invitation to
do it and see if my guesses are accurate (or maybe to do a bit of
searching and learn that someone has done this kind of survey, and see
what they found).

First, I should again confirm the importance of the very important
distinction you've made: to say that a major third is preferred in a
closing sonority (if a third is present, as it is a very large
percentage of the time although not invariably by the era of Zarlino)
doesn't mean that the arithmetic division (or minor triad) isn't very
much in demand for pieces in a range of modes!

Some of this can be subjective: a piece in the Dorian mode, for
example (and some of this might apply to plainsong as well as 16th-century
polyphony) might have a "minor feel" for many modern listeners where I
might hear more of a balance or alternation of the divisions, and also
focus on nuances like the often fluid sixth degree. However, I do
think it is possible to note certain trends.

If we start around the later 15th century, where we get increasing
tertian saturation, say the era of Ockeghem and Josquin, then indeed
we often get a weighting toward the arithmetic division. I think of a
piece like Ockeghem's _Missa Mi Mi_ in Phrygian, or maybe one of the
chansons of the Netherlands school around 1500.

One sign of a certain direction in taste by the 1550's is Zarlino's
advice that goes with his general guideline of alternating the
harmonic and arithmetic divisions, so that one does not too long
remain in either: if one must write several divisions of the same kind,
it's better to stay in the harmonic division, since otherwise the
music might become overly "melancholy" or "languid" or the like.
It's expected that the quality of divisions will be influenced by the
mode, one of the expressive dimensions that Zarlino addresses: but
strive for variety and balance, and if in doubt, lean a bit toward the
more "natural" harmonic division (in other words, the major triad).

While I haven't done a census, I would guess that if we survey some
pieces in the Phrygian modes (authentic and plagal) around 1500, we
might find a lot fewer sonorities of E-G#-B than around 1550 or 1600.
I also seem to notice a tendency often to _open_ a piece with a major
third, regardless of the mode.

Having emphasized the role of variety and alternation facilitated by
the structure of the modes themselves but also wonderfully in line
with the rules and patterns of counterpoint and harmony, I should add
that around 1600 we sometimes encounter a kind of "coloristic
otonality" which does not fit strictly within the framework of any
mode, but indeed exemplifies the winning qualities of the era: smooth
melodic movement, saturated 5-limit harmony, and contrary motion:

E F# G
B D E
G# A C
E D C

Here we have three successive harmonic divisions, with the progression
"guided" in part by what might be called classic 14th-century
progressions: from the lower major third E-G# of the first sonority to
the fifth D-A, and then from the major tenth D-F# between the outer
voices of the second sonority to the 12th C-G. Of course, in this
5-limit setting, the texture, color, and total effect are radically
different! It's fascinating how Gothic two-voice progressions can
influence smoothly flowing motion between 5-limit sonorities which has
its own unique logic, distinct from the more "directed" ethos of
either the Gothic or the 18th-century tonality.

We have indeed an abundance of harmonic divisions or "major triads,"
but in a fashion quite different than a typical later major tonality!
And although there is no direct chromaticism, the contrast between G#
in the opening sonority and G in the third sonority is another characteristic touch.

More generally, I'd certainly agree in guessing that around 1600, as
well as around 1500, we'll find a lot more arithmetic divisions (minor
triads) in modes having a major third above the final than we would in
a later major tonality.

Zarlino's advice on alternating the two diversions is a fine summary,
and might be one starting point for looking at what's going on
harmonically in the 16th century from its own perspective. Of course,
that is always an ideal rather than a reality: the main difference is
that lots of modern authors seem to apply 18th-century expectations,
while I may indulge in some 14th-century expectations!

>> (In a Gothic setting, the
>> same ideal of "richness"
>> or "perfection" applies,
>> but with 2:3:4 rather
>> than 4:5:6 as the
>> "threefold perfection
>> of harmony."

> Was that related to theology by any chance?

Definitely, at least for some of the most important theorists in this
department. Thus Johannes de Grocheio around 1300 speaks of a
trinitarian perfection in music, drawing a direct analogy with
theology, in describing 2:3:4; while Johannes Lippius in 1610 and 1612
does the same thing in describing 4:5:6 as the _trias harmonica_ or
"harmonic triad." For both Grocheio and Lippius, this recognition of
an ideally concordant three-voice sonority is the key to intelligent
compositional technique.

Not every reference is necessarily theological. Thus an anonymous
author around 1300 very likely to be Jacobus of Liege (as I agree with
the modern editor of this treatise) notes that in polyphony, one should
follow a "natural series" of numbers 2-3-4 by composing the octave so
that the 2:3 fifth comes before the 3:4 fourth. As in Zarlino's later
theory of 5-limit sonority, this concept leads to results similar to
those based on a harmonic series. Since a similar preference for 2:3:4
as compared to 3:4:6 is also discussed at some length by Jacobus in
his monumental _Speculum musicae_, although without specifically
invoking the 2-3-4 series, his authorship of this smaller treatise
indeed seems likely.

But Grocheio (or Grocheo) and Lippius were indeed very explicitly
theological, and the latter had an influence on later German theorists
including Werckmeister, for whom the _trias harmonica_ was a focus at
once musical and mystical.

With many thanks,

Margo

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> What I've been suggesting is that chords with a single
> dominant virtual fundamental are slightly more concordant
> than a general concordance rule would indicate IFF the
> lowest note in the chord is in the same pitch class as
> (music theory jargon for "is an octave duplicate of") that
> virtual fundamental. You can ask Paul if that accurately
> captures his idea.
>
> There are algorithms (e.g. Terhardt's) for calculating a
> chord's VF. I don't have much experience with them.
> Harmonic entropy and Tenney height will be low if a chord
> has a single dominant VF. Provided you get through all
> that, I don't know how much of a boost to grant.

Okay, so you don't have much experience with actually calculating VF. How do you know that H.e. and Tenney height will be low if a chord has a single dominant VF? Have VFs been calculated by other people for a variety of chords, sufficient to make a correlation?

> Tenney height works for any size chord as long as the
> geomean doesn't exceed 10 or so. See my plentiful recent
> examples. It's fundamentally NOT a dyadic measure.

So how do you calculate Tenney height for a chord? Do you put it in harmonic series terms (i.e. a minor chord would be 10:12:15) and then multiply all the harmonics (so the minor chord would be 1800)? That can't be right. That method would also rank 3:5:7 as more concordant than 4:5:6. But I don't see how else it could be calculated for triads....

> Harmonic entropy is also fundamentally NOT a dyadic measure,
> though to date only the dyadic version has been computed.
> In lieu of real higher-adic entropy values, Paul used average
> dyadic entropy of chords and scales in the thread Michael
> originally pointed to. That was an expedient only, but the
> results were still interesting.

So basically, when I asked you what other property of a chord, beyond the concordance of its dyads, might be useful for determining the chord's concordance, and you asked, "it's fit to a harmonic series", you were actually suggesting a property for which there is currently no established effective way to quantify. H.e. has only been worked out for dyads, and Tenney height (assuming I've understood its triadic version correctly) cannot effectively predict that a well-known concord like a minor chord should be concordant. Are you assuming that "fit to a harmonic series" might be a good predictor just because utonal chords tend to be less concordant than otonal chords?

I wonder if there aren't maybe some utonal chords that are more concordant than their otonal counterparts? I expected 1/(9:10:12) and 1/(16:18:21) to be more concordant, since the utonal versions both put a less complex interval closer to the lowest note, but oddly enough, I found the otonal versions still sounded more relaxed. But still I wonder...if only there was a good way to calculate rootedness.

> > Oh, gosh. I haven't exactly kept a record of every time
> > some has used "consonance" to mean "concordance".
>
> Well I was just doing that, but I wasn't conflating the
> two concepts, I was trying to distinguish them!
>
> Generally you should know what you're talking about if
> you're going to accuse published scientists, or mailing
> list colleagues who may be reading, of doing something
> wrong.

Well, I didn't accuse anyone by name, because I didn't see the need to. I figured that anyone for whom my accusation was valid would know I was talking to them, and anyone for whom it wasn't valid would know I was not talking to them. If my accusation was completely baseless--if there is in fact no one here that considers "concordance" to be equivalent to "consonance"--I apologize.

> Yes but also it's not terribly important. The single
> parameter does a good enough job for musical purposes;

You really think so? I may actually have to disagree.

What I'm looking for is a good guide to finding the most concordant triads in any given tuning, and for tunings where there is no approximate 4:5:6 or 6:7:9, I haven't found a good guiding principle except my own ears. Which is all fine and good for the purposes of my own music-making, but for the purposes of trying to explain to "outsiders" how to approach microtonal tunings, I feel like I need to be able to give some explanation. Scales are easy, but harmony is a mess. So far, I've not seen any theory that reliably predicts the "next most concordant thing" to a 4:5:6 triad. It sure isn't a 5:6:7, or a 3:5:7. Dyadic measures are more successful, but what good are they if they don't translate into triadic measures?

-Igs

Hi Igs,

>> There are algorithms (e.g. Terhardt's) for calculating a
>> chord's VF. I don't have much experience with them.
>> Harmonic entropy and Tenney height will be low if a chord
>> has a single dominant VF.
>
> Okay, so you don't have much experience with actually
> calculating VF. How do you know that H.e. and Tenney height
> will be low if a chord has a single dominant VF? Have VFs
> been calculated by other people for a variety of chords,
> sufficient to make a correlation?

Algorithms for telling if there is a single dominant VF are
different from algorithms for finding VFs. Tenney height
and harmonic entropy are examples of the former.

>> Tenney height works for any size chord as long as the
>> geomean doesn't exceed 10 or so. See my plentiful recent
>> examples. It's fundamentally NOT a dyadic measure.
>
> So how do you calculate Tenney height for a chord?

Did you see my examples here:
/tuning/topicId_91555.html#91730
?

> That can't be right. That method would also rank 3:5:7 as
> more concordant than 4:5:6. But I don't see how else it
> could be calculated for triads....

Yeah... keep in mind that 3:5:7 spans more than an octave.
It's a very consonant chord, and it's definitely more
consonant than 4:5:6 as you go down the keyboard.

> So basically, when I asked you what other property of a
> chord, beyond the concordance of its dyads, might be useful
> for determining the chord's concordance, and you asked,
> "it's fit to a harmonic series", you were actually suggesting
> a property for which there is currently no established
> effective way to quantify.

Would you prefer I suggest a property that's easy to compute
but wrong? :P

> H.e. has only been worked out for dyads, and Tenney height

Keep in mind, h.e. is proportional to Tenney height for just
chords. It only comes into its own in the case of tempered
chords. And if they're fairly accurate tempered chords, you
can assume they have the concordance of the just chord they
approximate. I realize in your work with ETs you have often
come across chords that don't fit this bill. So perhaps you
would like to collaborate on computing triadic h.e. for the
first time? Or at least collaborate on nagging Paul to do so?

> cannot effectively predict that a well-known concord like a
> minor chord should be concordant.

You mean 10:12:15? Paul's interpretation there is that we
basically hear a 3:2 with some junk in the middle, and the
junk has low roughness so it doesn't bother us much.
Similarly, he predicts the best inversion of the utonal 7th
chord should be 1/1 6/5 3/2 12/7 (70:84:105:120) even though
1/1 7/6 7/5 7/4 (60:70:84:105) has lower Tenney height.
Since it has a 3:2 above the root and that's a strong VF cue.

> Are you assuming that "fit to a harmonic series" might be
> a good predictor just because utonal chords tend to be less
> concordant than otonal chords?

That's one reason, yes. Harmonic entropy, by the way,
*assumes this* from the start.

> I wonder if there aren't maybe some utonal chords that are
> more concordant than their otonal counterparts? I expected
> 1/(9:10:12) and 1/(16:18:21) to be more concordant, since
> the utonal versions both put a less complex interval closer
> to the lowest note, but oddly enough, I found the otonal
> versions still sounded more relaxed.

Once you leave the regime of consecutive integer or odd
harmonics, the otonal/utonal distinction ceases to be
meaningful. For instance, chords like 10:12:15:18 are
their own inverses...

> But still I wonder...
> if only there was a good way to calculate rootedness.

Odd limit will at least erase such distinctions, and lets
you work along customary octave-equivalent lines. e.g.
3:4:5 and 4:5:6 are equally concordant under odd limit vs.
the former being more concordant under Tenney height.
On the other hand, octave equivalence gives loony results
beyond the 7-limit.

> If my accusation was completely baseless--if there is in
> fact no one here that considers "concordance" to be equivalent
> to "consonance"--I apologize.

I can't remember anyone ever claiming that on this list.
The idea that sensory and musical consonance are different
things is very widely accepted. Certainly by all the authors
of psychoacoustics research I have ever read.

>> Yes but also it's not terribly important. The single
>> parameter does a good enough job for musical purposes;
>
> You really think so? I may actually have to disagree.
> What I'm looking for is a good guide to finding the most
> concordant triads in any given tuning,

Normally what we do is find the most concordant tuning for
desired chords. This reverse approach is interesting to me,
and is one I've actually tried myself. See
http://lumma.org/music/theory/ets-program/ETsProgram3.xls

That's telling you the best-approximated chords (of 3-6
notes) for each ET < 100, where the ET is allowed to have
a stretched (or compressed) octave.

> and for tunings where there is no approximate 4:5:6 or 6:7:9,
> I haven't found a good guiding principle except my own ears.

I must say, I don't think by-ear is all that bad. It's a
little slow but the work itself might be inspiring. And I
think you can tell beginners about it. What's wrong with,
"I tried all these chords and this one sounded the most
concordant"?

> So far, I've not seen any theory that reliably predicts the
> "next most concordant thing" to a 4:5:6 triad. It sure
> isn't a 5:6:7, or a 3:5:7.

What is it then? Don't keep us in suspense!

-Carl

On Thu, Aug 19, 2010 at 8:30 PM, Carl Lumma <carl@...> wrote:
>
> You mean 10:12:15? Paul's interpretation there is that we
> basically hear a 3:2 with some junk in the middle, and the
> junk has low roughness so it doesn't bother us much.
> Similarly, he predicts the best inversion of the utonal 7th
> chord should be 1/1 6/5 3/2 12/7 (70:84:105:120) even though
> 1/1 7/6 7/5 7/4 (60:70:84:105) has lower Tenney height.
> Since it has a 3:2 above the root and that's a strong VF cue.

What is his take on 14:18:21? That one's pretty out there. Or 18:22:27?

Although 22:24:33 is higher than both of those and sounds fine to my ears.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Aug 19, 2010 at 8:30 PM, Carl Lumma <carl@...> wrote:
> >
> > You mean 10:12:15? Paul's interpretation there is that we
> > basically hear a 3:2 with some junk in the middle, and the
> > junk has low roughness so it doesn't bother us much.
> > Similarly, he predicts the best inversion of the utonal 7th
> > chord should be 1/1 6/5 3/2 12/7 (70:84:105:120) even though
> > 1/1 7/6 7/5 7/4 (60:70:84:105) has lower Tenney height.
> > Since it has a 3:2 above the root and that's a strong VF cue.
>
> What is his take on 14:18:21? That one's pretty out there.
> Or 18:22:27?

I don't want to speak for him. You may some stuff in the
archives. Or ask him.

Personally I'd expect both of them to sound similar, and
very much like 'utonal' chords. Low tonalness low roughness.
And I fancy that is how they sound to me.

> Although 22:24:33 is higher than both of those and sounds
> fine to my ears.

22:24 is pretty narrow...

-C.

On Thu, Aug 19, 2010 at 9:14 PM, Carl Lumma <carl@...> wrote:
>
> > What is his take on 14:18:21? That one's pretty out there.
> > Or 18:22:27?
>
> I don't want to speak for him. You may some stuff in the
> archives. Or ask him.
>
> Personally I'd expect both of them to sound similar, and
> very much like 'utonal' chords. Low tonalness low roughness.
> And I fancy that is how they sound to me.

18:22:27 is I think the 11-limit neutral triad... Is it utonal? The
supermajor triad is definitely utonal but sounds terrible to me and I
hate it. I can "get used to it," as in the case of the superpyth
tunings I've been talking about recently. But, I hate it. I'll ask
Paul about it.

> > Although 22:24:33 is higher than both of those and sounds
> > fine to my ears.
>
> 22:24 is pretty narrow...

It's just the utonal equivalent of 8:11:12. It's not narrow enough to
bother me. I like the sound of C-Db-G though (intone it however think
best), so maybe I'm just more used to stuff like that.

-Mike

Mike wrote:

> > > What is his take on 14:18:21? That one's pretty out there.
> > > Or 18:22:27?
> >
> > Personally I'd expect both of them to sound similar, and
> > very much like 'utonal' chords. Low tonalness low roughness.
> > And I fancy that is how they sound to me.
>
> 18:22:27 is I think the 11-limit neutral triad... Is it utonal?

No, but I'm saying it should sound like a utonal chord,
i.e. a chord with low tonalness and low roughness.

> The supermajor triad is definitely utonal

It isn't. The term is meaningless unless you have consecutive
odd or integer harmonics/subharmonics.

-Carl

On Thu, Aug 19, 2010 at 9:33 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > The supermajor triad is definitely utonal
>
> It isn't. The term is meaningless unless you have consecutive
> odd or integer harmonics/subharmonics.

Skipping one isn't allowed? So does 6:7:9 not count as otonal, or
4:5:7, or so on?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> 18:22:27 is I think the 11-limit neutral triad... Is it utonal? The
> supermajor triad is definitely utonal but sounds terrible to me and I
> hate it.

Why is it OK to hate the supermajor triad but not the 4edo tuning of dim7 I wonder?

> It's just the utonal equivalent of 8:11:12. It's not narrow enough to
> bother me.

It's narrow enough to add considerable roughness.

On Thu, Aug 19, 2010 at 10:13 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > 18:22:27 is I think the 11-limit neutral triad... Is it utonal? The
> > supermajor triad is definitely utonal but sounds terrible to me and I
> > hate it.
>
> Why is it OK to hate the supermajor triad but not the 4edo tuning of dim7 I wonder?

What?

> > It's just the utonal equivalent of 8:11:12. It's not narrow enough to
> > bother me.
>
> It's narrow enough to add considerable roughness.

Roughness is something I have built a high tolerance to and like. As I
said, I'm used to throwing minor seconds in chords, so neutral seconds
don't bother me in the least. C D Eb G Bb is a great voicing for a
minor 9 chord and it has more roughness than (8:11:12).

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > It isn't. The term is meaningless unless you have consecutive
> > odd or integer harmonics/subharmonics.
>
> Skipping one isn't allowed? So does 6:7:9 not count as otonal, or
> 4:5:7, or so on?

Partch only used the terms for complete tetrads, pentads, etc.
as far as I know. Other than that, how would you define it?
Presumably, they are sets of chords composed of the same dyads
but having (maximally?) different Tenney heights.

-Carl

On Thu, Aug 19, 2010 at 10:35 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > It isn't. The term is meaningless unless you have consecutive
> > > odd or integer harmonics/subharmonics.
> >
> > Skipping one isn't allowed? So does 6:7:9 not count as otonal, or
> > 4:5:7, or so on?
>
> Partch only used the terms for complete tetrads, pentads, etc.
> as far as I know. Other than that, how would you define it?
> Presumably, they are sets of chords composed of the same dyads
> but having (maximally?) different Tenney heights.

I've been using the term in the sense that if for a:b:c:d:... If the
Tenney height of 1/(a:b:c:d:...) is lower than the Tenney height of
a:b:c:d:..., then it's utonal, and if not, it's otonal. I've never
seen the term properly defined, and that's just what I had thought
everyone was saying by the concept. The distinction would in either
case become less and less useful as the Tenney height gets higher and
higher, and as more and more notes are added.

What I always thought would be interesting would be to generalize it
to some kind of scalar "xtonality" value in which chords like
4:5:6:7:8:9 have a large positive value, chords like 1/(4:5:6:7:8:9)
have a large negative value, ASS's have a value of zero, and things
like major 6 chords would have a lesser positive value.

There are a few ways I could imagine going about this:
- Calculating the xtonal value of each dyad as a -1 (utonal) or a 1
(otonal) by determining whether the dyad or its inverse, has a greater
Tenney height, and then averaging them in some way (RMS, linear
average, geometric mean, whatever)
- Calculating the xtonal value of each dyad by assigning it a
positive/negative value based on some weighting or complexity factor
(so that 3/2 is "more otonal" than 132/131, for example), and then
doing the averagine
- Calculating the harmonic entropy of each dyad and its inverse,
picking whichever is lower, and then working some clever math so that
an entropy of infinity is normalized to "0" and an entropy of 0 is
normalized to "1", and then averaging them with the average du jour
- Doing any of the above, but instead of just doing it with dyads,
doing it with every subset of the chord - e.g. all of the triads,
dyads, and what have you, and weighting and averaging them together in
some way

Mind you that this is stuff I just came up with on the spot in
response to your question, so I don't know if it's already been done
yet or if there's a better way (Gene?). But if I were king, something
like 8:9:10:12:13 would definitely be otonal, and something like
8:9:32/3:10:12:13 would definitely be "mostly otonal," except for the
one utonal 4/3 dyad.

-Mike

Hi Carl, thanks for these very helpful and informative replies!

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Algorithms for telling if there is a single dominant VF are
> different from algorithms for finding VFs. Tenney height
> and harmonic entropy are examples of the former.

Do you have any ready-at-hand sources on virtual fundamentals?

> Yeah... keep in mind that 3:5:7 spans more than an octave.
> It's a very consonant chord, and it's definitely more
> consonant than 4:5:6 as you go down the keyboard.

I suppose that must be true...never thought about it that way. I suppose the larger pitch spacing must have some critical band ramifications, especially in lower registers. But in terms of "instantaneously producing a sense of relaxation", I really don't get that from 3:5:7. If 3:5:7 was really more concordant than 4:5:6, shouldn't BP music be a lot more popular?

> Would you prefer I suggest a property that's easy to compute
> but wrong? :P

Without quantification, how can we even know if "fit to a harmonic series" is right? But I take your point. It seems to be a more promising direction than the alternatives.

> Keep in mind, h.e. is proportional to Tenney height for just
> chords. It only comes into its own in the case of tempered
> chords. And if they're fairly accurate tempered chords, you
> can assume they have the concordance of the just chord they
> approximate. I realize in your work with ETs you have often
> come across chords that don't fit this bill. So perhaps you
> would like to collaborate on computing triadic h.e. for the
> first time? Or at least collaborate on nagging Paul to do so?

My lack of math skills precludes me being of much assistance in working it out, but I'd be happy to nag Paul about it. Has he changed e-mail addresses in the last five years? Or is that one of his, the pun on "polyester walrus", still current?

> You mean 10:12:15? Paul's interpretation there is that we
> basically hear a 3:2 with some junk in the middle, and the
> junk has low roughness so it doesn't bother us much.

Isn't this interpretation basically an admission that Tenney height can be discounted in certain circumstances, like when the dyadic roughness for some intervals in the chord are low enough?

> Similarly, he predicts the best inversion of the utonal 7th
> chord should be 1/1 6/5 3/2 12/7 (70:84:105:120) even though
> 1/1 7/6 7/5 7/4 (60:70:84:105) has lower Tenney height.
> Since it has a 3:2 above the root and that's a strong VF cue.

So this is a good example of when "rootedness" may be a trump card over Tenney height?

> > Are you assuming that "fit to a harmonic series" might be
> > a good predictor just because utonal chords tend to be less
> > concordant than otonal chords?
>
> That's one reason, yes. Harmonic entropy, by the way,
> *assumes this* from the start.

Obviously, since all utonal chords can be translated into otonal form with a significant increase in Tenney height.

> > I wonder if there aren't maybe some utonal chords that are
> > more concordant than their otonal counterparts? I expected
> > 1/(9:10:12) and 1/(16:18:21) to be more concordant, since
> > the utonal versions both put a less complex interval closer
> > to the lowest note, but oddly enough, I found the otonal
> > versions still sounded more relaxed.
>
> Once you leave the regime of consecutive integer or odd
> harmonics, the otonal/utonal distinction ceases to be
> meaningful. For instance, chords like 10:12:15:18 are
> their own inverses...

Yeah, I figured that out.

> Odd limit will at least erase such distinctions, and lets
> you work along customary octave-equivalent lines. e.g.
> 3:4:5 and 4:5:6 are equally concordant under odd limit vs.
> the former being more concordant under Tenney height.
> On the other hand, octave equivalence gives loony results
> beyond the 7-limit.

Ugh. I feel like we've come full circle. It's amazing how we have all these measures of concordance, yet none is completely effective or consistent at describing and predicting the phenomenon of concordance...yet they are widely-used anyway. I am starting to think I ought to just leave out concordance entirely and take the Easley Blackwood approach.

> > If my accusation was completely baseless--if there is in
> > fact no one here that considers "concordance" to be equivalent
> > to "consonance"--I apologize.
>
> I can't remember anyone ever claiming that on this list.
> The idea that sensory and musical consonance are different
> things is very widely accepted. Certainly by all the authors
> of psychoacoustics research I have ever read.
>

Then I must apologize for my false accusations.

> Normally what we do is find the most concordant tuning for
> desired chords. This reverse approach is interesting to me,
> and is one I've actually tried myself. See
> http://lumma.org/music/theory/ets-program/ETsProgram3.xls

Awesome. Thanks! I like this.

> That's telling you the best-approximated chords (of 3-6
> notes) for each ET < 100, where the ET is allowed to have
> a stretched (or compressed) octave.
>
> > and for tunings where there is no approximate 4:5:6 or 6:7:9,
> > I haven't found a good guiding principle except my own ears.
>
> I must say, I don't think by-ear is all that bad. It's a
> little slow but the work itself might be inspiring. And I
> think you can tell beginners about it. What's wrong with,
> "I tried all these chords and this one sounded the most
> concordant"?

And this is probably what I'm going to end up going with. At least it's consistent.

> > So far, I've not seen any theory that reliably predicts the
> > "next most concordant thing" to a 4:5:6 triad. It sure
> > isn't a 5:6:7, or a 3:5:7.
>
> What is it then? Don't keep us in suspense!

At a guess, I'd say 6:7:9, but I can't say for sure what it is, only what it *isn't*.

-Igs

MikeB>> It's just the utonal equivalent of 8:11:12. It's not narrow enough to
>> bother me.
Gene>It's narrow enough to add considerable roughness.
I figure anything under 9/8 you'll hear a bit of extra roughness (not much to
bother most people, I'm betting)...but I find it fairly slight and level up to
about 12/11 and then become exponentially higher from 12/11 the 13/12 and so
on. Add Plomp and Llevelt's dissonance curve and the Harmonic Entropy curve up
and they also say the 10/9-12/11 range is quite flat far as discordance.

Igs>"If 3:5:7 was really more concordant than 4:5:6, shouldn't BP music be a lot
more popular?"

IMVHO, simply stacking triads is often a lousy way to make scales, because so
many of the interesting options involve suspended, diminished, add2, etc. chords
and if you make the triads too pure it often sends everything else down the
toilet for the triads' sake. Call me a skeptic, but I swear I could make up a
few "odd-harmonic-optimized" scales that sound better all-around than BP to a
good percentage of casual/"everyman" listeners: I'm not impressed by BP at all
even with suggested timbres used.

Hi Michael~
I agree.
In general i find i can deal with the harmonies of a given scale more than the otherway around.
In fact with a good scale once i have it in my head, higher numbers or strange spacings and configurations don't bother me at all. I have no problem with using any three notes in say a just diatonic, or most diatonics for that matter. I don't think i would pull those harmonies out odf the air though
I feel cramped not liberated by BP.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Igs>"If 3:5:7 was really more concordant than 4:5:6, shouldn't BP music be a lot
> more popular?"
>
> IMVHO, simply stacking triads is often a lousy way to make scales, because so
> many of the interesting options involve suspended, diminished, add2, etc. chords
> and if you make the triads too pure it often sends everything else down the
> toilet for the triads' sake. Call me a skeptic, but I swear I could make up a
> few "odd-harmonic-optimized" scales that sound better all-around than BP to a
> good percentage of casual/"everyman" listeners: I'm not impressed by BP at all
> even with suggested timbres used.
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I've been using the term in the sense that if for a:b:c:d:...
> If the Tenney height of 1/(a:b:c:d:...) is lower than the Tenney
> height of a:b:c:d:...

Sounds ok to me, though I'm not sure why the 1/ transformation
would be special vs. any other that kept the total list of
dyads the same. And probably there should be a new name for it.

> What I always thought would be interesting would be to generalize it
> to some kind of scalar "xtonality" value in which chords like
> 4:5:6:7:8:9 have a large positive value, chords like 1/(4:5:6:7:8:9)
> have a large negative value, ASS's have a value of zero, and things
> like major 6 chords would have a lesser positive value.
[snip]

Looks promising, in case you are motivated to develop it
futher...

-Carl

On Fri, Aug 20, 2010 at 4:37 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I've been using the term in the sense that if for a:b:c:d:...
> > If the Tenney height of 1/(a:b:c:d:...) is lower than the Tenney
> > height of a:b:c:d:...
>
> Sounds ok to me, though I'm not sure why the 1/ transformation
> would be special vs. any other that kept the total list of
> dyads the same. And probably there should be a new name for it.

What other transformation is there that keeps the total list the same?

-Mike

Hi Igs!

> > Algorithms for telling if there is a single dominant VF are
> > different from algorithms for finding VFs. Tenney height
> > and harmonic entropy are examples of the former.
>
> Do you have any ready-at-hand sources on virtual fundamentals?

Not to hand. Try a web search. You might also try poking
around Terhardt's site (I linked to a couple days ago).

> > Yeah... keep in mind that 3:5:7 spans more than an octave.
> > It's a very consonant chord, and it's definitely more
> > consonant than 4:5:6 as you go down the keyboard.
>
> I suppose that must be true...never thought about it that way.
> I suppose the larger pitch spacing must have some critical band
> ramifications, especially in lower registers. But in terms of
> "instantaneously producing a sense of relaxation", I really
> don't get that from 3:5:7.

Sure you're not just balking at 7 due to underexposure?

> If 3:5:7 was really more concordant
> than 4:5:6, shouldn't BP music be a lot more popular?

BP uses 3:1 instead of 2:1... seems anti- Tenney height to me.

Keep in mind that 4:5:6 chords aren't as common in music as
they are in music theory. I mean, how often is music harmonized
strictly in close-position triads? 2:3:5 seems more common.
In ragtime piano, there's a lot of 1:2:5:6:8 type stuff. And
I think a variety of voicings pop out of guitar fingerings,
but you know much more about that than I.

Conclusion: even in the 5-limit where octave equivalence works
pretty well, "voicing" is still incredibly important. In jazz
where larger chords reign, people pay even closer attention
to it. In extended just intonation, all hell breaks loose.

> > Would you prefer I suggest a property that's easy to compute
> > but wrong? :P
>
> Without quantification, how can we even know if "fit to a
> harmonic series" is right? But I take your point. It seems
> to be a more promising direction than the alternatives.

It's pretty clear it's right for dyads. For larger chords,
informal comments and polls seem to support it. We have
handwaving arguments from evolutionary biology that it should
be true. And it's compatible with known neuroanatomy.

> My lack of math skills precludes me being of much assistance
> in working it out, but I'd be happy to nag Paul about it. Has
> he changed e-mail addresses in the last five years? Or is
> that one of his, the pun on "polyester walrus", still current?

I think it's current.

> > You mean 10:12:15? Paul's interpretation there is that we
> > basically hear a 3:2 with some junk in the middle, and the
> > junk has low roughness so it doesn't bother us much.
>
> Isn't this interpretation basically an admission that Tenney
> height can be discounted in certain circumstances, like when
> the dyadic roughness for some intervals in the chord are
> low enough?

Paul's approach led me to suggest that different subsets of
a chord should compete, weighted by their Tenney height and
the portion of the chord they 'explain'. For 10:12:15 for
instance, we might have

interpretation -> portion explained * 1/geomean = result

2:3 -> 2/3 * 1/sqrt(6) = 0.27
10:12:15 -> 3/3 * 1/cubert(1800) = 0.08

So 2:3 is the preferred interpretation.

> Ugh. I feel like we've come full circle. It's amazing how
> we have all these measures of concordance, yet none is
> completely effective or consistent at describing and
> predicting the phenomenon of concordance...

Human hearing is actually a pretty complex phenomenon.
It's not like calculating the orbit of the moon. We shouldn't
expect a single concise formula to explain it completely.

> > Normally what we do is find the most concordant tuning for
> > desired chords. This reverse approach is interesting to me,
> > and is one I've actually tried myself. See
> > http://lumma.org/music/theory/ets-program/ETsProgram3.xls
>
> Awesome. Thanks! I like this.

Let me know (onlist or off) if you need help interpreting it.

-Carl

Mike wrote:

> > > I've been using the term in the sense that if for a:b:c:d:...
> > > If the Tenney height of 1/(a:b:c:d:...) is lower than the Tenney
> > > height of a:b:c:d:...
> >
> > Sounds ok to me, though I'm not sure why the 1/ transformation
> > would be special vs. any other that kept the total list of
> > dyads the same. And probably there should be a new name for it.
>
> What other transformation is there that keeps the total list the
> same?

I guess you're right, it's the only one. So the one with
lower Tenney height we'll call utonal, and if the Tenney height
is unchanged we'll just call them "symmetric chords" perhaps?

-Carl

Hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Not to hand. Try a web search. You might also try poking
> around Terhardt's site (I linked to a couple days ago).

Will do.

> > > Yeah... keep in mind that 3:5:7 spans more than an octave.
> > > It's a very consonant chord, and it's definitely more
> > > consonant than 4:5:6 as you go down the keyboard.
> >
> > I suppose that must be true...never thought about it that way.
> > I suppose the larger pitch spacing must have some critical band
> > ramifications, especially in lower registers. But in terms of
> > "instantaneously producing a sense of relaxation", I really
> > don't get that from 3:5:7.
>
> Sure you're not just balking at 7 due to underexposure?

Yeah. I find 4:5:7, 3:6:7, 4:6:7, and 3:4:5:7 all more "relaxing". My guess is it's the rootedness thing (or something).

> > If 3:5:7 was really more concordant
> > than 4:5:6, shouldn't BP music be a lot more popular?
>
> BP uses 3:1 instead of 2:1... seems anti- Tenney height to me.

Interesting. Could be an argument for Bohpier temperament, eh?

> Keep in mind that 4:5:6 chords aren't as common in music as
> they are in music theory. I mean, how often is music harmonized
> strictly in close-position triads? 2:3:5 seems more common.
> In ragtime piano, there's a lot of 1:2:5:6:8 type stuff. And
> I think a variety of voicings pop out of guitar fingerings,
> but you know much more about that than I.

True. On guitar, the major barre chord voicing is basically 2:3:4:5:6:8 (with the root on the low 'E' string) or 2:3:4:5:6 (with the root on the 'A' string. However, the open-position G major chord is 4:5:6:8:10:16, and open-position C major (with the fifth in the bass) is 3:4:5:6:8:10. Of course, it's pretty rare (except among acoustic/folk guitarists) to consistently use chords voiced like this. Electric guitarists in most genres rarely use all 6 strings at once. 4:5:6:8 chords, or sometimes just 5:6:8 chords, are common when played on the high strings as "partial" barre chords. Still, the 4:5:6 is often at the heart of the triad. It's rare to drop the 4 or the 6. It's also rare to put anything but the fundamental in the bass, though there is one voicing of a D major, usually used to transition from a G major to an E minor, that is voiced 5:6:8:12:16:20, but since it's 12-tET putting the third in the bass is pretty unstable (hence its use as a transition chord and not a final).

One wonders, though, if close-position triads wouldn't be more popular if we weren't using 12-tET.

Side note: I remember having trouble figuring out a few punk songs back in the day, because the guitarist used chords voiced 3:4:6:8, which created the illusion of a virtual 2 lower in pitch than the lowest-tuned string. Putting the fifth in the bass like that is sometimes used to thicken the sound because of that phenomenon.

> > My lack of math skills precludes me being of much assistance
> > in working it out, but I'd be happy to nag Paul about it. Has
> > he changed e-mail addresses in the last five years? Or is
> > that one of his, the pun on "polyester walrus", still current?
>
> I think it's current.

I'll drop him a line, then.

> > Isn't this interpretation basically an admission that Tenney
> > height can be discounted in certain circumstances, like when
> > the dyadic roughness for some intervals in the chord are
> > low enough?
>
> Paul's approach led me to suggest that different subsets of
> a chord should compete, weighted by their Tenney height and
> the portion of the chord they 'explain'. For 10:12:15 for
> instance, we might have
>
> interpretation -> portion explained * 1/geomean = result
>
> 2:3 -> 2/3 * 1/sqrt(6) = 0.27
> 10:12:15 -> 3/3 * 1/cubert(1800) = 0.08
>
> So 2:3 is the preferred interpretation.

Now this...this is interesting. If a triad can be thought of as a sandwich, this suggests that as long as the bread makes a good enough "fifth", that you should be able to get away with putting almost any sort of "meat" in the middle. Even with a 4:5:6 chord, this formula puts the 3/2 above the full triad by about 0.07.

So then, shouldn't octaves be even stronger than fifths? Looking at traditional "power chords" of 2:3:4, the octave takes up 0.47, while the whole triad is 0.35, suggesting that the 2/1 is the preferred interpretation. It's a narrower lead, to be sure, but in other chords it would be much larger. Maybe this also explains a phenomenon/"rule" I've noticed in playing microtonal guitar: adding a 2/1 above the root makes any dyad more concordant. In a tuning like 16-EDO, I use octave-doubling a LOT, making power-chords like ~4:5:8, ~4:7:8, and ~6:11:12...maybe the reason they work so well is that the octave is such a strong cue?

> Human hearing is actually a pretty complex phenomenon.
> It's not like calculating the orbit of the moon. We shouldn't
> expect a single concise formula to explain it completely.

And yet, so much evidence seems to point that there is *some* kind of pattern or consistency at work, even with all the variations between genera and culture. It surely can't be the case that *any* chord can be a consonance given the right context, can it?

-Igs

BP isn't actually a scale made by just "stacking triads". Well, perhaps the JI version is, but the ET version is a MOS scale with a generator of a slightly sharpened 9/7 and a period of an approximate 3/1. You can basically play BP in 19-EDO, too, if you don't mind the tempered 3/1, since 19-EDO gives an acceptable tuning of 3:5:7 and allows the same BP 9-note MOS scale. However, just as scales other than the diatonic can give 4:5:6 triads, there are bound to be scales other than the BP diatonic that give 3:5:7 triads. Especially if we use a different period. Bohpier comes to mind.

-Igs

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Igs>"If 3:5:7 was really more concordant than 4:5:6, shouldn't BP music be a lot
> more popular?"
>
> IMVHO, simply stacking triads is often a lousy way to make scales, because so
> many of the interesting options involve suspended, diminished, add2, etc. chords
> and if you make the triads too pure it often sends everything else down the
> toilet for the triads' sake. Call me a skeptic, but I swear I could make up a
> few "odd-harmonic-optimized" scales that sound better all-around than BP to a
> good percentage of casual/"everyman" listeners: I'm not impressed by BP at all
> even with suggested timbres used.
>

Igs wrote:

>> Paul's approach led me to suggest that different subsets of
>> a chord should compete, weighted by their Tenney height and
>> the portion of the chord they 'explain'. For 10:12:15 for
>> instance, we might have
>>
>> interpretation -> portion explained * 1/geomean = result
>>
>> 2:3 -> 2/3 * 1/sqrt(6) = 0.27
>> 10:12:15 -> 3/3 * 1/cubert(1800) = 0.08
>>
>> So 2:3 is the preferred interpretation.
>
> Now this...this is interesting. If a triad can be thought
> of as a sandwich, this suggests that as long as the bread
> makes a good enough "fifth", that you should be able to get
> away with putting almost any sort of "meat" in the middle.

...as long as the meat doesn't spoil the broth. I mean,
add roughness. It still has to make simple ratios on
either side.

> Even with a 4:5:6 chord, this formula puts the 3/2 above
> the full triad by about 0.07.

Yes, please note I haven't worked on this at all, and just
wrote the simplest implementation of the idea. There are
other ways to weight the interpretations besides multiplying
by the fraction of notes used. Still, the margin of
dominance of 3:2 over 4:5:6 is much smaller than over
10:12:15, and in neither case is 5:4 or 6:5 winning.

> In a tuning like 16-EDO, I use octave-doubling a LOT, making
> power-chords like ~4:5:8, ~4:7:8, and ~6:11:12...maybe the
> reason they work so well is that the octave is such a
> strong cue?

Yes, possibly. We're firmly in the realm of speculative
theory here. FWIW Paul has responded favorably to the idea
when I've suggested it.

>> Human hearing is actually a pretty complex phenomenon.
>> It's not like calculating the orbit of the moon.
>
> And yet, so much evidence seems to point that there is *some*
> kind of pattern or consistency at work, even with all the
> variations between genera and culture.

Sure there's consistency at work, but how much of what we're
interested in can it explain? Newton's laws aren't perfect
either, but they explain almost 100% of what you need to
navigate around the solar system. You'll never get anything
that good in music theory.

Of rhythm, melody and harmony, harmony is probably the least
important in music, and it's by far the best understood.
Only Western music really uses harmony in a way that makes
discordance formulas useful. That's probably because only
Westerners had ubiquitous, accurately-tuned instruments with
harmonic spectra. When people from Africa and elsewhere
encountered these instruments, they produced music (jazz etc)
in the same vein.

> It surely can't be the case that *any* chord can be a
> consonance given the right context, can it?

Musical consonance, yes. Concordance, no.

-Carl

On Fri, Aug 20, 2010 at 4:38 PM, cityoftheasleep
<igliashon@...> wrote:
>
> > Human hearing is actually a pretty complex phenomenon.
> > It's not like calculating the orbit of the moon. We shouldn't
> > expect a single concise formula to explain it completely.
>
> And yet, so much evidence seems to point that there is *some* kind of pattern or consistency at work, even with all the variations between genera and culture. It surely can't be the case that *any* chord can be a consonance given the right context, can it?

To chime in here, if you haven't read Rothenberg - you might want to
take a gander at it.

I have basically come to the same conclusion that you have - that the
"cultural" aspect to consonance is something that can be dissected and
explored. After all, what could the "cultural" aspect to it be, rather
than just ways we have learned to listen to music? And those "ways"
must be specific things that we can figure out.

I think that visuo-spatial reasoning is a form of reasoning that
develops with age. And in this case, there is some kind of "auditory"
reasoning that we are also developing. People who are used to
listening to 12-tet, or meantone music or what have you, will have
certain aspects of that reasoning be really developed, and other
aspects be not so developed. People in 12-tet who are used to hearing
the octatonic scale, and modal harmonies, and such, will have
different aspects developed than the camp that listens to
Classical-era music as a rule.

What Rothenberg has done by viewing scales as "mental reference
frames" and coming up with properties is, I think a good first step.
Cultural differences can be partly explained by the maps that we have,
and how forming a new map requires repeated listens of a piece of
music so as to figure out how the notes relate to each other (note the
visuo-spatial reasoning theme again). Until then, you'll just keep
using the same mental algorithm for figuring things out as you're used
to, until you can form a new map.

I'm not sure that Rothenberg has figured out the holy grail of music
theory, but I think a lot of his ideas are right on the money and you
might be interested. Either way, I think all of this is on a different
level of processing than the periodicity/JI mechanism, and that the
exact intonation of notes (even between low-integer JI ratios) is much
less important to the end gestalt produced than this stuff.

-Mike

On Fri, Aug 20, 2010 at 4:17 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> I guess you're right, it's the only one. So the one with
> lower Tenney height we'll call utonal, and if the Tenney height
> is unchanged we'll just call them "symmetric chords" perhaps?

Sure, although isn't that what the "anomalous suspended saturation"
term is for? Although I prefer "symmetric chords" to that.

-Mike

Hi Mike,

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I'm not sure that Rothenberg has figured out the holy grail of music
> theory, but I think a lot of his ideas are right on the money and you
> might be interested. Either way, I think all of this is on a different
> level of processing than the periodicity/JI mechanism, and that the
> exact intonation of notes (even between low-integer JI ratios) is much
> less important to the end gestalt produced than this stuff.

I haven't read Rothenberg in great depth, but I've skimmed a lot. In many regards, I do believe that the exact intonation of chords is less important than the scalar/tonal structure which supports them, but there are definitely a few regards where I think intonation is important. I know that you can convey the same "meaning" in, say, diatonic music regardless of whether you use sharp fifths (and the associated supermajor thirds) or flat fifths (and the associated sharp-neutral thirds); yet the actual "sensation" of the chords will vary with the intonation, sounding more (or less) stable/relaxed depending. I.e., they are like the same words spoken in a different voice. And I think that anyone looking at a variety of possible tunings will want to have some idea of 1) what sort of chords are possible and 2) how those chords will sound. I'm not exactly sure how much JI or psychoacoustic information is necessary to give people sufficient understanding of these properties, but suffice to say knowing that one scale has sharper thirds than another won't tell you which has the more stable third, unless you know what constitutes a stable third.

IOW, intonation is really just a way to evaluate the "character" of a tuning, telling whether it has a warbly active sound or a calm stable sound, or what chords in the tuning make you go "ah" and what chords make you go "eh" (and maybe even what chords make you go "augh!", as the case may be). These *are* important considerations, are they not?

-Igs

Igs>"IOW, intonation is really just a way to evaluate the "character" of a
tuning, telling whether it has a warbly active sound or a calm stable sound, or
what chords in the tuning make you go "ah" and what chords make you go "eh" (and
maybe even what chords make you go "augh!", as the case may be). These *are*
important considerations, are they not?"

Right. And actually I am trying to make a set of scales at this point to
test just that...the importance of intonation and "tonal stability" vs.
importance of having flexible musical usage IE "this can be used as an
X,Y,Z...-type chord and not just an X-type chord". All dyads in the scale are
within about 8 cents of the desired dyads they are "trying to capture" and very
similar in terms of melodic feel.

One of the scales has almost all 5-limit dyads and a few 7-limit and conforms
fairly well to JI diatonic, only with more "Just Compliant" chords possible
(again within 8 cents or so). It also has all 5ths except one virtually
perfect...just like 12TET diatonic. Consider it an "irregularly tempered
diatonic scale" that aims to maximize "average accuracy" for low limit just
chords.

...that scale is...
1.12
1.2
1.338
1.5
1.674
1.792
2

The second has 5 and 7-limit a handful of 9-limit and 11-limit dyads available
and leans more toward having chords with good and flexible "character/use" than
being "just". It also has a couple of odd 5ths (about 22/15).

........that scale is..............

1.12
1.2222*
1.338
1.5
1.674
1.792
2

The third scale has about half its dyads are in 7+ limit with maybe 25% in
11-limit. It has a few odd 5ths (about 22/15) and tons of neutral chords that
can be used as "major or minor". It aims to give maximum use flexibility at the
expense of "pure intonation".

....That scale is......

1.12
1.222222*
1.338;
1.5;
1.674
1.83*
2

Try them...do you hear/feel your preference leaning more toward purer intonation
(scale #1), a mix of qualities (scale #2), or musical flexibility and chords
that can have more multiple uses (scale #3)?

On Fri, Aug 20, 2010 at 9:16 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Hi Mike,
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > I'm not sure that Rothenberg has figured out the holy grail of music
> > theory, but I think a lot of his ideas are right on the money and you
> > might be interested. Either way, I think all of this is on a different
> > level of processing than the periodicity/JI mechanism, and that the
> > exact intonation of notes (even between low-integer JI ratios) is much
> > less important to the end gestalt produced than this stuff.
>
> I haven't read Rothenberg in great depth, but I've skimmed a lot. In many regards, I do believe that the exact intonation of chords is less important than the scalar/tonal structure which supports them, but there are definitely a few regards where I think intonation is important. I know that you can convey the same "meaning" in, say, diatonic music regardless of whether you use sharp fifths (and the associated supermajor thirds) or flat fifths (and the associated sharp-neutral thirds); yet the actual "sensation" of the chords will vary with the intonation, sounding more (or less) stable/relaxed depending.

Sure, of course.

> I.e., they are like the same words spoken in a different voice. And I think that anyone looking at a variety of possible tunings will want to have some idea of 1) what sort of chords are possible and 2) how those chords will sound. I'm not exactly sure how much JI or psychoacoustic information is necessary to give people sufficient understanding of these properties, but suffice to say knowing that one scale has sharper thirds than another won't tell you which has the more stable third, unless you know what constitutes a stable third.

I think that it all matters. Intonation matters, the scale matters,
and timbre matters too. But I think that in terms of the end result,
certain things matter more than others. And I think they matter in
this order:

1) The map
2) Intonation
3) Roughness

I'm just pointing out that, as you've noticed, you can mess with
wide-fifth and narrow-fifth tunings to the point where completely
different JI ratios are used for the thirds - and yet the overall
effect on the gestalt of these tunings is that they are "variations on
a theme" for general diatonic/chromatic hearing.

There is absolutely nothing wrong with this, or with experimenting
with "warped maps" in general. This is one of the zillions of new
compositional techniques afforded by microtonal music and it should be
explored. However, in my view, if we're moving towards some kind of
xenharmonic ideal, people should also consider messing around with the
map and trying to figure out whatever "theory" works there. I don't
have a clue what it is.

Even within 12-tet, screwing with the map can create all sorts of
sounds that are "xenharmonic" - or were definitely xenharmonic when
they first came out. It was fairly groundbreaking, for example, when
Debussy started messing around with modes like lydian dominant and
such. And we all know what happened when people started messing around
with the octatonic scale (was Stravinsky the first)? And when you have
classic rock bands writing songs in Mixolydian as though it were on
equal footing with major and minor - when was that sound heard
previously?

So in my view, if we're assuming this is the 1700s, it would be more
overall "xenharmonic" for someone to start messing around with modes
and symmetric scales than for someone to mess around with retunings of
the diatonic scale to get closer to 5/4, closer to 3/2, or make the
fifths wider so the minor thirds hit 7/6. The latter is certainly
interesting, but the former is what epouses the xenharmonic ideal more
strongly to me.

> IOW, intonation is really just a way to evaluate the "character" of a tuning, telling whether it has a warbly active sound or a calm stable sound, or what chords in the tuning make you go "ah" and what chords make you go "eh" (and maybe even what chords make you go "augh!", as the case may be). These *are* important considerations, are they not?

They certainly are, and if you are interested in exploring that side
of it you should. I'm just communicating a new perspective I've been
considering recently. To be honest, whatever you're doing, you're
doing it right, as your albums indicate.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Hi Igs!
> > Do you have any ready-at-hand sources on virtual fundamentals?
> Not to hand. Try a web search. You might also try poking
> around Terhardt's site (I linked to a couple days ago).

I have collected over 100 files in a folder on virtual pitch, although I am still ploughing through them so can't say which are the best (or if they are all actually about virtual pitch). Right now I can't find details of Terhardt's or Parncutt's or Hofmann-Engl's calculations, but here are two docs that may be of interest.

Pitch Perception models - a historical review
Alain de Cheveigne'

Commentary on Cook and Fujisawa ...
Richard Parncutt

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
> ... Right now I can't find details of Terhardt's or Parncutt's or Hofmann-Engl's calculations ...

Just remembered - see:
/tuning/topicId_84080.html#84280

This in turn reminds me that I'm sure Carl said he would consider scanning the relevant pages of van Eck when he had the time. How about it, Carl? As a quid pro quo let me say I am planning to implement the dyadic H.E. calculation as a possible precursor to trying triadic H.E.

Steve M.

MikeB>"
1) The map
2) Intonation
3) Roughness
" (most important determinants of "resolvedness" in a scale)

Agreed, if I understand the concept of map correctly.
Like...if a swap a 4/3 with a 11/8 or a 15/8 to a 11/6...I'm effectively
swapping the map?
Or only if I do something like make a scale, say, with both a 4/3 and 9/7 (IE
two different versions of the same type/class of tone)?

Could you define to concept of "map" further?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I think that it all matters. Intonation matters, the scale matters,
> and timbre matters too. But I think that in terms of the end result,
> certain things matter more than others. And I think they matter in
> this order:
>
> 1) The map
> 2) Intonation
> 3) Roughness
>
[snip]

> There is absolutely nothing wrong with this, or with experimenting
> with "warped maps" in general. This is one of the zillions of new
> compositional techniques afforded by microtonal music and it should be
> explored. However, in my view, if we're moving towards some kind of
> xenharmonic ideal, people should also consider messing around with the
> map and trying to figure out whatever "theory" works there. I don't
> have a clue what it is.

Have you tried making *literal* maps of various scales/temperaments? I'm speaking, of course, of triangular harmonic lattices. I wish I could remember their proper name, or where I first encountered them (I think in one of Paul's papers?), but basically they're like fokker periodicity blocks but instead of a square grid where the y-axis is a major third and the x-axis is fifths, it's a triangular grid where going horizontal is a fifth, going up and to the right is a major third, and going down and to the right is a minor third. So C major would be a triangle that points up, with the base being a line C---G, and the point on top being an E. Alternatively, C minor would be a triangle that points down, with the base still being C---G but the bottom point being Eb. For a more general form, use scale degrees instead of letter names.

Of course, you don't have to use major and minor thirds bisecting a perfect fifth; the x-axis can be any interval that can be subdivided into two intervals. For 18-EDO Father temperament, where 16:18:21 is the most stable concord, I use 21/16 in place of 3/2, and 9/8 in place of 6/5, and 7/6 in place of 5/4. Works just as well! And of course, they don't have to be Just intervals either (I actually never use JI with maps like these).

These maps help me immensely when dealing with alternative scales and temperaments, because they show you all the possible common-tone progressions at a glance. I can post a few in the files section if a visual illustration helps. I wouldn't have been able to write "Map of an Internal Landscape" without these maps as a guide. It's interesting how, when you look at these sorts of maps, you realize that certain chord progressions have a specific "shape", and any scale that has that shape has that chord progression.

-Igs

On Sat, Aug 21, 2010 at 1:31 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"
> 1) The map
> 2) Intonation
> 3) Roughness
> " (most important determinants of "resolvedness" in a scale)
>
> Agreed, if I understand the concept of map correctly.
> Like...if a swap a 4/3 with a 11/8 or a 15/8 to a 11/6...I'm effectively swapping the map?
> Or only if I do something like make a scale, say, with both a 4/3 and 9/7 (IE two different versions of the same type/class of tone)?
>
> Could you define to concept of "map" further?

For example, the diatonic map is LLsLLLs. This MOS arises if you use a
fifth as a generator and the fifth lies anywhere between the 7-et
fifth and the 5-et fifth.

For 1/3 comma meantone, the minor third is exactly 6/5. For a
1/3-septimal comma superpyth tuning, where the fourths are wide by 1/3
of 64/63, 3 of them will make exactly 7/6. This has an interesting
effect on the scale, but doesn't radically change anything like
switching from meantone[7] to porcupine [7] does, even though the
thirds are closer to 9/7 and 7/6.

The major thirds in the above scale will be even sharper than 9/7, and
yet STILL sound like "major thirds" in the sense that we're used to...
except sometimes they won't sound quite as "stable," or resolve
properly.

I keep repeating this example to show that multiple JI ratios can
substitute for each other in the same map, but the map is what's
important. How exactly the map is constructed, mentally, is something
I wish I knew but don't.

And remember my earlier example - the 22-equal superpyth harmonic
minor scale. That augmented second is going to be a 5/4 (!) and the
minor thirds will be 7/6.

In 31-equal, the augmented seconds are 7/6 and the minor thirds are ~6/5.

Something interesting to think about.

-Mike

Hi Steve!

> Pitch Perception models - a historical review
> Alain de Cheveigne'
>
> Commentary on Cook and Fujisawa ...
> Richard Parncutt

I've read the latter, but not the former.

> Just remembered - see:
> /tuning/topicId_84080.html#84280

Yes, I recalled that article you linked to and considered
pointing Igs to it. But I didn't recall it presenting the
algorithms in enough detail to do the calculations...

> This in turn reminds me that I'm sure Carl said he would
> consider scanning the relevant pages of van Eck when he had
> the time. How about it, Carl? As a quid pro quo let me
> say I am planning to implement the dyadic H.E. calculation
> as a possible precursor to trying triadic H.E.

Excellent! I still have had the time. :( I have a pile
of stuff to scan here. Basically, if it's made out of
paper, I don't have the time. Sad but true. Maybe when
I get back from burning man (I have a week of down time)...

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Sad but true. Maybe when
> I get back from burning man (I have a week of down time)...
>
> -Carl
>

Well, I never woulda pegged you for a Burner, Carl! One of these years, I swear I'm gonna do a "detwelvulation" camp out there, offering cheap fishing-wire-refretted ukuleles and an EDO personality test. This is my first year not going since 2005, so I'm missing it badly. You must be pretty busy right about now, eh?

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Well, I never woulda pegged you for a Burner, Carl!

It's not the first time he's gone, however.

Igs wrote:

> Well, I never woulda pegged you for a Burner, Carl! One of
> these years, I swear I'm gonna do a "detwelvulation" camp out
> there, offering cheap fishing-wire-refretted ukuleles and an
> EDO personality test. This is my first year not going since
> 2005, so I'm missing it badly. You must be pretty busy right
> about now, eh?

2005 is the only year I missed since 2004! I do have an extra
ticket I'm currently trying to dispose of...

Detwelvulation camp would be great. My dream has been to get
a quartet together that would spontaneously burst into baudy
Elizabethan madrigals as they went around... alas, I haven't
done much work on it. In fact this year I'm being lazy and
having my campmates bring all the gear. My only contribution
will be the portable laser show (now with 5 kinds of lasers!).

-Carl

On Sat, Aug 21, 2010 at 2:18 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Have you tried making *literal* maps of various scales/temperaments? I'm speaking, of course, of triangular harmonic lattices. I wish I could remember their proper name, or where I first encountered them (I think in one of Paul's papers?), but basically they're like fokker periodicity blocks but instead of a square grid where the y-axis is a major third and the x-axis is fifths, it's a triangular grid where going horizontal is a fifth, going up and to the right is a major third, and going down and to the right is a minor third. So C major would be a triangle that points up, with the base being a line C---G, and the point on top being an E. Alternatively, C minor would be a triangle that points down, with the base still being C---G but the bottom point being Eb. For a more general form, use scale degrees instead of letter names.

Well, isn't this just equivalent to the Tenney lattice? Whether the
map is triangular or square will make no difference. In a square map,
a major chord would be a right triangle pointing up, and a minor chord
would be the same shape rotated 180 degrees. In a triangular map, you
just slant the axes so that it makes a triangular map.

But either way, I see what you're getting at. I suppose that's the
same thing as describing the diatonic scale as LLsLLLs, except with an
emphasis on different intervals. The "LLsLLLs" description basically
tells you how the scale is formed in terms of differing sizes of
seconds, and your map tells you how it's formed in terms of differing
sizes of thirds - and how the notes can be grouped triadically, not
just dyadically.

I often wonder what about the diatonic scale that is the "defining"
factor to the brain in determining the overall sound produced. At one
point I thought it had something to do with 3-limit hearing, but the
fact that in mavila[7] the 3-limit relationship for each note is the
same as major - but that the whole scale sounds so much different,
overall - would seem to disprove that.

> Of course, you don't have to use major and minor thirds bisecting a perfect fifth; the x-axis can be any interval that can be subdivided into two intervals. For 18-EDO Father temperament, where 16:18:21 is the most stable concord, I use 21/16 in place of 3/2, and 9/8 in place of 6/5, and 7/6 in place of 5/4. Works just as well! And of course, they don't have to be Just intervals either (I actually never use JI with maps like these).

This is actually a really interesting idea. I guess it's like Tenney
space, but using a different basis for the axes than the primes (or
octave-equivalent primes, as in the example above). What is the proper
term for this, a subgroup temperament or something?

> These maps help me immensely when dealing with alternative scales and temperaments, because they show you all the possible common-tone progressions at a glance. I can post a few in the files section if a visual illustration helps. I wouldn't have been able to write "Map of an Internal Landscape" without these maps as a guide. It's interesting how, when you look at these sorts of maps, you realize that certain chord progressions have a specific "shape", and any scale that has that shape has that chord progression.

If you have them, post them up! I would find that very useful.

-Mike

On Sat, Aug 21, 2010 at 7:14 PM, Carl Lumma <carl@...> wrote:
>
> Igs wrote:
>
> > Well, I never woulda pegged you for a Burner, Carl! One of
> > these years, I swear I'm gonna do a "detwelvulation" camp out
> > there, offering cheap fishing-wire-refretted ukuleles and an
> > EDO personality test. This is my first year not going since
> > 2005, so I'm missing it badly. You must be pretty busy right
> > about now, eh?
>
> 2005 is the only year I missed since 2004! I do have an extra
> ticket I'm currently trying to dispose of...
>
> Detwelvulation camp would be great. My dream has been to get
> a quartet together that would spontaneously burst into baudy
> Elizabethan madrigals as they went around... alas, I haven't
> done much work on it. In fact this year I'm being lazy and
> having my campmates bring all the gear. My only contribution
> will be the portable laser show (now with 5 kinds of lasers!).

Haha, that would be awesome. I've never been to burning man but I need
to get there at some point. The EDO personality test thing sounds
awesome.

-Mike

MikeB>"For example, the diatonic map is LLsLLLs."

Ah ok. But then how to you decide if, say, a step is far enough away from
being a both "small" and a large to be a "medium" (or do you just "force"
everything to be either small or large by rounding to find the "nearest MOS
mapping")?
Funny example...in Wikipedia JI diatonic is listed as having small, medium,
and large sizes. Odd....

The above would seem to work fine for pure MOS scales, but especially not
scales with, say, 4+ different interval sizes. For example, what should I do if
a scale I have has consecutive sizes of 1.09,1.095,1.11, 1.09, 1.12, and 1.11?

>"I keep repeating this example to show that multiple JI ratios can
substitute for each other in the same map, but the map is what's
important."
Got it...but my point of confusion is how far do you have to go far as
difference before you effectively "have" to change the map?

>"And remember my earlier example - the 22-equal superpyth harmonic
minor scale. That augmented second is going to be a 5/4 (!) and the
minor thirds will be 7/6.
In 31-equal, the augmented seconds are 7/6 and the minor thirds are ~6/5."

Oh man am I confused here....it sounds like the interval classes not just
slightly but completely overlap each other IE

"the minor thirds will be 7/6." (in 22TET)
"the minor thirds are ~6/5" (in 31TET)
How, pray tell, do I figure out what is in what class with this sort of
overlap?

>"your map tells you how it's formed in terms of differing
sizes of thirds - and how the notes can be grouped triadically, not
just dyadically."
So, for example...Ls would be a major triad in that system and sL a minor one?

>"I often wonder what about the diatonic scale that is the "defining"
factor to the brain in determining the overall sound produced. At one
point I thought it had something to do with 3-limit hearing, but the
fact that in mavila[7] the 3-limit relationship for each note is the
same as major - but that the whole scale sounds so much different,
overall - would seem to disprove that."

Agreed. Based on experience I have huge doubt that how closely a system is
related to 5th ultimately improves its "resolved-ness"...at least beyond the
apparent fact that having a very low-limit dyad in a chord seems to give more
"slack" for other dyads to be further off pure while still having the chord feel
"stable".

It seems obvious to me that

1) The brain likes having the larger dyad first (IE major vs. minor)

2) The brain seems to choke when it sees Just equal dyads stacked (IE 5/4 times
5/4)...perhaps because they point toward two different roots with almost
absolute authority without putting the focus on one over the other. This may
also explain why 4-tone chords tend to sound best as LsL or sLs instead of Lss
or ssL.

3) If anything(s) seem to help across the board...they are
A) Having a low number of intervals from each class possible (IE one type of
third, fourth, etc. beats two type by a little and three by a whole lot)
B) Having most types of dyadic intervals be low limit (IE 80% or more in
7-odd-limit or less)

I had - since the early days of studying harmony - struggled to
understand how major and minor thirds that formed the backbone of all
common-practice triadic chord progressions could be considered
"dissonant" in Renaissance and in ages before. This was way back when
I had no notion of "alternate tunings" and took 12 equal or quasi-
equal steps per octave on the piano as granted.

About nine years ago, as I discovered and improvised in diverse
tunings - especially 12-tone temperaments and JI subsets easily
adaptable to an electronic keyboard, I noticed how a particular escape
from 12-tET boosted my musical innovativeness in Baroque and Classical
styles which did not arise otherwise. Later on, I had the priviledge
of hearing myself joyously play those Pythagorean and super-
Pythagorean intervals that were so "bright" and fun to employ in
regular Major and minor chords, but never caused much relief to the
accustomed listener toward the end. How much more so from the
perspective of those bygone Renaissance composers who employed a
wholly different texture and counterpoint of modality relying on
fourths, fifths and octaves as scaffolds?

Margo's eludicidation of the whereabouts of 5/4 - arising "naturally"
in a capella singing - as an "unintended consonance" could mark a
breaking point in the history of Western music. Could it be that,
theory had propelled 3-limit Renaissance harmony only so far and the
reality of these "unintended consonances in practice" had to be re-
incorporated into theory?

With little knowledge on the matter, I'd like to ask Margo: Was it not
that Halberstadt keyboard organs in 1300s were ordinarily tuned in
such a way that the naturals bore Pythagorean pitches and the
accidentals (black keys) were tuned midway of the wholetones? If such
was exactly the case, the novel habit in 1400s of tuning the black
keys to flats in the fifths cycle to acquire the schismatic harmonic
Majors on D, E and A would follow a desire to reflect the "consonantia
per accidens" of singers on the keyboard, no?

Might I be brave enough to make the following conjecture? Could it be
possible that the transition from Pythagorean to Meantone was not as
swift as generally assumed and spanned as considerable a time as it
took 12-tone Equal-Temperament to arise from Meantone? And in such a
manner that the trend was far from universal in every corner of
Europe? (attested by examples such as "Sumer is icumen in"?)

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Aug 18, 2010, at 9:35 AM, Margo Schulter wrote:

>
>
>> --- In tuning@yahoogroups.com, "genewardsmith"
>> <genewardsmith@...> wrote:
>>>
>>>
>>>
>>> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
>>>>
>>>> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@>
>>>> wrote:
>>>>>
>>>>> Note also that 4/3 has sometimes been classed as a dissonance,
>>>>> whereas 5/4 always seems to be seen as consonant.
>>>>
>>>> Maybe Margo can correct or qualify this, but in the medieval
>>>> period, I believe 5/4 was considered dissonant.
>>>
>>> I think it was 81/64 which was considered dissonant.
>
> (reply by Carl)
>> Yes, ok, but that doesn't mean that 5/4 was considered consonant.
>> If it had been, they probably wouldn't have tuned 3rds 81/64.
>
> Hi, Gene and Carl.
>
> Really there are two levels on which to answer this question, at least
> as far as Continental European music of the 13th-14th centuries is
> generally concerned.
>
> First, there's a simply categorical statement: _thirds are unstable_.
> Sooner or later, either a major or minor third is going to be followed
> by some stable sonority, often a full 2:3:4 trine or a 3:2 fifth, for
> example. This holds regardless of the fine points of tuning, just as
> a closing minor seventh would be out of place in Bach whether tuned
> 9/5, 16/9, or 7/4.
>
> If we want to use typical 13th-14th century language, we can call
> thirds "imperfect concords" -- although some Italian authors refer
> to them at times as "tolerable dissonances." Note that categories
> of this kind can also apply, especially in a 13th-century framework
> or a 14th-century one admitting lots of 13th-century liberties
> (e.g. Guillaume de Machaut) to things like 4:6:9, 9:12:16, 6:8:9,
> or 8:9:12 -- pleasing, and with a degree of blend, but definitely
> unstable.
>
> From personal experience, I can tell you that having some very
> well-meaning and skillful singers read a piece in 13th-century
> style from site and accidentally conclude a phrase with a major
> third and fifth above the lowest note where a stable fifth was
> intended sounds "wrong" -- whether this sonority was tuned
> 4:5:6, 0-400-700 cents, 64:81:96, or whatever (this was
> a capella, so lots of things were possible).
>
> Carl, I much agree that tuning major thirds at 81/64 -- or often
> in my favorite neo-medieval temperaments somewhere around 14/11
> or 33/26, in a way a "modern" variation on Pythagorean -- fits
> nicely with the stylistic assumption that thirds are active
> and unstable intervals.
>
> However, Gene, there is the interesting question of 5/4 in this
> kind of medieval Continental European setting, up to around 1400
> or so (you make an interesting observation on the following era
> which I'll address below).
>
> One view is that it, or the almost identical Pythagorean
> diminished fourth at a schisma narrower (8192/6561), or 384.36
> cents, is in the same general category as other augmented or
> diminished intervals. This is the view of Johannes Boen in 1357,
> who considers such a diminished fourth, a comma narrow of a
> ditone at 81/64, as a "consonance by situation" (_consonantia
> per accidens_) if supported by a regular major third below,
> e.g. E-G#-C. The same rule applies to the augmented fourth with
> a minor third below, e.g. D-F-B. (Boen notes that the perfect
> fourth, likewise, becomes more concordant when supported by
> an appropriate interval below, here the fifth, e.g. D-A-D.
> While 2:3:4 can thus continue to be seen as an array of the
> three simplest and most concordant non-unisonal dyads not
> exceeding an octave, the fourth has a more problematic and
> debated status than in the 13th century, when it ranks
> with but after the fifth.)
>
> Curiously, one meets a similar kind of principle in Renaissance
> music where a diminished fourth might be around 32:25 (precisely
> so in 1/4-comma meantone): it is acceptable in certain
> circumstances, but most typically occurs between two upper voices
> both forming usual concords with the lowest voice.
>
> Around 1400 or so, we do have a shift, as written sharps are
> often tuned on keyboards as flats, so that schismatic thirds
> become the norm in certain positions, and contrast with usual
> Pythagorean thirds between diatonic steps. And by sometime
> around 1450, experimenting with such systems leads to meantone
> temperament making approximate 5/4 thirds the general norm.
>
> And if we think of the early to middle of the 15th century
> as the early Renaissance, this transition ties in directly
> with your following comment, Gene.
>
>> Though in early Renaissance music I hear a marked tendency to
>> end on chords without thirds, which seems to disappear in the
>> later Renaissance. Whether that reflects a view that they
>> are still somehow dissonant, even though the
>> sweetness of the otonal thirds was admired, I don't know.
>
> This is an interesting question, and I might nuance my language
> a bit to describe the situation in the young Dufay's epoch around
> 1420-1440, the situation around 1500, and the situation in the
> later 16th century (say Lasso or Palestrina).
>
> In the early and middle 15th century, while sonorities with thirds
> (and sixths) become more and pervasive, there's still the
> expectation that we'll eventually cadence on a trine or fifth.
> This expectation, which in many ways is coupled with directed
> progressions and organizational schemes not too different from
> "classic" ones of the 14th century (which I still find myself
> using in the 21st century), feels very basic to me in the
> appeal of many of Dufay's early chansons, fauxbourdon settings,
> and some Mass settings and motets.
>
> As things move on, and likely we are moving into the early
> meantone era of keyboard tuning in more and more places,
> sonorities with thirds are reaching a status of "almost
> stable." In lots of the music of Busnois and Ockeghem,
> for example, I'd still expect to hear a final sonority
> of 2:3:4 (or, with the wider spacing that becomes common,
> often 1:2:3:4 or the like), or maybe even a simple octave
> (common in some three-voice writing).
>
> Around 1500, say in a popular four-voice style like the
> Cancionero de Palacio in Spain, I'd expect to hear
> a third-plus-fifth-or-sixth type of sonority (as Zarlino
> puts it later, in 1558) just about anywhere within a
> phrase, but with a final cadence on 2:3:4 or the like
> still very common and delightful -- yes, in keyboard
> performance, in 1/4-comma meantone or the like where
> those concluding fifths are indeed "bouncy."
>
> In this epoch, we also encounter a minor third plus
> a fifth as a closing sonority, for example in
> Josquin, which in meantone would approximate
> 10:12:15. One clue reinforcing the conclusion
> that raising the minor third to major as an
> implicit performance practice is _not_ necessarily
> meant is the advocacy of the Ab-C# meantone tuning
> by Ramos (1482, whose 5-limit JI monochord is
> distinct from his discussion of a practical
> keyobard, which Mark Lindley persuasively
> analyzes as a meantone instrument). If a
> closing third were meant to be major by
> unwritten alteration -- as will soon become
> true for the most part -- then G# would
> hardly be omissible for closes in the
> Third and Fourth Modes (E Phrygian with
> an authentic or plagal range). Yet Ramos
> considers it optional, and of lower
> priority than Ab, while Arnold Schlick
> (1511) describes an irregular temperament
> where Ab is acceptable but G# marginal,
> meant for an ornamental E-G# leading to
> a cadence on A, for example, rather than
> a fully harmonious E-G# as a closing
> major third.
>
> By 1523, the date of Pietro Aron's
> first edition of the _Toscanello in
> Musica_, the "otonal" preference
> of which you speak does seem in
> place, at least in Aron's milieu:
> altering a minor third to major
> is seen as enhancing the beauty
> of the music. And in his treatise
> on the modes of 1525, Aron shows
> the alteration of closing thirds
> from minor to major, a typical
> alteration of the 16th and early
> 17th centuries from this point on.
> Such a situation fits nicely with
> Eb-G# meantone, which supplies
> that E-G#-B sonority for Phrygian.
>
> However, there still seems a
> feeling, which can sometimes
> hold to a degree even in the
> late 16th century when Zarlino's
> _harmonia perfetta_ based on
> the division of the 3:2 fifth
> into 4:5:6 (or better yet with
> a voicing "in the order of
> the sonorous numbers" like
> 2:3:4:5, with the major tenth
> preferred to the simple third),
> that a close on a simple unison
> or octave in two-voice writing,
> or on 2:3:4 or 1:2:3:4 or the
> like in multi-voice writing,
> is somehow in a sense more
> "conclusive" or "formal."
> This may occur in liturgical
> music, for example, but also
> in secular pieces such as
> certain madrigals.
>
> By the 1550's, we have
> statements from Vicentino
> (1555) and Zarlino (1558)
> in favor of "richness of
> harmony" or "perfect
> harmony" including a
> third plus a fifth or
> sixth -- or their octave
> extensions -- wherever
> possible. And many final
> closes follow this
> axiom. However, the taste
> at times for an especially
> conclusive final sonority
> makes itself felt.
>
> (In a Gothic setting, the
> same ideal of "richness"
> or "perfection" applies,
> but with 2:3:4 rather
> than 4:5:6 as the
> "threefold perfection
> of harmony." They are
> the harmonic or otonal
> divisions of the octave
> and fifth respectively.)
>
> Thus we move from the
> active and unstable thirds
> of the Gothic Era to the
> mixture of regular Pythagorean
> and schismatic thirds around
> the early 15th century, the
> adoption of meantone in the
> middle to later portion of
> the century, and the
> acceptance of thirds as
> conclusive by the early
> 16th century, which seems
> to lead to the "otonal"
> preference expressed by
> some observations and
> examples of Aron in the
> 1520's, and in a more
> developed way by Vicentino
> and Zarlino.
>
> And this isn't a complete
> account, because we have
> accounts of English music
> in the 13th and early 14th
> centuries, as well as some
> surviving pieces including
> the famous _Sumer is icumen
> in_, which do point both
> to a treatment of thirds as
> "the best concords," upon
> which one may end as well as
> begin, and to vocal tunings
> "mollifying" these thirds
> by performing them at or
> near 5/4 or 6/5 (as
> Christopher Page puts it).
>
> Nor have I gotten into
> 14th-century variations also
> discussed and advocated by
> Page where cadential major
> thirds and sixths tend to
> be considerably _wider_ than
> Pythagorean, something that
> may apply not only to
> Marchettus of Padua and his
> early 14th-century Italian
> style, but to lots of French
> music including that of Machaut
> (Page's view and also mine).
>
> Anyway, Gene, your observations
> seem perceptive to me as far
> as Renaissance trends; and Carl,
> I much agree with your point
> that tuning a major third at
> 81/64 (or 14/11 or 33/26) may
> itself signal a style where
> such thirds are regarded as
> "_imperfect_ concords" with
> lots of stress on the first
> word of that expression.
>
> Best,
>
> Margo
>
>
>
>
>
>
>
>
>
>
>
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On Sat, Aug 21, 2010 at 11:07 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"For example, the diatonic map is LLsLLLs."
>
>    Ah ok.  But then how to you decide if, say, a step is far enough away from being a both "small" and a large to be a "medium" (or do you just "force" everything to be either small or large by rounding to find the "nearest MOS mapping")?

I'm just saying that with the usual diatonic scale, there are two
sizes of step - a larger and a smaller - and the larger I called "L"
and the smaller I called "s". This is just the usual terminology when
describing MOS's. I could have also said aabaaab, but I prefer LLsLLLs
because diatonic sounds quite different from mavila, which is ssLsssL
(but might also be called aabaaab).

There are MOS's which involve "neutral" seconds. Mohajira[7], for
example, is LssLsLs, except the "s" intervals are closer to 12/11
"neutral" territory.

>    Funny example...in Wikipedia JI diatonic is listed as having small, medium, and large sizes.  Odd....

Right, because 5-limit JI is a rank 3 system, and so the equivalent
would be that every interval has 3 sizes instead of 2. This is some
kind of generalization of the MOS concept that I don't fully
understand yet. But the map for 5-limit JI diatonic is

LMsLMLs

Where L is large, M is medium, and S is small. But the truth is that
we still tend to fit even the JI major scale to our LLsLLLs diatonic
template - set your keyboard to just-major and see how many times you
accidentally hit that D-A wolf fifth because you think it should still
be a perfect fifth, not some other interval.

That being said, the LMsLMLs template and LLsLLLs template are clearly
very close to each other perceptually, and related in some way - I am
not quite sure how yet. Hopefully after this 4th read of Rothenberg's
paper I will have a better understanding of how maps work.

>
>    The above would seem to work fine for pure MOS scales, but especially not scales with, say, 4+ different interval sizes.  For example, what should I do if a scale I have has consecutive sizes of 1.09,1.095,1.11, 1.09, 1.12, and 1.11?

It is 3 in the morning, and I really don't want to do the math here.
Can't you just convert that into cents or JI ratios or something? I
don't have a cents value for 1.095 just memorized into my head.

> >"I keep repeating this example to show that multiple JI ratios can
> substitute for each other in the same map, but the map is what's
> important."
>   Got it...but my point of confusion is how far do you have to go far as difference before you effectively "have" to change the map?

I don't know. What I have realized is that you can make the fifths for
LLsLLLs as wide as you want, and until you get up to 5-equal it
doesn't really make a difference. The fifths can also be as flat as
you want, and until you get down to the fifth of 7-equal (where it all
becomes one step) it still preserves the map structure. When you get
the fifths flatter than 7-equal, it flips around and becomes ssLsssL,
which is mavila[7]. And that is totally unrecognizable as a "diatonic"
mode to my ears.

What is more interesting is mavila[5] vs meantone[5]. Both of them are
ssLsL, but the mavila has the fifths so flat that 4 of them make 6/5
instead of 5/4. This is definitely noticeable, but if you start
playing with chord shapes and transposing them within the scale, the
general "pentatonic" pattern still remains the same. I think the main
difference between mavila[5] and meantone[5] is that you can hear
meantone[5] as a subset of meantone[7] - that is, you can hear the
major pentatonic scale as a subset of the major diatonic scale. But,
the "major" pentatonic mavila scale is actually "minor" from a
diatonic standpoint, because the thirds are 6/5. And hence mavila[5]
is imbued with a bit of "diatonic minorness."

But as I said, in 22-equal the augmented seconds are 5/4 and in
31-equal they're 7/6... Makes not too much difference (unless you're
looking for it). The main difference it makes is that the 22-equal
harmonic minor scale is Rothenberg improper, which means that wide
augmented second can have some interesting ambiguous properties.

> >"And remember my earlier example - the 22-equal superpyth harmonic
> minor scale. That augmented second is going to be a 5/4 (!) and the
> minor thirds will be 7/6.
> In 31-equal, the augmented seconds are 7/6 and the minor thirds are ~6/5."
>
>    Oh man am I confused here....it sounds like the interval classes not just slightly but completely overlap each other IE
> "the minor thirds will be 7/6." (in 22TET)
> "the minor thirds are ~6/5" (in 31TET)
>   How, pray tell, do I figure out what is in what class with this sort of overlap?

Right. So it all breaks down. You have a second that is bigger than a
third, which makes the scale Rothenberg improper. But if you play C
harmonic minor, you will note that it still definitely sounds just
like harmonic minor.

Try lydian #2 as well. C D# E F# G A B C. You'll note that that C-D#
is about 5/4. So you play C-D#, and your brain hears "major third."
And yet, as you keep playing the scale - C-D#-E-F#-G -- you will note
that the whole thing gets "reframed" so that that C-D# dyad, which is
an approximate 5/4, stops sounding like a C-E major third and starts
sounding like a C-D# augmented second.

You might want to check out Rothenberg's paper... it's pretty intense,
but very good. Has a bit of a classical slant to it (I didn't much
care for his explanation of why "only major and minor" can be used,
which was all over the place) but is very enlightening nonetheless.

-Mike

Addressing Mike & Michael in the same post:

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"your map tells you how it's formed in terms of differing
> sizes of thirds - and how the notes can be grouped triadically, not
> just dyadically."
> So, for example...Ls would be a major triad in that system and sL a minor one?

Sort of, if you take L to mean a major third and s to mean a minor third. Except that the "L and s" system is fundamentally one-dimensional, whereas the lattice is two-dimensional.

Mike B. said:

> >"I often wonder what about the diatonic scale that is the "defining"
> factor to the brain in determining the overall sound produced. At one
> point I thought it had something to do with 3-limit hearing, but the
> fact that in mavila[7] the 3-limit relationship for each note is the
> same as major - but that the whole scale sounds so much different,
> overall - would seem to disprove that."

I really think it has to do with shapes on the lattice. 3-limit is too one dimensional to explain things. Consider that the mavila[7] lattice is the a reflection of the diatonic[7] lattice across the 3-limit line. Tonality consists of the way each note is related to the others, the route you have to take to arrive back at the starting note. Consider that if you just look at the 3-limit, the tiniest variance in the size of the fifth makes a huge difference in how far along the 3-limit line you have to go to get back to the tonic; but even a large variance in the size of the fifth doesn't change the two-dimensional vector that takes you back to the tonic on the 3,5 lattice. It's only when you pass a critical point (3\5-EDO or 4\7-EDO) that the lattice shifts into Father[8] or Mavila[7].

Michael said:

> Agreed. Based on experience I have huge doubt that how closely a system is
> related to 5th ultimately improves its "resolved-ness"...at least beyond the
> apparent fact that having a very low-limit dyad in a chord seems to give more
> "slack" for other dyads to be further off pure while still having the chord feel
> "stable".

The resolvedness of individual triads may not be affected by having a scale related to fifths; a temperament like Porcupine [7 or 8] can produce some excellent 5-limit triads, but its generator is an approximate 11/10--at 165 cents or so, it's a loooooong way from a fifth. But having a generator of a fifth ensures that a maximum number of chords in the scale will have a fifth; no other generator will produce more chords that have fifths.

> It seems obvious to me that
>
> 1) The brain likes having the larger dyad first (IE major vs. minor)
>
> 2) The brain seems to choke when it sees Just equal dyads stacked (IE 5/4 times
> 5/4)...perhaps because they point toward two different roots with almost
> absolute authority without putting the focus on one over the other. This may
> also explain why 4-tone chords tend to sound best as LsL or sLs instead of Lss
> or ssL.

Ever try stacking 8/7's? Of all the stacked dyads of equal size, I think this forms the nicest chords. Especially if you temper them a little so that three of them spans a perfect fifth. It's sort of like a compressed 5-EDO. You can actually play 5 of them on top of each other, and it still sounds at least somewhat concordant. I can't think of any other dyad that works better for this purpose.

> 3) If anything(s) seem to help across the board...they are
> A) Having a low number of intervals from each class possible (IE one type of
> third, fourth, etc. beats two type by a little and three by a whole lot)
> B) Having most types of dyadic intervals be low limit (IE 80% or more in
> 7-odd-limit or less)

Your latter point seems less important to me. If 12-tET has taught me anything, it's that you really need only ONE dyadic interval of low limit, so long as it's REALLY low limit. Of course, there's nothing near the fifth in Tenney height--4/3 is the nearest, and it's twice as complex. So if you drop having good fifths, I think you need at least two dyads of relatively-low limit. Since you can't have 5/4 and 6/5 (or 7/6 and 9/7) without having good 3/2's, I'd say 5/4 and 7/4 are good candidates. Lemba temperament (I think?) gives 4 good 4:5:7 chords out of a 6-note MOS scale, which isn't too shabby. This is one approach I've used to good effect in 16-EDO (which has decent 5/4's and pretty good 7/4's, but terrible 3/2's).

This is, of course, from a strictly "concordance" point of view. I'm still not entirely certain whether a bad 4:5:6 sounds more or less resolved than a good 4:5:7. I know I'm accustomed to hearing a 4:5:(6):7 on the dominant chord resolve to a minor (10:12:15) tonic chord, as in an E major-->A minor cadence, and the idea of trying to resolve to a 4:5:7 chord seems kind of absurd from a tonal standpoint. But then again, that could just be my cultural/12-tET bias talking.

-Igs

On Sun, Aug 22, 2010 at 3:43 AM, cityoftheasleep
<igliashon@...> wrote:
>
> Mike B. said:
>
> > >"I often wonder what about the diatonic scale that is the "defining"
> > factor to the brain in determining the overall sound produced. At one
> > point I thought it had something to do with 3-limit hearing, but the
> > fact that in mavila[7] the 3-limit relationship for each note is the
> > same as major - but that the whole scale sounds so much different,
> > overall - would seem to disprove that."
>
> I really think it has to do with shapes on the lattice. 3-limit is too one dimensional to explain things. Consider that the mavila[7] lattice is the a reflection of the diatonic[7] lattice across the 3-limit line.

How is that so? The reflection of, say, the major scale across the
3-limit line seems like it would be aeolian, rather than mavila. If
you have 3 triangles pointing up - Fmaj, Cmaj, and G maj, then you'd
now have 3 of them pointing down, so Fmin, Cmin, and Gmin, which makes
aeolian.

> Tonality consists of the way each note is related to the others, the route you have to take to arrive back at the starting note. Consider that if you just look at the 3-limit, the tiniest variance in the size of the fifth makes a huge difference in how far along the 3-limit line you have to go to get back to the tonic; but even a large variance in the size of the fifth doesn't change the two-dimensional vector that takes you back to the tonic on the 3,5 lattice. It's only when you pass a critical point (3\5-EDO or 4\7-EDO) that the lattice shifts into Father[8] or Mavila[7].

OK, but does that route have anything to do with JI at all? Like I
said, we can replace 5/4 with 9/7 and still give 90% of the same
effect. Replacing 5/4 with 6/5 gives 0% of the same effect, because
then LLsLLLs switches into ssLsssL.

But assuming that you are saying it doesn't have to do with JI - how
do you derive the axes then? It seems like you're saying that tonality
has to do with how every note gets back to the root via this map that
you draw. And with this map, there's a 3-limit line, and a 5-limit
line. And how do you derive what the fundamental axes are that really
"determine" perception, if it has nothing to do with JI? Why 5/4 and
3/2, and why not 9/7 and 3/2, or 16/15 and 9/8 or something?

-Mike

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> MikeB>"For example, the diatonic map is LLsLLLs."
>
> Ah ok. But then how to you decide if, say, a step is far enough away from
> being a both "small" and a large to be a "medium" (or do you just "force"
> everything to be either small or large by rounding to find the "nearest MOS
> mapping")?
> Funny example...in Wikipedia JI diatonic is listed as having small, medium,
> and large sizes. Odd....

The JI diatonic is not an MOS scale, it is not "generated" by a series of approximate 3/2's. In that sense, it is actually a very different scale from the equal-tempered, meantone, or Pythagorean diatonic. The whole "L/s" way of looking at scales is purely for MOS scales. If there are more than 2 step sizes, then yes, you can use an "m" for medium. Beyond that, there's not much use in using letters at all.

> >"I keep repeating this example to show that multiple JI ratios can
> substitute for each other in the same map, but the map is what's
> important."
> Got it...but my point of confusion is how far do you have to go far as
> difference before you effectively "have" to change the map?

It depends on the map.

> >"And remember my earlier example - the 22-equal superpyth harmonic
> minor scale. That augmented second is going to be a 5/4 (!) and the
> minor thirds will be 7/6.
> In 31-equal, the augmented seconds are 7/6 and the minor thirds are ~6/5."
>
> Oh man am I confused here....it sounds like the interval classes not just
> slightly but completely overlap each other IE
>
> "the minor thirds will be 7/6." (in 22TET)
> "the minor thirds are ~6/5" (in 31TET)
> How, pray tell, do I figure out what is in what class with this sort of
> overlap?

This confusion comes because in the diatonic scale, "interval class" is defined by "number of fifths (or fourths) from the tonic", not by frequency ratio or pitch value. It gets confusing when you try to make diatonic names correspond to JI ratios--this doesn't work, because the diatonic scale spans multiple temperaments, and thus multiple mappings. Thus, in diatonic terms, you simply *can't* think of a minor third as a specific ratio; you have to think of *only* it as a being "3 fourths minus one octave (or 3 fifths below, plus two octaves) above the tonic". This may be 6/5, it may be 13/11, it may be 7/6, it may be 17/14, it all depends on the size of the fourth (and thus the size of the fifth as well).

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > I really think it has to do with shapes on the lattice. 3-limit is too one dimensional to explain things. Consider that the mavila[7] lattice is the a reflection of the diatonic[7] lattice across the 3-limit line.
>
> How is that so? The reflection of, say, the major scale across the
> 3-limit line seems like it would be aeolian, rather than mavila. If
> you have 3 triangles pointing up - Fmaj, Cmaj, and G maj, then you'd
> now have 3 of them pointing down, so Fmin, Cmin, and Gmin, which makes
> aeolian.

Draw it out, after you get some sleep. You have to draw at least two octaves of the scale to get it. Remember that A Aeolian = C major. The Aeolian mode is the same lattice shape as the major mode, just with the I in a different position. Yes, in F Mavila [7], the I, IV, and V will all be minor, as in the F Aeolian. But the VII will be augmented. In F aeolian, the VII is Eb major. In F Mavila, also, Fmin= F A C, not F Ab C. Circle-of-fifths-wise, F Mavila is spelled the same as F major.

> OK, but does that route have anything to do with JI at all? Like I
> said, we can replace 5/4 with 9/7 and still give 90% of the same
> effect. Replacing 5/4 with 6/5 gives 0% of the same effect, because
> then LLsLLLs switches into ssLsssL.

You're still thinking one-dimensionally, only now you're just looking at the thirds. You have to think of the relationship between the third and the fifth, i.e. how many fifths it takes to make which type of third. In the diatonic scale, going up four fifths makes a major third and going down three fifths gives a minor third; in Mavila, the opposite is the case. So you're not just replacing 5/4 with 6/5 per se; you're changing the value of 4 fifths. And just as there are diatonic scales where 4 fifths gives you 9/7 instead of 5/4, there are Mavila scales where 4 fifths gives you 7/6 instead of 6/5 (I think?).

What this really has to do with JI, I imagine, is that we are conditioned to hear 6/5 as having a distinct musico-semantic content from 5/4, but not so much 9/7 from 5/4. You can interpret the 22-EDO diatonic in 2 ways: one, as mapping 5/4 to an augmented 2nd, having low error and high complexity, or two, as mapping 5/4 to a major third--four fifths up minus two octaves, i.e. 9/7 in this case--and having high error and low complexity. If we were actually conditioned to regard 9/7 as having a different musico-semantic content from 5/4 (i.e., if we all grew up with 31-EDO instead of 12-EDO), we might NOT hear 22-EDO's diatonic as meaning the same thing as 31-EDO's. It might be as different to us as is Mavila.

> But assuming that you are saying it doesn't have to do with JI - how
> do you derive the axes then? It seems like you're saying that tonality
> has to do with how every note gets back to the root via this map that
> you draw. And with this map, there's a 3-limit line, and a 5-limit
> line. And how do you derive what the fundamental axes are that really
> "determine" perception, if it has nothing to do with JI? Why 5/4 and
> 3/2, and why not 9/7 and 3/2, or 16/15 and 9/8 or something?

Commas, Mike. Or rather, unison vectors. These are temperaments, not JI. In JI, you *never* get back to the tonic. Even in 22-EDO, where the major third *sounds like* a 9/7, we're still treating it as 5/4 in the map. Or at least, it's still the larger of the two intervals dividing the fifth.

I suppose in a sense, you *can* reduce it to one dimension--the generator--you just have to look at how x generators modulo the period relates back to 1 generator. With meantone, 4 fifths modulo an octave makes an interval that is more than half a fifth; in mavila, 4 generators modulo an octave makes an interval that is less than half a fifth.

So strictly speaking, the "fundamental axes" are arbitrary. You could just as well use 9/8, and 16/15 spanning a 6/5. The map would look different, of course, but it would contain the same information. It's still a JI grid warped according to the unison vectors to make parallel lines intersect.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Ever try stacking 8/7's? Of all the stacked dyads of equal size, I think this forms the nicest chords. Especially if you temper them a little so that three of them spans a perfect fifth.

Gamelismic tempering--tempering out 1029/1024. It doesn't involve 5 at all, just 2, 3, and 7.

It's sort of like a compressed 5-EDO. You can actually play 5 of them on top of each other, and it still sounds at least somewhat concordant.

Well...I would suggest stacking four of them and filling out the octave, giving a 1029/1024-tempered 8/7-8/8-8/7-8/7-7/6 chord.

> I can't think of any other dyad that works better for this purpose.

In meantone (among other tunings) we have the augmented triad, a 225/224 tempered 5/4-5/4-9/7, and the dim7, a 126/125-tempered 6/5-6/5-6/5-7/6, and they not only have historical clout, they have fewer notes more widely spaced and would doubtless be preferred by many. Alas, the 12et versions pretty much piss all over the tuning.
There's also a schismatic (32805/32768-tempered) chord which is a very slightly tempered 9/8-9/8-9/8-9/8-9/8-10/9. Less regular but a nice chord is the 1728/1715-tempered 7/6-7/6-7/6-5/4.

On Sun, Aug 22, 2010 at 4:43 AM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > > I really think it has to do with shapes on the lattice. 3-limit is too one dimensional to explain things. Consider that the mavila[7] lattice is the a reflection of the diatonic[7] lattice across the 3-limit line.
> >
> > How is that so? The reflection of, say, the major scale across the
> > 3-limit line seems like it would be aeolian, rather than mavila. If
> > you have 3 triangles pointing up - Fmaj, Cmaj, and G maj, then you'd
> > now have 3 of them pointing down, so Fmin, Cmin, and Gmin, which makes
> > aeolian.
>
> Draw it out, after you get some sleep. You have to draw at least two octaves of the scale to get it. Remember that A Aeolian = C major. The Aeolian mode is the same lattice shape as the major mode, just with the I in a different position. Yes, in F Mavila [7], the I, IV, and V will all be minor, as in the F Aeolian. But the VII will be augmented. In F aeolian, the VII is Eb major. In F Mavila, also, Fmin= F A C, not F Ab C. Circle-of-fifths-wise, F Mavila is spelled the same as F major.

OK, I have drawn it, but I'm still not seeing it. The lattice to me
represents JI, not any temperament - mavila or meantone or otherwise.
Are you saying to draw out unison vectors, and that if draw 135/128
and 25/24 as unison vectors instead of 81/80 and 25/24 that the PB
will flip symmetrically? Because just flipping the triangles around
still yields to me another mode of the major scale.

> > OK, but does that route have anything to do with JI at all? Like I
> > said, we can replace 5/4 with 9/7 and still give 90% of the same
> > effect. Replacing 5/4 with 6/5 gives 0% of the same effect, because
> > then LLsLLLs switches into ssLsssL.
>
> You're still thinking one-dimensionally, only now you're just looking at the thirds. You have to think of the relationship between the third and the fifth, i.e. how many fifths it takes to make which type of third. In the diatonic scale, going up four fifths makes a major third and going down three fifths gives a minor third; in Mavila, the opposite is the case. So you're not just replacing 5/4 with 6/5 per se; you're changing the value of 4 fifths. And just as there are diatonic scales where 4 fifths gives you 9/7 instead of 5/4, there are Mavila scales where 4 fifths gives you 7/6 instead of 6/5 (I think?).

Right, I understand that. But note that in 22-EDO, a superpyth
augmented second gets mapped to 5/4... And also notice that whether or
not that major third is mapped to 5/4 or 16/13 doesn't seem to destroy
the map, as in the case of 31-tet vs 26-tet.

I had thrown out the idea that diatonic hearing really meant "3-limit"
hearing in a previous message, noticing that for superpyth vs
1/4-comma meantone - every note retains the same 3-limit relationship
with the root, but the direct 5- or 7- or higher limit relationship
changes. However, note that mavila[5] is recognizable as an altered
form of meantone[5], but mavila[7] is not. And superpyth[5] and
superpyth[7] are both recognizable as alternate forms of meantone[5]
and meantone[7]. So the more direct correlation here seems to be
whether the maps are equivalent to each other, which Rothenberg has
worked out with his "equivalence class" concept. Not that I am an
expert on any of this, but just trying to see what the pattern is.

> What this really has to do with JI, I imagine, is that we are conditioned to hear 6/5 as having a distinct musico-semantic content from 5/4, but not so much 9/7 from 5/4. You can interpret the 22-EDO diatonic in 2 ways: one, as mapping 5/4 to an augmented 2nd, having low error and high complexity, or two, as mapping 5/4 to a major third--four fifths up minus two octaves, i.e. 9/7 in this case--and having high error and low complexity. If we were actually conditioned to regard 9/7 as having a different musico-semantic content from 5/4 (i.e., if we all grew up with 31-EDO instead of 12-EDO), we might NOT hear 22-EDO's diatonic as meaning the same thing as 31-EDO's. It might be as different to us as is Mavila.

Right, but we are DEFINITELY used to hearing 5/4 as having a different
musico-semantic content from 6/5, or 7/6, and in 22-tet that augmented
second is 5/4, and in 31-tet it's 7/6, and in 12-tet it's 300 cents.
You may be onto something with your last statement, since mavila[5] is
very distinguishable from meantone[5], even though superpyth [5] is a
bit less so. But note that mavila[5] can be heard as some kind of
warped diatonic subset as well, and your brain sort of snaps it to
"minor."

What I think is going on is that when we hear a major third, we aren't
really thinking of it in terms of any specific scale anymore. We're
used to hearing that interval as the root for major, and mixolydian,
and aeolian dominant, and it being in the octatonic scale, that we
have a generic "schema" built up for "major third" at this point. But
consider the possibility that "5/4" isn't actually part of that
schema... since 9/7 seems to substitute for it just fine. And our
schema for "augmented second" seems to be even more flexible than
major third, although perhaps that's just because we're used to
thinking of an augmented second as the interval that's a diatonic
semitone less than the major third, no matter how it's intoned. Or, if
you're into modern harmony, you're used to thinking of it as a major
third above the major 7, a la C E F# B D#. And in 22-tet, both of
those functions are fulfilled by 5/4, not 7/6.

> > But assuming that you are saying it doesn't have to do with JI - how
> > do you derive the axes then? It seems like you're saying that tonality
> > has to do with how every note gets back to the root via this map that
> > you draw. And with this map, there's a 3-limit line, and a 5-limit
> > line. And how do you derive what the fundamental axes are that really
> > "determine" perception, if it has nothing to do with JI? Why 5/4 and
> > 3/2, and why not 9/7 and 3/2, or 16/15 and 9/8 or something?
>
> Commas, Mike. Or rather, unison vectors. These are temperaments, not JI. In JI, you *never* get back to the tonic. Even in 22-EDO, where the major third *sounds like* a 9/7, we're still treating it as 5/4 in the map. Or at least, it's still the larger of the two intervals dividing the fifth.
>
> I suppose in a sense, you *can* reduce it to one dimension--the generator--you just have to look at how x generators modulo the period relates back to 1 generator. With meantone, 4 fifths modulo an octave makes an interval that is more than half a fifth; in mavila, 4 generators modulo an octave makes an interval that is less than half a fifth.

Ah, I see. That is an interesting correlation. Why do you think that
this might be the case - that it all gets related back to the
generator?

-Mike

Hi Mike. First of all, see the file I uploaded here:
/tuning/files/IgliashonJones/Lattices.pdf

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> OK, I have drawn it, but I'm still not seeing it. The lattice to me
> represents JI, not any temperament - mavila or meantone or otherwise.
> Are you saying to draw out unison vectors, and that if draw 135/128
> and 25/24 as unison vectors instead of 81/80 and 25/24 that the PB
> will flip symmetrically? Because just flipping the triangles around
> still yields to me another mode of the major scale.

The lattice *represents* JI, but is *not* JI, if you are actually drawing what I suggested you draw (i.e. what I drew in the file I uploaded). In a major scale, you go up four fifths and then up a minor third to get back to the tonic note; in mavila, you go up four fifths and then up a major third; thus in diatonic, if you repeat the lattice, it skews down to the right (or up to the left, if you're drawing in that direction). The mavila lattice skews up to the right. IOW, the VII chord in C diatonic ionian is diminished (i.e. 3 minor thirds in a row), but in C mavila ionian, the VII chord is augmented (i.e. 3 major thirds in a row).

> Right, I understand that. But note that in 22-EDO, a superpyth
> augmented second gets mapped to 5/4... And also notice that whether or
> not that major third is mapped to 5/4 or 16/13 doesn't seem to destroy
> the map, as in the case of 31-tet vs 26-tet.

The problem is, you're conflating two types of "mapping". Yes, you can take 22-EDO's superpyth to map 5/4 to an augmented second...but you don't *have to*. You can take it to map 5/4 to a major third, just with a really high error. IOW you can "call" the 9/7 third made by 4 fifths of 22-EDO a "bad" 5/4. In fact, that is exactly what you are doing when you say that you "hear" 22-EDO diatonic as being the same as 31-EDO or 26-EDO diatonic.

The point being, you have to stop conflating the "absolute" JI ratio with the "approximated" JI ratio. Just because the absolute JI ratio is closer to a 9/7 or a 16/13, that doesn't mean it's not interpreted as an approximate 5/4. At this point, it's a matter of harmonic entropy: 9/7 and 16/13 are near enough to the gravity-well of 5/4 to get "sucked in".

Also, let me assure you that if you tried to use 22-EDO in a chromatic way, i.e. a 12-note MOS scale, it WOULDN'T sound or work the same way that 31-EDO does, because while they have the same 7-note MOS, they have inverse 12-note MOS's (i.e. s's and L's switch places, as between diatonic and mavila). Also, 26 and 31 diverge when you take a 19-note MOS. So just as mavila and diatonic share the same 5-note MOS but not the same 7-note MOS, the various diatonic scales you're comparing share the same 7-note MOS but NOT the same higher-number MOS's.

> What I think is going on is that when we hear a major third, we aren't
> really thinking of it in terms of any specific scale anymore. We're
> used to hearing that interval as the root for major, and mixolydian,
> and aeolian dominant, and it being in the octatonic scale, that we
> have a generic "schema" built up for "major third" at this point. But
> consider the possibility that "5/4" isn't actually part of that
> schema... since 9/7 seems to substitute for it just fine.

Again, this is probably more to do with harmonic entropy than anything else. What makes 6/5 different is that it's its own local minimum of harmonic entropy, separated from 5/4 by the local maximum of 347 cents. At 347 cents, as you've noted on MMM, you can sound like major AND/OR minor at once, but slide away from it in either direction and you're either in major land or minor land.

> And our
> schema for "augmented second" seems to be even more flexible than
> major third, although perhaps that's just because we're used to
> thinking of an augmented second as the interval that's a diatonic
> semitone less than the major third, no matter how it's intoned. Or, if
> you're into modern harmony, you're used to thinking of it as a major
> third above the major 7, a la C E F# B D#. And in 22-tet, both of
> those functions are fulfilled by 5/4, not 7/6.

You should compare some chromatic music in 22-EDO vs. 31-EDO and see if you still hear them as "saying the same thing". Augmented seconds sort of imply a 12-tone chromatic framework, or at least an addition to the lattice that makes one chord both a major and a minor. In 22-EDO, if you want to "map" 5/4 to the augmented second, you have to use a very different lattice than the standard diatonic-7 lattice. It no longer looks the same as 31-EDO's lattice.

> > I suppose in a sense, you *can* reduce it to one dimension--the generator--you just have to look at how x generators modulo the period relates back to 1 generator. With meantone, 4 fifths modulo an octave makes an interval that is more than half a fifth; in mavila, 4 generators modulo an octave makes an interval that is less than half a fifth.
>
> Ah, I see. That is an interesting correlation. Why do you think that
> this might be the case - that it all gets related back to the
> generator?

Because these scales are all generated by a generator and a period (or I guess just "two generators" is the fashionable way to say it). Each generator itself implies a unison vector (or vectors), which implies that two JI intervals (or more) are "merged" by the temperament. So the generator itself actually tells us the entire mapping, presuming that we can read from it the commas that are tempered out of it.

Note that in meantone, if ONLY 81/80 is tempered out, then four 3/2's minus two 2/1's EXACTLY equals 5/4. But scales like 12, 19, and 26 temper other commas out of the fifth as well, giving different results. The commas "fight" with each other, you could say. But all that information is encoded in the generators. But in all meantones, four approximate 3/2's minus two exact 2/1's always approximates a 5/4.

The upshot is that you are free to interpret the word "approximates" however you please; if you can accept 9/7 as an "approximate" 5/4, you can call superpyth a meantone (though you'll be sharpening by a fraction of 81/80 instead of flattening). As long as you stick to the 7-note MOS, you don't run into trouble...but when you extend the MOS far enough to get augmented seconds to appear, which are clearly better approximations to 5/4, you'll probably want to stop calling the 9/7 an approximation to 5/4, or else you'll have two intervals approximating one interval, which is sort of the opposite of tempering.

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Gamelismic tempering--tempering out 1029/1024. It doesn't involve 5 at all, just 2, 3, and 7.

Yep! I quite like it. Great way to use 21-, 26-, and 31-EDO, and 36-EDO.

> Well...I would suggest stacking four of them and filling out the octave, giving a 1029/1024-tempered 8/7-8/8-8/7-8/7-7/6 chord.

Yeah, that sounds a bit better.

>
> > I can't think of any other dyad that works better for this purpose.
>
> In meantone (among other tunings) we have the augmented triad, a 225/224 tempered 5/4-5/4-9/7, and the dim7, a 126/125-tempered 6/5-6/5-6/5-7/6, and they not only have historical clout, they have fewer notes more widely spaced and would doubtless be preferred by many. Alas, the 12et versions pretty much piss all over the tuning.
> There's also a schismatic (32805/32768-tempered) chord which is a very slightly tempered 9/8-9/8-9/8-9/8-9/8-10/9. Less regular but a nice chord is the 1728/1715-tempered 7/6-7/6-7/6-5/4.
>

I dunno, I think a big advantage of the gamelismic chord is that it has a fifth in it. I find it more "restful" than any of the alternatives, myself. Boring as I find the fifth, it is undeniably a "restful" interval. Am I correct in asserting that 8/7 is the only interval with Tenney heigh <70 that approximately subdivides the fifth?

-Igs

On Sun, Aug 22, 2010 at 3:20 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Hi Mike. First of all, see the file I uploaded here:
> /tuning/files/IgliashonJones/Lattices.pdf
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > OK, I have drawn it, but I'm still not seeing it. The lattice to me
> > represents JI, not any temperament - mavila or meantone or otherwise.
> > Are you saying to draw out unison vectors, and that if draw 135/128
> > and 25/24 as unison vectors instead of 81/80 and 25/24 that the PB
> > will flip symmetrically? Because just flipping the triangles around
> > still yields to me another mode of the major scale.
>
> The lattice *represents* JI, but is *not* JI, if you are actually drawing what I suggested you draw (i.e. what I drew in the file I uploaded). In a major scale, you go up four fifths and then up a minor third to get back to the tonic note; in mavila, you go up four fifths and then up a major third; thus in diatonic, if you repeat the lattice, it skews down to the right (or up to the left, if you're drawing in that direction). The mavila lattice skews up to the right. IOW, the VII chord in C diatonic ionian is diminished (i.e. 3 minor thirds in a row), but in C mavila ionian, the VII chord is augmented (i.e. 3 major thirds in a row).

Ahhh, I get it now, I didn't realize you were flipping the note names
around as well. That makes sense.

> > Right, I understand that. But note that in 22-EDO, a superpyth
> > augmented second gets mapped to 5/4... And also notice that whether or
> > not that major third is mapped to 5/4 or 16/13 doesn't seem to destroy
> > the map, as in the case of 31-tet vs 26-tet.
>
> The problem is, you're conflating two types of "mapping". Yes, you can take 22-EDO's superpyth to map 5/4 to an augmented second...but you don't *have to*. You can take it to map 5/4 to a major third, just with a really high error. IOW you can "call" the 9/7 third made by 4 fifths of 22-EDO a "bad" 5/4. In fact, that is exactly what you are doing when you say that you "hear" 22-EDO diatonic as being the same as 31-EDO or 26-EDO diatonic.

Ah, but are you sure? That is what I'm contesting. I think that there
is something more fundamental that a major third "is" to us -
something which isn't tied down to 5/4. Perhaps I'm wrong, but
consider my comments below.

> The point being, you have to stop conflating the "absolute" JI ratio with the "approximated" JI ratio. Just because the absolute JI ratio is closer to a 9/7 or a 16/13, that doesn't mean it's not interpreted as an approximate 5/4. At this point, it's a matter of harmonic entropy: 9/7 and 16/13 are near enough to the gravity-well of 5/4 to get "sucked in".

Right, but what about 7/6? 7/6 is not near enough to the gravity-well
of 6/5 to get sucked in. Compare the sound of 22-tet superpyth aeolian
to 19-tet superpyth aeolian, and that still also "works." And to my
ears, at least, 7/6 and 6/5 are two different intervals.

But try an experiment - play a neutral third and you will notice that
it will "flip flop" between a major third and a minor third. Is this
because it's at an HE maximum between 5/4 and 6/5? Or is it that we
can't figure out whether it fits better, literally, to a "major third"
or a "minor third" in the diatonic sense? I will note that when I hear
scales in which neutral thirds make their own kind of "sense" in some
kind of new map - like the 7-tet example I posted on MMM, or like
porcupine[7] where the minor 7 is almost 11/6, or with certain uses of
mohajira, this no longer happens.

But to be honest, I thought the same thing as you for a while - that
meaning was in fact tied down to JI ratios, and that I was just
perceiving the 9/7 in 22-tet to be a mistuned 5/4. Seeing the minor
thirds be 7/6 in aeolian, and having it all work the same, destroyed
that hypothesis for me. You could make the argument that in superpyth,
we just interpret the minor thirds as being 6/5 even though they're
7/6 - but why would that be the case? You can clearly hear the triads
switch from "minor" to "subminor" between 19-tet and 22-tet -- the
whole mood changes. BUT, the "function," or "feeling," or "meaning,"
or whatever really embodies the "essence" of what's going on doesn't
change.

> Also, let me assure you that if you tried to use 22-EDO in a chromatic way, i.e. a 12-note MOS scale, it WOULDN'T sound or work the same way that 31-EDO does, because while they have the same 7-note MOS, they have inverse 12-note MOS's (i.e. s's and L's switch places, as between diatonic and mavila). Also, 26 and 31 diverge when you take a 19-note MOS. So just as mavila and diatonic share the same 5-note MOS but not the same 7-note MOS, the various diatonic scales you're comparing share the same 7-note MOS but NOT the same higher-number MOS's.

I haven't messed around with that - I will assume you're right. My
approach to using meantone[12] is to use modes as chords, and write
chord progressions where the mode keeps changing, and hence fleshing
out a chunk of meantone[12] (maybe meantone[9] or something). So like
||: Em7 | A7 | Dm7 | G7 :||, where it starts out in E dorian and moves
to D dorian. Or you could have ||: Bmaj7/C# | Cmaj7 | Emaj/F# | Bmaj7
:|| or something. Sorry I can't come up with anything sexier now, it's
like 1 AM here, haha. I'm beat.

I'll have to mess around with 22-EDO and see if modal harmonies are
completely destroyed, haha. You have made a very good point, although
I'm not sure I'm familiar enough with the 31-edo 12-note MOS to tell
the difference... Perhaps this means I don't have a proper chromatic
"map" to BE violated.

> > What I think is going on is that when we hear a major third, we aren't
> > really thinking of it in terms of any specific scale anymore. We're
> > used to hearing that interval as the root for major, and mixolydian,
> > and aeolian dominant, and it being in the octatonic scale, that we
> > have a generic "schema" built up for "major third" at this point. But
> > consider the possibility that "5/4" isn't actually part of that
> > schema... since 9/7 seems to substitute for it just fine.
>
> Again, this is probably more to do with harmonic entropy than anything else. What makes 6/5 different is that it's its own local minimum of harmonic entropy, separated from 5/4 by the local maximum of 347 cents. At 347 cents, as you've noted on MMM, you can sound like major AND/OR minor at once, but slide away from it in either direction and you're either in major land or minor land.

Right, but I'm not entirely sure that this is HE at work here. I
actually had a conversation with Paul about this, and he was the one
who pointed out to me that the effect of hearing a neutral triad flip
flop might actually be the effect of trying to figure out what map it
fits to, and have nothing to do at all with HE. Then he pointed out
this 7/6 vs 6/5 superpyth example. And suddenly I'm in a situation
where the guy who invented HE is giving me examples to counter what
I'd been using HE to do this whole time.

I think I'm starting to finally understand what HE models, from a
perceptual standpoint... it just models how resonant, or periodic, or
harmonic a dyad or triad is. But there is something totally different
going on with the map here.

> > And our
> > schema for "augmented second" seems to be even more flexible than
> > major third, although perhaps that's just because we're used to
> > thinking of an augmented second as the interval that's a diatonic
> > semitone less than the major third, no matter how it's intoned. Or, if
> > you're into modern harmony, you're used to thinking of it as a major
> > third above the major 7, a la C E F# B D#. And in 22-tet, both of
> > those functions are fulfilled by 5/4, not 7/6.
>
> You should compare some chromatic music in 22-EDO vs. 31-EDO and see if you still hear them as "saying the same thing". Augmented seconds sort of imply a 12-tone chromatic framework, or at least an addition to the lattice that makes one chord both a major and a minor. In 22-EDO, if you want to "map" 5/4 to the augmented second, you have to use a very different lattice than the standard diatonic-7 lattice. It no longer looks the same as 31-EDO's lattice.

In what way? If your axes are 3/2 and 9/7, and you temper out whatever
the difference is between 9/7*3/2*9/7 and 5/2 - how would that be any
different of a lattice between making the axes 3/2 and 5/4 and
tempering out 225/224?

> Because these scales are all generated by a generator and a period (or I guess just "two generators" is the fashionable way to say it). Each generator itself implies a unison vector (or vectors), which implies that two JI intervals (or more) are "merged" by the temperament. So the generator itself actually tells us the entire mapping, presuming that we can read from it the commas that are tempered out of it.
>
> Note that in meantone, if ONLY 81/80 is tempered out, then four 3/2's minus two 2/1's EXACTLY equals 5/4. But scales like 12, 19, and 26 temper other commas out of the fifth as well, giving different results. The commas "fight" with each other, you could say. But all that information is encoded in the generators. But in all meantones, four approximate 3/2's minus two exact 2/1's always approximates a 5/4.

OK, but why do you think that it is of particular psychoacoustic
significance how each interval matches up to the size of the
generator? Note that in meantone the generators can be taken to be a
tempered 3/2 and 2/1, or they can also be taken to be an approximate
16/15 and something between 9/8 and 10/9...

> The upshot is that you are free to interpret the word "approximates" however you please; if you can accept 9/7 as an "approximate" 5/4, you can call superpyth a meantone (though you'll be sharpening by a fraction of 81/80 instead of flattening). As long as you stick to the 7-note MOS, you don't run into trouble...but when you extend the MOS far enough to get augmented seconds to appear, which are clearly better approximations to 5/4, you'll probably want to stop calling the 9/7 an approximation to 5/4, or else you'll have two intervals approximating one interval, which is sort of the opposite of tempering.

I do agree that 9/7 can be used to approximate 5/4... I'm not so
certain that 7/6 is stuck as a type of 6/5 though. They approximate
different fundamentals.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So I think that 8:10:12:15 is much prettier
> sounding than 4:5:6:7. But, 4:5:6:7 is definitely more resonant and
> periodic and concordant and whatever you want to call it. So which > is more consonant in the end?

Hi Mike,
in order to investigate that really subtle, bust answerable, quest
analyze and compare that both chords by the methods as given in:

http://en.wikipedia.org/wiki/Binaural_beats
http://en.wikipedia.org/wiki/Brainwave_synchronization

realize 8:10:12:15 for instance as the four concrete sinus-tones
with the absolute frequencies

800Hz : 1000Hz : 1200Hz : 1500Hz

Distribute out of that for the just two separate pitches
distributed on an ear-phone, so that each side is restricted to
just one single tone:

1. left ear: 800Hz
2. right ear: 1000Hz

here one percives an dichotic beat-frequency of about ~1.12 Hz
or when compared with an Metronome: ~67 beats/min.
But it is possible get completely rid of the beatings when tuning
the 3rd downwards, departing 5/4,
by an 5-lim-schisma 5*3^8/2^15 = 32805/32768 flattend donwards to

(998 + 5722/6561)Hz = ~998.872...Hz

hence in the artificially primed binaural case the true binaural consonant 3rd appears physiologically not at the usual assuemd

5/4 = 1.25 = ~386.31...Cents

but instead of that weakly out of tune
rather at neighbouring located Pythagorean 3-limit resonance

8192/6561 = 2^13/3^8 = ~1.2471... = ~384.36..Cents.

The same strange effect can be observed too,
by exchangeing ears, when turning the headphones reverse.

Respecively the (15/8) 7th @ 800Hz:1500Hz beats with ~1.69...Hz
or when compared against the metronome @ ~102 beats/min.

That corresponding mismatch vanishes too,
when lowering the upper pitch carefully downwards to zero-beating @

(1498+674/2187)Hz =~ 1498.31... Hz

That procedure yield the intermediate resulting data for makeing 8:10:12:15 binaurally just towards the true binaural consonances:

left_ear 8: right_ear 10/(5*3^8/2^15) : l_e 12 : r_e 15/(5*3^8/2^15)

or when calculated into decimal approximation

l_e 8 : r_e ~9.98872... : l_e 12 : r_e ~14.9831...

The so gained 4-chord sounds now binaural full resonant without the slightest remains of residual beats. Gotcha!

Extend the binaural procedure by the obvious next step:
Now consider the septimal case 4:5:6:7 or 8:10:12:14 in 7-limit.
Use the same dichotic divided earphone conditions as above

Present simultaneous sine-tones separaltely distributed on each single ear at the same time

left_ear : 800Hz
right_ear : 1400Hz

Observation
Here one can detect and percive an even larger amount of dichotic beating against the initial 1400Hz of the seize

~3.1Hz or metronome ~185 beats/min

that vanishes respectively when departing the initial 7/4
about an 7-limit-schisma, that is defined as the quotient of
Archytas-Comma 64/63 over the Pythagorean-Comma 3^12/2^19

2^25/(7*3^14) = 33554432/33480783 := (64/63)/(3^12/2^19)

arised sharpend ~3.8...Cents in upwards direction.

Express that modification arithmetically en detail,
in order to get rid of the bigger detuning as found for 7/4
inbetween the separated ears:
Here it turns out that the beatings against the fundamental
800Hz do disappear effectively at the true dichotic resonance:

(1400*2^25/3^14/7)Hz
= (1403 + 380893/4782969)Hz
= ~1403.08...Hz

Hence in the 7-limit 4:5:6:7 chord case,
it turns out as impossible to detune the initial r4:l5:r6:l7 binaurally so that it becomes simultan beatless consistent,
so that the two problematic intervals:
the 3rd: 5/4 and the: 7th 7/4
got both involved resonant at once in an 4-tone chord.

Even if one tries, there exist no unadultered clear fitting as

l=4 : r=5/(5*3^8/2^15) : l=6 : r=7*(2^25/3^14/7)

l=4 : r=2^15/3^8 : l=6 : r=2^25/3^14

l=4 : r=(5-37/6561)=(4+6524/6561) : l=6 : r=(7+73649/4782969)

or the same can be expressed barely in plain 3-limit

l=4 : r=2^15/3^8=8192/6561 : l=6 : r=2^25/3^14=33554432/4782969

There remains always a small residue in beatings, that amounts:

5120/5103 := (64/63)/(3^12/2^19)*(32805/32768) = ~5.76...Cents

http://www.huygens-fokker.org/docs/intervals.html
" 5120/5103 Beta 5"

due to the opposite reverse algebraic prefix sign,
that can be deduced from the observed experimental deviations.

For the nerds as horrible inflated periodically fraction in decimal expansion:

l=4

r=4.994360615759792714525224813290656912056089010821521109^_\
58695320835238530711781740588324950464868160341411370217^_\
95458009449778997104099984758420972412741960067062947721^_\
38393537570492303002591068434689833866788599298887364730^_\
98613016308489559518366102728242645938119189147995732357^_\
87227556774881877762536198750190519737844840725499161713^_\
15348270080780368846212467611644566377076665142508763907^_\
94086267337296143880506020423715896966925773510135650053^_\
34552659655540313976527968297515622618503276939490931260^_\
47858558146623990245389422344154854442920286541685718640^_\
45115073921658283798201493674744703551287913427831123304^_\
37433318091754305746075293400396281054717268709038256363^_\
35924401768023167200121932632220698064319463496418228928^_\
516^_(period 729)

l=6

r=7.01539817632102570599976709027384455136548031149689659289031561776795960835205078686481137552846359656522967219733182464699227613643324888787696512354564706566151693644679695812370935291447634304132015072646299819212710766053470135.......^_(period 531441=3^12)

or when again reduced to reasonable approximation

l=4 : r=~4.994... : l=6 : r=~7.015...

that is sufficient in precision for human ears accuracy
for the true underlying ratio:
http://www.huygens-fokker.org/docs/intervals.html
"4782969/4194304 Pythagorean double augmented prime" 3^14/2^22

Conclusion
The suggested 5&7-limit surrogate replacements of the 3rds and 7ths

5/4 = 1.25 = ~386.31...Cents and
7/4 = 1.75 = ~968.82...Cents

do represent in deed very good numerically approximations
of the true beatless sounding binaural 3-limit resonances
at the bottom reasons:

2^13/3^61 = 8182/6561 = ~1.2471... = ~384.36...Cents
2^23/3^14 = 8388608/4782969 = ~1.7538... = ~972.63...Cents

Appearently uses the human brain just that both ones internally
at least for the internal communicationinbetween the both cerebral hemispheres.
Hence we may assume all further steps of evaluations in
the proceeding deeper analytic regions of the
auditory-cortex can be regarded as comletely restricted to
barely high powers of plain 3-limit,
as shown in the above binaural executed amazing experiment.

So far Mike, when returning back to yours initial quest,
from my own personal experimental point of view I can
present for you as final result, certainly without any doubt:
I can confirm yours "thinking that 8:10:12:15 has much prettier
sounding than 4:5:6:7", at least in the binaural case,
because there is no escape from the inherent 5120/5103
residual disturbance intrinsic trapped within cloudy 4:5:6:7
blue-chord, when the brain converts 5&7-limit input-data
down into the true underlying high powered 3-limit Pythagoreism.

http://news.softpedia.com/news/Pythagoras-Was-Right-Scientists-Devise-the-Geometrical-Music-Theory-83839.shtml

bye
Andy

Okay, Mike, here goes. Please note that I don't actually have this all worked out, I'm just sort of spinning hypotheses here as I go. But this is getting kind of more long-winded than it should, so maybe we can splinter some of these ideas off into separate threads?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Right, but what about 7/6? 7/6 is not near enough to the gravity-well
> of 6/5 to get sucked in. Compare the sound of 22-tet superpyth aeolian
> to 19-tet superpyth aeolian, and that still also "works." And to my
> ears, at least, 7/6 and 6/5 are two different intervals.

Well, the H.E. curve is much flatter between 6/5 and 7/6 than it is between 6/5 and 5/4. Below 6/5, it seems like the transition between "minor third" and "major second" is a very smooth one, without a significant peak of "what the hell IS that interval?". IOW, that whole region is already pretty high in discordance. Using the "adjusted and inverted scale" of H.E. that Steve M. just helped me work out, where a unison is 100 and a 51-cent quartertone is 0 (and an octave is 85, a fifth 50, etc.), 7/6 is about 23.74, 6/5 is about 25.91, and the "dip" between them is about 22.24 at about 288 cents. So the difference between local maxima and minima of H.E. in that region is very slight. OTOH, 5/4 scores about 30.3, and the valley (or peak, with the original H.E. numbers) between that and 6/5 is 18.73 at 350 cents; much more significant.

But at any rate, none of this may actually matter, and what may be more important--as I've tried to suggest--is that when you bring it back to a question of generators, in the diatonic range it's always going to be the case that (4g-2p)>(-3g+2p), whereas in Mavila, (4g-2p)<(-3g+2p). At 7-EDO, (4g-2p)=(-3g+2p). At 5-EDO, (4g-2p)=(-1g+1p), and (-3g+2p)=(2g-1p). As long as you can distinguish 4g-2p from -1g+1p, it doesn't matter what the exact ratio is. However, in order to know whether you can distinguish between two intervals, you have to look at harmonic entropy, which drags JI back into the equation.

> But try an experiment - play a neutral third and you will notice that
> it will "flip flop" between a major third and a minor third. Is this
> because it's at an HE maximum between 5/4 and 6/5? Or is it that we
> can't figure out whether it fits better, literally, to a "major third"
> or a "minor third" in the diatonic sense? I will note that when I hear
> scales in which neutral thirds make their own kind of "sense" in some
> kind of new map - like the 7-tet example I posted on MMM, or like
> porcupine[7] where the minor 7 is almost 11/6, or with certain uses of
> mohajira, this no longer happens.

The thing that really hammers me about neutrals is that I've never found a good way to map them on a lattice, except as BOTH major and minor thirds. I have to say, I don't like doing it that way, but it sort of makes sense. I really think that with 7-EDO, there are multiple maps coinciding at once, and with the right chord progressions and melodic phrasing, you can suggest different maps.

Mohajira, OTOH, has a unique map that is different from 7-EDO. Unlike 7-EDO, thirds other than neutral will appear in Mohajira, and they can remove the ambiguity of the mapping. However, mapping Mohajira using thirds and fifths isn't the best way to do it; I use major 2nds and neutral 2nds to span one neutral third; this gives a lattice in the same shape as the diatonic scale, actually, just with letters in different places.

> But to be honest, I thought the same thing as you for a while - that
> meaning was in fact tied down to JI ratios, and that I was just
> perceiving the 9/7 in 22-tet to be a mistuned 5/4. Seeing the minor
> thirds be 7/6 in aeolian, and having it all work the same, destroyed
> that hypothesis for me. You could make the argument that in superpyth,
> we just interpret the minor thirds as being 6/5 even though they're
> 7/6 - but why would that be the case? You can clearly hear the triads
> switch from "minor" to "subminor" between 19-tet and 22-tet -- the
> whole mood changes. BUT, the "function," or "feeling," or "meaning,"
> or whatever really embodies the "essence" of what's going on doesn't
> change.

5/4 and 6/5 are only important in the diatonic scale because they are psychoacoustically as far as possible from each other AND from 4/3 (which is -1g+1p) and 9/8 (which is 2g-1p). It is their psychoacoustic spacing that matters, not so much their actual low harmonic entropy. This is what makes them ideal. 12-tET could be another sort of ideal, since the space between 200, 300, 400, and 500 cents is all equal and as wide as possible. But the point is that the spacing doesn't HAVE to be ideal, it just has to be enough for us to tell the difference. As long as the minor third is above the entropic peak around 240 cents, and the major third is below the entropic peak around 453 cents, we can get a recognizable diatonic scale.

Of course, I actually have no idea how to deal with the approaching-7-EDO-case; how close can the thirds get together before we stop being able to tell them apart? That interests me a bit more than seeing how wide we can make the thirds.

> I haven't messed around with that - I will assume you're right. My
> approach to using meantone[12] is to use modes as chords, and write
> chord progressions where the mode keeps changing, and hence fleshing
> out a chunk of meantone[12] (maybe meantone[9] or something). So like
> ||: Em7 | A7 | Dm7 | G7 :||, where it starts out in E dorian and moves
> to D dorian. Or you could have ||: Bmaj7/C# | Cmaj7 | Emaj/F# | Bmaj7
> :|| or something. Sorry I can't come up with anything sexier now, it's
> like 1 AM here, haha. I'm beat.

Just carry the circle of fifths out to 12 iterations in both 17-EDO and 19-EDO, and compare the pitches you end up with.

> Right, but I'm not entirely sure that this is HE at work here. I
> actually had a conversation with Paul about this, and he was the one
> who pointed out to me that the effect of hearing a neutral triad flip
> flop might actually be the effect of trying to figure out what map it
> fits to, and have nothing to do at all with HE. Then he pointed out
> this 7/6 vs 6/5 superpyth example. And suddenly I'm in a situation
> where the guy who invented HE is giving me examples to counter what
> I'd been using HE to do this whole time.

Yeah, I'm not certain I interpret H.E. to mean the same things that everyone else does. I'm not sure if that means I'm wrong, or if I'm just seeing something new in it. I'll have to quiz Paul on this myself, I suppose. FWIW, I don't hear neutral thirds "flip flop" until I start playing chord progressions with them. If I just sit and listen to a neutral third sustaining, it's got it's own unique character to me. I hear it as sounding "disappointed", or maybe "reluctant". It expresses something quite different than either a major or minor, but it loses that effect the minute I start playing a I-IV-V or something.

I strongly believe that to understand why certain scale-families have defining characteristics, we have to look at both the MOS structure AND the relationship of the generator and period (and their multiples) to JI. To sum up (hopefully more understandably) what I tried to say above: a scale family is defined by the relationship that different multiples of the generator, modulo the period, have to each other pitch-wise.

For all I know, I'm coming at this all bass-ackwards, though. I'll bet looking at commas might be more revealing.

> I think I'm starting to finally understand what HE models, from a
> perceptual standpoint... it just models how resonant, or periodic, or
> harmonic a dyad or triad is. But there is something totally different
> going on with the map here.

Concordance is strongly correlated to intervallic identity. The closer an interval is to something you can tune by ear, the easier it is to identify. The further an interval is between two tunable-by-ear intervals, the harder. Intervals like the neutral third are about as far from two easily-identifiable intervals as possible, so they are hard to identify. So there is that component to it....

> > You should compare some chromatic music in 22-EDO vs. 31-EDO and see if you still hear them as "saying the same thing". Augmented seconds sort of imply a 12-tone chromatic framework, or at least an addition to the lattice that makes one chord both a major and a minor. In 22-EDO, if you want to "map" 5/4 to the augmented second, you have to use a very different lattice than the standard diatonic-7 lattice. It no longer looks the same as 31-EDO's lattice.
>
> In what way? If your axes are 3/2 and 9/7, and you temper out whatever
> the difference is between 9/7*3/2*9/7 and 5/2 - how would that be any
> different of a lattice between making the axes 3/2 and 5/4 and
> tempering out 225/224?

It's the difference between the 31-EDO and 22-EDO diatonic[12] I was talking about earlier. An augmented 2nd is 9g-5p; in 22-EDO, this is higher in pitch than -3g+2p, while in 31-EDO, it is lower.

> OK, but why do you think that it is of particular psychoacoustic
> significance how each interval matches up to the size of the
> generator? Note that in meantone the generators can be taken to be a
> tempered 3/2 and 2/1, or they can also be taken to be an approximate
> 16/15 and something between 9/8 and 10/9...

It's like holograms. Anything that can generate a meantone scale contains within it all the information of the entire scale. I'm not familiar with using major and minor seconds to generate a meantone, though...how does that work? Are you talking about treating meantone as a rank-3 temperament? I don't see how you could use a major second and a minor second as sole inputs to an MOS algorithm and come out with a diatonic scale. One has to be the modulo.

> I do agree that 9/7 can be used to approximate 5/4... I'm not so
> certain that 7/6 is stuck as a type of 6/5 though. They approximate
> different fundamentals.

Again, in the case of minor thirds, I'm not sure any interval in that pitch-range is significantly stronger than its neighbors such that it stands out enough to be the musico-semantic ideal. 7/6 is so close to 6/5 in both pitch and complexity that there's practically a plateau between them, entropically-speaking. Neither one is a "type" of the other; the whole minor third class is just sort of indistinct. There's certainly no point of distinct ambiguity between them.

-Igs

Ozan wrote:

> With little knowledge on the matter, I'd like to ask Margo: Was it
> not that Halberstadt keyboard organs in 1300s were ordinarily tuned
> in such a way that the naturals bore Pythagorean pitches and the
> accidentals (black keys) were tuned midway of the wholetones? If
> such was exactly the case, the novel habit in 1400s of tuning the
> black keys to flats in the fifths cycle to acquire the schismatic
> harmonic Majors on D, E and A would follow a desire to reflect the
> "consonantia per accidens" of singers on the keyboard, no?

Dear Ozan,

Scholars such as Mark Lindley have suggested, very reasonably in my
view, that the Halberstadt organs of Continental Europe in the 1300's
often used a regular Pythagorean tuning of Eb-G#, with all naturals at
accidentals at their usual Pythagorean pitches.

As Lindley observes, the Robertsbridge Codex, a collection of dances
and other instrumental pieces dating somewhere around 1325-1365, calls
for all 12 notes on a Halberstadt keyboard, and nicely fits Eb-G#,
with most other choices raising lots of problems.

For 14th-century singers like Marchettus of Padua, who wasn't writing
about keyboards but favored placing cadential sharps notably _higher_
than their regular Pythagorean positions, the Eb-G# tuning would
likely be preferable to an early 15th=century tuning where some or all
of those sharps would be lowered by a comma from their usual pitches.

The fact that Johannes Boen in 1357 describes a Pythagorean diminished
fourth like G#-C as dissonant in itself but a _consonantia per
accidens_ in the special sonority of E-G#-C may be a clue that singers
and keybaordists alike on the Continent did not yet favor the kind of
schismatic tuning that caught on by around 1400-1420. In E-G#-C, the
upper voices form regular intervals of a major third and minor sixth
with the lowest voice. It may be notable that Boen does not recommend
something like C#-F-G# or B-Eb-F# at an approximate 4:5:6.

Whatever the interplay between the tastes of singers and keyboard
players, by 1400-1420 the schismatic tunings had come into vogue. Lindley suggests that around 1420-1430, the young Dufay was much
influenced in his vocal music by the Gb-B tuning, favoring prolonged
sonorities such as D-Gb-A (written D-F#-A) or A-Db-E (written A-C#-E).
And by 1450-1480, the desire to have these smoother thirds in as many
locations as possible evidently led to meantone temperament.

> Might I be brave enough to make the following conjecture? Could it
> be possible that the transition from Pythagorean to Meantone was not
> as swift as generally assumed and spanned as considerable a time as
> it took 12-tone Equal-Temperament to arise from Meantone? And in
> such a manner that the trend was far from universal in every corner
> of Europe? (attested by examples such as "Sumer is icumen in"?)

Here I should clarify that the above remarks are about Continental
European music, and the English _Sumer is icumen in_ is a fine example
of how thirds were treated as pervasive and sometimes stable concords
in some English styles at least as early as the 13th century. Whether
or how medieval English keyboards may have been tuned to fit such
vocal styles remains an open question. It's easy to imagine either a
schismatic Pythagorean tuning or meantone catching on under such
conditions, but we don't have much evidence.

Curiously, an English manuscript from around 1275 -- tha age of Safi
al-Din al-Urmavi -- describes a full 17-note Pythagorean division of
the monochord! If one wanted to acquire lots of schismatic thirds,
this would be a way to do it!

In 1373, a treatise on the organ advocates using a pipe length for an
accidental which is the average or arithmetic mean of the two adjacent
pipes at the 9:8 tone it divides into two semitones. For example:

18 17 16
C C# D
1/1 18/17 9/8
0 99 204

This arrangement, like a schismatic tuning, would produce thirds
involving accidentals which are closer to 5:4 and 6:5.

While I might guess that meantone had come into vogue in England at
around the same era of 1450-1500 are on the Continent, the first
theoretical evidence of which I know is a statement by Thomas Morley
in 1597 that on keyboards, G# would be too low to serve as Ab. so that
keybaords called _chromatica_ have both. The nature of 16th-century
English music, for keyboard or otherwise, suggests to me that meantone
may have been in use for a century or more by the time Morley came to
give us this theoretical hint as to its presence.

A fine point is that 12-EDO may have actually developed as a lute
tuning in the early to middle 15th century as a variation on
Pythagorean tuning permitting the use of the same fret as either a
diatonic or chromatic semitone, in an age when a Pythagorean tuning on
keyboard was still widespread.

Your comment, of course, focuses on the question of 12-EDO for
keyboards in later eras when meantone or unequal well-temperaments
were the norm. In the 1580's, we hear a few opinions that it would be
convenient if keyboards could be tuned to match the 12-EDO common on
lutes; while in 1643, Jean Denis replies that it would be much better
if lutes were improved by being tuned in a proper meantone (as had
sometimes been done in the earlier 16th century, but with 12-EDO as
the theoretical norm starting around the mid-16th century despite some
interesting irregular tunings such as that of Dowland around 1600).

Best,

Margo

Hi Andy,

Thanks for all of this. I've never heard of anything tying binaural
beating into the 3-limit, but I'll check it out and let you know what
I find.

-Mike

On Mon, Aug 23, 2010 at 3:35 PM, Andy <a_sparschuh@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > So I think that 8:10:12:15 is much prettier
> > sounding than 4:5:6:7. But, 4:5:6:7 is definitely more resonant and
> > periodic and concordant and whatever you want to call it. So which > is more consonant in the end?
>
> Hi Mike,
> in order to investigate that really subtle, bust answerable, quest
> analyze and compare that both chords by the methods as given in:
>
> http://en.wikipedia.org/wiki/Binaural_beats
> http://en.wikipedia.org/wiki/Brainwave_synchronization
>
> realize 8:10:12:15 for instance as the four concrete sinus-tones
> with the absolute frequencies
>
> 800Hz : 1000Hz : 1200Hz : 1500Hz
>
> Distribute out of that for the just two separate pitches
> distributed on an ear-phone, so that each side is restricted to
> just one single tone:
>
> 1. left ear: 800Hz
> 2. right ear: 1000Hz
>
> here one percives an dichotic beat-frequency of about ~1.12 Hz
> or when compared with an Metronome: ~67 beats/min.
> But it is possible get completely rid of the beatings when tuning
> the 3rd downwards, departing 5/4,
> by an 5-lim-schisma 5*3^8/2^15 = 32805/32768 flattend donwards to
>
> (998 + 5722/6561)Hz = ~998.872...Hz
>
> hence in the artificially primed binaural case the true binaural consonant 3rd appears physiologically not at the usual assuemd
>
> 5/4 = 1.25 = ~386.31...Cents
>
> but instead of that weakly out of tune
> rather at neighbouring located Pythagorean 3-limit resonance
>
> 8192/6561 = 2^13/3^8 = ~1.2471... = ~384.36..Cents.
>
> The same strange effect can be observed too,
> by exchangeing ears, when turning the headphones reverse.
>
> Respecively the (15/8) 7th @ 800Hz:1500Hz beats with ~1.69...Hz
> or when compared against the metronome @ ~102 beats/min.
>
> That corresponding mismatch vanishes too,
> when lowering the upper pitch carefully downwards to zero-beating @
>
> (1498+674/2187)Hz =~ 1498.31... Hz
>
> That procedure yield the intermediate resulting data for makeing 8:10:12:15 binaurally just towards the true binaural consonances:
>
> left_ear 8: right_ear 10/(5*3^8/2^15) : l_e 12 : r_e 15/(5*3^8/2^15)
>
> or when calculated into decimal approximation
>
> l_e 8 : r_e ~9.98872... : l_e 12 : r_e ~14.9831...
>
> The so gained 4-chord sounds now binaural full resonant without the slightest remains of residual beats. Gotcha!
>
> Extend the binaural procedure by the obvious next step:
> Now consider the septimal case 4:5:6:7 or 8:10:12:14 in 7-limit.
> Use the same dichotic divided earphone conditions as above
>
> Present simultaneous sine-tones separaltely distributed on each single ear at the same time
>
> left_ear : 800Hz
> right_ear : 1400Hz
>
> Observation
> Here one can detect and percive an even larger amount of dichotic beating against the initial 1400Hz of the seize
>
> ~3.1Hz or metronome ~185 beats/min
>
> that vanishes respectively when departing the initial 7/4
> about an 7-limit-schisma, that is defined as the quotient of
> Archytas-Comma 64/63 over the Pythagorean-Comma 3^12/2^19
>
> 2^25/(7*3^14) = 33554432/33480783 := (64/63)/(3^12/2^19)
>
> arised sharpend ~3.8...Cents in upwards direction.
>
> Express that modification arithmetically en detail,
> in order to get rid of the bigger detuning as found for 7/4
> inbetween the separated ears:
> Here it turns out that the beatings against the fundamental
> 800Hz do disappear effectively at the true dichotic resonance:
>
> (1400*2^25/3^14/7)Hz
> = (1403 + 380893/4782969)Hz
> = ~1403.08...Hz
>
> Hence in the 7-limit 4:5:6:7 chord case,
> it turns out as impossible to detune the initial r4:l5:r6:l7 binaurally so that it becomes simultan beatless consistent,
> so that the two problematic intervals:
> the 3rd: 5/4 and the: 7th 7/4
> got both involved resonant at once in an 4-tone chord.
>
> Even if one tries, there exist no unadultered clear fitting as
>
> l=4 : r=5/(5*3^8/2^15) : l=6 : r=7*(2^25/3^14/7)
>
> l=4 : r=2^15/3^8 : l=6 : r=2^25/3^14
>
> l=4 : r=(5-37/6561)=(4+6524/6561) : l=6 : r=(7+73649/4782969)
>
> or the same can be expressed barely in plain 3-limit
>
> l=4 : r=2^15/3^8=8192/6561 : l=6 : r=2^25/3^14=33554432/4782969
>
> There remains always a small residue in beatings, that amounts:
>
> 5120/5103 := (64/63)/(3^12/2^19)*(32805/32768) = ~5.76...Cents
>
> http://www.huygens-fokker.org/docs/intervals.html
> " 5120/5103 Beta 5"
>
> due to the opposite reverse algebraic prefix sign,
> that can be deduced from the observed experimental deviations.
>
> For the nerds as horrible inflated periodically fraction in decimal expansion:
>
> l=4
>
> r=4.994360615759792714525224813290656912056089010821521109^_\
> 58695320835238530711781740588324950464868160341411370217^_\
> 95458009449778997104099984758420972412741960067062947721^_\
> 38393537570492303002591068434689833866788599298887364730^_\
> 98613016308489559518366102728242645938119189147995732357^_\
> 87227556774881877762536198750190519737844840725499161713^_\
> 15348270080780368846212467611644566377076665142508763907^_\
> 94086267337296143880506020423715896966925773510135650053^_\
> 34552659655540313976527968297515622618503276939490931260^_\
> 47858558146623990245389422344154854442920286541685718640^_\
> 45115073921658283798201493674744703551287913427831123304^_\
> 37433318091754305746075293400396281054717268709038256363^_\
> 35924401768023167200121932632220698064319463496418228928^_\
> 516^_(period 729)
>
> l=6
>
> r=7.01539817632102570599976709027384455136548031149689659289031561776795960835205078686481137552846359656522967219733182464699227613643324888787696512354564706566151693644679695812370935291447634304132015072646299819212710766053470135.......^_(period 531441=3^12)
>
> or when again reduced to reasonable approximation
>
> l=4 : r=~4.994... : l=6 : r=~7.015...
>
> that is sufficient in precision for human ears accuracy
> for the true underlying ratio:
> http://www.huygens-fokker.org/docs/intervals.html
> "4782969/4194304 Pythagorean double augmented prime" 3^14/2^22
>
> Conclusion
> The suggested 5&7-limit surrogate replacements of the 3rds and 7ths
>
> 5/4 = 1.25 = ~386.31...Cents and
> 7/4 = 1.75 = ~968.82...Cents
>
> do represent in deed very good numerically approximations
> of the true beatless sounding binaural 3-limit resonances
> at the bottom reasons:
>
> 2^13/3^61 = 8182/6561 = ~1.2471... = ~384.36...Cents
> 2^23/3^14 = 8388608/4782969 = ~1.7538... = ~972.63...Cents
>
> Appearently uses the human brain just that both ones internally
> at least for the internal communicationinbetween the both cerebral hemispheres.
> Hence we may assume all further steps of evaluations in
> the proceeding deeper analytic regions of the
> auditory-cortex can be regarded as comletely restricted to
> barely high powers of plain 3-limit,
> as shown in the above binaural executed amazing experiment.
>
> So far Mike, when returning back to yours initial quest,
> from my own personal experimental point of view I can
> present for you as final result, certainly without any doubt:
> I can confirm yours "thinking that 8:10:12:15 has much prettier
> sounding than 4:5:6:7", at least in the binaural case,
> because there is no escape from the inherent 5120/5103
> residual disturbance intrinsic trapped within cloudy 4:5:6:7
> blue-chord, when the brain converts 5&7-limit input-data
> down into the true underlying high powered 3-limit Pythagoreism.
>
> http://news.softpedia.com/news/Pythagoras-Was-Right-Scientists-Devise-the-Geometrical-Music-Theory-83839.shtml
>
> bye
> Andy

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I've never heard of anything tying binaural
> beating into the 3-limit,
> but I'll check it out and let you know what
> I find....
> > > So I think that 8:10:12:15 is much prettier
> > > sounding than 4:5:6:7....

Good idea Mike,
simply to convince yourself by repeating the binaural experiments
by paying attention to the inherent beats with yours own ears.
Then you should also try out the even more "prettier" twao penta-chords extensions by investigating the both 5-fold ratios:

r8:l10:r12:l15:r18 in 5-limit (begin best with r4:l5:r6)

and also its corresponding 7-limit counterpart

r8:r12:l14:r18:l21 in 7-limit (begin best with 4:6:7)

because the problem in binaural dichotic listening arises
from mixing that both incompatible limits simultaneous at once.
Don't forget to exchange the right versus left side, simply by
turning yours earphones rotated.

Have a nice time when performing that amazing crucial experiments.
I'm eagery looking forward to yours personal observations and remarks.

bye
Andy

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
> Was it not
> that Halberstadt keyboard organs in 1300s were ordinarily tuned in
> such a way that the naturals bore Pythagorean pitches and the
> accidentals (black keys) were tuned midway of the wholetones?

Hi Oz,
here comes an link to the famous Praetorius of
the ~1361 Halberstadt-organ keyboard and pedal desingn

http://upload.wikimedia.org/wikipedia/commons/c/cc/Syntagma14.png

> If such
> was exactly the case, the novel habit in 1400s of tuning the black
> keys to flats in the fifths cycle to acquire the schismatic harmonic....

One of the probaly oldest still playable organ dates back about to the year ~1435 is located in Sion, Switzerland

http://de.wikipedia.org/wiki/Orgel_der_Basilique_de_Val%C3%A8re
(sorry, that is only in German or Esparanto available)
http://eo.wikipedia.org/wiki/Valeria-orgeno_Sion

the history and properties of that fine instrument is well documented
in the public accessible book:

http://books.google.de/books?id=Qf3y8LXGjUEC&printsec=frontcover&dq=valeria+orgel+sion&source=bl&ots=4egXuqNey6&sig=__VP144t9YappiLpAzT8DCKCeRE&hl=de&ei=34l2TLrNB4rDswaxkfilBg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CB8Q6AEwAg#v=onepage&q&f=false

see there in that reference, especially on page 30
quote:
"... ursprüngliche Stimmung wahr wahrscheinlich die Pythagoräische..."

tr:
'... original tuning was probably Pythagorean...'

Attend in that document some pages later the drawings of even older precursing models, alike especially the no-more-playable relicts of the 'Norrlanda'-organ ~1400~

http://lh5.ggpht.com/_NCfVE93sADw/Sd-9hS-8UGI/AAAAAAAAH68/RO2L0B14-rw/hela+mappen+26+mars+146.jpg

http://lh6.ggpht.com/_NCfVE93sADw/Sd-9iDVy4vI/AAAAAAAAH7I/TRB2qSu7P_8/hela+mappen+26+mars+148.jpg

which remains are today located, preserved and exhibited in the
National-Museum-Stockholm, Sweden.

I had the pleasure to play one of the modern attempts of an
so called 'historically-informed' replica in the organ-museum:
http://www.gdo.de/fileadmin/gdo/images/Ostheim-Norrlanda.jpg

bye
Andy

Another belated reply:

Dear Margo, thank you for this highly informative note. Much
speculation abounds it seems as to the true nature of the tuning(s) of
keyboards across Europe throughout the Age of Renaissance. At the very
least, I feel that the Islamic connection awaits discovery -
especially since it was Al-Farabi who first transported the Ptolemaic
literature on music theory into Arabic and found a place for 18/17 in
some of his scale formulations.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Aug 24, 2010, at 8:01 AM, Margo Schulter wrote:

> Ozan wrote:
>
>> With little knowledge on the matter, I'd like to ask Margo: Was it
>> not that Halberstadt keyboard organs in 1300s were ordinarily tuned
>> in such a way that the naturals bore Pythagorean pitches and the
>> accidentals (black keys) were tuned midway of the wholetones? If
>> such was exactly the case, the novel habit in 1400s of tuning the
>> black keys to flats in the fifths cycle to acquire the schismatic
>> harmonic Majors on D, E and A would follow a desire to reflect the
>> "consonantia per accidens" of singers on the keyboard, no?
>
> Dear Ozan,
>
> Scholars such as Mark Lindley have suggested, very reasonably in my
> view, that the Halberstadt organs of Continental Europe in the 1300's
> often used a regular Pythagorean tuning of Eb-G#, with all naturals at
> accidentals at their usual Pythagorean pitches.
>
> As Lindley observes, the Robertsbridge Codex, a collection of dances
> and other instrumental pieces dating somewhere around 1325-1365, calls
> for all 12 notes on a Halberstadt keyboard, and nicely fits Eb-G#,
> with most other choices raising lots of problems.
>
> For 14th-century singers like Marchettus of Padua, who wasn't writing
> about keyboards but favored placing cadential sharps notably _higher_
> than their regular Pythagorean positions, the Eb-G# tuning would
> likely be preferable to an early 15th=century tuning where some or all
> of those sharps would be lowered by a comma from their usual pitches.
>
> The fact that Johannes Boen in 1357 describes a Pythagorean diminished
> fourth like G#-C as dissonant in itself but a _consonantia per
> accidens_ in the special sonority of E-G#-C may be a clue that singers
> and keybaordists alike on the Continent did not yet favor the kind of
> schismatic tuning that caught on by around 1400-1420. In E-G#-C, the
> upper voices form regular intervals of a major third and minor sixth
> with the lowest voice. It may be notable that Boen does not recommend
> something like C#-F-G# or B-Eb-F# at an approximate 4:5:6.
>
> Whatever the interplay between the tastes of singers and keyboard
> players, by 1400-1420 the schismatic tunings had come into vogue.
> Lindley suggests that around 1420-1430, the young Dufay was much
> influenced in his vocal music by the Gb-B tuning, favoring prolonged
> sonorities such as D-Gb-A (written D-F#-A) or A-Db-E (written A-C#-E).
> And by 1450-1480, the desire to have these smoother thirds in as many
> locations as possible evidently led to meantone temperament.
>
>> Might I be brave enough to make the following conjecture? Could it
>> be possible that the transition from Pythagorean to Meantone was not
>> as swift as generally assumed and spanned as considerable a time as
>> it took 12-tone Equal-Temperament to arise from Meantone? And in
>> such a manner that the trend was far from universal in every corner
>> of Europe? (attested by examples such as "Sumer is icumen in"?)
>
> Here I should clarify that the above remarks are about Continental
> European music, and the English _Sumer is icumen in_ is a fine example
> of how thirds were treated as pervasive and sometimes stable concords
> in some English styles at least as early as the 13th century. Whether
> or how medieval English keyboards may have been tuned to fit such
> vocal styles remains an open question. It's easy to imagine either a
> schismatic Pythagorean tuning or meantone catching on under such
> conditions, but we don't have much evidence.
>
> Curiously, an English manuscript from around 1275 -- tha age of Safi
> al-Din al-Urmavi -- describes a full 17-note Pythagorean division of
> the monochord! If one wanted to acquire lots of schismatic thirds,
> this would be a way to do it!
>
> In 1373, a treatise on the organ advocates using a pipe length for an
> accidental which is the average or arithmetic mean of the two adjacent
> pipes at the 9:8 tone it divides into two semitones. For example:
>
> 18 17 16
> C C# D
> 1/1 18/17 9/8
> 0 99 204
>
> This arrangement, like a schismatic tuning, would produce thirds
> involving accidentals which are closer to 5:4 and 6:5.
>
> While I might guess that meantone had come into vogue in England at
> around the same era of 1450-1500 are on the Continent, the first
> theoretical evidence of which I know is a statement by Thomas Morley
> in 1597 that on keyboards, G# would be too low to serve as Ab. so that
> keybaords called _chromatica_ have both. The nature of 16th-century
> English music, for keyboard or otherwise, suggests to me that meantone
> may have been in use for a century or more by the time Morley came to
> give us this theoretical hint as to its presence.
>
> A fine point is that 12-EDO may have actually developed as a lute
> tuning in the early to middle 15th century as a variation on
> Pythagorean tuning permitting the use of the same fret as either a
> diatonic or chromatic semitone, in an age when a Pythagorean tuning on
> keyboard was still widespread.
>
> Your comment, of course, focuses on the question of 12-EDO for
> keyboards in later eras when meantone or unequal well-temperaments
> were the norm. In the 1580's, we hear a few opinions that it would be
> convenient if keyboards could be tuned to match the 12-EDO common on
> lutes; while in 1643, Jean Denis replies that it would be much better
> if lutes were improved by being tuned in a proper meantone (as had
> sometimes been done in the earlier 16th century, but with 12-EDO as
> the theoretical norm starting around the mid-16th century despite some
> interesting irregular tunings such as that of Dowland around 1600).
>
> Best,
>
> Margo
>
>
>
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I have forwarded this work to those who can deliver it to Erv.
I am sure he will be as trilled as i was in hearing it.
It is always a delightful and rare treat when we get to hear Margo's work

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again