Scales, Analysis, Accidentals and Performance

>
> Message: 3
> Date: Thu, 11 Apr 2002 17:49 +0100 (BST)
> From: graham@microtonal.co.uk
> Subject: Re: Decatonics
>
> In-Reply-To: <B8C71E03.38EA%mark.gould@argonet.co.uk>
> Mark Gould wrote (27th March):
>
>> This is inconsistent with my rule ii: it contains segments of 3 and
>> more adjacent PCs
>>
>> 0 1 2, 4 5 6 7, 9 10 11 12, etc
>>
>> So it will show up as intervallically inchoerent (as defined by
>> Balzano).
>
>>From the discussion that followed, it emerged that Balzano's
>>intervallic
> coherence is the same as Rothenberg's propriety.
[snip]
> Which looks proper to me. So this rule ii remains a spectacularly bad
> way of predicting propriety.

I don't think I was trying for propriety, only some measure of whether the
scale bears similarity to the diatonic scale that I started from. I still
think that scales of this form are not 'diatonic' scales. They may be
perfectly useable, after all the 7ET scale is perfectly useable with all of
its pitch-classes, so all seven PCs are adjacent there.

Like all rules it was designed for a specific purpose. Maybe there are
other ways of analysing scales, but analysis alone does not make music.
This may be a list for tuning math, but at the end of the day we must be
sure of our focus.

My question is why you use 'spectacularly'? Does it mean you take personal
issue? Does it mean that you feel that your own statements carry more
weight than any of the other posters to this list? From your exchanges I
sense a certain garrulous sense of personal injustice should anyone appear
to suggest anything that you would not. DO you 'own' tuning-math?

If this is a disccusion group, I would keep personally directed comments
out. And, if anything, one rule is as good as another. Who states one rule
for analysing scales carries ultimate weight. These 'rules' are entirely
subjective, from all that I have read. How we lay out lattices is also
subjective, as I prefer one way of constructing them that is different from
those that I have seen. That does not make mine 'wrong' nor the
others 'wrong'. All are equally valid views on the complex problem of
determining the nature of relationships in music.

For my own part I find the endless discussion of increasingly complex
analyses of ET scales with many hundreds of tones a pointless persuit of
mathematics beyond the call of music. Who has played music in 311 ET? Where
has it been written. And that is just one scale from the many discussed
here. How does a musician react to the many descriptions of vectors and
matrices? They are merely tools, not the understanding.

As for accidentals. Tell me, how would a string player react to

_
| \
_/| |
|
|
|

and
_
_| \
| |
|
|
|

in a passage of semiquavers in Allegro tempo?

When you can tell the difference in performance, and that difference has a
musical effect, then it is important. I'd like to see a decent analysis of
the intonational variance of a string player or vocalist on successive
performances of the same passage, or different performers on the same
passage. This would be a useful correlative to the proliferation of
accidentals. If was merely the act of playing or singing 'in tune' then
often the ear will guide the performer more than any accidental. Only if
you want to be 'out of tune' do you want to 'alter' the note by some micro
amount. After all, if the ET is designed to duplicate some n-limit tuning
then surely an indication of the desired ratio by number would be more
appropriate?

Some may ask why I am on this list. Because I had the temerity (as a
musician and a composer) to write about a mathematical view on scales and
tonality. But only from the viewpoint of a musician. I am still interested
in all everyone has to say, but only if it has musical relevance.

Mark Gould

>Some may ask why I am on this list. Because I had the temerity (as a
>musician and a composer) to write about a mathematical view on scales and
>tonality. But only from the viewpoint of a musician. I am still
>interested in all everyone has to say, but only if it has musical
>relevance.
>
>Mark Gould

Heya Mark,

I for one am glad you're on this list. So far, I've read all your posts.
I doubt Graham meant anything by it (probably just colorful metaphor, not
uncommon on the other side of the pond), but I'll let him speak to it.
Which isn't to say there isn't a healthy dose of oneupsmanship at work on
these lists, and even some amount of anti-academic sentiment...

I think propriety and the attached model is generally important in music,
but it certainly isn't the whole picture, and there are many valid ways
to approach the subject. Anyway, I still have the explanatory document
for my diatonicity "shopping list" in my stack of things to do in the
next month...

-Carl

--- In tuning-math@y..., "Mark Gould" <mark.gould@a...> wrote:
> For my own part I find the endless discussion of increasingly
complex
> analyses of ET scales with many hundreds of tones a pointless
persuit of
> mathematics beyond the call of music. Who has played music in 311
ET? Where
> has it been written. And that is just one scale from the many
discussed
> here.

Hey Mark, relax. Did you miss where we were talking about 1600-ET. :-)

Just as when one talks about music in 7-limit JI, one doesn't mean it
uses the infinity of notes in the 7-limit, one might only use a
handful of notes, and yet they might meaningfully be said to be in say
217-ET, such as Paul's adaptive-JI chord-progression request.

And such large ETs are mainly being employed as a mathematical
convenience in designing a general purpose notation, after which its
high ET origins will likely be forgotten.

> How does a musician react to the many descriptions of vectors and
> matrices? They are merely tools, not the understanding.

Indeed. I guess that's why we started the tuning-math list. So we
wouldn't have to put up with the endless whining of musicians who
didn't want to see all that stuff, but just wanted to know the final
results.

> As for accidentals. Tell me, how would a string player react to
>
>
> _
> | \
> _/| |
> |
> |
> |
>
> and
> _
> _| \
> | |
> |
> |
> |
>
> in a passage of semiquavers in Allegro tempo?

In allegro semiquavers, they could simply ignore them and no-one would
be any the wiser. If they suceeded in recognising that, as kinds of
full-headed arrows pointing upwards they indicated a sharpening of
somewhere between an eighth and a quarter of a tone, they would be
doing exceedingly well.

In a long-sustained large otonal chord however, a more detailed
knowledge might be of benefit. They would then need to know if the
piece is in JI or which ET, and preferably the composer would offer a
legend on the score.

> When you can tell the difference in performance, and that difference
has a
> musical effect, then it is important.

I totally agree.

> I'd like to see a decent
analysis of
> the intonational variance of a string player or vocalist on
successive
> performances of the same passage, or different performers on the
same
> passage.

So would I. I have no doubt that we are going to ridiculous lengths to
design a notation that will suit every conceivable tuning, but so long
as we are having fun, and doing it in a way that does not complicate
the notation of the most common tunings, where's the harm?

> This would be a useful correlative to the proliferation of
> accidentals. If was merely the act of playing or singing 'in tune'
then
> often the ear will guide the performer more than any accidental.

Don't count on it.

> Only if
> you want to be 'out of tune' do you want to 'alter' the note by some
micro
> amount. After all, if the ET is designed to duplicate some n-limit
tuning
> then surely an indication of the desired ratio by number would be
more
> appropriate?

I think it's been tried. It just gets too cluttered.

> Some may ask why I am on this list. Because I had the temerity (as a
> musician and a composer) to write about a mathematical view on
scales and
> tonality. But only from the viewpoint of a musician. I am still
interested
> in all everyone has to say, but only if it has musical relevance.

I feel the same way, believe me. I think we all greatly value your
contribution to this list.

Regards,

In-Reply-To: <3969.194.203.13.66.1018597443.squirrel@email.argonet.co.uk>
Mark Gould wrote:

> I don't think I was trying for propriety, only some measure of whether
> the
> scale bears similarity to the diatonic scale that I started from.

What you originally said was "...it will show up as intervallically
incoherent (as defined by Balzano)." The consensus of the list is that
Balzano's intervallic coherence is the same as Rothenberg's propriety. If
that wasn't what you were trying for, why didn't you say what you meant?

> I still
> think that scales of this form are not 'diatonic' scales. They may be
> perfectly useable, after all the 7ET scale is perfectly useable with
> all of
> its pitch-classes, so all seven PCs are adjacent there.

Well, yes, there are a whole load of first rate scales that fail
completely as generalized diatonics. I don't want to give the impression
that the criteria we give for diatonics are important for music in
general. But even so, I don't see why this three-adjacent-pitch-classes
rule has anything to do with the important properties of the diatonic
scale -- the ease of establishing a key centre, clarity of modulation, the
sense of roughly equal melodic intervals and the clever things you can do
with symmetric chords and tritone substitutions.

One very promising diatonic that breaks this rule is the 7 from 10 neutral
third MOS:

0 1 3 4 6 7 9 0
1 2 1 2 1 2 1

Bill Sethares has a fair amount on it in the chapter on 10-equal music
theory in his book. It's also something I've found works well as a fuzzy
subset of the unequal Decimal scale. Well enough that it is a priority
for me to get some music together illustrating this.

It's also similar to the classic diatonic in a number of ways. It has 7
notes. It can be generated by alternating the best approximations to 6:5
and 5:4 (which happen to be identical). The half-octave is an ambiguous
interval.

The three adjacent pitch classes are enough to uniquely specify the key.
The middle note (pitch class) is the obvious tonic because it has a small
leading tone from both directions. I find this to be a very useful
property and, given that it has to be different to the diatonic somehow,
it may as well be like this.

> Like all rules it was designed for a specific purpose. Maybe there are
> other ways of analysing scales, but analysis alone does not make music.
> This may be a list for tuning math, but at the end of the day we must be
> sure of our focus.

What purpose was it designed for? You haven't actually said. If your
focus is on music, where's the music?

> My question is why you use 'spectacularly'? Does it mean you take
> personal
> issue? Does it mean that you feel that your own statements carry more
> weight than any of the other posters to this list? From your exchanges I
> sense a certain garrulous sense of personal injustice should anyone
> appear
> to suggest anything that you would not. DO you 'own' tuning-math?

I say 'spectacularly' because it is spectacular. You put forward a rule
of thumb to identify proper scales (although you now say that wasn't the
intention at all). It fails to say that this 21-from-26 "pentatonic" is
proper, it fails to say that the classic diatonic in a positive
temperament (eg 17, 22, 41) is improper, and it fails to say that the
classic pentatonic in 7-equal is proper.

Even the rule you give in your paper about the relative sizes of the
generalized thirds doesn't fail as spectacularly as this. Assuming both
thirds should be the same number of diatonic scale steps, it should only
give false positives.

A better rule, as it happens, is to consider the relative sizes of the
diatonic scale steps. If the larger step is more then twice the size of
the smaller one, you have an improper scale. If the large step is exactly
two small steps, you have a proper scale. If the large step is less than
two small steps, the scale is strictly proper. If the scale is an MOS (as
all your examples are) I think this relationship is exact, although I
can't prove it.

Hence I didn't have to go to all the trouble of generating the propriety
grid to know that the 21 note "pentatonic" is proper. If I were on a
personal vendetta, I could have sent an immediate reply asserting such,
and taken the risk of you disproving it.

Of course my statements don't carry special weight here. Have you heard
my music? I have to compensate for not having any inherent credibility by
making sure what I say is true.

> If this is a disccusion group, I would keep personally directed comments
> out. And, if anything, one rule is as good as another. Who states one
> rule
> for analysing scales carries ultimate weight. These 'rules' are entirely
> subjective, from all that I have read. How we lay out lattices is also
> subjective, as I prefer one way of constructing them that is different
> from
> those that I have seen. That does not make mine 'wrong' nor the
> others 'wrong'. All are equally valid views on the complex problem of
> determining the nature of relationships in music.

Well, you're actually the one who introduced the personal comments here.
And that's fine with me -- if you have a problem it's best to air it. The
big list has definitely suffered from people simmering in disagreement,
and getting more and more upset, but not saying so.

I don't think one rule is as good as another. Your next sentence
mystifies me.

The rules Paul gave in his paper are objective. The rules Rothenberg gave
(propriety, efficiency and stability -- reading the original papers is not
the ideal way of learning about them ;) are also objective. The rules you
give in your paper are objective, and correctly applied.

Carl's rules, of which we might soon have a discussion, aren't quite
objective. He says things like "high efficiency" without putting a number
to how high. Still, they could be made objective if that's what you
really want. But perhaps he's putting forward objective rules, but
allowing use to make subjective judgements about their relative
importance, which sounds right to me.

The layout of lattices is also objective.

> When you can tell the difference in performance, and that difference
> has a
> musical effect, then it is important. I'd like to see a decent analysis
> of
> the intonational variance of a string player or vocalist on successive
> performances of the same passage, or different performers on the same
> passage. This would be a useful correlative to the proliferation of
> accidentals. If was merely the act of playing or singing 'in tune' then
> often the ear will guide the performer more than any accidental. Only if
> you want to be 'out of tune' do you want to 'alter' the note by some
> micro
> amount. After all, if the ET is designed to duplicate some n-limit
> tuning
> then surely an indication of the desired ratio by number would be more
> appropriate?

Johnny Reinhard has a large amount of experience performing microtonal
music. He insists that the most helpful way of notating such music is
writing the deviations from 24-equal in cents. If I'm channelling him
correctly, his response to the "the ear will guide the performer" argument
is that the ear can't do any guiding until you've started to sing/play the
wrong note. But that time, it may be too late.

Some microtonal music is indeed intended to sound "out of tune" in a
specific way. For people who actually write 31-limit music, that's the
most likely intention.

Johnny's also been very dismissive of the kind of experiments you propose.
Although he welcomes a precision of 1 cent, he doesn't claim to reproduce
arbitrary scores with that level of accuracy. The performer's intentions
can't be reduced to a set of numbers. And perhaps they'll have different
intentions from one performance to another.

Ratios on scores have a very bad record in performance. Partch used them,
the scores are usually translated to some other system for modern
performance. The big problem is that the ratio doesn't strongly indicate
pitch height. You can work it out, but performers fairly obviously aren't
going to do that when sight reading. Compositions based on a single
harmonic series, like those of LaMonte Young and the European spectralist
school, are a niche where the numbers are more appropriate, and LaMonte
does use them. In this case, they are numbers rather than ratios, and so
are in pitch order. The compositional style also makes them the simplest
way of describing the music.

We'll have to see how this profusion of accidentals is received when it
reaches the big list. For now, it seems they're working out the best
system that fulfils a set of criteria. It's best to save outside comment
until the best such system's decided on, rather than a continuous stream
of comments on the working drafts.

> Some may ask why I am on this list. Because I had the temerity (as a
> musician and a composer) to write about a mathematical view on scales
> and
> tonality. But only from the viewpoint of a musician. I am still
> interested
> in all everyone has to say, but only if it has musical relevance.

If you're a musician, where's the music? I don't remember hearing any,
but perhaps I wasn't paying attention. Mine's out there for anybody to
chuckle over at <http://x31eq.com/music/>.

Graham

--- In tuning-math@y..., graham@m... wrote:

> What you originally said was "...it will show up as intervallically
> incoherent (as defined by Balzano)."

in other words, mark . . .

no one is saying your criteria are poor. maybe they're ultimately
the "best" (according to God, or whatever) possible criteria for
generalizing diatonicity -- that's not the issue. all we're trying to
point out is that you seem to have made an error in your statement
that any scale that doesn't follow a certain one of your rules "will
show up as intervallically incoherent (as defined by Balzano)."
Balzano's definition was clear, and whether the implication you put
forth is correct or not is a matter of logical analysis. it should be
possible to pursue such logical analysis without hurting one
another's feelings, shouldn't it? we all make mistakes sometimes!

ok, so maybe there were other issues some of us had with other
aspects of your paper. at one point i had a long philosophical
discussion with John Clough about the musical meaningfulness
of "pitch universe sets" with respect to which properties such as
maximal evenness are defined. it seems you might agree with clough
rather than with me. is there any harm in fleshing this out? surely
we can engage in heated debate over such ideas without getting
defensive about our musicianship, or casting aspersions about others'
unrelated endeavors on this list, or whatnot.

this shouldn't be a contest. this should be a bunch of people,
bouncing ideas off one another with open ears, correcting one
another's logical and mathematical errors, and above all, learning
from one another. personally, the last six years on these lists have
been an incredible voyage for me, particularly for the sharing and
cooperation that has taken place. i hope you might feel the same way
six years from now!

peace,
paul

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> ok, so maybe there were other issues some of us had with other
> aspects of your paper.

Where is this paper? Is there an on-line version?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> > ok, so maybe there were other issues some of us had with other
> > aspects of your paper.

> Where is this paper? Is there an on-line version?

I think I found it: Volume 38, Number 2 (Summer 2000) of Perspectives of New Music.

Graham wrote...

>The rules Paul gave in his paper are objective. The rules Rothenberg
>gave (propriety, efficiency and stability -- reading the original
>papers is not the ideal way of learning about them ;) are also
>objective. The rules you give in your paper are objective, and
>correctly applied.
>
>Carl's rules, of which we might soon have a discussion, aren't quite
>objective. He says things like "high efficiency" without putting a
>number to how high.

My rules aren't mine at all -- they were all created by other people,
chiefly Rothenberg and Erlich. If rules are objective in one place,
they're objective in another (at least for your definition of objective).
Rothenberg sorts all subsets of 12-et by stability. That's about as
trivial and non-special a definition of "high" as you can get. Erlich
accepts 6 out of 10 tetrads vs. 6 out of 7 triads. His rule is a
"majority", but he hasn't shown why > 50% has any special meaning. I
do the same.

>But perhaps he's putting forward objective rules, but allowing us to
>make subjective judgements about their relative importance, which
>sounds right to me.

http://lumma.org/gd.txt

The relative importance of the rules are given by their order in the
list, per thing I generalized. I generalized three things, having to
do with the encoding of information in:

() series of absolute pitches.

() series of relative pitches, in particular,
the ability to change the absolute pitches
but keep the relationships the same.

() the simultaneity of two or more series
that do one or both of the above.

Let's look at each:

() series of absolute pitches.

This is equivalent to the first item in the Legend, "Pitches of the
scale are trackable". To work with a group of things, we use short-
term, or working memory. According to George Miller, human working
memory can hold about 5 to 9 things. Properties 1 and 2 work together
so that the pitch set can fit in working memory.

I'll mention now that implicit in all of this is the assumption that
we want the scale to function as an autonomous thing. The diatonic
scale has interesting subsets, such as the pentatonic scale. But for
our purposes, if we were interested in the pent. scale, we would put
it through the list separately.

5-9 things says Miller, but what is a thing? In keeping with the
parallel approach of the brain, a thing can probably be almost
anything. We've said we want them to be pitches, but we allow pitches
an octave apart to be "chunked" into a single thing. Paul's idea
is that the 3:2 provides a weaker version of this, and because I have
found tetrachordal scales to be very singable, I allow it. It sounds
to me like there may be some chunking going on in arpeggiated triads...
In selecting 12-tone 7-limit scales for my piano, Paul Hahn searched
all rotations of the hexany 2-plex. I found the ones with chord
coverage to be more singable than ones without.

But in general no more than 9, 'cause you'll start dropping stuff, and
no fewer than 5, because that would too simple. You want to have to
work a little bit. Mind-expanding, dude.

Pitch tracking is the simplest form of melody. Rothenberg says any
scale can do it. Improper scales require a drone to measure things
from, and work best when they are not very efficient. Proper scales
can work too. I love pitch-tracking -- Persian chamber music is all
about it. It's clear there's more going on in diatonic music than
just pitch-tracking, though.

() series of relative pitches, in particular,
the ability to change the absolute pitches
but keep the relationships the same.

This is what I call modal transposition. You can play happy birthday
in the relative minor, and it's still happy birthday. This is what
propriety is all about -- when is it no longer happy birthday?
Rothenberg says it becomes not happy birthday rather suddenly with
respect to smooth, random changes in the pitches of the scale. Top
on my list of Experiments That Should Be Funded are the ones that R.
designed to test this.

Modal transposition is made up of items 2 and 3 in the Legend. Item
2, "Modes of the scale function independently". Efficiency is how
many notes of a scale you need to hear before you know what mode
you're in. When it's low, it's bad for mode independence, because
the composer has to work harder to get you to accept a New Scale
Degree #1, because you still know where the old #1 is. You want to
be a little lost in it. Similarly, humans can recognize consonant
intervals. If every mode contains a 5:4 as a type of 3rd, then when
you hear a 5:4, you know the :4 could be a New Scale Degree #1. Item
3 in the Legend is propriety, which I won't spill more ink about now.

Notice that I don't generalize things about tonal music, try to pick
which modes are tonal, etc., as Erlich has done. It's all fine and
good, but to insist on stuff like characteristic dissonances is to
put too much stock in the historical evolution of Western music, in
my opinion.

() the simultaneity of two or more series
that do one or both of the above.

This is the big one, which I call the Diatonic property, because it
weeds the large number of proper, efficient, 5-10 tone scales down
to very few indeed. Pay close attention.

Why are parallel fifths forbidden in common-practice harmony?
"They're not!" you may say, or certainly not by the time Jazz comes
around. But you don't often get parallel fifths in the melody, say
a vocal duet in a song, at least not for very long. My answer is:
because it sounds like a timbre, which isn't harmony at all.

If you harmonize in 6ths, however, you get consonance, but not
synthesis. I call it "series-breaking" or "timbre-breaking" harmony.
It's the last item in the Legend.

Could we sing one voice in the scale, and create a timbre-breaking
harmony for it _outside_ the scale? Yes, we could. And I'd love
us to. But this probably would interfere with following series
of absolute and relative pitches.

-Carl

G,

--- In tuning-math@y..., graham@m... wrote:
> Johnny's also been very dismissive of the kind of experiments you
> propose. Although he welcomes a precision of 1 cent, he doesn't
> claim to reproduce arbitrary scores with that level of accuracy.

Excuse me?! JR has not only claimed such, but purported to have a personal accuracy to tolerances *smaller* than 1 cent. And claimed to reproduce said accuracy, in pieces of music, on a regular basis.

Cheers,
Jon

Carl Lumma wrote:

> My rules aren't mine at all -- they were all created by other people,
> chiefly Rothenberg and Erlich. If rules are objective in one place,
> they're objective in another (at least for your definition of
> objective).

Rothenberg stability is an objective measure, but "highly
Rothenberg-stable" isn't an objective criterion. That's all I meant.

> http://lumma.org/gd.txt

> () series of absolute pitches.
>
> This is equivalent to the first item in the Legend, "Pitches of the
> scale are trackable". To work with a group of things, we use short-
> term, or working memory. According to George Miller, human working
> memory can hold about 5 to 9 things. Properties 1 and 2 work together
> so that the pitch set can fit in working memory.
>
> I'll mention now that implicit in all of this is the assumption that
> we want the scale to function as an autonomous thing. The diatonic
> scale has interesting subsets, such as the pentatonic scale. But for
> our purposes, if we were interested in the pent. scale, we would put
> it through the list separately.

Does it have to be a fixed set of notes, or are we allowed statistical
definitions?

> 5-9 things says Miller, but what is a thing? In keeping with the
> parallel approach of the brain, a thing can probably be almost
> anything. We've said we want them to be pitches, but we allow pitches
> an octave apart to be "chunked" into a single thing. Paul's idea
> is that the 3:2 provides a weaker version of this, and because I have
> found tetrachordal scales to be very singable, I allow it. It sounds
> to me like there may be some chunking going on in arpeggiated triads...
> In selecting 12-tone 7-limit scales for my piano, Paul Hahn searched
> all rotations of the hexany 2-plex. I found the ones with chord
> coverage to be more singable than ones without.

That all looks reasonable. I don't know if it's for pitch trackability or
not, but you certainly want chord coverage in a diatonic scale.

> But in general no more than 9, 'cause you'll start dropping stuff, and
> no fewer than 5, because that would too simple. You want to have to
> work a little bit. Mind-expanding, dude.

Except you're also allowing 10?

> Pitch tracking is the simplest form of melody. Rothenberg says any
> scale can do it. Improper scales require a drone to measure things
> from, and work best when they are not very efficient. Proper scales
> can work too. I love pitch-tracking -- Persian chamber music is all
> about it. It's clear there's more going on in diatonic music than
> just pitch-tracking, though.

You could also say that the flexibility in tuning of a pitch should be
smaller than the difference between pitches. So far, you're assuming the
tuning isn't flexible.

> () series of relative pitches, in particular,
> the ability to change the absolute pitches
> but keep the relationships the same.
>
> This is what I call modal transposition. You can play happy birthday
> in the relative minor, and it's still happy birthday. This is what
> propriety is all about -- when is it no longer happy birthday?
> Rothenberg says it becomes not happy birthday rather suddenly with
> respect to smooth, random changes in the pitches of the scale. Top
> on my list of Experiments That Should Be Funded are the ones that R.
> designed to test this.

And both stability definitions imply propriety. Is that right?

> Modal transposition is made up of items 2 and 3 in the Legend. Item
> 2, "Modes of the scale function independently". Efficiency is how
> many notes of a scale you need to hear before you know what mode
> you're in. When it's low, it's bad for mode independence, because
> the composer has to work harder to get you to accept a New Scale
> Degree #1, because you still know where the old #1 is. You want to
> be a little lost in it. Similarly, humans can recognize consonant
> intervals. If every mode contains a 5:4 as a type of 3rd, then when
> you hear a 5:4, you know the :4 could be a New Scale Degree #1. Item
> 3 in the Legend is propriety, which I won't spill more ink about now.

Do you have code for calculating efficiency? I never got the hang of it.
I'd rather use some measure involving the minimal sufficient sets. A
full efficiency calculation should include the probability of a note being
used in any given context. The minimal sufficient sets let you work
backwards to see what notes should be most common to establish the key.
I'd like to see some measure based on inspecting them.

The classic diatonic has lots of 3 note MSSs (any 3 notes including a
tritone) which means it's easy to establish the key when you want to. But
it also has two 6 note insufficient sets. That is, take out either note
not involving the tritone and you don't know what key you're in. That
means you can also leave the key ambiguous when you want to, which is good
for modulation. Any octave-based MOS will have such an n-1 not maximal
insufficient subset.

The "in a majority of a scale's modes" means it belongs to the majority of
intervals of that diatonic class, does it?

> Notice that I don't generalize things about tonal music, try to pick
> which modes are tonal, etc., as Erlich has done. It's all fine and
> good, but to insist on stuff like characteristic dissonances is to
> put too much stock in the historical evolution of Western music, in
> my opinion.

So you're really after modality rather than tonality?

> () the simultaneity of two or more series
> that do one or both of the above.
>
> This is the big one, which I call the Diatonic property, because it
> weeds the large number of proper, efficient, 5-10 tone scales down
> to very few indeed. Pay close attention.

I still don't understand what that means, in the light of the explanation.

> Why are parallel fifths forbidden in common-practice harmony?
> "They're not!" you may say, or certainly not by the time Jazz comes
> around. But you don't often get parallel fifths in the melody, say
> a vocal duet in a song, at least not for very long. My answer is:
> because it sounds like a timbre, which isn't harmony at all.
>
> If you harmonize in 6ths, however, you get consonance, but not
> synthesis. I call it "series-breaking" or "timbre-breaking" harmony.
> It's the last item in the Legend.

This is one thing Mark Gould's process leads to, when the alternating
intervals are consonant. But it's also stricter because it means all the
intervals have to be consonant, and they have to alternate. And it
doesn't have the criterion about the consonances not being ambiguous.

So his 11 from 41 note scale may be worth a look. It can be based on a
7:8:9 chord. It's outside the Miller limit, but I couldn't find anything
better in the "diatonics". There may be something in the "pentatonics".
The consonances of this class don't have to be the only ones adding up to
the criterion (5) consonance, do they? So the generalized fifth doesn't
have to be an odd number of diatonic scale steps.

> Could we sing one voice in the scale, and create a timbre-breaking
> harmony for it _outside_ the scale? Yes, we could. And I'd love
> us to. But this probably would interfere with following series
> of absolute and relative pitches.

But how about using contrapuntal harmonisation? This one looks like it
isn't so useful in that context.

Graham

Me:

> > Johnny's also been very dismissive of the kind of experiments you
> > propose. Although he welcomes a precision of 1 cent, he doesn't
> > claim to reproduce arbitrary scores with that level of accuracy.

Jon Szanto:
> Excuse me?! JR has not only claimed such, but purported to have a
> personal accuracy to tolerances *smaller* than 1 cent. And claimed to
> reproduce said accuracy, in pieces of music, on a regular basis.

I found an exchange on the big list, from the 8th of March. This bit
seems to be Carl Lumma:

> If we pretend the "holy grail" of auditory scene analysis, the
> "unmixer" (which may just have been realized by these guys:
> www.appliedneurosystems.com) exists, feed it a recording of say,
> a string quartet, apply pitch tracking to each part, and do a
> statistical analysis on the vertical relationships, I wager
> we'd find that they are centered around JI, not 72 or 31, EVEN
> IF THE MUSIC WAS NOTATED IN 72 OR 31, and NO MATTER HOW GOOD THE
> PERFORMERS ARE. This is basically the assumption behind my
> recent posts. I probably should have said this earlier. :)

And this is Johnny Reinhard:

"""
I really do not want to be insulting, but this is idiocy. Assumptions are
a
poor basis for preaching on the internet.
Feeding a recording proves nothing. It is a bit shocking that Carl does
not
understand this. For myself, I refuse to be reduced to Carl's
expectations.
He is now doing real harm regarding what people feel they should believe
about microtonal performance and he has practically no experience as a
performing musician. Maybe he tells his doctors how to diagnose his
ailments
rather than accept professional medical advice.
"""

And this from 10th June last year, in reply to you

"""
This has become funny, now. There is no measurement you will be able to
do
based on released music. Sorry it doesn't work that way. The onus is not
on
me to prove anything. And with this I stop discussing what was originally
a
question to me about notation. With notation I use 1 cent deviation and
have
explained why.
"""

If that isn't "very dismissive" of experiments based on recordings, then
sue me. He has claimed 1 cent accuracy, but not in an objective way,
following a score.

Graham

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> http://lumma.org/gd.txt

The rules on this page are fairly useless unless there are definitions to go with them. Maybe you and Monzo could get together and define Rothenberg efficient, and so forth?

--- In tuning-math@y..., graham@m... wrote:
> If that isn't "very dismissive" of experiments based on recordings,
> then sue me.

Hey, Graham, I agree with that completely! Dismissive it is! But...

> He has claimed 1 cent accuracy, but not in an objective way,
> following a score.

My reason for writing was that I remembered Johnny claiming
performance accuracy of < 1 cent, indeed! And he doesn't put up
anything more than "I say I can do it, and I do", so I'm not
suggesting it *is* accurate, only that he claims even finer degrees
of accuracy than you ascribed to him.

In scored music, since he and his groups do a lot of written
material, to boot. But he totally dispelled the notion of having any
acuracy in pitch measured which, after making such claims, put an end
to my interest in that thread.

Cheers,
Jon

>>But in general no more than 9, 'cause you'll start dropping stuff, and
>>no fewer than 5, because that would too simple. You want to have to
>>work a little bit. Mind-expanding, dude.
>
>Except you're also allowing 10?

Yes; my idea is that tetrachordality and chord coverage could buy you
something, but 11 just seems too many.

>> I'll mention now that implicit in all of this is the assumption that
>> we want the scale to function as an autonomous thing. The diatonic
>> scale has interesting subsets, such as the pentatonic scale. But for
>> our purposes, if we were interested in the pent. scale, we would put
>> it through the list separately.
>
>Does it have to be a fixed set of notes, or are we allowed statistical
>definitions?

The entire model assumes a fixed set of notes, but could probably be
successfully extended with statistical definitions. I don't see how
it would buy you anything when searching for scales, though.

>You could also say that the flexibility in tuning of a pitch should be
>smaller than the difference between pitches.

Sounds like a good idea. Lumma stability should catch some of this
where flexibility is defined as not breaking propriety. What did
you have in mind?

>>() series of relative pitches, in particular,
>>the ability to change the absolute pitches
>>but keep the relationships the same.
>>
>>This is what I call modal transposition. You can play happy birthday
>>in the relative minor, and it's still happy birthday. This is what
>>propriety is all about -- when is it no longer happy birthday?
>>Rothenberg says it becomes not happy birthday rather suddenly with
>>respect to smooth, random changes in the pitches of the scale. Top
>>on my list of Experiments That Should Be Funded are the ones that R.
>>designed to test this.
>
>And both stability definitions imply propriety. Is that right?

Yes.

>>Modal transposition is made up of items 2 and 3 in the Legend. Item
>>2, "Modes of the scale function independently". Efficiency is how
>>many notes of a scale you need to hear before you know what mode
>>you're in. When it's low, it's bad for mode independence, because
>>the composer has to work harder to get you to accept a New Scale
>>Degree #1, because you still know where the old #1 is. You want to
>>be a little lost in it. Similarly, humans can recognize consonant
>>intervals. If every mode contains a 5:4 as a type of 3rd, then when
>>you hear a 5:4, you know the :4 could be a New Scale Degree #1. Item
>>3 in the Legend is propriety, which I won't spill more ink about now.
>
>Do you have code for calculating efficiency? I never got the hang of
>it. I'd rather use some measure involving the minimal sufficient sets.

That is efficiency. R. claims to have trick of enumerating the sets
easily. Manuel might know more. I forget if I ever was able to get
him R.'s code, but he obviously can do it in scala.

>The classic diatonic has lots of 3 note MSSs (any 3 notes including a
>tritone) which means it's easy to establish the key when you want to.
>But it also has two 6 note insufficient sets. That is, take out either
>note not involving the tritone and you don't know what key you're in.
>That means you can also leave the key ambiguous when you want to, which
>is good for modulation.

Agree, if you meant "either note involving the tritone".

>The "in a majority of a scale's modes" means it belongs to the majority
>of intervals of that diatonic class, does it?
>
>> Notice that I don't generalize things about tonal music, try to pick
>> which modes are tonal, etc., as Erlich has done. It's all fine and
>> good, but to insist on stuff like characteristic dissonances is to
>> put too much stock in the historical evolution of Western music, in
>> my opinion.
>
>So you're really after modality rather than tonality?

Yes. Once you can do modality, I suspect tonality or something
equally interesting could follow in any number of ways, and I don't
want to cut anything good out by mistake. But it is possible I
just don't understand tonality.

>>() the simultaneity of two or more series
>>that do one or both of the above.
>>
>>This is the big one, which I call the Diatonic property, because it
>>weeds the large number of proper, efficient, 5-10 tone scales down
>>to very few indeed. Pay close attention.
>
>I still don't understand what that means, in the light of the
>explanation.

I might even say that I see diatonicity as making very precise the
difference between additive synthesis with fancy envelopes and music.
We change the fundamental to get melody. That's the first break.

If we want to play two melodies at once, we have to make sure the
virtual fundamentals don't move in parallel with one of them or the
voices will timbre-fuse. Power chords on electric guitars lead to
a melodic style, for example. One might say the diatonic scale
prevents the virtual fundamental mechanism from being used to chunk
the two voices, by scrambling the melody there in the time domain
a bit.

>>If you harmonize in 6ths, however, you get consonance, but not
>>synthesis. I call it "series-breaking" or "timbre-breaking" harmony.
>>It's the last item in the Legend.
>
>This is one thing Mark Gould's process leads to, when the alternating
>intervals are consonant. But it's also stricter because it means all
>the intervals have to be consonant, and they have to alternate.

Only the majority must be consonant, and they don't have to strictly
alternate -- just trade places once in a while, in some fashion.

>And it doesn't have the criterion about the consonances not being
>ambiguous.

In the propriety-breaking sense? I don't have Mark's paper, but
I just remembered Gene implying he found it on-line. Anyway, I
deal with ambiguities when determining the stability of the scale,
but can't think what they would hurt by occurring on the interval
class(es) responsible for the diatonic property.

>So his 11 from 41 note scale may be worth a look. It can be based
>on a 7:8:9 chord. It's outside the Miller limit, but I couldn't find
>anything better in the "diatonics". There may be something in the
>"pentatonics".

It's two big to claim it as diatonic, in my book. You're going to
hear melodies as a subset of the 41 whether you like it or not.
If some stable subsets have the diatonic property (unstable ones
wouldn't be the subsets you'd wind up hearing), then they would be
candidates. Else it goes into the category of making a diatonic
harmony with pitches outside the core melodic scale, which damages
something in my opinion, but is still interesting.

>The consonances of this class don't have to be the only ones adding
>up to the criterion (5) consonance, do they? So the generalized fifth
>doesn't have to be an odd number of diatonic scale steps.

God, no.

>>Could we sing one voice in the scale, and create a timbre-breaking
>>harmony for it _outside_ the scale? Yes, we could. And I'd love
>>us to. But this probably would interfere with following series
>>of absolute and relative pitches.
>
>But how about using contrapuntal harmonisation? This one looks like
>it isn't so useful in that context.

Either you're hearing a lot of the same relationship or you're not.
Try writing contrapuntal music in the wholetone scale and get back
to me.

-Carl

>> http://lumma.org/gd.txt
>
>The rules on this page are fairly useless unless there are definitions
>to go with them. Maybe you and Monzo could get together and define
>Rothenberg efficient, and so forth?

I thought he had. Anyway, turnabout is fair play when it comes to
posting things dependent on external definitions. :)

Message # 4044 on the big list may help.

-Carl

In-Reply-To: <4.2.2.20020414225815.01eae230@lumma.org>
Carl Lumma wrote:

> The entire model assumes a fixed set of notes, but could probably be
> successfully extended with statistical definitions. I don't see how
> it would buy you anything when searching for scales, though.

One way of interpreting the 7 from 10 MOS, with propriety grid

1 2 1 2 1 2 1
3 3 3 3 3 3 2
4 5 4 5 4 4 4
6 6 6 6 5 6 5
7 8 7 7 7 7 7
9 9 8 9 8 9 8

is that the 3 could be either 5:4 or 6:5. That means it counts as two
consonances although it only looks like one. You could say 7:6 is allowed
as a consonance, but only in the diatonic pitch class with 3. Or that 6:5
can be consonant, but only if it isn't in the root, like fourths are
treated in traditional counterpoint. I don't know how it's going to end
up, but other scales might do the same kind of things.

> >You could also say that the flexibility in tuning of a pitch should
> be >smaller than the difference between pitches.
>
> Sounds like a good idea. Lumma stability should catch some of this
> where flexibility is defined as not breaking propriety. What did
> you have in mind?

In the above example, allowing 5:4 and 6:5 to inhabit the same chromatic
pitch class would break that rule. But it still looks okay for propriety
-- in fact, it all the examples of fixed scales I've worked out are
strictly proper. So the statistical stability would increase.

> That is efficiency. R. claims to have trick of enumerating the sets
> easily. Manuel might know more. I forget if I ever was able to get
> him R.'s code, but he obviously can do it in scala.

How do you calculate it in Scala, for an ET subset?

> Yes. Once you can do modality, I suspect tonality or something
> equally interesting could follow in any number of ways, and I don't
> want to cut anything good out by mistake. But it is possible I
> just don't understand tonality.

I certainly don't understand tonality. But there may be ways of getting
it to work without all the modal criteria being fulfilled. Ambiguous
intervals look important.

> I might even say that I see diatonicity as making very precise the
> difference between additive synthesis with fancy envelopes and music.
> We change the fundamental to get melody. That's the first break.
>
> If we want to play two melodies at once, we have to make sure the
> virtual fundamentals don't move in parallel with one of them or the
> voices will timbre-fuse. Power chords on electric guitars lead to
> a melodic style, for example. One might say the diatonic scale
> prevents the virtual fundamental mechanism from being used to chunk
> the two voices, by scrambling the melody there in the time domain
> a bit.

I'm not sure virtual fundamentals should be brought into this. Terhardt
relates the traditional root to virtual pitch, so major and minor triads
come out the same.

> >And it doesn't have the criterion about the consonances not being
> >ambiguous.
>
> In the propriety-breaking sense? I don't have Mark's paper, but
> I just remembered Gene implying he found it on-line. Anyway, I
> deal with ambiguities when determining the stability of the scale,
> but can't think what they would hurt by occurring on the interval
> class(es) responsible for the diatonic property.

Gene implied he found it, but as he can get at a university library it
might have been there. I'm not familiar with Balzano, but it looks like
he does the same thing building scales with alternating intervals. What
Mark adds is the interpretation of those intervals as ratios.

If those consonances can be ambiguous, it does open up new ways of
interpreting the 7 from 10 scale, which is what I'm looking at now.

> >So his 11 from 41 note scale may be worth a look. It can be based
> >on a 7:8:9 chord. It's outside the Miller limit, but I couldn't find
> >anything better in the "diatonics". There may be something in the
> >"pentatonics".
>
> It's two big to claim it as diatonic, in my book. You're going to
> hear melodies as a subset of the 41 whether you like it or not.
> If some stable subsets have the diatonic property (unstable ones
> wouldn't be the subsets you'd wind up hearing), then they would be
> candidates. Else it goes into the category of making a diatonic
> harmony with pitches outside the core melodic scale, which damages
> something in my opinion, but is still interesting.

When I originally tried it out, after he mentioned it on the big list, I
found myself not using notes too far from the tonic. So it might break
down tetrachordally -- except it doesn't have tetrachords. The definition
of 9:8 is such that there can be no 4:3. So the almost-periodicity would
have to be about 9:7. Perhaps each part could stay in one such period.

4 3 4 4 4 3 4 4 4 3 4

4 3 4 4 is the "tetrachord". 4+4 4+3 is the 7:8:9. Another consonance is
4+4+3 as 6:5.

I can't find anything else promising by this method. 6:7:8 works with the
classic pentatonic. 7:11 is only an even number of steps in 3-equal,
below 43-equal. 5:7 gives a 9 from 11 scale, but it's nearly 50 cents
from 5:6:7 with the optimal tuning. The ideal defining chord is large and
simple enough to be comprehensible, and has internal intervals close
enough to be the same diatonic class, but also different enough to have a
good approximation in the chromatic.

> >The consonances of this class don't have to be the only ones adding
> >up to the criterion (5) consonance, do they? So the generalized fifth
> >doesn't have to be an odd number of diatonic scale steps.
>
> God, no.

In that case less equal chords, like 7:8:11 might work.

> >But how about using contrapuntal harmonisation? This one looks like
> >it isn't so useful in that context.
>
> Either you're hearing a lot of the same relationship or you're not.
> Try writing contrapuntal music in the wholetone scale and get back
> to me.

As long as we have a variety of consonances to hand, it shouldn't matter
if they're all the same diatonic interval class or not.

Graham

>> That is efficiency. R. claims to have trick of enumerating the sets
>> easily. Manuel might know more. I forget if I ever was able to get
>> him R.'s code, but he obviously can do it in scala.

>How do you calculate it in Scala, for an ET subset?

I haven't understood how Rothenberg calculated it for ET subsets.
And I was only interested in the general method, which I've
implemented. So for ET subsets the calculation can be done more
efficiently, perhaps not in all cases, but I don't do that.
The only exception is for Myhill scales, for which there's an
easy analytical result.

Manuel

>>The entire model assumes a fixed set of notes, but could probably be
>>successfully extended with statistical definitions. I don't see how
>>it would buy you anything when searching for scales, though.
>
>One way of interpreting the 7 from 10 MOS, with propriety grid
>
>1 2 1 2 1 2 1
>3 3 3 3 3 3 2
>4 5 4 5 4 4 4
>6 6 6 6 5 6 5
>7 8 7 7 7 7 7
>9 9 8 9 8 9 8
>
>is that the 3 could be either 5:4 or 6:5.

The interval between 5:4 and 6:5 is very high in harmonic
entropy. I don't think any single interval can reasonably
be said to serve as a 5:4 and 6:5. If you claim 11:9 is
consonant, I'd consider it.

>That means it counts as two consonances although it only looks
>like one.

Something has got to change, to change the context in which
the single interval is heard. Triads where the other interval
changes would count. Maybe even octave registration changes
could count.

>You could say 7:6 is allowed as a consonance,

And I do.

>but only in the diatonic pitch class with 3.

?

>Or that 6:5 can be consonant, but only if it isn't in the root, like
>fourths are treated in traditional counterpoint. I don't know how
>it's going to end up, but other scales might do the same kind of
>things.

Phooey!

>>>You could also say that the flexibility in tuning of a pitch should
>>>be smaller than the difference between pitches.
>>
>>Sounds like a good idea. Lumma stability should catch some of this
>>where flexibility is defined as not breaking propriety. What did
>>you have in mind?
>
>In the above example, allowing 5:4 and 6:5 to inhabit the same chromatic
>pitch class would break that rule. But it still looks okay for propriety
>-- in fact, it all the examples of fixed scales I've worked out are
>strictly proper. So the statistical stability would increase.

Oh, you mean to actually re-tune it one way or the other, not just
have it be re-interpreted in the Erlich sense. If you can do this
without breaking propriety, fine. But then there would be no point
in expressing the scale without the adjustments in the first place.

>>That is efficiency. R. claims to have trick of enumerating the sets
>>easily. Manuel might know more. I forget if I ever was able to get
>>him R.'s code, but he obviously can do it in scala.
>
>How do you calculate it in Scala, for an ET subset?

show data

>>I might even say that I see diatonicity as making very precise the
>>difference between additive synthesis with fancy envelopes and music.
>>We change the fundamental to get melody. That's the first break.
>>
>>If we want to play two melodies at once, we have to make sure the
>>virtual fundamentals don't move in parallel with one of them or the
>>voices will timbre-fuse. Power chords on electric guitars lead to
>>a melodic style, for example. One might say the diatonic scale
>>prevents the virtual fundamental mechanism from being used to chunk
>>the two voices, by scrambling the melody there in the time domain
>>a bit.
>
>I'm not sure virtual fundamentals should be brought into this.
>Terhardt relates the traditional root to virtual pitch, so major
>and minor triads come out the same.

The triads come out the same, because of the strength of the fifth,
and/or the 19:16 approximation of the minor third in 12-et. It
definitely has to do with the vf mechanism. I'm not sure if the
vf changing in the way I described is right.

>>>And it doesn't have the criterion about the consonances not being
>>>ambiguous.
>>
>>In the propriety-breaking sense? I don't have Mark's paper, but
>>I just remembered Gene implying he found it on-line. Anyway, I
>>deal with ambiguities when determining the stability of the scale,
>>but can't think what they would hurt by occurring on the interval
>>class(es) responsible for the diatonic property.
>
>Gene implied he found it, but as he can get at a university library it
>might have been there. I'm not familiar with Balzano, but it looks like
>he does the same thing building scales with alternating intervals. What
>Mark adds is the interpretation of those intervals as ratios.

That sounds good -- Balzano's alternating intervals are useless for
what I want to do because they're not consonant.

>If those consonances can be ambiguous, it does open up new ways of
>interpreting the 7 from 10 scale, which is what I'm looking at now.

You mean Rothenberg ambiguous, or ambiguous in the sense that they
may have more than one harmonic series representation?

>>It's two big to claim it as diatonic, in my book. You're going to
>>hear melodies as a subset of the 41 whether you like it or not.
>>If some stable subsets have the diatonic property (unstable ones
>>wouldn't be the subsets you'd wind up hearing), then they would be
>>candidates. Else it goes into the category of making a diatonic
>>harmony with pitches outside the core melodic scale, which damages
>>something in my opinion, but is still interesting.
>
>When I originally tried it out, after he mentioned it on the big list, I
>found myself not using notes too far from the tonic.

Any subsets in particular?

>>>The consonances of this class don't have to be the only ones adding
>>>up to the criterion (5) consonance, do they? So the generalized fifth
>>>doesn't have to be an odd number of diatonic scale steps.
>>
>> God, no.
>
>In that case less equal chords, like 7:8:11 might work.

You bet!

>>>But how about using contrapuntal harmonisation? This one looks like
>>>it isn't so useful in that context.
>>
>>Either you're hearing a lot of the same relationship or you're not.
>>Try writing contrapuntal music in the wholetone scale and get back
>>to me.
>
>As long as we have a variety of consonances to hand, it shouldn't matter
>if they're all the same diatonic interval class or not.

Wrong! That's what diatonicity is all about -- tying scale objects
to harmonic objects. There are a million ways to have lots of
different consonances, but very few to have them make sense in terms
of scale intervals.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> In the propriety-breaking sense? I don't have Mark's paper, but
> I just remembered Gene implying he found it on-line.

I found on-line how to find it in the library. I suspect the UC library has Perspectives on New Music, so you can find it also.

>>In the propriety-breaking sense? I don't have Mark's paper, but
>>I just remembered Gene implying he found it on-line.
>
>I found on-line how to find it in the library. I suspect the UC
>library has Perspectives on New Music, so you can find it also.

They do; I need to get over there.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >>The entire model assumes a fixed set of notes, but could
probably be
> >>successfully extended with statistical definitions. I don't see
how
> >>it would buy you anything when searching for scales, though.
> >
> >One way of interpreting the 7 from 10 MOS, with propriety grid
> >
> >1 2 1 2 1 2 1
> >3 3 3 3 3 3 2
> >4 5 4 5 4 4 4
> >6 6 6 6 5 6 5
> >7 8 7 7 7 7 7
> >9 9 8 9 8 9 8
> >
> >is that the 3 could be either 5:4 or 6:5.
>
> The interval between 5:4 and 6:5 is very high in harmonic
> entropy. I don't think any single interval can reasonably
> be said to serve as a 5:4 and 6:5.

if you use an s of 1.2% or 1.5% or so, 11:9 shows up as more
consonant than 9:7 -- but at 11:9 the two most likely interpretations
are '5:4' and '6:5'. this in fact seems to account for the interval
preferences of correspondents such as dan stearns and brian mclaren.

> Phooey!

does that 'phooey' apply to 22-equal scales, where the same interval
approximates both 7:5 and 10:7?

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> >So his 11 from 41 note scale may be worth a look. It can be based
> >on a 7:8:9 chord. It's outside the Miller limit, but I couldn't
find
> >anything better in the "diatonics". There may be something in the
> >"pentatonics".
>
> It's two big to claim it as diatonic, in my book. You're going to
> hear melodies as a subset of the 41 whether you like it or not.

this is where i depart from both mark and carl, maybe. i don't think
11 notes will be enough for the listener to extrapolate a 41-note
universe. and (directed to balzano and his followers), for similar
reasons, i don't think the scale's important properties should depend
in any way upon its tuning as a subset of 41. the scale should be
viewable in its own terms, comprising intervals with some allowed
ranges of values, etc. . . . the reference to a 41-universe, aside
from convenience, should not play any role in evaluating the scale's
suitability or unsuitability. (i make a similar argument about the
diatonic scale vis-a-vis the 12-tone "universe").

> >But how about using contrapuntal harmonisation? This one looks
like
> >it isn't so useful in that context.
>
> Either you're hearing a lot of the same relationship or you're not.
> Try writing contrapuntal music in the wholetone scale and get back
> to me.

the wholetone scale, whether used contrapuntally or not,
sounds "floaty" because there's no built-in preference for any tonal
center, or possibility of establishing such a preference through
some 'grammatical' style. however, over a drone, the wholetone scale
becomes just another mode, with about as many opportunities for
tension and release as your typical hindu raga.

not sure if that supports/disputes/is_tangential_to anything carl
said, just thought i'd throw that in there.

--- In tuning-math@y..., graham@m... wrote:

> Terhardt
> relates the traditional root to virtual pitch, so major and minor
triads
> come out the same.

they don't tend to come out the same in first inversion, though --
the perception of a first-inversion minor triad is very close to the
perception of the root-position major triad with two notes (bottom
major third) in common.

> > Either you're hearing a lot of the same relationship or you're
not.
> > Try writing contrapuntal music in the wholetone scale and get back
> > to me.
>
> As long as we have a variety of consonances to hand, it shouldn't
matter
> if they're all the same diatonic interval class or not.

it matters in tonal music. this is why the symmetrical decatonic
scale in 22-equal, which is an MOS (we now know), 'wants' to
be 'broken' into the non-MOS pentachordal decatonic.
omnitetrachordality implies that not only can one find one's place
within the octave, but that the correct placement of all the pitches
in the scale can be grasped in as simple and perceptually direct a
manner as possible.

>> The interval between 5:4 and 6:5 is very high in harmonic
>> entropy. I don't think any single interval can reasonably
>> be said to serve as a 5:4 and 6:5.
>
>if you use an s of 1.2% or 1.5% or so, 11:9 shows up as more
>consonant than 9:7 -- but at 11:9 the two most likely interpretations
>are '5:4' and '6:5'. this in fact seems to account for the interval
>preferences of correspondents such as dan stearns and brian mclaren.

What prefs are those?

>> Phooey!
>
>does that 'phooey' apply to 22-equal scales, where the same interval
>approximates both 7:5 and 10:7?

It means phooey to this being important:

>Or that 6:5 can be consonant, but only if it isn't in the root, like
>fourths are treated in traditional counterpoint. I don't know how
>it's going to end up, but other scales might do the same kind of
>things.

-Carl

>>It's two big to claim it as diatonic, in my book. You're going to
>>hear melodies as a subset of the 41 whether you like it or not.
>
>this is where i depart from both mark and carl, maybe. i don't think
>11 notes will be enough for the listener to extrapolate a 41-note
>universe. and (directed to balzano and his followers), for similar
>reasons, i don't think the scale's important properties should depend
>in any way upon its tuning as a subset of 41. the scale should be
>viewable in its own terms, comprising intervals with some allowed
>ranges of values, etc. . . . the reference to a 41-universe, aside
>from convenience, should not play any role in evaluating the scale's
>suitability or unsuitability. (i make a similar argument about the
>diatonic scale vis-a-vis the 12-tone "universe").

That doesn't depart from me at all. Did you see:

>I'll mention now that implicit in all of this is the assumption that
>we want the scale to function as an autonomous thing. The diatonic
>scale has interesting subsets, such as the pentatonic scale. But for
>our purposes, if we were interested in the pent. scale, we would put
>it through the list separately.

I see I said "hear melodies as a subset of the 41", which isn't
what I meant. I meant "the melodies you hear will be in subsets
of the 41; you won't know there are 41.".

>the wholetone scale, whether used contrapuntally or not,
>sounds "floaty" because there's no built-in preference for any tonal
>center, or possibility of establishing such a preference through
>some 'grammatical' style. however, over a drone, the wholetone scale
>becomes just another mode, with about as many opportunities for
>tension and release as your typical hindu raga.

Agree 100%.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >> The interval between 5:4 and 6:5 is very high in harmonic
> >> entropy. I don't think any single interval can reasonably
> >> be said to serve as a 5:4 and 6:5.
> >
> >if you use an s of 1.2% or 1.5% or so, 11:9 shows up as more
> >consonant than 9:7 -- but at 11:9 the two most likely
interpretations
> >are '5:4' and '6:5'. this in fact seems to account for the
interval
> >preferences of correspondents such as dan stearns and brian
mclaren.
>
> What prefs are those?

considering 11:9 as more consonant than any ratio n:d where n*d>34 or
so.

>
> >> Phooey!
> >
> >does that 'phooey' apply to 22-equal scales, where the same
interval
> >approximates both 7:5 and 10:7?
>
> It means phooey to this being important:
>
> >Or that 6:5 can be consonant, but only if it isn't in the root,
like
> >fourths are treated in traditional counterpoint. I don't know how
> >it's going to end up, but other scales might do the same kind of
> >things.

hmm . . . this seems to agree with some of your own comments, that
whether 5:4 or 6:5 is implied can depend on chordal context or
such . . . no?

>>It means phooey to this being important:
>>
>>>Or that 6:5 can be consonant, but only if it isn't in the root,
>>>like fourths are treated in traditional counterpoint. I don't
>>>know how it's going to end up, but other scales might do the same
>>>kind of things.
>
>hmm . . . this seems to agree with some of your own comments, that
>whether 5:4 or 6:5 is implied can depend on chordal context or
>such . . . no?

Maybe I'm not reading Graham correctly here.

-C.

>the wholetone scale, whether used contrapuntally or not,
>sounds "floaty" because there's no built-in preference for any tonal
>center, or possibility of establishing such a preference through
>some 'grammatical' style. however, over a drone, the wholetone scale
>becomes just another mode, with about as many opportunities for
>tension and release as your typical hindu raga.

Here's one attempt at contrapuntal music in the wholetone scale:

http://www.sibeliusmusic.com/cgi-bin/showscore.pl?scoreid=6331

-Carl

In-Reply-To: <a9fiuo+lh4v@eGroups.com>
Me:
> > Terhardt
> > relates the traditional root to virtual pitch, so major and minor
> triads
> > come out the same.

Paul:
> they don't tend to come out the same in first inversion, though --
> the perception of a first-inversion minor triad is very close to the
> perception of the root-position major triad with two notes (bottom
> major third) in common.

The root finding algorithm given in
<http://www.mmk.ei.tum.de/persons/ter/top/harmony.html> is clearly
octave-equivalent. He actually says "by definition, octave-equivalent
pitches are not distinguished".

Me:
> > As long as we have a variety of consonances to hand, it shouldn't
> matter
> > if they're all the same diatonic interval class or not.

Paul:
> it matters in tonal music. this is why the symmetrical decatonic
> scale in 22-equal, which is an MOS (we now know), 'wants' to
> be 'broken' into the non-MOS pentachordal decatonic.
> omnitetrachordality implies that not only can one find one's place
> within the octave, but that the correct placement of all the pitches
> in the scale can be grasped in as simple and perceptually direct a
> manner as possible.

That's an argument for a tetrachordality, not one diatonic interval class
having all the consonance.

Given that special casing fourths is phooey, that means Palestrina
counterpoint recognises 5 diatonic consonances within the octave --
unisons, thirds, fourths, fifths and sixths. Why does pitch placement go
to the four winds when you increase it to 7?

Graham

In-Reply-To: <4.2.2.20020415110352.01eac2e8@lumma.org>
Carl Lumma wrote:

> >>The entire model assumes a fixed set of notes, but could probably be
> >>successfully extended with statistical definitions. I don't see how
> >>it would buy you anything when searching for scales, though.
> >
> >One way of interpreting the 7 from 10 MOS, with propriety grid
> >
> >1 2 1 2 1 2 1
> >3 3 3 3 3 3 2
> >4 5 4 5 4 4 4
> >6 6 6 6 5 6 5
> >7 8 7 7 7 7 7
> >9 9 8 9 8 9 8
> >
> >is that the 3 could be either 5:4 or 6:5.
>
> The interval between 5:4 and 6:5 is very high in harmonic
> entropy. I don't think any single interval can reasonably
> be said to serve as a 5:4 and 6:5. If you claim 11:9 is
> consonant, I'd consider it.

If 11:9 is consonant, so is almost everything else, so there's no way (5)
can be fulfilled.

> Something has got to change, to change the context in which
> the single interval is heard. Triads where the other interval
> changes would count. Maybe even octave registration changes
> could count.

Yes, the tuning changes.

> >You could say 7:6 is allowed as a consonance,
>
> And I do.

Well, not in another message you don't. Still the problem is that if 7:6
counts, so should 7:5 and 10:7 and we lose (5) again.

> >but only in the diatonic pitch class with 3.
>
> ?

Back to that grid

1 2 1 2 1 2 1
3 3 3 3 3 3 2
4 5 4 5 4 4 4
6 6 6 6 5 6 5
7 8 7 7 7 7 7
>9 9 8 9 8 9 8

In the top row, 2 could be 9:8 or 8:7, and in the second row it could be
8:7 or 7:6. So that means 2 and 3 are both consonances, and 2/7 is the
only interval class where 2/10 is this consonance. It also runs into
problems when intervals add up, hence:

> >Or that 6:5 can be consonant, but only if it isn't in the root, like
> >fourths are treated in traditional counterpoint. I don't know how
> >it's going to end up, but other scales might do the same kind of
> >things.
>
> Phooey!

If major thirds and perfect fifths are both legal consonances, why not put
them together?

> Oh, you mean to actually re-tune it one way or the other, not just
> have it be re-interpreted in the Erlich sense. If you can do this
> without breaking propriety, fine. But then there would be no point
> in expressing the scale without the adjustments in the first place.

Yes. There are problems in getting thirds and fourths to work together.
The scale also ends up strictly proper, and I don't want that. The main
reason is that it comes out distinctly unexciting:

0 1 3v 4^ 6 7 9v 0
3 5 5 5 3 5 5
4 7 6 7 4 7 6

The propriety grids are like this:

3 5 5 5 3 5 5
8 10 10 8 8 10 8
13 15 13 13 13 13 13
18 18 18 18 16 18 18
21 23 23 21 21 23 23
26 28 26 26 26 28 26

4 7 6 7 4 7 6
11 13 13 11 11 13 10
17 20 17 18 17 17 17
24 24 24 24 21 24 23
28 31 30 28 28 30 30
35 37 34 35 34 37 34

so it's strictly proper. I don't have Scala to hand, so I can't work out
the efficiency. But it's similar to, and will probably be heard as,
12-equal melodic minor. Rothenberg gives an efficiency of 63% for that.
Why isn't it on your list?

> >If those consonances can be ambiguous, it does open up new ways of
> >interpreting the 7 from 10 scale, which is what I'm looking at now.
>
> You mean Rothenberg ambiguous, or ambiguous in the sense that they
> may have more than one harmonic series representation?

Rothenberg ambiguous.

> >When I originally tried it out, after he mentioned it on the big
> list, I >found myself not using notes too far from the tonic.
>
> Any subsets in particular?

The notes nearest the tonic. One down, four up, or something.

> >In that case less equal chords, like 7:8:11 might work.
>
> You bet!

It should be possible to automate the search. I'll think about it.

> >As long as we have a variety of consonances to hand, it shouldn't
> matter >if they're all the same diatonic interval class or not.
>
> Wrong! That's what diatonicity is all about -- tying scale objects
> to harmonic objects. There are a million ways to have lots of
> different consonances, but very few to have them make sense in terms
> of scale intervals.

But isn't that what stability gives us?

Graham

In-Reply-To: <a9fife+lpo6@eGroups.com>
emotionaljourney22 wrote:

> this is where i depart from both mark and carl, maybe. i don't think
> 11 notes will be enough for the listener to extrapolate a 41-note
> universe. and (directed to balzano and his followers), for similar
> reasons, i don't think the scale's important properties should depend
> in any way upon its tuning as a subset of 41. the scale should be
> viewable in its own terms, comprising intervals with some allowed
> ranges of values, etc. . . . the reference to a 41-universe, aside
> from convenience, should not play any role in evaluating the scale's
> suitability or unsuitability. (i make a similar argument about the
> diatonic scale vis-a-vis the 12-tone "universe").

Where is Mark these days? The 11 from 41 should work fine so long as you
can distinguish 3 and 4 steps. You can temper it a bit to improve that.
It'd actually work in 19-equal if you turn

4 3 4 4 4 3 4 4 4 3 4

into

2 1 2 2 2 1 2 2 2 1 2

Now, I'm wondering how

7 4 4 7 4 4 7 4

or

3 2 2 3 2 2 3 2

would do. Here's the grid

7 4 4 7 4 4 7 4
11 8 11 11 8 11 11 11
15 15 15 15 15 15 18 15
22 19 19 22 19 22 22 19
26 23 26 26 26 26 26 26
30 30 30 33 30 30 33 30
37 34 37 37 34 37 37 34

9:7 is still the primary consonance and the secondaries are 6:5 and 8:7.
There isn't a 9-limit consonance at 18 steps from 41 is there? Oh,
0-11-15 doesn't work unless you allow 16:15 and 15:14.

The grid in 19=

3 2 2 3 2 2 3 2
5 4 5 5 4 5 5 5
7 7 7 7 7 7 8 7
10 9 9 10 9 10 10 9
12 11 12 12 12 12 12 12
14 14 14 15 14 14 15 14
17 16 17 17 16 17 17 16

Graham

In-Reply-To: <memo.579417@cix.compulink.co.uk>
I wrote:

> so it's strictly proper. I don't have Scala to hand, so I can't work
> out the efficiency. But it's similar to, and will probably be heard
> as, 12-equal melodic minor. Rothenberg gives an efficiency of 63% for
> that. Why isn't it on your list?

It's even worse! It's the usual diatonic, at least in 31-equal. But
still, why isn't melodic minor on the list?

Graham

>> The interval between 5:4 and 6:5 is very high in harmonic
>> entropy. I don't think any single interval can reasonably
>> be said to serve as a 5:4 and 6:5. If you claim 11:9 is
>> consonant, I'd consider it.
>
>If 11:9 is consonant, so is almost everything else, so there's no
>way (5) can be fulfilled.

We agree, then, that 11:9 is not really consonant by itself.

>>>You could say 7:6 is allowed as a consonance,
>>
>>And I do.
>
>Well, not in another message you don't. Still the problem is that if 7:6
>counts, so should 7:5 and 10:7 and we lose (5) again.

I generally think of 7:6, 7:5 as barely strong enough, and 10:7 too weak,
to be considered consonant dyads.

Which scale are you talking about loosing (5) in?

>>>but only in the diatonic pitch class with 3.
>>
>> ?
>
>Back to that grid
>
> 1 2 1 2 1 2 1
> 3 3 3 3 3 3 2
> 4 5 4 5 4 4 4
> 6 6 6 6 5 6 5
> 7 8 7 7 7 7 7
> 9 9 8 9 8 9 8
>
>In the top row, 2 could be 9:8 or 8:7, and in the second row it could be
>8:7 or 7:6.

Granted for the sake of argument...

>So that means 2 and 3 are both consonances, and 2/7 is the
>only interval class where 2/10 is this consonance.

??

>It also runs into problems when intervals add up, hence:
>
>>>Or that 6:5 can be consonant, but only if it isn't in the root, like
>>>fourths are treated in traditional counterpoint. I don't know how
>>>it's going to end up, but other scales might do the same kind of
>>>things.
>>
>> Phooey!
>
>If major thirds and perfect fifths are both legal consonances, why not
>put them together?

You've lost me completely.

>>Oh, you mean to actually re-tune it one way or the other, not just
>>have it be re-interpreted in the Erlich sense. If you can do this
>>without breaking propriety, fine. But then there would be no point
>>in expressing the scale without the adjustments in the first place.
>
>Yes. There are problems in getting thirds and fourths to work together.
>The scale also ends up strictly proper, and I don't want that. The
>main reason is that it comes out distinctly unexciting:
>
>0 1 3v 4^ 6 7 9v 0
> 3 5 5 5 3 5 5
> 4 7 6 7 4 7 6
>
>The propriety grids are like this:
>
> 3 5 5 5 3 5 5
> 8 10 10 8 8 10 8
>13 15 13 13 13 13 13
>18 18 18 18 16 18 18
>21 23 23 21 21 23 23
>26 28 26 26 26 28 26

Isn't this just the diatonic scale?

> 4 7 6 7 4 7 6
>11 13 13 11 11 13 10
>17 20 17 18 17 17 17
>24 24 24 24 21 24 23
>28 31 30 28 28 30 30
>35 37 34 35 34 37 34

Doesn't look very consonant.

>so it's strictly proper. I don't have Scala to hand, so I can't work out
>the efficiency. But it's similar to, and will probably be heard as,
>12-equal melodic minor. Rothenberg gives an efficiency of 63% for that.
>Why isn't it on your list?

The Harmonic and Hungarian minors should be in there too. I'll add
them.

>>>If those consonances can be ambiguous, it does open up new ways of
>>>interpreting the 7 from 10 scale, which is what I'm looking at now.
>>
>> You mean Rothenberg ambiguous, or ambiguous in the sense that they
>> may have more than one harmonic series representation?
>
>Rothenberg ambiguous.

I'll be anxious to see the results.

>> >As long as we have a variety of consonances to hand, it shouldn't
>> matter >if they're all the same diatonic interval class or not.
>>
>> Wrong! That's what diatonicity is all about -- tying scale objects
>> to harmonic objects. There are a million ways to have lots of
>> different consonances, but very few to have them make sense in terms
>> of scale intervals.
>
>But isn't that what stability gives us?

No, stability ties acoustic objects with scales objects. Diatonicity
restricts things further to acoustic objects which are harmonious.

-Carl

In-Reply-To: <4.2.2.20020416105902.00a9ad20@lumma.org>
Carl Lumma wrote:

> I generally think of 7:6, 7:5 as barely strong enough, and 10:7 too
> weak,
> to be considered consonant dyads.
>
> Which scale are you talking about loosing (5) in?

The 7 from decimal. The two tritones are the characteristic dissonances
in the fourth/fifth category. There could always be a statistical
definition where only two of the 6-steps are considered to be 3:2 fifths,
but not always the same two. But the same would then have to apply to the
3-steps being 5:4.

> > 1 2 1 2 1 2 1
> > 3 3 3 3 3 3 2
> > 4 5 4 5 4 4 4
> > 6 6 6 6 5 6 5
> > 7 8 7 7 7 7 7
> > 9 9 8 9 8 9 8
> >
> >In the top row, 2 could be 9:8 or 8:7, and in the second row it could
> be >8:7 or 7:6.
>
> Granted for the sake of argument...
>
> >So that means 2 and 3 are both consonances, and 2/7 is the
> >only interval class where 2/10 is this consonance.
>
> ??

You said one interval class has to contain two consonances that aren't
consonant anywhere else. So, the thirds here could be either 5:4 or 7:6.
6:5 isn't allowed because it's too close to 7:6, which is a different
interval class, and you lose tonalness. 8:7 isn't allowed because it's
ambiguous with the seconds.

This is actually fairly reasonable in practice. The 2-step third will
tend to be larger than the 2-steps second for melodic reasons. This will
increase the statistical stability, but still leave the ambiguity where
you need it.

> >If major thirds and perfect fifths are both legal consonances, why not
> >put them together?
>
> You've lost me completely.

I've decided above that 5:4 and 3:2 are the consonances. Tertian harmony
does seem to work best for 7 note diatonicity. However, put them together
and you see a 6:5 as well. With a fixed decimal scale this wouldn't
happen. With flexible notes, the 6:5 pulls against the decimalness of the
scale, because it's a diminished interval. It doesn't affect propriety,
but should still be avoided melodically. In harmony, you have to make
sure it isn't too obvious. So, like fourths in Common Practice harmony,
don't put it in the bass.

The 7:9 in a 6:7:9 chord is even worse, because it isn't even a kind of
third. But 4:6:7 looks okay, shame it only exists in one place. Although
that is useful for cadences.

> >The propriety grids are like this:
> >
> > 3 5 5 5 3 5 5
> > 8 10 10 8 8 10 8
> >13 15 13 13 13 13 13
> >18 18 18 18 16 18 18
> >21 23 23 21 21 23 23
> >26 28 26 26 26 28 26
>
> Isn't this just the diatonic scale?

Yes, but not the just diatonic scale.

> > 4 7 6 7 4 7 6
> >11 13 13 11 11 13 10
> >17 20 17 18 17 17 17
> >24 24 24 24 21 24 23
> >28 31 30 28 28 30 30
> >35 37 34 35 34 37 34
>
> Doesn't look very consonant.

It's consonant enough.

I've found quite a few alternatives. They all have something wrong with
them. None have high efficiency. The advantage of using 10 fuzzy pitch
classes is that you can have all of them at once, and maintain the sense
of a 10 note chromatic. Also that you have all those 11-limit intervals
when you want them.

> >>>If those consonances can be ambiguous, it does open up new ways of
> >>>interpreting the 7 from 10 scale, which is what I'm looking at now.
> >>
> >> You mean Rothenberg ambiguous, or ambiguous in the sense that they
> >> may have more than one harmonic series representation?
> >
> >Rothenberg ambiguous.
>
> I'll be anxious to see the results.

If we allow 7-limit consonance, thirds and fourths become the intervals
with two kinds of consonance. That means 7:8 could be single consonance
for seconds, unfortunately it isn't in the majority. But now you've
dropped the requirement for a characteristic dissonance, that isn't a
problem. However, the way you've worded (2b) means 7:8, 5:7 and 7:10
still drop out because they're ambiguous.

> >> Wrong! That's what diatonicity is all about -- tying scale objects
> >> to harmonic objects. There are a million ways to have lots of
> >> different consonances, but very few to have them make sense in terms
> >> of scale intervals.
> >
> >But isn't that what stability gives us?
>
> No, stability ties acoustic objects with scales objects. Diatonicity
> restricts things further to acoustic objects which are harmonious.

Why are you assuming harmonious acoustic objects can't be ambiguous? The
tritone plays an important part in tonal harmony because it's rare and
ambiguous. Why can't a consonance take on that role, and dissonances tie
the harmony to the scale?

Graham

--- In tuning-math@y..., graham@m... wrote:

> Me:
> > > As long as we have a variety of consonances to hand, it
shouldn't
> > matter
> > > if they're all the same diatonic interval class or not.
>
> Paul:
> > it matters in tonal music. this is why the symmetrical decatonic
> > scale in 22-equal, which is an MOS (we now know), 'wants' to
> > be 'broken' into the non-MOS pentachordal decatonic.
> > omnitetrachordality implies that not only can one find one's
place
> > within the octave, but that the correct placement of all the
pitches
> > in the scale can be grasped in as simple and perceptually direct
a
> > manner as possible.
>
> That's an argument for a tetrachordality, not one diatonic interval
class
> having all the consonance.

sorry, i thought you were talking about the whole-tone scale. can you
refresh my memory as to what it is you were really talking about?

> Given that special casing fourths is phooey, that means Palestrina
> counterpoint recognises 5 diatonic consonances within the octave --
> unisons, thirds, fourths, fifths and sixths. Why does pitch
placement go
> to the four winds when you increase it to 7?

i've lost you completely. what goes to the four winds? sorry --
please start from the beginning, and remind me what it is we were
discussing here.

This has been about 10 hours without appearing, so I'm posting again.

-C.

>>> 1 2 1 2 1 2 1
>>> 3 3 3 3 3 3 2
>>> 4 5 4 5 4 4 4
>>> 6 6 6 6 5 6 5
>>> 7 8 7 7 7 7 7
>>> 9 9 8 9 8 9 8
>>>
>>>In the top row, 2 could be 9:8 or 8:7, and in the second row it could
>>>be 8:7 or 7:6.
>>
>>Granted for the sake of argument...
>
>>>So that means 2 and 3 are both consonances, and 2/7 is the
>>>only interval class where 2/10 is this consonance.
>>
>> ??
>
>You said one interval class has to contain two consonances that aren't
>consonant anywhere else.

I did, but I've changed this. Only the 2b interval class must still
be un-ambiguous, since this is supposed to aid mode recognition. The
"diatonic" interval class can now be ambiguous.

I'm also in the process of fixing things so the 3rds here wouldn't
qualify as the "diatonic" interval class because there's only one
instance of the other interval. The 3rds= 5:4 are instead a bet
for 2b, along with 5ths= 3:2.

The best bets for a diatonic interval class here would be 4ths=
4:3 or 7:5, and 7ths= 7:4 or 15:8. If you want to make triads,
I would think 1-3-7= 4:5:7 or 4:5:15 would be your best bet.

>>> 4 7 6 7 4 7 6
>>>11 13 13 11 11 13 10
>>>17 20 17 18 17 17 17
>>>24 24 24 24 21 24 23
>>>28 31 30 28 28 30 30
>>>35 37 34 35 34 37 34
>>
>> Doesn't look very consonant.
>
>It's consonant enough.

I'll put it on my list of scales to play.

>>>>That's what diatonicity is all about -- tying scale objects
>>>>to harmonic objects. There are a million ways to have lots of
>>>>different consonances, but very few to have them make sense in
>>>>terms of scale intervals.
>>>
>>>But isn't that what stability gives us?
>>
>>No, stability ties acoustic objects with scales objects. Diatonicity
>>restricts things further to acoustic objects which are harmonious.
>
>Why are you assuming harmonious acoustic objects can't be ambiguous?

I'm not! I just meant stability is ignorant of harmony. I said "ties
scale objects to harmonic objects". You said stability does that. It
does, but it doesn't guarantee it, since there are plenty of dissonant,
stable scales.

I still want the 2b interval to be un-ambiguous, because when you
hear the interval, you're supposed to know right away what scale
position it's in -- that's what 2b is for.

>Why can't a consonance take on that role, and dissonances tie
>the harmony to the scale?

Because you can't tell dissonances apart from eachother, in the
same way that you can tell 5:4, 6:5 apart from eachother.

-Carl

In-Reply-To: <4.2.2.20020417084504.01e58740@lumma.org>
Carl Lumma wrote:

> The best bets for a diatonic interval class here would be 4ths=
> 4:3 or 7:5, and 7ths= 7:4 or 15:8. If you want to make triads,
> I would think 1-3-7= 4:5:7 or 4:5:15 would be your best bet.

There can't be a 2b interval that way. Thirds are the obvious one, but
it's a fudge to expect enough of them to be 5:4. Neutral thirds work in
another context, but we don't need the miracle approximations then. When
the characteristic dissonances were being enforced, the 8:7 or 7:6
couldn't be it in a 7-limit context. So there are two different diatonic
intervals classes in the 7-limit, but no 2b.

> I still want the 2b interval to be un-ambiguous, because when you
> hear the interval, you're supposed to know right away what scale
> position it's in -- that's what 2b is for.

That sounds reasonable. I'd like it to be separate points:

A consonance exists
- in the majority of instances of a diatonic interval class
- in a diatonic interval class with no other consonances
- in no other diatonic interval classes
- and is strong

So if a scale fails on one of them, it only loses one point. I think in
general each criterion should only enforce one property.

> >Why can't a consonance take on that role, and dissonances tie
> >the harmony to the scale?
>
> Because you can't tell dissonances apart from eachother, in the
> same way that you can tell 5:4, 6:5 apart from eachother.

That looks like the heart of the issue. I find it much easier to tell two
dissonances apart than a dissonance from a consonance. With my schismic
keyboard setup I found it remarkably difficult to distinguish 5:4 and 6:5
what with them both being well tuned consonances and not far apart.

Graham

In-Reply-To: <a9klsv+hjop@eGroups.com>
emotionaljourney22 wrote:

> sorry, i thought you were talking about the whole-tone scale. can you
> refresh my memory as to what it is you were really talking about?

It's the generalisation of what thirds do in a diatonic scale. Carl says
a particular diatonic interval class should have two distinct consonances.
There are other criteria like the majority of intervals in that class
should be consonant, that there have to be more than one of each, and that
the same consonances can't occur in other interval classes, although that
one's been dropped.

I say none of this matters so long as the scale has a variety of
consonances that work with the primary consonance. Carl brought up the
whole tone scale but I don't think it's a good one because that doesn't
have the variety of consonances to start with.

Graham

>>The best bets for a diatonic interval class here would be 4ths=
>>4:3 or 7:5, and 7ths= 7:4 or 15:8. If you want to make triads,
>>I would think 1-3-7= 4:5:7 or 4:5:15 would be your best bet.
>
>There can't be a 2b interval that way. Thirds are the obvious one,
>but it's a fudge to expect enough of them to be 5:4.

Don't look at me -- I said the scale fails. I'm just trying to
show how it might work.

>>I still want the 2b interval to be un-ambiguous, because when you
>>hear the interval, you're supposed to know right away what scale
>>position it's in -- that's what 2b is for.
>
>That sounds reasonable. I'd like it to be separate points:
>
>A consonance exists
> - in the majority of instances of a diatonic interval class
> - in a diatonic interval class with no other consonances
> - in no other diatonic interval classes
> - and is strong
>
>So if a scale fails on one of them, it only loses one point. I
>think in general each criterion should only enforce one property.

Note "diatonic interval class" should just be "interval class"
here.

My new scheme keeps the last two, but does away with the second,
and turns the first from a binary to more continuous value:

| The ratio f/k, where f is the number of modes having a strong
| consonance, which appears in only one interval class throughout
| the scale, and forms a consonant triad with the equivalence
| interval.

>>>Why can't a consonance take on that role, and dissonances tie
>>>the harmony to the scale?
>>
>>Because you can't tell dissonances apart from eachother, in the
>>same way that you can tell 5:4, 6:5 apart from eachother.
>
>That looks like the heart of the issue. I find it much easier to tell
>two dissonances apart than a dissonance from a consonance.

You do? Have you got that backward? Anyway, that isn't the question.
The question is: if I randomly play you either 6:5 or 5:4, harmonically,
and ask you to identify them, would you perform better than if I had
used 11:9 and 9:7?

>With my schismic keyboard setup I found it remarkably difficult to
>distinguish 5:4 and 6:5 what with them both being well tuned
>consonances and not far apart.

Really? Maybe I just find consonances easier to recognize because
I've trained myself to do it... maybe it isn't innate.

-Carl

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a9klsv+hjop@e...>
> emotionaljourney22 wrote:
>
> > sorry, i thought you were talking about the whole-tone scale. can
you
> > refresh my memory as to what it is you were really talking about?
>
> It's the generalisation of what thirds do in a diatonic scale.
Carl says
> a particular diatonic interval class should have two distinct
consonances.
> There are other criteria like the majority of intervals in that
class
> should be consonant, that there have to be more than one of each,
and that
> the same consonances can't occur in other interval classes,
although that
> one's been dropped.
>
> I say none of this matters so long as the scale has a variety of
> consonances that work with the primary consonance. Carl brought up
the
> whole tone scale but I don't think it's a good one because that
doesn't
> have the variety of consonances to start with.

ok . . . well my opinion is that tonality is influenced in a very
strong way by the dissonances that occur in sometimes-consonant
interval classes. there are the characteristic dissonances,
the 'signposts' of tonality. tonal music does a remarkable job of
convincing the ear that such intervals 'make sense' (or 'sound
beautiful'), despite their dissonance, by structuring the important
resolving progressions around them -- and by making it impossible for
them to occur any time a melody occurs over the tonic chord.

In-Reply-To: <4.2.2.20020418115051.01e58740@lumma.org>
Me:
> >That looks like the heart of the issue. I find it much easier to tell
> >two dissonances apart than a dissonance from a consonance.

Carl:
> You do? Have you got that backward? Anyway, that isn't the question.
> The question is: if I randomly play you either 6:5 or 5:4, harmonically,
> and ask you to identify them, would you perform better than if I had
> used 11:9 and 9:7?

Um, yes, wrong way round. I don't have very good relative pitch at all.
The real test would be 9:7 and 11:8, because they're close enough to be
confused. The easiest way of telling intervals apart is by how consonant
they are, so I should easily be able to differentiate 5:4 from either 11:9
or 9:7 if they're tuned well enough.

I'm sure I can hear the in-tune-ness of 11:8 as well, in the right
circumstances. So an interval class containing both 9- and 11-limit
"consonances" should have more audible variety than one with only 5-limit
consonances.

> >With my schismic keyboard setup I found it remarkably difficult to
> >distinguish 5:4 and 6:5 what with them both being well tuned
> >consonances and not far apart.
>
> Really? Maybe I just find consonances easier to recognize because
> I've trained myself to do it... maybe it isn't innate.

Really, I think it's your training at work here. Fifths and thirds are
easy to tell a part, of course, because of the change in consonance.
Thirds can also be distinguished in 12- or 31-equal because major thirds
are better tuned. I was surprised at how the nature of a minor third
changes in 19-equal, and how similar major and minor triads (to be
specific, rather than thirds) become when the two thirds are well tuned,
and slightly closer in pitch than they would be in JI.

Two dissonant chords with no particular rationalisation may be hard to
tell apart. But there are plenty of dissonances, like 11-limit intervals,
which have their own quality distinct from other dissonances. And miracle
tuning is specifically optimized for them.

I'm certainly not going to reject scales because they don't have enough
unambiguous consonances. We should be collecting scales that have most
reasonable properties, and see how well they work as diatonics. That's
going to take a long time, because it means writing fairly complex music
for each.

Graham

>Two dissonant chords with no particular rationalisation may be hard to
>tell apart. But there are plenty of dissonances, like 11-limit intervals,
>which have their own quality distinct from other dissonances. And miracle
>tuning is specifically optimized for them.
>
>I'm certainly not going to reject scales because they don't have enough
>unambiguous consonances. We should be collecting scales that have most
>reasonable properties, and see how well they work as diatonics. That's
>going to take a long time, because it means writing fairly complex music
>for each.

It would be fun to discuss this, but I'm not sure what it has to do with
where we started from. Do we agree that a principle thing about the
diatonic scale, that is not a common thing among scales, is the ability
to harmonize a melody inside the scale and have the harmony voice
still sound like the same melody? If we do, then we can proceed to figure
out which properties have anything to do with it, and which don't.

Regardless of if we agree about this, we can ask if there are any
other principle things that differentiate the diatonic scale from the
wide variety of possible scales. What might they be?

-Carl

Carl Lumma wrote:

> It would be fun to discuss this, but I'm not sure what it has to do with
> where we started from. Do we agree that a principle thing about the
> diatonic scale, that is not a common thing among scales, is the ability
> to harmonize a melody inside the scale and have the harmony voice
> still sound like the same melody? If we do, then we can proceed to
> figure
> out which properties have anything to do with it, and which don't.

Maybe, I'm not sure if that's the most important property.

> Regardless of if we agree about this, we can ask if there are any
> other principle things that differentiate the diatonic scale from the
> wide variety of possible scales. What might they be?

Having a large proportion of consonant triads with a consistent pattern is
one of the hardest to find. But that may be at too low a level of
abstraction. It may be easier to list the rare things about tonality, to
see if they can be applied to other scales.

- Stable but complex melody. It has a sense of being 7 almost equal
notes, but they're not so equal as to be boring.

- Different melodies can work together with consonant harmony.

- Key center can be established through harmony.

- Melodies and chord sequences have a sense of direction. You can often
predict what will come next, but still be surprised.

- Notes from outside the key can be used to reinforce it.

Graham