Ranking Eikosany tetrads and hexanies

Has anyone (Paul, Gene, Monz?) come up with a method for ranking tetrads and
hexanies according to their consonance/dissonance? In otherwords if I have a
list of Eikosany tetrads, 1.3.7.9, 1.3.5.11, etc., how, other than by ear,
can I rank them from more to less consonant?

I seem to remember various formulae for achieving this, ie. "plug in your
chord and calculate".

Kraig, in figures 38 and 39 of the d'Alessandro paper it seems as if Erv
Wilson's charts of Eikosany rank the harmonic tetrads (and hexanies)
according to their consonance down the right hand side? Is this so and do
you know how the subharmonic chords fit in to this ranking?

Thanks in anticipation.

Sincerely
a.m.

Hi Alison,

>Has anyone (Paul, Gene, Monz?) come up with a method for ranking
>tetrads and hexanies according to their consonance/dissonance? In
>otherwords if I have a list of Eikosany tetrads, 1.3.7.9, 1.3.5.11,
>etc., how, other than by ear, can I rank them from more to less
>consonant?

The short answer is yes, there are several methods for rating the
consonance/dissonance of chords that have been 'tuning list
approved'. They are all based on the size of the numbers in the
chord, and thus they are all similar. I forget exactly which one
Paul's currently advocating; IIRC it's log(1/abcd) where the chord
is a:b:c:d. We once did a blind by-ear ranking of some tetrads and
compared it to the method, on the harmonic_entropy list.

I'll leave the Wilson question to Kraig.

-Carl

>

Hi Alison!
Erv just list them in order of smaller numbers to higher w/o any thought to consonant /disonant. he seems to be not much interested in this (too many varibles? but i am not sure)

My own method (which this list has more than once glossed over, if i may be kind) is to take the harmonics numbers as they occur in the actual spacing and add them, including the 1st order difference tones, but any number that is repeated i count only once. So only unique numbers are counted. This takes into consideration that inversions can vary quite drastically according to spacing. In fact it is the only method i know that even takes this into account at all. It is not perfect but is an advancement over treating all inversions equally, which seems a bit absurd. It lacks something with subharmonics as it does lean them as being more dissonant than they seem to be. For Harmonic 11 limit chords the formula has been worked out and was
used in the following composition
http://anaphoria.com/lullaby.html
where the numbers correspond to the above formula.

One has only to compare the sound of some of the latter with some of the earlier chords with the same notes in a difference spacing to realize how little we know but just looking at the factors.
which i will call Grady's consonance scale II.

Maybe it needed a name for those on this list to think it as real.

Grady's consonance scale 1 was where repeated numbers where not omited but it seemed to not be as good t to my ear.

In the 80's, i did quite a few compositions on single tetrads and this formula seemed to work quite well.
it was quite practical in that instrumentalist only have to learn a few pitches and their octives and since the piece would stay on this chord, they could use there ears. a few though did alternate between roots a 5th away from each other.

I would have hope that those so geared in math could come up with a quick method to figure this out as well as taking a set of ratios and going through the inversions. to run through all 720 inversions of a hexany takes a while. I did only one with the inversions of a hexany as it was way too time consuming for me. I did not use all the inversions thpough but selected those that really grabbed me but using them in an order for con. to dis. i would be more than happy to investigate this further for with any one with the know-how and ear to do so.

The thing I got from all of these experiments is that consonances is not necessarily determined by lower number ratios but by 'coincidence' in your tones and difference tones. If one tries to do the most dissonant sounds such as Phi, we run across difference tones that generate tone of your scale or series, hence making them sound often quite consonant. Obviously the method above is not quite capable of dealing with these type of chords. Helmholtz's method is good in that it also is quite effected by range, where the same chord will be more of less, depending on where it is. And Sethares approach points to also the influence of timbre.
It is probably this level of complexity that had me stop where i did, one could spend a lifetime on this.

Novaro on the other Hand made quite a bit to do about chords where the tones are seperated by a repeated difference tone number . An example might be clearer as in 11,17, 23,29 with 6 being a common difference tone between successive tones. Chord of such construction often sound quite consonant, even high harmonics.

>
> From: Alison Monteith <alison.monteith3@which.net>
> Subject: Ranking Eikosany tetrads and hexanies
>
> Has anyone (Paul, Gene, Monz?) come up with a method for ranking tetrads and
> hexanies according to their consonance/dissonance? In otherwords if I have a
> list of Eikosany tetrads, 1.3.7.9, 1.3.5.11, etc., how, other than by ear,
> can I rank them from more to less consonant?
>
> I seem to remember various formulae for achieving this, ie. "plug in your
> chord and calculate".
>
> Kraig, in figures 38 and 39 of the d'Alessandro paper it seems as if Erv
> Wilson's charts of Eikosany rank the harmonic tetrads (and hexanies)
> according to their consonance down the right hand side? Is this so and do
> you know how the subharmonic chords fit in to this ranking?
>
> Thanks in anticipation.
>
> Sincerely
> a.m.
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Hi Kraig,

> My own method (which this list has more than once glossed over, if i
>may be kind) is to take the harmonics numbers as they occur in the
>actual spacing and add them, including the 1st order difference tones,
>but any number that is repeated i count only once.

I must have missed this. Can you do an example, say 5:6:7:9?

>This takes into consideration that inversions can vary quite
>drastically according to spacing. In fact it is the only method i know
>that even takes this into account at all.

In the method I just described, different inversions will be different.
So for 5:6:7:9, log(1/1890) = log(1)-log(1890) = -3.276. The higher
the number here, the more consonant, though I don't recall there being
negative numbers... maybe I've forgotten something.

Anyway, if you have instead 6:7:9:10, it'll be different.

> I would have hope that those so geared in math could come up with a
>quick method to figure this out as well as taking a set of ratios and
>going through the inversions. to run through all 720 inversions of a
>hexany takes a while.

720 inversions??

> Novaro on the other Hand made quite a bit to do about chords where
>the tones are seperated by a repeated difference tone number. An example
>might be clearer as in 11,17, 23,29 with 6 being a common difference
>tone between successive tones. Chord of such construction often sound
>quite consonant, even high harmonics.

Hmm, intrestin'. . .

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The short answer is yes, there are several methods for rating the
> consonance/dissonance of chords that have been 'tuning list
> approved'. They are all based on the size of the numbers in the
> chord, and thus they are all similar. I forget exactly which one
> Paul's currently advocating; IIRC it's log(1/abcd) where the chord
> is a:b:c:d.

You can put the log in there if you want; it won't affect the rank-
ordering. If you're *not* assuming octave-equivalence, so you put in
all the appropriate factors of 2 instead of ignoring them, abcd is a
decent measure as long as it's not too large. The utonal chords, when
expressed in otonal form, will exceed this usefulness limit, so it
won't be a useful way of comparing the utonal chords, but the utonal
chords will show up as more dissonant than the otonal chords. Someday
I hope to be able to calculate tetradic harmonic entropy, to get
beyond this 'limit' and be able to evaluate any tetrad, tempered or
just. If you use a Plomp-Levelt-Sethares approach alone, you will
predict that each utonal chord is almost equally consonant as its
otonal inverse -- which contradicts experience for the majority of
timbres, durations, and loudnesses.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> This takes into consideration that inversions can vary quite
>drastically according to spacing. In fact it is the only method i
>know that even takes this into account at all.

Virtually all methods do distinguish between different inversions and
voicings -- including the methods I mentioned in my last post on this
topic.

>It is not perfect but is an advancement over treating all inversions
>equally, which seems a bit absurd.

It's only not absurd when you're facing the problem of designing an
octave-repeating scale and finding a way for a composer to think
about and approach the scale without undue complications -- such as
those you cite below. Of course any real composer should ultimately
be able to shape his or her materials by ear without reliance on
oversimplified formulae, which ultimately they all are. But can make
very useful simplifications and generalizations when one is climbing
the steep ladder of first working with an unfamiliar tuning system. I
have no objections to the ones you're using if they're useful for
your purposes.

> I would have hope that those so geared in math could come up with
>a quick method to figure this out as well as taking a set of ratios
>and going through the inversions. to run through all 720 inversions
>of a hexany takes a while. I did only one with the inversions of a
>hexany as it was way too time consuming for me. I did not use all
>the inversions thpough but selected those that really grabbed me but
>using them in an order for con. to dis. i would be more than happy
>to investigate this further for with any one with the know-how and
>ear to do so.

> The thing I got from all of these experiments is that
>consonances is not necessarily determined by lower number ratios but
>by 'coincidence' in your tones and difference tones.

I believe this has importance in some musical circumstances. I hope
Bill Sethares will take note of this because he's currently claiming
that difference tones are irrelevant to the perception of
consonance/dissonance.
>
> Novaro on the other Hand made quite a bit to do about chords
>where the tones are seperated by a repeated difference tone number .
>An example might be clearer as in 11,17, 23,29 with 6 being a common
>difference tone between successive tones. Chord of such construction
>often sound quite consonant, even high harmonics.

Yes, Fokker lavished a lot of attention on such chords as well, and
more recently I've heard George Secor do the same.

>
> From: Carl Lumma <ekin@lumma.org>
>
> Hi Kraig,
>
> > My own method (which this list has more than once glossed over, if i
> >may be kind) is to take the harmonics numbers as they occur in the
> >actual spacing and add them, including the 1st order difference tones,
> >but any number that is repeated i count only once.
>
> I must have missed this. Can you do an example, say 5:6:7:9?

It is quite convoluted to do so in this format
but here it goes
5+6+7+9
+
6-5=1
7-6=1
9-7=2
7-5=2
9-5=4
9-6=3
the unique terms added
1+2+3+4+5+6+7+9=37

>
> .
>
> > I would have hope that those so geared in math could come up with a
> >quick method to figure this out as well as taking a set of ratios and
> >going through the inversions. to run through all 720 inversions of a
> >hexany takes a while.
>
> 720 inversions??

permutations of 6. i believe this is correct

>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>
>
> Message: 16
> Date: Fri, 28 Nov 2003 02:47:00 -0000
> From: "Paul Erlich" <paul@stretch-music.com>
> Subject: Re: Ranking Eikosany tetrads and hexanies
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > The short answer is yes, there are several methods for rating the
> > consonance/dissonance of chords that have been 'tuning list
> > approved'. They are all based on the size of the numbers in the
> > chord, and thus they are all similar. I forget exactly which one
> > Paul's currently advocating; IIRC it's log(1/abcd) where the chord
> > is a:b:c:d.
>
> You can put the log in there if you want; it won't affect the rank-
> ordering.

I glad you mentioned that in that such additions i always find excessive. I believe you formula would come out the same as my formula number I?

>
> Message: 17
> Date: Fri, 28 Nov 2003 02:58:58 -0000
> From: "Paul Erlich" <paul@stretch-music.com>
> Subject: Re: Ranking Eikosany tetrads and hexanies
>
> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
> > This takes into consideration that inversions can vary quite
> >drastically according to spacing. In fact it is the only method i
> >know that even takes this into account at all.
>
> Virtually all methods do distinguish between different inversions and
> voicings -- including the methods I mentioned in my last post on this
> topic.
>
> >It is not perfect but is an advancement over treating all inversions
> >equally, which seems a bit absurd.
>
> It's only not absurd when you're facing the problem of designing an
> octave-repeating scale and finding a way for a composer to think
> about and approach the scale without undue complications -- such as
> those you cite below. Of course any real composer should ultimately
> be able to shape his or her materials by ear without reliance on
> oversimplified formulae, which ultimately they all are. But can make
> very useful simplifications and generalizations when one is climbing
> the steep ladder of first working with an unfamiliar tuning system. I
> have no objections to the ones you're using if they're useful for
> your purposes.

In the 80's there was little or no exploration into tuning, aleast available. Like i mentioned it also made it possible to get instrumentalist to hear and play a piece with the least amount of difficulty. Now i pay absolutely no attention to such things except intuitively. I do think i gained alot by the experience and no matter how one thinks , it is easy to forget about the possibility of and inversion or two,and one discovers what one likes.

> I believe this has importance in some musical circumstances. I hope
> Bill Sethares will take note of this because he's currently claiming
> that difference tones are irrelevant to the perception of
> consonance/dissonance.

this is the absolute opposite point of view i hold. Difference tones have been shown to influence on our fundemental tracking of pitches. As illustrated to Roederer

>
> >
> > Novaro on the other Hand made quite a bit to do about chords
> >where the tones are seperated by a repeated difference tone number .
> >An example might be clearer as in 11,17, 23,29 with 6 being a common
> >difference tone between successive tones. Chord of such construction
> >often sound quite consonant, even high harmonics.
>
> Yes, Fokker lavished a lot of attention on such chords as well, and
> more recently I've heard George Secor do the same.

I meant to mention to Fokker and George is always in this category of theorist

>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > The short answer is yes, there are several methods for rating the
> > consonance/dissonance of chords that have been 'tuning list
> > approved'. They are all based on the size of the numbers in the
> > chord, and thus they are all similar. I forget exactly which one
> > Paul's currently advocating; IIRC it's log(1/abcd) where the chord
> > is a:b:c:d.
>
> You can put the log in there if you want; it won't affect the rank-
> ordering.

That is, the ordinal property of the method will remain intact. The
cardinal information (how much difference there is between two
chords' dissonance) will differ.

> [...] The utonal chords, when expressed in otonal form, will exceed
this usefulness limit, so it
> won't be a useful way of comparing the utonal chords, but the
utonal
> chords will show up as more dissonant than the otonal chords.
Someday
> I hope to be able to calculate tetradic harmonic entropy, to get
> beyond this 'limit' and be able to evaluate any tetrad, tempered or
> just. If you use a Plomp-Levelt-Sethares approach alone, you will
> predict that each utonal chord is almost equally consonant as its
> otonal inverse -- which contradicts experience for the majority of
> timbres, durations, and loudnesses.

Paul, I understand here that, apart from the octave equiv. point, you
mean the system log(1/abcd) predicts 4:5:6:7 as more consonant than
8:9:10:12, for instance, and that discriminating between these types
of chords is important (and accomplished more or less with the cited
method). Have I got you right?
I agree with the point of differentiating those structures.

Hi Paul!
I think i am mistaken in this and that your formula would be the
same as if you just added the harmonic numbers together?
The way i judged the effectivenes of a method was by playing an
entire sequence through and subjectively went with the one i thought
flowed the best overall. Empirical totally. My girlfriend at the time
had a certain tolerance of what she could take and i could narrow it
down to within a narrow range. I would play alot on just below her
threshold,but as soon as i crossed the line, she would comment (although
not necessarily make me stop).
There were some other fringe benefits to this in that i found i
could do quite humorous things alternated between the the two extremes.

> > The short answer is yes, there are several methods for rating the
> > consonance/dissonance of chords that have been 'tuning list
> > approved'. They are all based on the size of the numbers in the
> > chord, and thus they are all similar. I forget exactly which one
> > Paul's currently advocating; IIRC it's log(1/abcd) where the chord
> > is a:b:c:d.
>
> You can put the log in there if you want; it won't affect the rank-
> ordering.

I glad you mentioned that in that such additions i always find
excessive. I believe you formula would come out the same as my formula
number I?
-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Oops, that's not a minor (pun intended) mistake!

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:
> Paul, I understand here that, apart from the octave equiv. point,
you
> mean the system log(1/abcd) predicts 4:5:6:7 as more consonant than
> 8:9:10:12, for instance, and that discriminating between these
types
> of chords is important (and accomplished more or less with the
cited
> method).

It should read 10 12 15 18 or (4:5:6:7) or any other representation
of a minor chord. I don't know why I wrote that.

Max.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> > >hexany takes a while.
> >
> > 720 inversions??
>
> permutations of 6. i believe this is correct

That's not correct. If you truly have enough octaves of range so that
you can order the notes in any way you please, there will be more
than 720 possibilities since there is still possibility for different
voicings to have the same ordering. However, you normally have a much
more limited range to consider, and there will actually end up being
fewer than 720 possibilities.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> I glad you mentioned that in that such additions i always find
>excessive. I believe you formula would come out the same as my
>formula number I?

Not quite, due to your attention to the first-order difference tones
of the fundamentals. In practice, the higher-order difference tones
(especially second-order difference tones) will often be even louder,
and for harmonic timbres, one has to consider the difference tones
the harmonics form against the fundamentals, and against one another,
as well. All these complications tend to 'average out', though, so
that a*b*c*d (or (a*b*c*d)^(1/4), which is the geometric mean, which
is what was discussed on the harmonic entropy list) is not a bad
formula, as long as the result is a relatively small number.

> > I believe this has importance in some musical circumstances. I
hope
> > Bill Sethares will take note of this because he's currently
claiming
> > that difference tones are irrelevant to the perception of
> > consonance/dissonance.
>
> this is the absolute opposite point of view i hold. Difference
>tones have been shown to influence on our fundemental tracking of
>pitches. As illustrated to Roederer

Kraig, Roederer as well as all the other modern references I know of
do *not* equate or relate the fundamental tracking (a.k.a. virtual
pitch, etc.) mechanism to difference tones or any other combinational
tones. There is clear evidence that the two mechanisms lead to
different results for the fundamental, as I've explicated in other
posts to this list -- experiments have shown that a series of
inharmonic partials with a common difference between the frequencies
will evoke a fundamental which does *not* match this difference
frequency, but rather is such that *its* harmonics would be as good a
fit as possible to the inharmonic partials. However, I still feel
Bill Sethares is unfairly dismissing combinational tones in his
discussion of sensory consonance/dissonance.

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:

> Paul, I understand here that, apart from the octave equiv. point,
you
> mean the system log(1/abcd) predicts 4:5:6:7 as more consonant than
> 8:9:10:12, for instance, and that discriminating between these
types
> of chords is important (and accomplished more or less with the
cited
> method). Have I got you right?

Maybe, though the utonal inverse of 4:5:6:7 that I was referring to
would be (1/7):(1/6):(1/5):(1/4). Now express *that* in otonal terms.
You should get a 105 for the highest note.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> Hi Paul!
> I think i am mistaken in this and that your formula would be the
> same as if you just added the harmonic numbers together?

Almost, but not quite. Multiplying leads to a slightly different
ranking than adding. It's basically a generalization of the Tenney
harmonic distance function from dyads to n-ads. It already begins to
fall apart when the numbers are as large as 9:11:13:15, though. And
there's definitely a roughness component that's missing, that would
be well handled by something similar to the Plomp-Levelt-Sethares
formula. But the latter alone seems to do a much, much poorer job of
ranking than just multiplying, at least when dealing with a set of JI
tetrads. See the archives of the harmonic entropy list for more
details, including a few different listeners' rankings of 36
different tetrads.

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:
> Oops, that's not a minor (pun intended) mistake!
>
> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:
> > Paul, I understand here that, apart from the octave equiv. point,
> you
> > mean the system log(1/abcd) predicts 4:5:6:7 as more consonant
than
> > 8:9:10:12, for instance, and that discriminating between these
> types
> > of chords is important (and accomplished more or less with the
> cited
> > method).
>
> It should read 10 12 15 18 or (4:5:6:7) or any other representation
> of a minor chord. I don't know why I wrote that.
>
> Max.

I'm really not understanding you. 10:12:15:18 would normally be
considered a "minor seventh" chord, while 4:5:6:7 would normally be
considered similar to a "dominant seventh" or "German augmented sixth
chord". Meanwhile, 1/(4:5:6:7) would be considered similar to a "half-
diminished seventh chord" or an inversion of a "minor sixth" chord.
So I don't know what you're getting at. Care to clarify (only if it's
still relevant, of course)?

-Paul

on 11/27/03 6:58 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
>> This takes into consideration that inversions can vary quite
>> drastically according to spacing. In fact it is the only method i
>> know that even takes this into account at all.
>
> Virtually all methods do distinguish between different inversions and
> voicings -- including the methods I mentioned in my last post on this
> topic.
>
>> It is not perfect but is an advancement over treating all inversions
>> equally, which seems a bit absurd.
>
> It's only not absurd when you're facing the problem of designing an
> octave-repeating scale and finding a way for a composer to think
> about and approach the scale without undue complications -- such as
> those you cite below. Of course any real composer should ultimately
> be able to shape his or her materials by ear without reliance on
> oversimplified formulae, which ultimately they all are. But can make
> very useful simplifications and generalizations when one is climbing
> the steep ladder of first working with an unfamiliar tuning system. I
> have no objections to the ones you're using if they're useful for
> your purposes.
>
>> I would have hope that those so geared in math could come up with
>> a quick method to figure this out as well as taking a set of ratios
>> and going through the inversions. to run through all 720 inversions
>> of a hexany takes a while. I did only one with the inversions of a
>> hexany as it was way too time consuming for me. I did not use all
>> the inversions thpough but selected those that really grabbed me but
>> using them in an order for con. to dis. i would be more than happy
>> to investigate this further for with any one with the know-how and
>> ear to do so.
>
>> The thing I got from all of these experiments is that
>> consonances is not necessarily determined by lower number ratios but
>> by 'coincidence' in your tones and difference tones.
>
> I believe this has importance in some musical circumstances. I hope
> Bill Sethares will take note of this because he's currently claiming
> that difference tones are irrelevant to the perception of
> consonance/dissonance.

Well it would be interesting to try an "entirely" irrational chord with very
just differences.

It occurs to me however, that testing a predictor of consonance/dissonance
with electronically-generated sound will be *very* problematic due to the
conversion of difference tones to faint *actual* tones by almost any speaker
(never mind amplifiers), due to intermodulation distortion. Probably
intermodulation also occurs to some extent in any material (e.g. so that
playing chords on a single guitar would yield suspect results), though
probably it is vainishingly small when separately generated sounds are mixed
in the air, so that playing several solo instruments in the same room
(without vibrating windows) would yield valid results.

Mind you I'm not saying a theory about guitar chords (for example) is not
relevant, just that that has to be put aside if the relevance/irrelevance of
difference tones (without intermodulation that makes them *real*) is being
established.

But something about this whole question surprises me: the fact that it came
up in a context in which (unless I missed something) timbre was not
mentioned once. Surely you can't be talking about chords built up from pure
sinusoidal harmonics can you? But if not, how can you disregard timbre
(although putting inharmonicity side seems reasonable). But if so, then you
are talking about a different kind of consonance than what we I have been
assuming in most of the discussions I have followed on this list. Certainly
in some discussions (of piano tuning for example) inharmonicity played a big
role. So it would seem to follow that even without inharmonicity being
involved timbre must enter into the question of consonance.

-Kurt

>>
>> Novaro on the other Hand made quite a bit to do about chords
>> where the tones are seperated by a repeated difference tone number .
>> An example might be clearer as in 11,17, 23,29 with 6 being a common
>> difference tone between successive tones. Chord of such construction
>> often sound quite consonant, even high harmonics.
>
> Yes, Fokker lavished a lot of attention on such chords as well, and
> more recently I've heard George Secor do the same.

on 11/28/03 3:01 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
>> I glad you mentioned that in that such additions i always find
>> excessive. I believe you formula would come out the same as my
>> formula number I?
>
> Not quite, due to your attention to the first-order difference tones
> of the fundamentals. In practice, the higher-order difference tones
> (especially second-order difference tones) will often be even louder,
> and for harmonic timbres, one has to consider the difference tones
> the harmonics form against the fundamentals, and against one another,
> as well. All these complications tend to 'average out', though, so
> that a*b*c*d (or (a*b*c*d)^(1/4), which is the geometric mean, which
> is what was discussed on the harmonic entropy list) is not a bad
> formula, as long as the result is a relatively small number.
>
>>> I believe this has importance in some musical circumstances. I
> hope
>>> Bill Sethares will take note of this because he's currently
> claiming
>>> that difference tones are irrelevant to the perception of
>>> consonance/dissonance.
>>
>> this is the absolute opposite point of view i hold. Difference
>> tones have been shown to influence on our fundemental tracking of
>> pitches. As illustrated to Roederer
>
> Kraig, Roederer as well as all the other modern references I know of
> do *not* equate or relate the fundamental tracking (a.k.a. virtual
> pitch, etc.) mechanism to difference tones or any other combinational
> tones. There is clear evidence that the two mechanisms lead to
> different results for the fundamental, as I've explicated in other
> posts to this list -- experiments have shown that a series of
> inharmonic partials with a common difference between the frequencies
> will evoke a fundamental which does *not* match this difference
> frequency, but rather is such that *its* harmonics would be as good a
> fit as possible to the inharmonic partials. However, I still feel
> Bill Sethares is unfairly dismissing combinational tones in his
> discussion of sensory consonance/dissonance.

Sorry, I had not read the last post when I replied to this thread regarding
timbre. (I thought I had, but didn't notice that was not scrolled to the
top.) Some of my points still hold though.

It would be interesting to take a large harmonic consonance and tweak all
the pitches by the same (smallish) amount, so that the same differences are
retained, and see over what ranges of variation the harmonic versus
difference-tone model has the most relevance. Has anyone tried anything
like this? It would be pretty easy to set this up in Max/MSP, but I don't
know when I'll get around to actually doing it.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Well it would be interesting to try an "entirely" irrational chord
with very
> just differences.

I'm not following you. Can you give an example?

>
> It occurs to me however, that testing a predictor of
consonance/dissonance
> with electronically-generated sound will be *very* problematic due
to the
> conversion of difference tones to faint *actual* tones by almost
any speaker
> (never mind amplifiers), due to intermodulation distortion.
Probably
> intermodulation also occurs to some extent in any material (e.g. so
that
> playing chords on a single guitar would yield suspect results),

BTW, I think that an even more important effect on the guitar is
sympathetic resonance between different modes of vibration on
different strings.

>though
> probably it is vainishingly small when separately generated sounds
are mixed
> in the air, so that playing several solo instruments in the same
room
> (without vibrating windows) would yield valid results.

Actually, difference tones in the ear can be loud, almost
overwhelming, for two soprano recorders playing together, for example.

> Mind you I'm not saying a theory about guitar chords (for example)
is not
> relevant, just that that has to be put aside if the
relevance/irrelevance of
> difference tones (without intermodulation that makes them *real*)
is being
> established.
>
> But something about this whole question surprises me: the fact
that it came
> up in a context in which (unless I missed something) timbre was not
> mentioned once. Surely you can't be talking about chords built up
from pure
> sinusoidal harmonics can you? But if not, how can you disregard
timbre
> (although putting inharmonicity side seems reasonable). But if so,
then you
> are talking about a different kind of consonance than what we I
have been
> assuming in most of the discussions I have followed on this list.
Certainly
> in some discussions (of piano tuning for example) inharmonicity
played a big
> role. So it would seem to follow that even without inharmonicity
being
> involved timbre must enter into the question of consonance.
>
> -Kurt

Timbre does enter into the question. With harmonic timbres, a greater
amplitude in the higher harmonics will translate into clearer pitch
perceptions, less confusion of complex ratios with simpler ratios,
and by all theories, a greater propensity to find local minima of
dissonance at ratios of higher numbers. With inharmonic timbres (and
I did specify period timbres, which are not inharmonic), all theories
predict that using whatever intervals make the inharmonic partials
coincide will tend to produce more consonance than one would
otherwise predict. But in Western common-practice music theory (which
is based almost entirely on harmonic timbres, plus a few nearly
harmonic ones like the piano, guitar, and specially shaped
metallophones), the dichotomy between consonances and dissonances is
*not* dependent on timbre. This is a nice property that is
convenient, and possible to some extent, to extend to experimental
musics involving harmonies derived from ratios of somewhat larger
intervals.

The experiments on the harmonic entropy list were done with a
particular periodic timbre -- a rectified sine wave -- which has a
fairly "average" amplitude in its higher harmonics. I'd be happy to
repeat the experiment with a different timbre if you post to that
effect over on that list, but I doubt that any periodic timbre would
yield tetrad rankings showing marked deviations from the general
pattern we found.

Keep up the good thinking,
Paul

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 11/28/03 3:01 PM, Paul Erlich <paul@s...> wrote:

> > Kraig, Roederer as well as all the other modern references I know
of
> > do *not* equate or relate the fundamental tracking (a.k.a. virtual
> > pitch, etc.) mechanism to difference tones or any other
combinational
> > tones. There is clear evidence that the two mechanisms lead to
> > different results for the fundamental, as I've explicated in other
> > posts to this list -- experiments have shown that a series of
> > inharmonic partials with a common difference between the
frequencies
> > will evoke a fundamental which does *not* match this difference
> > frequency, but rather is such that *its* harmonics would be as
good a
> > fit as possible to the inharmonic partials. However, I still feel
> > Bill Sethares is unfairly dismissing combinational tones in his
> > discussion of sensory consonance/dissonance.
>
> Sorry, I had not read the last post when I replied to this thread
regarding
> timbre. (I thought I had, but didn't notice that was not scrolled
to the
> top.) Some of my points still hold though.

Not sure which of your points *wouldn't* hold just because of the
above. Can you elaborate?

> It would be interesting to take a large harmonic consonance

Meaning . . . A very wide dyad? A consonant chord with a lot of
notes? A ratio of large numbers?

> and tweak all
> the pitches by the same (smallish) amount,

You mean all the sinusoidal components, or do you really mean all the
pitches?

> so that the same differences are
> retained, and see over what ranges of variation the harmonic versus
> difference-tone model has the most relevance. Has anyone tried
anything
> like this? It would be pretty easy to set this up in Max/MSP, but
I don't
> know when I'll get around to actually doing it.

I'd be happy to indulge everyone, using Matlab, if I knew exactly
what you were after. What I wrote above doesn't concern harmonic
consonances (directly) at all, but rather our pitch perception
mechanism, which extracts a heard pitch from a set of partials even
when there is no physical energy at or near the frequency
corresponding to the heard pitch.

>

Hi Paul!
I believe i stated where no two tones are more than an octive apart which puts it within 4 octives which is quite easy to work with.

>
> From: "Paul Erlich" <paul@stretch-music.com>
>
> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
> > > >hexany takes a while.
> > >
> > > 720 inversions??
> >
> > permutations of 6. i believe this is correct
>
> That's not correct. If you truly have enough octaves of range so that
> you can order the notes in any way you please, there will be more
> than 720 possibilities since there is still possibility for different
> voicings to have the same ordering. However, you normally have a much
> more limited range to consider, and there will actually end up being
> fewer than 720 possibilities.
>

>
> ________________________________________________________________________
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>

Hi Paul!
If you put two different pitches in different ears isolated from each other and you can hear difference tones, this seems to point to tracking as it is a product produced in the brain

>
> From: "Paul Erlich" <paul@stretch-music.com>
> Subject: Re: Ranking Eikosany tetrads and hexanies
>
>
> Kraig, Roederer as well as all the other modern references I know of
> do *not* equate or relate the fundamental tracking (a.k.a. virtual
> pitch, etc.) mechanism to difference tones or any other combinational
> tones. There is clear evidence that the two mechanisms lead to
> different results for the fundamental, as I've explicated in other
> posts to this list -- experiments have shown that a series of
> inharmonic partials with a common difference between the frequencies
> will evoke a fundamental which does *not* match this difference
> frequency, but rather is such that *its* harmonics would be as good a
> fit as possible to the inharmonic partials. However, I still feel
> Bill Sethares is unfairly dismissing combinational tones in his
> discussion of sensory consonance/dissonance.
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>

Hi Paul!
So what would make you feel that multiplication would be better than addition?
I already have a problem with the assumptions of additions such as is a 7 more that 2 times a disonant than a 3?

>
> From: "Paul Erlich" <paul@stretch-music.com>
> Subject: Re: Ranking Eikosany tetrads and hexanies
>
> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> > Hi Paul!
> > I think i am mistaken in this and that your formula would be the
> > same as if you just added the harmonic numbers together?
>
> Almost, but not quite. Multiplying leads to a slightly different
> ranking than adding. It's basically a generalization of the Tenney
> harmonic distance function from dyads to n-ads. It already begins to
> fall apart when the numbers are as large as 9:11:13:15, though. And
> there's definitely a roughness component that's missing, that would
> be well handled by something similar to the Plomp-Levelt-Sethares
> formula. But the latter alone seems to do a much, much poorer job of
> ranking than just multiplying, at least when dealing with a set of JI
> tetrads. See the archives of the harmonic entropy list for more
> details, including a few different listeners' rankings of 36
> different tetrads.
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
>
> Hi Paul!
> If you put two different pitches in different ears isolated from
>each other and you can hear difference tones, this seems to point to
>tracking as it is a product produced in the brain

If I'm not mistaken, these binaural difference tones that are not the
product of nonlinearities in each ear are all of the form 2*f1 - f2,
so they're not even the ones you were referring to. But the
fundamental tracking experiments I was referring to were unambiguous
in ruling out difference tones. I'd be more than happy to recreate
some of these experiments for all our ears to hear when I get back to
the office; unfortunately I doubt I can rely on many list members
using equipment of high enough fidelity to avoid introducing
intermodulation distortion, as Kurt mentioned, and thus difference
tones, hence spoiling the results. Needless to say, when difference
tones are present, they have to enter the equation as to what
frequency components are being heard, and thus they will have a large
influence on the virtual (or now, not so virtual) fundamental that
ends up being heard. Large, but not defining.

Anyway, since you referenced Roederer, I wonder how exactly you got
this idea which appears to contradict what's right there in
Roederer's book . . .

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
>
> Hi Paul!
> So what would make you feel that multiplication would be better
>than addition?

Various harmonic entropy studies.

> I already have a problem with the assumptions of additions such as
>is a 7 more that 2 times a disonant than a 3?

Kraig, recall that we were talking about *rank order*, and throwing
out the details of the formula would allow one to say what is 2 times
more dissonant than what. Tenney's actual formula for harmonic
distance of a ratio n/d is log(n*d), and this agrees with relative
harmonic entropy as long as n*d isn't too large, so using either
dissonance measure, 7:1 would end up showing up as *less* than twice
as dissonant as 3:1.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
>
> Hi Paul!
> I believe i stated where no two tones are more than an octive
>apart

Oh, sorry I missed that! Then 720 would be the correct number. But
the vast majority of these, if not all of them, are probably too
complex for simple numerical rules to have much effectiveness.
Alison, are you actually using hexanies as six notes simultaneously
sounding, as *chords*?

on 11/28/03 9:47 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> Well it would be interesting to try an "entirely" irrational chord
> with very
>> just differences.
>
> I'm not following you. Can you give an example?

Sorry for the vagueness. I was thinking along the lines of picking numbers
whose closest rational approximation involves very large integers. But even
that is too vague. It all started with kraig grady's reference to Novaro,
and his 11:17:23:29 (difference of 6) example. I was thinking don't use
integers at all, neglecting that rational approximations would get in the
way. But just to complete the original thought, suppose you take instead:

pi : pi + 6 : pi + 12 : pi + 18

where pi is the familiar circumference/diameter ratio. Potentially the 22/7
approximation for pi might make this not the best possible example though.
Also the choice of 6 may not be the best here.

To take the thought to its ultimate conclusion, I probably end up begging a
theory of dissonance/consonance rating for a means to pick the "most
dissonance possible" chord, rather than the "most irrational possible",
since after all there is no theory at all (or so I imagine) for ranking
degree of rationality. So if you have a dissonance ranking scheme, then
device a way to search for maximally dissonant chords with maximally
consonant difference tones. But I can't imagine what theoretical approach
would make this in to a finite problem without consequent assumptions that
might defeat the hypothesis. Pseudo-random numbers might have a problem in
this domain that is not a problem for general statistical purposes. But
maybe random is good enough to start with.

Does that clarify?

This brings up a terminology question: how do you deal with the concept if
an irrational ratio? Ratio tends to imply integers in the context of
tunings, does it not? Is there another word or a way of making the
distinction that is customary here?

>> It occurs to me however, that testing a predictor of
> consonance/dissonance
>> with electronically-generated sound will be *very* problematic due
> to the
>> conversion of difference tones to faint *actual* tones by almost
> any speaker
>> (never mind amplifiers), due to intermodulation distortion.
> Probably
>> intermodulation also occurs to some extent in any material (e.g. so
> that
>> playing chords on a single guitar would yield suspect results),
>
> BTW, I think that an even more important effect on the guitar is
> sympathetic resonance between different modes of vibration on
> different strings.
>
>> though
>> probably it is vainishingly small when separately generated sounds
> are mixed
>> in the air, so that playing several solo instruments in the same
> room
>> (without vibrating windows) would yield valid results.
>
> Actually, difference tones in the ear can be loud, almost
> overwhelming, for two soprano recorders playing together, for example.

Oh, yes, I've heard that. So it will be really hard to sort out
intermodulation caused "externally" vs "somewhat externally" in the
eardrum/bones vs internally in the nervous system. But if there is a range
over which amplitude can be reduced with no consequent reduction in the
"effect" (whatever consonance/dissonance effect is being examined), then
that might allow external factors to be eliminated, though admittedly it
still depends on an intuitive argument, I think, to make such a claim.

>> Mind you I'm not saying a theory about guitar chords (for example)
> is not
>> relevant, just that that has to be put aside if the
> relevance/irrelevance of
>> difference tones (without intermodulation that makes them *real*)
> is being
>> established.
>>
>> But something about this whole question surprises me: the fact
> that it came
>> up in a context in which (unless I missed something) timbre was not
>> mentioned once. Surely you can't be talking about chords built up
> from pure
>> sinusoidal harmonics can you? But if not, how can you disregard
> timbre
>> (although putting inharmonicity side seems reasonable). But if so,
> then you
>> are talking about a different kind of consonance than what we I
> have been
>> assuming in most of the discussions I have followed on this list.
> Certainly
>> in some discussions (of piano tuning for example) inharmonicity
> played a big
>> role. So it would seem to follow that even without inharmonicity
> being
>> involved timbre must enter into the question of consonance.
>>
>> -Kurt
>
> Timbre does enter into the question. With harmonic timbres, a greater
> amplitude in the higher harmonics will translate into clearer pitch
> perceptions, less confusion of complex ratios with simpler ratios,
> and by all theories, a greater propensity to find local minima of
> dissonance at ratios of higher numbers. With inharmonic timbres (and
> I did specify period timbres, which are not inharmonic), all theories
> predict that using whatever intervals make the inharmonic partials
> coincide will tend to produce more consonance than one would
> otherwise predict. But in Western common-practice music theory (which
> is based almost entirely on harmonic timbres, plus a few nearly
> harmonic ones like the piano, guitar, and specially shaped
> metallophones), the dichotomy between consonances and dissonances is
> *not* dependent on timbre. This is a nice property that is
> convenient, and possible to some extent, to extend to experimental
> musics involving harmonies derived from ratios of somewhat larger
> intervals.

Wonderful to have anything convenient at all!

-Kurt

> The experiments on the harmonic entropy list were done with a
> particular periodic timbre -- a rectified sine wave -- which has a
> fairly "average" amplitude in its higher harmonics. I'd be happy to
> repeat the experiment with a different timbre if you post to that
> effect over on that list, but I doubt that any periodic timbre would
> yield tetrad rankings showing marked deviations from the general
> pattern we found.
>
> Keep up the good thinking,
> Paul

on 11/28/03 9:53 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 11/28/03 3:01 PM, Paul Erlich <paul@s...> wrote:
>
>>> Kraig, Roederer as well as all the other modern references I know
> of
>>> do *not* equate or relate the fundamental tracking (a.k.a. virtual
>>> pitch, etc.) mechanism to difference tones or any other
> combinational
>>> tones. There is clear evidence that the two mechanisms lead to
>>> different results for the fundamental, as I've explicated in other
>>> posts to this list -- experiments have shown that a series of
>>> inharmonic partials with a common difference between the
> frequencies
>>> will evoke a fundamental which does *not* match this difference
>>> frequency, but rather is such that *its* harmonics would be as
> good a
>>> fit as possible to the inharmonic partials. However, I still feel
>>> Bill Sethares is unfairly dismissing combinational tones in his
>>> discussion of sensory consonance/dissonance.
>>
>> Sorry, I had not read the last post when I replied to this thread
> regarding
>> timbre. (I thought I had, but didn't notice that was not scrolled
> to the
>> top.) Some of my points still hold though.
>
> Not sure which of your points *wouldn't* hold just because of the
> above. Can you elaborate?

Not important.

>> It would be interesting to take a large harmonic consonance
>
> Meaning . . . A very wide dyad? A consonant chord with a lot of
> notes? A ratio of large numbers?

Sorry, I was thinking "large chord", meaning many notes.

>> and tweak all
>> the pitches by the same (smallish) amount,
>
> You mean all the sinusoidal components, or do you really mean all the
> pitches?

Well help me pick. I was at first inclined to dismiss experiments with
sinusoids as "notes" as being musically meaningless. Carl once played me
some music from someone (on this list?) who created music in 12et that was
contrived to have no beats by manipulation of the frequencies of the
partials. I can't remember it well enough to decide whether this means "no
dissonance", but I suspect that there was in fact still an active
consonance/dissonance axis in spite of the "beatless" kind of consonance
being a given. If so, then perhaps using sinusoids is best. Not using
sinusoids presents other problems I'd rather not deal with anyway. If you
are inclined to agree then I'd suggest starting with a few "favorite" 4-note
or larger chords like 4:5:6:7 or 8:9:10:11 as a base, use a sinusoid for
each note, and then add a variable value to each element of the ratio, as
in:

4+v : 5+v : 6+v : 7+v

and initially just provide a way of semi-continuously varying v over a range
like -.5 to +.5. And just listen to the results. How does the implied
"fundamental" vary, if at all, and does this depend on the range over which
v is varied?

And actually, I did not mean to pick chords where the ratios are an unbroken
series of integers, as both exaples above seemed to indicate. So including
5:7:9:11 adds some breadth. And including 4:7:9:10 adds even more. The
point would be to make as few assumptions as possible, and try some things,
and hear the results.

Control over the frequency corresponding to "1" would also be important.

>> so that the same differences are
>> retained, and see over what ranges of variation the harmonic versus
>> difference-tone model has the most relevance. Has anyone tried
> anything
>> like this? It would be pretty easy to set this up in Max/MSP, but
> I don't
>> know when I'll get around to actually doing it.
>
> I'd be happy to indulge everyone, using Matlab, if I knew exactly
> what you were after. What I wrote above doesn't concern harmonic
> consonances (directly) at all, but rather our pitch perception
> mechanism, which extracts a heard pitch from a set of partials even
> when there is no physical energy at or near the frequency
> corresponding to the heard pitch.

It would be great if a web-enabled applet could be created that allowed up
to 8 integers to be entered, and provided mouse control over the offset v,
with real-time audible results. Is that possible?

I suppose short of that, a few canned results on a web page would do the
trick.

Otherwise the Max/MSP approach might be good, but I've never used it to
create a standalone app, and I think that is tricky in some ways. And I
also don't have the Windows version, which would be very limiting.

-Kurt

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>

> > Novaro on the other Hand made quite a bit to
> > do about chords where the tones are seperated by
> > a repeated difference tone number . An example
> > might be clearer as in 11,17, 23,29 with 6 being
> > a common difference tone between successive tones.
> > Chord of such construction often sound quite consonant,
> > even high harmonics.
>
> Yes, Fokker lavished a lot of attention on
> such chords as well, and more recently I've heard
> George Secor do the same.

Franz Richter Herf had an interesting approach like this
which was more sopisticated.

. .. i'm too tired to look up my old posts on Richter
right now and will just summarize from memory. i might
have the order of numbers wrong, but the concept is right ...

he created patterns of intervallic differences which
he would notate between vertical bars preceded by the
starting harmonic number: thus, 6|4 3| would give the
series 6, 10, 13, 17, 20, 24, 27, 31, 34 ...

to limit the series and create a specific compositional
harmonic entity, he put a cardinality number in parentheses
after the sequence numbers. so the 13:17:20:24 subset
of the above example makes an interesting tetrad which
would in Herf notation be 13|4 3|(4).

Kraig's example of 11, 17, 23, 29 would be notated in
Herf notation as 11|6|(4). a "harmonic-7th" tetrad of
4:5:6:7 would be 4|1|(4).

there are some old posts of mine in the archives
about Herf ... you might want to look ...

-monz
(especially lazy tonight)

hi Maximiliano (and paul)

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:

> > Paul, I understand here that, apart from the octave
> > equiv. point, you mean the system log(1/abcd) predicts
> > 4:5:6:7 as more consonant than 8:9:10:12, for instance,
> > and that discriminating between these types of chords
> > is important (and accomplished more or less with the
> > cited method). Have I got you right?
>
> Maybe, though the utonal inverse of 4:5:6:7 that I was
> referring to would be (1/7):(1/6):(1/5):(1/4). Now express
> *that* in otonal terms.
> You should get a 105 for the highest note.

i'll save Maximiliano the trouble ...
you multiply all the ratios by 420 and get 60:70:84:105.

-monz

on 29/11/03 06:41, Paul Erlich at paul@stretch-music.com wrote:

> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>>>
>>
>> Hi Paul!
>> I believe i stated where no two tones are more than an octive
>> apart
>
> Oh, sorry I missed that! Then 720 would be the correct number. But
> the vast majority of these, if not all of them, are probably too
> complex for simple numerical rules to have much effectiveness.
> Alison, are you actually using hexanies as six notes simultaneously
> sounding, as *chords*?
>

First I'd like to thank you all for a most stimulating discussion on this
subject.

To answer your question, I don't intentionally use the hexanies as chords.
I'm more interested in the triads 'around the edges' so to speak, and the
way in which they allow one to modulate from one hexany to another and
indeed from one eikosany to another. But at times, with two and three note
suspensions in moving parts, all or most of a full hexany might sound at any
one time.

From the responses to my initial question, I think it would be too time
consuming for me to rank every possible inversion of every hexany and
tetrad. But with the various formulae proposed in this thread I have at
least found methods of ranking the structures that I wish to use in a given
piece of music.

There must be an infinite number of ways of planning a composition. I spend
a lot of time preparing the ground and I have a particular leaning in my
current work towards a consonance/dissonance ebb and flow, hence my
interest. You might remember me asking a similar question about 22 tet
intervals some time ago.

The problem now is comparing inversions of the least consonant harmonic
tetrads with the most consonant subharmonic tetrads. This will have to be
done by ear and, as we all know, by considering context.

Sincerely
a.m.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 11/28/03 9:47 PM, Paul Erlich <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >
> >> Well it would be interesting to try an "entirely" irrational
chord
> > with very
> >> just differences.
> >
> > I'm not following you. Can you give an example?
>
> Sorry for the vagueness. I was thinking along the lines of picking
numbers
> whose closest rational approximation involves very large integers.
But even
> that is too vague. It all started with kraig grady's reference to
Novaro,
> and his 11:17:23:29 (difference of 6) example. I was thinking
don't use
> integers at all, neglecting that rational approximations would get
in the
> way. But just to complete the original thought, suppose you take
instead:
>
> pi : pi + 6 : pi + 12 : pi + 18
>
> where pi is the familiar circumference/diameter ratio. Potentially
the 22/7
> approximation for pi might make this not the best possible example
though.
> Also the choice of 6 may not be the best here.

Well, thanks for sharing your incomplete thoughts. You may be
touching on the fact that any theory that uses the numbers in the
ratios is ultimately flawed. You are absolutely correct, and that is
one of the primary reasons for harmonic entropy theory. One recovers
the simple numerical formulae only for the very simplest just
intervals and chords.

> To take the thought to its ultimate conclusion, I probably end up
begging a
> theory of dissonance/consonance rating for a means to pick the "most
> dissonance possible" chord, rather than the "most irrational
possible",
> since after all there is no theory at all (or so I imagine) for
ranking
> degree of rationality.

If I understand you correctly, the latter is not even necessary. For
example, dyadic harmonic entropy with one particular choice for
the 'resolution' parameter shows its biggest hump at 70 cents, so
this would be the most dissonant dyad under this scheme.

> So if you have a dissonance ranking scheme, then
> device a way to search for maximally dissonant chords with maximally
> consonant difference tones. But I can't imagine what theoretical
approach
> would make this in to a finite problem without consequent
assumptions that
> might defeat the hypothesis. Pseudo-random numbers might have a
problem in
> this domain that is not a problem for general statistical
purposes. But
> maybe random is good enough to start with.
>
> Does that clarify?

Well, there is a beautiful answer to your question that,
unfortunately, relies on a somewhat unrealistic model, and that
*does* use the notion of "degree of rationality" that you mentioned
above. Ask a mathematician or physicist what the most irrational
ratio is, and they will tell you it's the golden ratio. The reasons
for this have been much explored on this list and on the web, and I
can repeat them later if necessary. The reasons these reasons are
pretty irrelevant for consonance/dissonance theory, I can go into as
well. But the golden ratio has the wonderful property that squaring
it yields the same result as adding 1 to it. So the difference tones
occuring in a series of tones having the golden ratio between
successive tones will all line up with tones in the series -- the
difference tone between tone x+1 and tone x+2 will be tone x. So this
satisfies your desire, in some (somewhat naive) sense, of a maximally
dissonant chord with maximally consonant difference tones. Kraig is a
big fan of such series and has posted about them in the past -- not
only this golden series, but lots of other interesting series where
certain difference tones coincide with certain scale tones.

>
> This brings up a terminology question: how do you deal with the
concept if
> an irrational ratio? Ratio tends to imply integers in the context
of
> tunings, does it not? Is there another word or a way of making the
> distinction that is customary here?

There's nothing wrong with saying "irrational ratio".

> > Actually, difference tones in the ear can be loud, almost
> > overwhelming, for two soprano recorders playing together, for
example.
>
> Oh, yes, I've heard that. So it will be really hard to sort out
> intermodulation caused "externally" vs "somewhat externally" in the
> eardrum/bones vs internally in the nervous system.

Not necessarily. As Kraig touched on, experiments have been done
which present tones separately to each ear, so that any
intermodulation can't be occuring in either ear. As I recall, out of
the whole series there's only one combinational tone, a cubic
difference tone in fact, that can't be explained by a simple
nonlinear response in each ear.

> But if there is a range
> over which amplitude can be reduced with no consequent reduction in
the
> "effect" (whatever consonance/dissonance effect is being examined),
then
> that might allow external factors to be eliminated, though
admittedly it
> still depends on an intuitive argument, I think, to make such a
claim.

Somewhat, although as shown in the Feynman lectures, all "physical"
combinational tones should show a dependence on external amplitude
that is "quadratic", "cubic", etc. according to their order, and in
fact, as I recall, only the one cubic difference tone fails to obey
this relationship.
> >

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> To take the thought to its ultimate conclusion, I probably end up
begging a
> theory of dissonance/consonance rating for a means to pick the "most
> dissonance possible" chord, rather than the "most irrational
possible",
> since after all there is no theory at all (or so I imagine) for
ranking
> degree of rationality.

In the sense you seem to mean, there actually is such a thing; the
golden ratio (1+sqrt(5))/2 is sometimes even called "most irrational".

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> >> It would be interesting to take a large harmonic consonance
> >
> > Meaning . . . A very wide dyad? A consonant chord with a lot of
> > notes? A ratio of large numbers?
>
> Sorry, I was thinking "large chord", meaning many notes.
>
> >> and tweak all
> >> the pitches by the same (smallish) amount,
> >
> > You mean all the sinusoidal components, or do you really mean all
the
> > pitches?
>
> Well help me pick. I was at first inclined to dismiss experiments
with
> sinusoids as "notes" as being musically meaningless. Carl once
played me
> some music from someone (on this list?) who created music in 12et
that was
> contrived to have no beats by manipulation of the frequencies of the
> partials. I can't remember it well enough to decide whether this
means "no
> dissonance", but I suspect that there was in fact still an active
> consonance/dissonance axis in spite of the "beatless" kind of
consonance
> being a given.

Well, if the music used any vertical dissonances such as minor
seconds, major seconds, etc., then there would have been dissonance
and beating regardless of how the partials were manipulated . . .

> If so, then perhaps using sinusoids is best. Not using
> sinusoids presents other problems I'd rather not deal with anyway.
If you
> are inclined to agree then I'd suggest starting with a
few "favorite" 4-note
> or larger chords like 4:5:6:7 or 8:9:10:11 as a base, use a
sinusoid for
> each note, and then add a variable value to each element of the
ratio, as
> in:
>
> 4+v : 5+v : 6+v : 7+v

So just 4 sinusoids -- rather than being heard as a chord, this will
tend to be heard as a single pitch, especially when v is an integer,
in which case the pitch will be 1 (in the above units).

>
> and initially just provide a way of semi-continuously varying v
over a range
> like -.5 to +.5. And just listen to the results. How does the
implied
> "fundamental" vary, if at all, and does this depend on the range
over which
> v is varied?

OK, these are the experiments which were actually published and which
I was referring to earlier. The implied fundamental, or virtual
pitch, *does* vary as a function of v. In your example, if v is
about .2, the pitch will be approximately 1.05. When v approaches .5,
there are about three different pitches evoked; though the sensation
isn't really that of a 'triad', similarity is found with sinusoids
with frequencies of about 1.1, about 0.9, and, of course, about 0.5.

> And actually, I did not mean to pick chords where the ratios are an
unbroken
> series of integers, as both exaples above seemed to indicate. So
including
> 5:7:9:11 adds some breadth.

Then you get some more interesting results, for example a small
positive v will lead the brain to hear this, with some weight, as
3:4:5:6.
>
> I suppose short of that, a few canned results on a web page would
do the
> trick.

That I'd be happy to provide . . . you have to all promise to use
ultra hi-fi equipment, though :)

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> I'm really not understanding you. 10:12:15:18 would normally be
> considered a "minor seventh" chord, while 4:5:6:7 would normally be
> considered similar to a "dominant seventh" or "German augmented
sixth
> chord". Meanwhile, 1/(4:5:6:7) would be considered similar to
a "half-
> diminished seventh chord" or an inversion of a "minor sixth" chord.
> So I don't know what you're getting at. Care to clarify (only if
it's
> still relevant, of course)?
>
> -Paul

Sorry for the mess. Just assume I should have sticked to, for
instance, 4:5:6:7 and (4:5:6:7) [60:70:84:105]. Thanks Monz for the
calc.

--- In tuning@yahoogroups.com, Alison Monteith

> The problem now is comparing inversions of the least consonant
harmonic
> tetrads with the most consonant subharmonic tetrads. This will have
to be
> done by ear

Not necessarily -- you might not be going too terribly wrong by
simply expressing the subharmonic tetrads in harmonic form and using
the same formulae. For example, the subharmonic tetrad 1/(7:6:5:4)
can be written as 60:70:84:105, and then the formula can be applied
to that.

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:

> Sorry for the mess. Just assume I should have sticked to, for
> instance, 4:5:6:7 and (4:5:6:7) [60:70:84:105]. Thanks Monz for the
> calc.

OK, so Plomp-Levelt-Sethares would say that these two chords are
about equally consonant, while harmonic entropy or Kraig's formula,
etc., would say that the first chord is much more consonant. Our
listening tests on the harmonic entropy list seem to support the
latter conclusion.

>There's nothing wrong with saying "irrational ratio".

Sounds hoaky to me. What about "irrational interval"?

-C.

>That I'd be happy to provide . . . you have to all promise to use
>ultra hi-fi equipment, though :)

Kurt's equipment isn't ultra hi-fi, but it comes closest of anybody
I know on this list!

-C.

on 11/29/03 11:39 AM, Carl Lumma <ekin@lumma.org> wrote:

>> That I'd be happy to provide . . . you have to all promise to use
>> ultra hi-fi equipment, though :)
>
> Kurt's equipment isn't ultra hi-fi, but it comes closest of anybody
> I know on this list!
>
> -C.

Send me your 24-bit 96Khz 30-channel sound files and I can render them
directly without any mix-down required. I use 3 Metric Halo MIO 2882's at
10 channels each for D/A conversion controlled by my dual 2GHz G5.

I have $3000 worth of speakers in my living room, and unfortunately (or
fortunately) those bucks got divided up into 30 separate speakers. A room
full of discontinued Wharfedale Valdus's bought on ebay - nothing to write
home about (except for the spectacle). Great for organ synthesis, and great
for synthetic reverb like you've probably never heard.

People in the SF bay area (or visiting) who want to hear it should just let
me know. My software organ has lost of scales and temperaments (many of
them from tunings list people) available, and dynamic (usually
pedalboard-cued) retuning based on "xmw" (xenharmonic moving windows) which
Carl and others (who?) worked on.

-Kurt

--- In tuning@yahoogroups.com, Alison Monteith

/tuning/topicId_48659.html#48659

<alison.monteith3@w...> wrote:
> Has anyone (Paul, Gene, Monz?) come up with a method for ranking
tetrads and
> hexanies according to their consonance/dissonance? In otherwords if
I have a
> list of Eikosany tetrads, 1.3.7.9, 1.3.5.11, etc., how, other than
by ear,
> can I rank them from more to less consonant?
>

***Hello Alison,

This is funny that this would come up right now, since I
just "revived" a page with Paul Erlich's experiment ranking tetrads
according to "Diadic Harmonic Entropy..."

http://www.soundclick.com/bands/5/tuninglabmusic.htm

Harmonic Entropy, though, is not for the "faint at heart" and, in
fact, a more accurate *Triadic* Harmonic Entropy is in the works but,
if I am understanding it correctly, so many calculations are involved
that computing power is not quite up to the task...

Joseph

Hi Joseph,

>This is funny that this would come up right now, since I
>just "revived" a page with Paul Erlich's experiment ranking tetrads
>according to "Diadic Harmonic Entropy..."
>
>http://www.soundclick.com/bands/5/tuninglabmusic.htm

Great find on soundclick! Looks a lot nicer than mp3.com anyway.

However, I'm confused... what order are the tetrads in on this
page, exactly? And which are "neccessarily tempered" -- the top
two?

Thanks,

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_48659.html#48829

> Hi Joseph,
>
> >This is funny that this would come up right now, since I
> >just "revived" a page with Paul Erlich's experiment ranking
tetrads
> >according to "Diadic Harmonic Entropy..."
> >
> >http://www.soundclick.com/bands/5/tuninglabmusic.htm
>
> Great find on soundclick! Looks a lot nicer than mp3.com anyway.
>
> However, I'm confused... what order are the tetrads in on this
> page, exactly? And which are "neccessarily tempered" -- the top
> two?
>
> Thanks,
>
> -Carl

***Hi Carl!

Yes, there is one thing quite a bit better about SoundClick than
mp3.com, and that is they are not "collecting" email addresses every
time somebody wants to hear something!! A lot of people resented
that, and for good reason, even though mp3.com didn't send much spam
(not *too* much, anyway... :)

Quite frankly, I'm a little confused about the ordering, too. I
maintained the original ordering from the mp3.com page. I *believe*
this was either a *subjective* ordering on *my* part, or a collective
consensus of the group. I don't remember, but maybe Paul will.

In any case, the *diadic* Harmonic Entropy order numbers are on the
examples, as you can see, so I could rearrange them like that as
well, but, as I recall, people didn't seem to *hear* them that
way! :)

So, basically, I'm open to any ordering suggestion that people feel
is most appropriate for the page. I'm not really sure, myself, what
that should be...

Thanks!

Joseph

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_48659.html#48721

>
> But something about this whole question surprises me: the fact
that it came
> up in a context in which (unless I missed something) timbre was not
> mentioned once. Surely you can't be talking about chords built up
from pure
> sinusoidal harmonics can you? But if not, how can you disregard
timbre
> (although putting inharmonicity side seems reasonable). But if so,
then you
> are talking about a different kind of consonance than what we I
have been
> assuming in most of the discussions I have followed on this list.
Certainly
> in some discussions (of piano tuning for example) inharmonicity
played a big
> role. So it would seem to follow that even without inharmonicity
being
> involved timbre must enter into the question of consonance.
>
> -Kurt
>

***This is a really interesting thought, since even in the study of
diadic harmonic entropy that Paul did a few years ago:

http://www.soundclick.com/bands/5/tuninglabmusic.htm

we never discussed what would happen to the rankings if the timbres
were different. I imagine that would influence the rankings, no??

J. Pehrson

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Hi Joseph,
>
> >This is funny that this would come up right now, since I
> >just "revived" a page with Paul Erlich's experiment ranking
tetrads
> >according to "Diadic Harmonic Entropy..."
> >
> >http://www.soundclick.com/bands/5/tuninglabmusic.htm
>
> Great find on soundclick! Looks a lot nicer than mp3.com anyway.
>
> However, I'm confused... what order are the tetrads in on this
> page, exactly?

Ah -- NB Joseph -- it's clear to me now -- they're in *alphabetical*
order according to some dumb computer program, most likely belonging
to mp3.com!

> And which are "neccessarily tempered" -- the top
> two?

No, but

Tetrad #2: 0__492__980__1472 cents

is, for example.

>> And which are "neccessarily tempered" -- the top
>> two?
>
>No, but
>
>Tetrad #2: 0__492__980__1472 cents
>
>is, for example.

Sorry, for some reason I thought you said the top
two, but looking at this thread I see you merely
said "some".

So is this a different chord than...

1/1 : 4/3 : 7:4 : 7:3

...and if not, does it sound any better?

http://lumma.org/tuning/a.mid

http://lumma.org/tuning/b.mid

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> And which are "neccessarily tempered" -- the top
> >> two?
> >
> >No, but
> >
> >Tetrad #2: 0__492__980__1472 cents
> >
> >is, for example.
>
> Sorry, for some reason I thought you said the top
> two, but looking at this thread I see you merely
> said "some".
>
> So is this a different chord than...
>
> 1/1 : 4/3 : 7:4 : 7:3

Yes, the interval between the middle two notes is much more consonant
in the non-JI rendition.

> ...and if not, does it sound any better?
>
> http://lumma.org/tuning/a.mid
>
> http://lumma.org/tuning/b.mid

All I can clearly hear is a lot more vibrato in version b.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> > ...and if not, does it sound any better?
> >
> > http://lumma.org/tuning/a.mid
> >
> > http://lumma.org/tuning/b.mid
>
> All I can clearly hear is a lot more vibrato in version b.

That only happened the first time; now both have a lot of vibrato.

>> >> And which are "neccessarily tempered" -- the top
>> >> two?
>> >
>> >No, but
>> >
>> >Tetrad #2: 0__492__980__1472 cents
>> >
>> >is, for example.
>>
>> Sorry, for some reason I thought you said the top
>> two, but looking at this thread I see you merely
>> said "some".
>>
>> So is this a different chord than...
>>
>> 1/1 : 4/3 : 7:4 : 7:3
>
>Yes, the interval between the middle two notes is much more consonant
>in the non-JI rendition.
>
>> ...and if not, does it sound any better?
>>
>> http://lumma.org/tuning/a.mid
>>
>> http://lumma.org/tuning/b.mid
>
>All I can clearly hear is a lot more vibrato in version b.

Version b sounds smoother and more locked to me.

The vibrato was probably due to the choice of cello patch
(my cello may have less than yours). Now the files are
in reed organ.

Maybe a different inversion would give different results?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> http://lumma.org/tuning/a.mid
> >>
> >> http://lumma.org/tuning/b.mid
> >
> >All I can clearly hear is a lot more vibrato in version b.
>
> Version b sounds smoother and more locked to me.
>
> The vibrato was probably due to the choice of cello patch
> (my cello may have less than yours). Now the files are
> in reed organ.

Thanks, it's now completely obvious which is which. The second one
does sound more 'locked', but still 'sour'. The first one is too
beaty and rough (especially in this timbre) and probably could
benefit from a little finer gradation than the 2 cents of the
original calculation. Try it in 22-equal -- the chord occurs in my
piece _TIBIA_.

> Maybe a different inversion would give different results?

In this voicing the 22-equal version is very near a local mimimum of
total diadic harmonic entropy. That won't be true for most, if any,
other voicings. The reason is the two 7:4 approximations and the one
7:3 approximation -- in most inversions these intervals will be too
complex to influence the three 4:3s (or 3:2s!) much, except to push
them in the *opposite* (meantone) direction since when 7:3 becomes
12:7, it gets tugged beyond 27:16 and toward 5:3.

Hi Alison (if you're still reading),

In light of Joseph's repost of part of your message below, I'm
wondering if you're actually planning to consider all the voicings of
each tetrad as a single entity.

In that case, I'd strongly recommend *not* using the product formula
over the odd factors (since the product formula is only intended to
be used for the terms, even & odd, in a specific voicing of a chord),
but instead using simply the largest odd number all by itself -- this
will represent a far better "average over voicings" discordance
measure.

Though again, the formula will work far better for otonal chords than
for utonal chords.

Let me know how you're progressing on this project, and of course
feel free to ignore me especially if you're getting your hands dirty
with the musical materials themselves -- a much more important
process than any calculations, especially with 'half-assed' measures
like these. I'd love to hear the music you come up with in any case.

-Paul

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, Alison Monteith
>
> /tuning/topicId_48659.html#48659
>
> <alison.monteith3@w...> wrote:
> > Has anyone (Paul, Gene, Monz?) come up with a method for ranking
> tetrads and
> > hexanies according to their consonance/dissonance? In otherwords
if
> I have a
> > list of Eikosany tetrads, 1.3.7.9, 1.3.5.11, etc., how, other
than
> by ear,
> > can I rank them from more to less consonant?
> >
>
>
> ***Hello Alison,
>
> This is funny that this would come up right now, since I
> just "revived" a page with Paul Erlich's experiment ranking tetrads
> according to "Diadic Harmonic Entropy..."
>
> http://www.soundclick.com/bands/5/tuninglabmusic.htm
>
> Harmonic Entropy, though, is not for the "faint at heart" and, in
> fact, a more accurate *Triadic* Harmonic Entropy is in the works
but,
> if I am understanding it correctly, so many calculations are
involved
> that computing power is not quite up to the task...
>
> Joseph

on 11/30/03 1:19 PM, Joseph Pehrson <jpehrson@rcn.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> /tuning/topicId_48659.html#48721
>
>> But something about this whole question surprises me: the fact
> that it came
>> up in a context in which (unless I missed something) timbre was not
>> mentioned once. Surely you can't be talking about chords built up
> from pure
>> sinusoidal harmonics can you? But if not, how can you disregard
> timbre
>> (although putting inharmonicity side seems reasonable). But if so,
> then you
>> are talking about a different kind of consonance than what we I
> have been
>> assuming in most of the discussions I have followed on this list.
> Certainly
>> in some discussions (of piano tuning for example) inharmonicity
> played a big
>> role. So it would seem to follow that even without inharmonicity
> being
>> involved timbre must enter into the question of consonance.
>>
>> -Kurt
>
> ***This is a really interesting thought, since even in the study of
> diadic harmonic entropy that Paul did a few years ago:
>
> http://www.soundclick.com/bands/5/tuninglabmusic.htm
>
> we never discussed what would happen to the rankings if the timbres
> were different. I imagine that would influence the rankings, no??

Yes, and I can't help but think of this also because of other recent
threads: if harmonic function can influence consonance/dissonance of dyads,
I'm staggered to think how it could influence tetrads. So what are the
experimental conditions under which consonance/dissonance is ranked so that
harmonic function is not interfering? When tasting wine you need to clear
the palette. Is this issue dealt with in subjective rating experiments?

-Kurt

>
> J. Pehrson

on 12/1/03 4:22 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
>>>> http://lumma.org/tuning/a.mid
>>>>
>>>> http://lumma.org/tuning/b.mid
>>>
>>> All I can clearly hear is a lot more vibrato in version b.
>>
>> Version b sounds smoother and more locked to me.
>>
>> The vibrato was probably due to the choice of cello patch
>> (my cello may have less than yours). Now the files are
>> in reed organ.
>
> Thanks, it's now completely obvious which is which. The second one
> does sound more 'locked', but still 'sour'. The first one is too
> beaty and rough (especially in this timbre) and probably could
> benefit from a little finer gradation than the 2 cents of the
> original calculation. Try it in 22-equal -- the chord occurs in my
> piece _TIBIA_.

In the first one I tend to hear a single entity, in spite of rough movement.
(I'm used to rough movement in "celeste" renderings from french romantic
organ.) In the second I am getting an impression that seems to conflict
with my ability to have an "overall" sense of pitch (and yes, I mean pitch,
not interval). It is as if I am more distracted by some of the upper notes
in a way that (to use your word from below) tugs my perception away from the
lower ones. Very curious that the just chord should create a more divided
perception and the tempered a more unified one. Perhaps the movement
creates some forgiveness; perhaps it also allows the perception to rest more
("lulled" by the ease of the movement) into a wholistic perception. Perhaps
the 7 is harder to unify than the 3 (no surprise), and perhaps there is a
unifying way of hearing which "kicks in" when sufficient ambiguity is
present, and which the just rendering does not allow because of its
explicitness.

>> Maybe a different inversion would give different results?
>
> In this voicing the 22-equal version is very near a local mimimum of
> total diadic harmonic entropy. That won't be true for most, if any,
> other voicings. The reason is the two 7:4 approximations and the one
> 7:3 approximation -- in most inversions these intervals will be too
> complex to influence the three 4:3s (or 3:2s!) much, except to push
> them in the *opposite* (meantone) direction since when 7:3 becomes
> 12:7, it gets tugged beyond 27:16 and toward 5:3.

It would be nice to have a tool (web page?) that displayed an N-et note
space of several octaves (and I don't mean to imply 2/1 octaves as a
requirement), and allowed chords of given ratios to be mapped on to the N-et
space. Does something like this exist? I realize the conflict between
rendering individual intervals in an et and rendering a chord. But you seem
to be assuming in what you say above that there is an understood method of
chosing the best N-et rendering for a given chord. Is this something that
scala even does in a sufficiently sophisticated way as to be generally
agreeable and not require hand-tweaking? (That is: rounding to the closest
note requires some anchoring. It is assumed that the chord note spelled as
"1/1" will fall on an exact N-et note?) And this also assumes no
restrictive horizontal dependencies are involved, something which I am still
questioning and haven't had enough experience with. Obviously I'm a little
staggerd by the implications of N-et renderings of just-spelled chords.

Perhaps the question of equal-beating should be brought into the picture?
This goes beyond N-et and I have no idea where to "start" with it. This
depends on understanding the composer's intention in chosing a chord such as
Carl's example

1/1 : 4/3 : 7:4 : 7:3

What is the best equal-beating tempered rendition of this chord?

If tempering is necessary, doesn't this mean the original intention of the
chord choice might be flawed?

-Kurt

>

I just LOVE the 21/16 even more than the 27/20

>
>
> From: "Paul Erlich" <paul@stretch-music.com>
> Subject: Re: Ranking Eikosany tetrads and hexanies
>
>
> >
> > 1/1 : 4/3 : 7:4 : 7:3
>
> Yes, the interval between the middle two notes is much more consonant
> in the non-JI rendition.
>
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> When tasting wine you need to clear
> the palette. Is this issue dealt with in subjective rating
> experiments?

It should be, but the experiment was not scientific, it was left up
to each subject to cleanse his or her palette. The bottom note was
common to every tetrad, which may not have been the best idea because
it might have led some pairs tetrads to be interpreted according
to 'diatonic function' and evaluated in that manner.

BTW, we used to use, after Blackwood, 'concordance/discordance' to
mean precisely this divorced-from-context judgment that we're hoping
to get at here -- maybe we should resume making this linguistic
distinction.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > In this voicing the 22-equal version is very near a local mimimum
of
> > total diadic harmonic entropy. That won't be true for most, if
any,
> > other voicings. The reason is the two 7:4 approximations and the
one
> > 7:3 approximation -- in most inversions these intervals will be
too
> > complex to influence the three 4:3s (or 3:2s!) much, except to
push
> > them in the *opposite* (meantone) direction since when 7:3 becomes
> > 12:7, it gets tugged beyond 27:16 and toward 5:3.
>
> It would be nice to have a tool (web page?) that displayed an N-et
note
> space of several octaves (and I don't mean to imply 2/1 octaves as a
> requirement), and allowed chords of given ratios to be mapped on to
the N-et
> space. Does something like this exist? I realize the conflict
between
> rendering individual intervals in an et and rendering a chord. But
you seem
> to be assuming in what you say above that there is an understood
method of
> chosing the best N-et rendering for a given chord.

No I'm not assuming anything of the sort, except for this one chord
and one ET, where there's no conceivable question which approximation
is best! Where did we lost each other?

> Is this something that
> scala even does in a sufficiently sophisticated way as to be
generally
> agreeable and not require hand-tweaking? (That is: rounding to the
closest
> note requires some anchoring. It is assumed that the chord note
spelled as
> "1/1" will fall on an exact N-et note?

I would never assume that, since that note might not even be present,
and even if it is, intervals not involving it might be more important
to approximate well -- as I pointed out with the 64-equal example
here recently)

> And this also assumes no
> restrictive horizontal dependencies are involved, something which I
>am still
> questioning and haven't had enough experience with.

Can you give an example?

> Obviously I'm a little
> staggerd by the implications of N-et renderings of just-spelled
>chords.

Normally, I insist on consistency

http://www.sonic-arts.org/dict/consiste.htm

through a high enough odd-limit to capture enough of the intervals of
the chord such that the chord can be built from those intervals
alone. But one could laxify these standards considerably before one
hit the muddy waters of ambiguity.

> If tempering is necessary, doesn't this mean the original intention
of the
> chord choice might be flawed?

What do you mean by that? I thought you appreciated that the
necessarily tempered chord sounded more 'unified' than Carl's
just 'de-tempering' of it. Is this 'unified' flavor not enough of a
justification for chord choice?

hi paul,

> BTW, we used to use, after Blackwood, 'concordance/discordance'
> to mean precisely this divorced-from-context judgment that
> we're hoping to get at here -- maybe we should resume making
> this linguistic distinction.

yes, absolutely! by all means!

you can see that i deliberately created Dictionary pages
for "accordance", "concordance", and "discordance" to
supercede what i used to have for "sonance", "consonance",
and "dissonance", which now have their very own
context-sensitive meanings.

since doing that, i've been careful to always use
the "...cordance" terms instead of "...sonance" terms
in my tuning-list posts, unless i specifically meant to
refer to compositional contexts which create or imply
some type of "...sonance".

-monz

>Obviously I'm a little
>staggerd by the implications of N-et renderings of just-spelled
>chords.

Not sure what you're asking. Normally it's just a matter of finding
the lowest error from JI. Usually Paul and I prefer RMS error. But
in this case the target chord isn't just -- JI is an approximation to
it. The target chord is given by a local minimum in the pairwise
harmonic entropy function for tetrads.

>If tempering is necessary, doesn't this mean the original intention of
>the chord choice might be flawed?

How could it?

-Carl

on 12/2/03 12:02 AM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>>> In this voicing the 22-equal version is very near a local mimimum
> of
>>> total diadic harmonic entropy. That won't be true for most, if
> any,
>>> other voicings. The reason is the two 7:4 approximations and the
> one
>>> 7:3 approximation -- in most inversions these intervals will be
> too
>>> complex to influence the three 4:3s (or 3:2s!) much, except to
> push
>>> them in the *opposite* (meantone) direction since when 7:3 becomes
>>> 12:7, it gets tugged beyond 27:16 and toward 5:3.
>>
>> It would be nice to have a tool (web page?) that displayed an N-et
> note
>> space of several octaves (and I don't mean to imply 2/1 octaves as a
>> requirement), and allowed chords of given ratios to be mapped on to
> the N-et
>> space. Does something like this exist? I realize the conflict
> between
>> rendering individual intervals in an et and rendering a chord. But
> you seem
>> to be assuming in what you say above that there is an understood
> method of
>> chosing the best N-et rendering for a given chord.
>
> No I'm not assuming anything of the sort, except for this one chord
> and one ET, where there's no conceivable question which approximation
> is best! Where did we lost each other?

I got confused by Carl's de-tempering (as you coined below) and your use of
the word "tempered" which by its nature had the connotation that the just
must have pre-dated the tempered, in spite of cues to the contrary. So I
forgot the "necessarily tempered" origin, forgot to ask what "necessarily
tempered" meant--that was some old disucssion from before my time on this
list. Searching the tunings list archives (just now) for the phrase did not
yield anything useful, but I think the meaning is already starting to gel,
from your response and Carl's.

>> Is this something that
>> scala even does in a sufficiently sophisticated way as to be
> generally
>> agreeable and not require hand-tweaking? (That is: rounding to the
> closest
>> note requires some anchoring. It is assumed that the chord note
> spelled as
>> "1/1" will fall on an exact N-et note?
>
> I would never assume that, since that note might not even be present,
> and even if it is, intervals not involving it might be more important
> to approximate well -- as I pointed out with the 64-equal example
> here recently)

Good.

>> And this also assumes no
>> restrictive horizontal dependencies are involved, something which I
>> am still
>> questioning and haven't had enough experience with.
>
> Can you give an example?

Well, actually, with the apparently strong consensus that horizontal
adjustment can be allowed to be pretty flexible (as long as it doesn't
change too quickly), as recently confirmed also by Werner Mohrlok, I can
perhaps let go of this point. That whole Hermode Tuning thread touches on
the issues I had in mind here, and clearly I need to assimilate more before
I can comment further.

>> Obviously I'm a little
>> staggerd by the implications of N-et renderings of just-spelled
>> chords.
>
> Normally, I insist on consistency
>
> http://www.sonic-arts.org/dict/consiste.htm

Aha. That idea mediates some of my concerns, some of which I have
expresssed in my response to Carl on this same thread, which for some silly
reason I have been writing in parallel to this response, and which I will
allow myself to send without rewriting (much).

> through a high enough odd-limit to capture enough of the intervals of
> the chord such that the chord can be built from those intervals
> alone. But one could laxify these standards considerably before one
> hit the muddy waters of ambiguity.

So Carl's example

1/1 : 4/3 : 7:4 : 7:3

chord is problematic because of the 21:16 interval which is made visible in
this respelling

3/4 : 1 : 21/16 : 7/4

because 21 requires too high of an et to assimilate it? The chart on the
consiste.htm only goes up to 20, so doesn't help here. I am rather
surprised at the consistency results and would have expected much more
"difficulty" in building JI chords in almost any et. So this is kind of
good numerical news.

>> If tempering is necessary, doesn't this mean the original intention
> of the
>> chord choice might be flawed?
>
> What do you mean by that? I thought you appreciated that the
> necessarily tempered chord sounded more 'unified' than Carl's
> just 'de-tempering' of it. Is this 'unified' flavor not enough of a
> justification for chord choice?

Yes, I heard that, but was confused - as explained above.

Yet again I am surprised in a way that tempering should clarify a JI chord.
And on another level I have not made sense of utonality at all and somehow
would not expect this unification in anything but a purely otonal chord
(assuming a JI chord in the first place). By "purely otonal" I also mean to
imply a "reasonably" low odd-limit, so that the implied fundamental would
not be too low.

And there I wonder whether limits of ordinary hearing enter into the
perception of implied fundamentals. It would be interesting to know for
example whether a perceived "unified flavor" of a chord depends on a single
implied fundamental for the entire chord being in the audible range. And it
would be interesting to extend the same question to tempered chords that are
sufficiently out-of-JI to destroy any "apparent" hearing of the implied
fundamental. (I'm taking some liberties with terminology that is fairly new
to me, so I hope it works out ok.)

-Kurt

on 12/2/03 1:02 AM, Carl Lumma <ekin@lumma.org> wrote:

>> Obviously I'm a little
>> staggerd by the implications of N-et renderings of just-spelled
>> chords.
>
> Not sure what you're asking. Normally it's just a matter of finding
> the lowest error from JI.

Yes, and even this still staggers me. But Paul's new "Apprx&Temperament"
post gives me more reasons to feel that "approximation" of JI (and therefore
definitions of error) can be complex. And it brings into question any
concept of numerical error as a primary criterion, and at least complicates
the notion when "stacking" of JI intervals that each have their own life is
involved.

> Usually Paul and I prefer RMS error.

Yes, and I should look back at what has been said about this. Minimize
error on all the intervals? Not just consecutive intervals? Not just
certain intervals perhaps considered more important or more dominant than
others? Recent experience suggests to me that certain intervals of a chord
*do* dominate the sense of consonance. Other recent experience indicates
that what I might now call "concordant beating" may be more globally (within
a chord) important than individual intervals, possibly even in making
certain intervals "hearable".

> But
> in this case the target chord isn't just -- JI is an approximation to
> it. The target chord is given by a local minimum in the pairwise
> harmonic entropy function for tetrads.

I think you are saying the target chord is the 0__492__980__1472 cents one.
I got confused on a subtle point in Paul's response to you:

on 12/1/03 2:57 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>>> And which are "neccessarily tempered" -- the top
>>>> two?
>>>
>>> No, but
>>>
>>> Tetrad #2: 0__492__980__1472 cents
>>>
>>> is, for example.
>>
[snip]
>>
>> So is this a different chord than...
>>
>> 1/1 : 4/3 : 7:4 : 7:3
>
> Yes, the interval between the middle two notes is much more consonant
> in the non-JI rendition.

In reading that, I started thinking of the 0__492__980__1472 cents as a
non-JI "rendition" of the "1/1 : 4/3 : 7:4 : 7:3" JI chord, in spite of the
fact that Paul said "Yes" to your question of whether it was a different
chord. I think that was what got me confused.

But also the use of "tempered" confused me. I think of tempering as
something that is done in designing a *scale*, compromising a more "ideal",
possibly unbounded or in any case larger scale in order to create a bounded
or smaller scale, mind you possibly creating unanticipated musical utility
as a result. But in spite of that understanding that tempering has musical
utility, I have not thought of tempering a chord before, in the absense of
other constraints. And I have certainly not thought of tempering as a way
of making a chord possibly more clear. (Though sometimes I wish I was *not*
hearing an implied fundamental and want to temper enough to get rid of it.)

>> If tempering is necessary, doesn't this mean the original intention of
>> the chord choice might be flawed?
>
> How could it?

I wrote this thinking that the target chord was a JI chord. Sorry.

Still the question remains based on all the information currently "flowing"
in this thread whether tempering a JI chord might help to clarify it or
whether that would necessarily imply an ambiguous relationship to JI in the
first place.

-Kurt

>
> -Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> So Carl's example
>
> 1/1 : 4/3 : 7:4 : 7:3
>
> chord is problematic

OK, stop right there -- it's problematic? Are you referring to the
original context in which the chord came up, or somethine else? I'll
assume the former in what follows . . .

> because of the 21:16 interval which is made visible in
> this respelling
>
> 3/4 : 1 : 21/16 : 7/4
>
> because 21 requires too high of an et to assimilate it?

No, ETs have nothing to do with this. A 21:16 dyad alone simply can't
be pinpointed by ear, the way the intervals used to tune just
intonation are, since it isn't an acoustical minimum of anything --
except in unusual timbral circumstances. In the chord above, it
occurs as a by-product of other interlocking concordant intervals,
much like 15:8 occurs as a by-product of five interlocking concordant
intervals in the 'just major seventh chord' 1/1 : 5/4 : 3/2 : 15/8 or
8:10:12:15. So this chord isn't much different, only it's harder to
fit into a single harmonic series; it would be 12:16:21:28 in those
terms -- kinda high up there.

> The chart on the
> consiste.htm only goes up to 20, so doesn't help here. I am rather
> surprised at the consistency results and would have expected much
more
> "difficulty" in building JI chords in almost any et. So this is
kind of
> good numerical news.

Paul Hahn has a better presentation of the data. This chart shows the
stepsize and the greatest error among the primary or "consonant"
intervals within the harmonic (odd-)limit, both in cents, for all ETs
up to 200TET, for all harmonic limits within which they are
consistent. So each line goes,

ET stpsz max-3-lmt-err max-5-lmt-err max-7-lmt-err max-9-limit-err . .

and peters out when inconsistency is reached:

http://library.wustl.edu/~manynote/consist3.txt

> >> If tempering is necessary, doesn't this mean the original
intention
> > of the
> >> chord choice might be flawed?
> >
> > What do you mean by that? I thought you appreciated that the
> > necessarily tempered chord sounded more 'unified' than Carl's
> > just 'de-tempering' of it. Is this 'unified' flavor not enough of
a
> > justification for chord choice?
>
> Yes, I heard that, but was confused - as explained above.
>
> Yet again I am surprised in a way that tempering should clarify a
JI chord.
> And on another level I have not made sense of utonality at all and
somehow
> would not expect this unification in anything but a purely otonal
chord
> (assuming a JI chord in the first place). By "purely otonal" I
also mean to
> imply a "reasonably" low odd-limit, so that the implied fundamental
would
> not be too low.

Aha -- now you're talking about a different chordal odd-limit
definition than what's explicitly in there now (which we can
call "intervallic limit"), let's call it the "otonal limit", which I
just hinted at in my last post to Alison. I think SCALA already
calculates both. I also think, assuming low enough numbers, that both
the "otonal limit" (since it models harmonic entropy) and
the "intervallic limit" (since it models roughness) are relevant to a
chord's concordance, but the "utonal limit" isn't -- if you think it
is, try calculating it for the chords below. See

http://www.cix.co.uk/~gbreed/erlichs.htm

and

http://www.cix.co.uk/~gbreed/ass.htm

First, let's look at the major triad -- intervallic limit and otonal
limit both 5. Don't get it? Read on . . .

Let's look at the minor triad; we can strip away the 2s since we're
assuming octave equivalence. Its otonal limit is 15 since it's
expressed as 3:5:15, but its intervallic limit is 5, since none of
its intervals exceed an odd-limit of 5.

Next, let's look at the major seventh chord again, 8:10:12:15, otonal
limit also 15. Its intervallic limit *would* be 5 except for that one
ratio of 15, 15:8 -- the other five intervals in the chord are much
more concordant. Thus, it's quite a bit more discordant than the
minor triad, but beautiful nonetheless.

Now look at the triad 7:9:13 -- both limits are 13. Otonally it's no
more complex than the minor triad, but intervallically it's far
rougher -- this is revealed by listening.

Finally, consider 1/9:1/7:1/1. Intervallic limit is 9, but otonally
the chord is 7:9:63 -- a doubtfully high stretch up the harmonic
series by any standard. How does it compare with 7:9:13, in your
opinion?

(of course the answer will probably depend on timbre . . .)

When speaking of specific voicings, hence in octave-specific terms,
it would seem that one should use an 'integer-limit' rather than 'odd-
limit' definition for the otonal and intervallic limits. That's fine.
But it's actually better to use *products*, rather than odd- or
integer-limits, to evaluate the two components of concordance (as
always, the caveat: for JI chords where the numbers are not too
large).

> And there I wonder whether limits of ordinary hearing enter into the
> perception of implied fundamentals. It would be interesting to
know for
> example whether a perceived "unified flavor" of a chord depends on
a single
> implied fundamental for the entire chord being in the audible range.

The stack of narrowed fourths -- the chord you identified as having
a 'unified flavor' -- certainly does not have a single, clear implied
fundamental -- if it is indeed interpreted that way, it can be heard
somewhat as 9:12:16:21, somewhat as 12:16:21:28, amybe somewhat as
27:36:48:64, etc. But any of these rendered as a JI chord would have
an intervallic limit of at least 21, while the stack of narrowed
fourths would have a supposed intervallic limit of 7 -- mitigated
somewhat (but not completely, if the tempering is successful) by the
impurity of the six supposed 7-limit concordant intervals. This,
then, appears to represent the 'unification' effected by the
tempering of the chord -- all the intervals are close to lining up
lots of partials, hence the sensation is an 'integrated' one, without
implying a fundamental -- much like the first inversion of the minor
triad (OK, getting octave-specific there), or better yet, like some
of Bill Sethares's chords where Tuning has been made to nearly match
some inharmonic Timbre.

> And it
> would be interesting to extend the same question to tempered chords
that are
> sufficiently out-of-JI to destroy any "apparent" hearing of the
implied
> fundamental. (I'm taking some liberties with terminology that is
fairly new
> to me, so I hope it works out ok.)

I think in this case, there is relatively little clarity to the
implied fundamental even in Carl's JI chord, or the first-inversion
minor triad, or in the 1/9:1/7:1/1 chord mentioned earlier. So chords
need not be tempered to exhibit this 'fundamentallessness' :) On the
other hand, if you start with a clear otonal chord, and move it
sufficiently out-of-JI to destroy the apparent fundamental, but no
further, you're going to end up with a not-unified, discordant
sonority.

:) Join the harmonic entropy list :)

-Paul

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> and at least complicates
> the notion when "stacking" of JI intervals that each have their own
life is
> involved.

That shouldn't complicate the notion, since the "stacking" takes care
of itself automatically as long as the "rungs" in the lattice satisfy
whatever 'concordance' (or other?) criterion that's deemed necessary
to make the language of, or a mapping from, ratios meaningful.

> > Usually Paul and I prefer RMS error.
>
> Yes, and I should look back at what has been said about this.
Minimize
> error on all the intervals? Not just consecutive intervals? Not
just
> certain intervals perhaps considered more important or more
dominant than
> others?

Well, we've occasionally used various "weighted RMS" schemes where
ratios of 3 are weighted differently than ratios of 5 . . . for
example, my table of optimal meantone tunings here:

http://www.sonic-arts.org/dict/meantone.htm

shows that if RMS is used, the optimal meantone (at least as far as
complete 5-limit triads are concerned) is 63/250-comma meantone, 7/26-
comma meantone, 175/634-comma meantone, or 7/25-comma meantone --
depending on whether the ratio of {the weight on the ratios of 5 to
the weight on the ratio of 3} is 3:5, 1:1, 5:3, or 1:0, respectively.

Where there is inconsistency, but only there, there's a chance that
changing the weighting could change your choice of the N-equal
rendition of a just chord.

> Recent experience suggests to me that certain intervals of a chord
> *do* dominate the sense of consonance. Other recent experience
indicates
> that what I might now call "concordant beating" may be more
globally (within
> a chord) important than individual intervals, possibly even in
making
> certain intervals "hearable".

Do you have an example in mind?

> > But
> > in this case the target chord isn't just -- JI is an
approximation to
> > it. The target chord is given by a local minimum in the pairwise
> > harmonic entropy function for tetrads.
>
> I think you are saying the target chord is the 0__492__980__1472
cents one.

Yes, though this was only calculated to a 2 cent resolution -- 22-
equal makes for a nicer comprimise with 0__491__982__1473.

> I got confused on a subtle point in Paul's response to you:

Yes; I'm glad you sorted that out (and kept me up all night with your
interesting posts)!

> But also the use of "tempered" confused me. I think of tempering as
> something that is done in designing a *scale*, compromising a
more "ideal",
> possibly unbounded or in any case larger scale in order to create a
bounded
> or smaller scale, mind you possibly creating unanticipated musical
utility
> as a result.

Yes.

> But in spite of that understanding that tempering has musical
> utility, I have not thought of tempering a chord [rather than a
>scale] before, in the absense of
> other constraints.

Think of a C 6/9 chord, C-E-G-A-D, vs. a C major pentatonic scale, C-
D-E-G-A. Doesn't meantone temperament make sense for both, for much
the same reasons?

Kurt wrote:

>Is this something that
>scala even does in a sufficiently sophisticated way as to be generally
>agreeable and not require hand-tweaking?

I think so, but whether it requires tweaking should be decided
after listening.

>(That is: rounding to the closest
>note requires some anchoring. It is assumed that the chord note spelled
as
>"1/1" will fall on an exact N-et note?)

The QUANTIZE/BEST command doesn't assume this.
There's also QUANTIZE/CONSISTENT for a prime-consistent ET approximation.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> The QUANTIZE/BEST command doesn't assume this.
> There's also QUANTIZE/CONSISTENT for a prime-consistent ET
>approximation.

What does prime-consistent mean? Consistency is normally defined
within some odd-limit; but any prime limit will eventually include
some interval whose approximation differs from the result of
splitting it into simpler intervals and adding the approximations of
those.

>but any prime limit will eventually include
>some interval whose approximation differs from the result of
>splitting it into simpler intervals and adding the approximations of
>those.

That's what I meant, the approximation is formed by factoring the
interval into simpler intervals and adding the approximations of
those.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> >but any prime limit will eventually include
> >some interval whose approximation differs from the result of
> >splitting it into simpler intervals and adding the approximations
of
> >those.
>
> That's what I meant, the approximation is formed by factoring the
> interval into simpler intervals and adding the approximations of
> those.

And how are the approximations of the simpler intervals determined?
Are they undefined if the ET is inconsistent in the highest odd-limit
not exceeding the next prime after the given prime-limit? Or if the
ET is inconsistent in the odd-limit equal to the given prime-limit?
Or is some consistent set of approximations, perhaps using 'best',
decided on by the program for one of these odd-limits? Or is it up to
the user? . . .

>And how are the approximations of the simpler intervals determined?

The nearest step is taken for each prime. Whether the ET is
consistent or inconsistent doesn't play a role.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> >And how are the approximations of the simpler intervals
determined?
>
> The nearest step is taken for each prime. Whether the ET is
> consistent or inconsistent doesn't play a role.

I'm not saying it should, only that there can sometimes be better
choices for the primes than the nearest interval to each prime, as
recently mentioned on tuning-math. And yes, these choices may differ
from the "nearest interval to each prime" choice only (but not
necessarily) if the ET is inconsistent.

>> Usually Paul and I prefer RMS error.
>
>Yes, and I should look back at what has been said about this. Minimize
>error on all the intervals? Not just consecutive intervals? Not just
>certain intervals perhaps considered more important or more dominant
>than others?

All the odd-limit intervals.

>Recent experience suggests to me that certain intervals of a chord
>*do* dominate the sense of consonance.

Yeah, with chords it's more tricky. Currently, the only mature tool
we have is looking at all the dyads in a chord (sometimes called
"pairwise ..."). But true tetradic harmonic entropy works on tetrads
as atoms. Pretty cool stuff.

>Other recent experience indicates
>that what I might now call "concordant beating" may be more globally
>(within a chord) important than individual intervals, possibly even
>in making certain intervals "hearable".

Well I don't know what you mean by concordant beating. If you mean
"equal beating", for me it doesn't significantly improve the
consonance of chords, although no thorough test of this has ever been
done (I've asked Gene for a list of chords to render for a test).

Charles Lucy has made claims that his meantone temperament has beat
rates that tend to entrain the brain, as it were, in alpha states.
YMMV.

As for making intervals more "hearable", there have been extensive
debates here about pure JI causing notes in a chord to phase-lock out
of existence. My take was that in practical terms the accuracy is
rarely great enough for this to be a problem, the timbre will prevent
it (inharmonicity on a piano, say), and most importantly, that voice
leading can indicate where the notes are (experiments have shown that
attention is crucial in directing what we hear, "priming", "pop-out",
and so on...).

>> But
>> in this case the target chord isn't just -- JI is an approximation
>> to it. The target chord is given by a local minimum in the pairwise
>> harmonic entropy function for tetrads.
>
>I think you are saying the target chord is the 0__492__980__1472 cents
>one. I got confused on a subtle point in Paul's response to you:

Yep.

>>> So is this a different chord than...
>>>
>>> 1/1 : 4/3 : 7:4 : 7:3
>>
>> Yes, the interval between the middle two notes is much more consonant
>> in the non-JI rendition.
>
>In reading that, I started thinking of the 0__492__980__1472 cents as a
>non-JI "rendition" of the "1/1 : 4/3 : 7:4 : 7:3" JI chord, in spite of
>the fact that Paul said "Yes" to your question of whether it was a
>different chord. I think that was what got me confused.
>
>But also the use of "tempered" confused me. I think of tempering as
>something that is done in designing a *scale*, compromising a more
>"ideal",

Yes, that is a more traditional usage. But here, and especially on
tuning math, we've introduced/refined a much more interesting definition!
Mwhahahahaha!

>And I have certainly not thought of tempering as a way
>of making a chord possibly more clear.

That's Paul's idea (at least I haven't seen it elsewhere). For a
while we were calling these "magic" chords. Except I've yet to hear
one that I think really works.

There are certainly irrational ("tempered") chords that are targets
in their own right. The diminished 7ths and augmented triads of
12-equal being classic examples.

>(Though sometimes I wish I was *not*
>hearing an implied fundamental and want to temper enough to get rid
>of it.)

Well, if you perturb the tuning of a chord enough to remove the
implied fundamental, it's safe to say you've got a different chord
on your hands. Finding chords with highly ambiguous fundamentals
is something you can do with harmonic entropy, though they're
usually quite discordant. Find-the-fundamental isn't a game the
ear quits often.

Chords without a base-2 on the bottom sometimes sound less rooted,
since the implied fundamental is not actually being sounded. Here
you have to be careful to check both the bottom note of the entire
chord and the bottom note of prominent dyads in the chord. The
5-limit minor triad for example is 10:12:15, but the outer 2:3
usually makes the root pretty strong.

Then there are chords whose fundamental is ambiguous, but only
between maybe 2 choices, and context can determine which you'll hear.

-Carl

I wrote...

>Finding chords with highly ambiguous fundamentals
>is something you can do with harmonic entropy, though they're
>usually quite discordant.

An exception might be high-limit utonal chords. With the right
timbre and voicing (esp. in the high register) they can often be
rendered fairly concordant.

-Carl

on 2/12/03 01:24, Paul Erlich at paul@stretch-music.com wrote:

> Hi Alison (if you're still reading),
>
> In light of Joseph's repost of part of your message below, I'm
> wondering if you're actually planning to consider all the voicings of
> each tetrad as a single entity.
>
> In that case, I'd strongly recommend *not* using the product formula
> over the odd factors (since the product formula is only intended to
> be used for the terms, even & odd, in a specific voicing of a chord),
> but instead using simply the largest odd number all by itself -- this
> will represent a far better "average over voicings" discordance
> measure.
>
> Though again, the formula will work far better for otonal chords than
> for utonal chords.
>
> Let me know how you're progressing on this project, and of course
> feel free to ignore me especially if you're getting your hands dirty
> with the musical materials themselves -- a much more important
> process than any calculations, especially with 'half-assed' measures
> like these. I'd love to hear the music you come up with in any case.
>
> -Paul
>

Thanks for the advice Paul. Yes, I am following the thread with great
interest and I take on board the points you make above. When all is said and
done I'm happy to have at least a hint of a working method.

Sincerely
a.m.

on 30/11/03 18:27, Joseph Pehrson at jpehrson@rcn.com wrote:

> --- In tuning@yahoogroups.com, Alison Monteith
>
> /tuning/topicId_48659.html#48659
>
> <alison.monteith3@w...> wrote:
>> Has anyone (Paul, Gene, Monz?) come up with a method for ranking
> tetrads and
>> hexanies according to their consonance/dissonance? In otherwords if
> I have a
>> list of Eikosany tetrads, 1.3.7.9, 1.3.5.11, etc., how, other than
> by ear,
>> can I rank them from more to less consonant?
>>
>
>
> ***Hello Alison,
>
> This is funny that this would come up right now, since I
> just "revived" a page with Paul Erlich's experiment ranking tetrads
> according to "Diadic Harmonic Entropy..."
>
> http://www.soundclick.com/bands/5/tuninglabmusic.htm
>
> Harmonic Entropy, though, is not for the "faint at heart" and, in
> fact, a more accurate *Triadic* Harmonic Entropy is in the works but,
> if I am understanding it correctly, so many calculations are involved
> that computing power is not quite up to the task...
>
> Joseph
>

Hi Joseph

I understand the general principles but the details I find daunting. Still
it's refreshing to be able to ask questions and to get a variety of informed
answers.

Sincerely
a.m.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> As for making intervals more "hearable", there have been extensive
> debates here about pure JI causing notes in a chord to phase-lock
out
> of existence.

hmm? maybe you mean "pure frequency components destructively
interfering"? These wouldn't comprise entire "notes", except in cases
where a note's fundamental is a harmonic of another note, and you get
all the phases and amplitudes just right . . . rare.

> >And I have certainly not thought of tempering as a way
> >of making a chord possibly more clear.
>
> That's Paul's idea (at least I haven't seen it elsewhere). For a
> while we were calling these "magic" chords. Except I've yet to hear
> one that I think really works.
>
> There are certainly irrational ("tempered") chords that are targets
> in their own right. The diminished 7ths and augmented triads of
> 12-equal being classic examples.

so why aren't these "magic"?

> Then there are chords whose fundamental is ambiguous, but only
> between maybe 2 choices,

I don't see why you'd say that. Some chords may be ambiguous among
quite a few choices, for example a cluster of several stacked minor
seconds in 12-equal.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> I wrote...
>
> >Finding chords with highly ambiguous fundamentals
> >is something you can do with harmonic entropy, though they're
> >usually quite discordant.
>
> An exception might be high-limit utonal chords. With the right
> timbre and voicing (esp. in the high register) they can often be
> rendered fairly concordant.
>
> -Carl

And don't they, in that case, have fairly *unambiguous* fundamentals?
So isn't it not an exception at all? Or am I misreading you again?

>> There are certainly irrational ("tempered") chords that are targets
>> in their own right. The diminished 7ths and augmented triads of
>> 12-equal being classic examples.
>
>so why aren't these "magic"?

Because they don't sound in any way more concordant than similar
JI chords.

>> Then there are chords whose fundamental is ambiguous, but only
>> between maybe 2 choices,
>
>I don't see why you'd say that. Some chords may be ambiguous among
>quite a few choices, for example a cluster of several stacked minor
>seconds in 12-equal.

Ah, I didn't mean the "then there are" - "but" that way. I just
meant, chords with only 2 (or a few) choices may be interesting.

-Carl

>> An exception might be high-limit utonal chords. With the right
>> timbre and voicing (esp. in the high register) they can often be
>> rendered fairly concordant.
>>
>> -Carl
>
>And don't they, in that case, have fairly *unambiguous* fundamentals?
>So isn't it not an exception at all? Or am I misreading you again?

They would be not an exception, but they can still have ambiguous
fundamentals.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> There are certainly irrational ("tempered") chords that are
targets
> >> in their own right. The diminished 7ths and augmented triads of
> >> 12-equal being classic examples.
> >
> >so why aren't these "magic"?
>
> Because they don't sound in any way more concordant than similar
> JI chords.

So why are they "targets"?

> >> Then there are chords whose fundamental is ambiguous, but only
> >> between maybe 2 choices,
> >
> >I don't see why you'd say that. Some chords may be ambiguous among
> >quite a few choices, for example a cluster of several stacked
minor
> >seconds in 12-equal.
>
> Ah, I didn't mean the "then there are" - "but" that way. I just
> meant, chords with only 2 (or a few) choices may be interesting.

Yes, for dyads typical in timbre and register, a neutral third evokes
primarily 2 choices: 5:4 and 6:5. 11:9 not as strongly as either,
unless you've got highly harmonic timbre, fairly high register, and a
really good ear -- not to mention extra "supporting" pitches.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> An exception might be high-limit utonal chords. With the right
> >> timbre and voicing (esp. in the high register) they can often be
> >> rendered fairly concordant.
> >>
> >> -Carl
> >
> >And don't they, in that case, have fairly *unambiguous*
fundamentals?
> >So isn't it not an exception at all? Or am I misreading you again?
>
> They would be not an exception, but they can still have ambiguous
> fundamentals.

Can you give an example? And say what is making them, when rendered
as you say, concordant, if not a clear fundamental?

>> >> An exception might be high-limit utonal chords. With the right
>> >> timbre and voicing (esp. in the high register) they can often be
>> >> rendered fairly concordant.
>> >>
>> >> -Carl
>> >
>> >And don't they, in that case, have fairly *unambiguous*
>> >fundamentals?
>> >
>> >So isn't it not an exception at all? Or am I misreading you again?
>>
>> They would be not an exception, but they can still have ambiguous
>> fundamentals.
>
>Can you give an example? And say what is making them, when rendered
>as you say, concordant, if not a clear fundamental?

I've gotten this with utonal 9th chords on my slide guitar and with
tetrads of the 11-limit eikosany. I think a coincidence of partials
and lack of beating makes them concordant. And above 4K the
periodicity mechanism falls out; though I'm not playing them that
high probably more of their strong harmonics are in that range than
would be otherwise.

-Carl

>> because of the 21:16 interval which is made visible in
>> this respelling
>>
>> 3/4 : 1 : 21/16 : 7/4
>>
>> because 21 requires too high of an et to assimilate it?
>
>No, ETs have nothing to do with this. A 21:16 dyad alone simply can't
>be pinpointed by ear, the way the intervals used to tune just
>intonation are, since it isn't an acoustical minimum of anything --

In particular, the numbers involved in this dyad are pretty high
*and* it is very close to another very low-numbered ratio, 4:3.
So it tends to be very sour; a wolf 4th.

>Now look at the triad 7:9:13 -- both limits are 13. Otonally it's no
>more complex than the minor triad, but intervallically it's far
>rougher -- this is revealed by listening.
>
>Finally, consider 1/9:1/7:1/1. Intervallic limit is 9, but otonally
>the chord is 7:9:63 -- a doubtfully high stretch up the harmonic
>series by any standard. How does it compare with 7:9:13, in your
>opinion?

http://lumma.org/tuning/28-36-63.mid
http://lumma.org/tuning/7-9-13.mid

>The stack of narrowed fourths -- the chord you identified as having
>a 'unified flavor' -- certainly does not have a single, clear implied
>fundamental -- if it is indeed interpreted that way, it can be heard
>somewhat as 9:12:16:21, somewhat as 12:16:21:28, maybe somewhat as
>27:36:48:64, etc. But any of these rendered as a JI chord would have
>an intervallic limit of at least 21, while the stack of narrowed
>fourths would have a supposed intervallic limit of 7 -- mitigated
>somewhat (but not completely, if the tempering is successful) by the
>impurity of the six supposed 7-limit concordant intervals.

Yeah but to me it just sounds like a restless version of 12:16:21:28
or 9:12:16:21...

http://lumma.org/tuning/0_491_982_1473.mid
http://lumma.org/tuning/12-16-21-28.mid
http://lumma.org/tuning/9-12-16-21.mid

-Carl

>> >> There are certainly irrational ("tempered") chords that are
>> >> targets in their own right. The diminished 7ths and augmented
>> >> triads of 12-equal being classic examples.
>> >
>> >so why aren't these "magic"?
>>
>> Because they don't sound in any way more concordant than similar
>> JI chords.
>
>So why are they "targets"?

I was thinking along more along the 'puns are natural' line than
the 'magic' line.

Anywho, here's a dim7 chord (in 12-equal) with added octave,
being inverted...

http://lumma.org/tuning/300.mid

...and here's the same thing with a chain of 6:5s and added octave
(and thus a kleisma-fudged 6:5)...

http://lumma.org/tuning/6-5.mid

-Carl

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48659.html#48871

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > Hi Joseph,
> >
> > >This is funny that this would come up right now, since I
> > >just "revived" a page with Paul Erlich's experiment ranking
> tetrads
> > >according to "Diadic Harmonic Entropy..."
> > >
> > >http://www.soundclick.com/bands/5/tuninglabmusic.htm
> >
> > Great find on soundclick! Looks a lot nicer than mp3.com anyway.
> >
> > However, I'm confused... what order are the tetrads in on this
> > page, exactly?
>
> Ah -- NB Joseph -- it's clear to me now -- they're in
*alphabetical*
> order according to some dumb computer program, most likely
belonging
> to mp3.com!
>

***Well, that's pretty funny... I guess someday we can put them in
some kind of order if we wish. At least I got them off the mp3.com
site in time. The site is now history... and 240,000 people are
probably trying to find alternative hosts...

JP

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Well I don't know what you mean by concordant beating. If you mean
> "equal beating", for me it doesn't significantly improve the
> consonance of chords, although no thorough test of this has ever
been
> done (I've asked Gene for a list of chords to render for a test).

I don't recall this. When and where?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> There are certainly irrational ("tempered") chords that are targets
> in their own right. The diminished 7ths and augmented triads of
> 12-equal being classic examples.

Or for example the augmented triad of 225/224-planar, or the
diminished 7th of 126/125-planar, whoich are considerably better in
tune.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> There are certainly irrational ("tempered") chords that are
> targets
> > >> in their own right. The diminished 7ths and augmented triads
of
> > >> 12-equal being classic examples.
> > >
> > >so why aren't these "magic"?
> >
> > Because they don't sound in any way more concordant than similar
> > JI chords.
>
> So why are they "targets"?

I think a more interesting comparison would be with the 225/224
augmented triad and the 126/125 diminished 7th.

>> Well I don't know what you mean by concordant beating. If you mean
>> "equal beating", for me it doesn't significantly improve the
>> consonance of chords, although no thorough test of this has ever
>been
>> done (I've asked Gene for a list of chords to render for a test).
>
>I don't recall this. When and where?

It was here, after the brat stuff hit. I asked if we could, for
a given chord, find a pair of approximations with brats as
extremely different as possible even though they both approximated
the target chord about as well.

-Carl

>> There are certainly irrational ("tempered") chords that are targets
>> in their own right. The diminished 7ths and augmented triads of
>> 12-equal being classic examples.
>
>Or for example the augmented triad of 225/224-planar, or the
>diminished 7th of 126/125-planar, whoich are considerably better in
>tune.

If you give the cent values I'll post comparo files.

-Carl

Paul wrote:
>I'm not saying it should, only that there can sometimes be better
>choices for the primes than the nearest interval to each prime, as
>recently mentioned on tuning-math. And yes, these choices may differ
>from the "nearest interval to each prime" choice only (but not
>necessarily) if the ET is inconsistent.

Yes, I could make it more elaborate someday.

Manuel

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> An exception might be high-limit utonal chords. With the
right
> >> >> timbre and voicing (esp. in the high register) they can often
be
> >> >> rendered fairly concordant.
> >> >>
> >> >> -Carl
> >> >
> >> >And don't they, in that case, have fairly *unambiguous*
> >> >fundamentals?
> >> >
> >> >So isn't it not an exception at all? Or am I misreading you
again?
> >>
> >> They would be not an exception, but they can still have ambiguous
> >> fundamentals.
> >
> >Can you give an example? And say what is making them, when
rendered
> >as you say, concordant, if not a clear fundamental?
>
> I've gotten this with utonal 9th chords on my slide guitar

Once again, I read you wrong! You wrote "utonal" above, and I
read "otonal". I think I've gotten too old to be trusted!

Anyway, guitar is about the only instrument I've gotten
real "specialness" from such utonalities. It might have something to
do with sympathetic resonance involving that common overtone.

> And above 4K the
> periodicity mechanism falls out;

You're playing chords where the fundamental is above 4KHz????

> though I'm not playing them that
> high probably more of their strong harmonics are in that range than
> would be otherwise.

It's the *overall fundamental* which represents the frequency of
periodicity, not the fundamentals of the individual pitches, let
along the harmonics. 4Khz is actually near the *best* range for
frequency components to evoke a precise virtual pitch response.

>
> -Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Yeah but to me it just sounds like a restless version of 12:16:21:28
> or 9:12:16:21...
>
> http://lumma.org/tuning/0_491_982_1473.mid
> http://lumma.org/tuning/12-16-21-28.mid
> http://lumma.org/tuning/9-12-16-21.mid
>
> -Carl

Thanks, If you listen to held, isolated chords like this, you're
closer to the 'minimalist' aesthetic where JI otonalities are quite
popular. In a context with movement, counterpoint, and voice leading,
the comparison might strike you in a very different way.

Regardless, could we try a lower register (like where your left hand
might be)?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> There are certainly irrational ("tempered") chords that are
> >> >> targets in their own right. The diminished 7ths and augmented
> >> >> triads of 12-equal being classic examples.
> >> >
> >> >so why aren't these "magic"?
> >>
> >> Because they don't sound in any way more concordant than similar
> >> JI chords.
> >
> >So why are they "targets"?
>
> I was thinking along more along the 'puns are natural' line than
> the 'magic' line.

What's the difference, pray tell?

> Anywho, here's a dim7 chord (in 12-equal) with added octave,
> being inverted...
>
> http://lumma.org/tuning/300.mid
>
> ...and here's the same thing with a chain of 6:5s and added octave
> (and thus a kleisma-fudged 6:5)...
>
> http://lumma.org/tuning/6-5.mid

I'm confused. A kleisma is 15635:15552, about 8 cents, but four 6:5s
differ from an octave by 648:625, about 63 cents. In any case, the
accordion timbre obscures tuning quite well, but the second midi
sounded 'bad' on first listening.

>You're playing chords where the fundamental is above 4KHz????
>
>> though I'm not playing them that
>> high probably more of their strong harmonics are in that range than
>> would be otherwise.
>
>It's the *overall fundamental* which represents the frequency of
>periodicity, not the fundamentals of the individual pitches, let
>along the harmonics.

Yes, but periodicity detection is based on all stimuli (harmonics
and all) *below about 4KHz.*. I've read that it's often hard to
assign a single pitch to complex tones played above 4KHz.

>4Khz is actually near the *best* range for
>frequency components to evoke a precise virtual pitch response.

Why's that?

-Carl

>> Yeah but to me it just sounds like a restless version of 12:16:21:28
>> or 9:12:16:21...
>>
>> http://lumma.org/tuning/0_491_982_1473.mid
>> http://lumma.org/tuning/12-16-21-28.mid
>> http://lumma.org/tuning/9-12-16-21.mid
>>
>> -Carl
>
>Thanks, If you listen to held, isolated chords like this, you're
>closer to the 'minimalist' aesthetic where JI otonalities are quite
>popular. In a context with movement, counterpoint, and voice leading,
>the comparison might strike you in a very different way.

Agreed.

>Regardless, could we try a lower register (like where your left hand
>might be)?

It's all done with smoke and mirrors (Scala). Look ma, no hands!

http://lumma.org/tuning/0_491_982_1473-lower.mid
http://lumma.org/tuning/12-16-21-28-lower.mid
http://lumma.org/tuning/9-12-16-21-lower.mid

-Carl

>> >> >> There are certainly irrational ("tempered") chords that are
>> >> >> targets in their own right. The diminished 7ths and augmented
>> >> >> triads of 12-equal being classic examples.
>> >> >
>> >> >so why aren't these "magic"?
>> >>
>> >> Because they don't sound in any way more concordant than similar
>> >> JI chords.
>> >
>> >So why are they "targets"?
>>
>> I was thinking along more along the 'puns are natural' line than
>> the 'magic' line.
>
>What's the difference, pray tell?

"If you listen to held, isolated chords like this, you're
closer to the 'minimalist' aesthetic where JI otonalities are quite
popular. In a context with movement, counterpoint, and voice leading,
the comparison might strike you in a very different way."

>> Anywho, here's a dim7 chord (in 12-equal) with added octave,
>> being inverted...
>>
>> http://lumma.org/tuning/300.mid
>>
>> ...and here's the same thing with a chain of 6:5s and added octave
>> (and thus a kleisma-fudged 6:5)...
>>
>> http://lumma.org/tuning/6-5.mid
>
>I'm confused. A kleisma is 15635:15552, about 8 cents, but four 6:5s
>differ from an octave by 648:625, about 63 cents.

Sorry, wrong comma name. I meant 'minor diesis', or whatever.

>In any case, the
>accordion timbre obscures tuning quite well, but the second midi
>sounded 'bad' on first listening.

Now wait just a minute, that's a reed organ I'll have you know, one
of the best GM patches for tuning comparisons. Because they're midi
files I can't predict how they'll sound on your hardware, but the
patch is easy enough to change with any sequencer software and the
benefits of distributing midi files (as opposed to mp3s) for quick
tests like this are many.

If you're listening with quicktime in your browser, I recommend
against that. At least on my system, it applies reverb and other
effects inconsistently (ie, the first time you play a file after
opening it, it'll have the effects, and subsequently it won't).

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >You're playing chords where the fundamental is above 4KHz????
> >
> >> though I'm not playing them that
> >> high probably more of their strong harmonics are in that range
than
> >> would be otherwise.
> >
> >It's the *overall fundamental* which represents the frequency of
> >periodicity, not the fundamentals of the individual pitches, let
> >along the harmonics.
>
> Yes, but periodicity detection is based on all stimuli (harmonics
> and all) *below about 4KHz.*.

No, it's the periodicity that needs to be below 4KHz, not the
frequencies of the components.

> I've read that it's often hard to
> assign a single pitch to complex tones played above 4KHz.

Right, because "complex tones above 4KHz" normally means that the
*fundamental*, and thus the *periodicity*, is above 4KHz.

> >4Khz is actually near the *best* range for
> >frequency components to evoke a precise virtual pitch response.
>
> Why's that?

I doubt anyone knows the real evolutionary reasons or whatever. 3KHz
was the best range according to Goldstein's experiments, and is where
the 0.6% accuracy figure I often cite was found. Outside that range,
no one's virtual pitch perceptions indicated anything anywhere near
that level of accuracy. This would indicate that EQing out the 3KHz
band could have a disastrous effect on harmonically subtle music.

>> >It's the *overall fundamental* which represents the frequency of
>> >periodicity, not the fundamentals of the individual pitches, let
>> >along the harmonics.
>>
>> Yes, but periodicity detection is based on all stimuli (harmonics
>> and all) *below about 4KHz.*.
>
>No, it's the periodicity that needs to be below 4KHz, not the
>frequencies of the components.

You have a source on this?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >Regardless, could we try a lower register (like where your left
hand
> >might be)?
>
> It's all done with smoke and mirrors (Scala). Look ma, no hands!
>
> http://lumma.org/tuning/0_491_982_1473-lower.mid
> http://lumma.org/tuning/12-16-21-28-lower.mid
> http://lumma.org/tuning/9-12-16-21-lower.mid
>
> -Carl

Now I like the first one the best, even in this 'minimalist' context!
The beating in the first one is slow enough not to bother me. I think
we might be approaching the range where the '1' corresponding to your
latter two chords is too low to be understood, if that matters . . .

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >> There are certainly irrational ("tempered") chords that
are
> >> >> >> targets in their own right. The diminished 7ths and
augmented
> >> >> >> triads of 12-equal being classic examples.
> >> >> >
> >> >> >so why aren't these "magic"?
> >> >>
> >> >> Because they don't sound in any way more concordant than
similar
> >> >> JI chords.
> >> >
> >> >So why are they "targets"?
> >>
> >> I was thinking along more along the 'puns are natural' line than
> >> the 'magic' line.
> >
> >What's the difference, pray tell?
>
> "If you listen to held, isolated chords like this, you're
> closer to the 'minimalist' aesthetic where JI otonalities are quite
> popular. In a context with movement, counterpoint, and voice
leading,
> the comparison might strike you in a very different way."

Carl, what does this quote from me have to do with what I was asking
you above?? What's the difference between the 'puns are natural' line
and the 'magic' line? The two seem completely identical to me.

> >> Anywho, here's a dim7 chord (in 12-equal) with added octave,
> >> being inverted...
> >>
> >> http://lumma.org/tuning/300.mid
> >>
> >> ...and here's the same thing with a chain of 6:5s and added
octave
> >> (and thus a kleisma-fudged 6:5)...
> >>
> >> http://lumma.org/tuning/6-5.mid
> >
> >I'm confused. A kleisma is 15635:15552, about 8 cents, but four
6:5s
> >differ from an octave by 648:625, about 63 cents.
>
> Sorry, wrong comma name. I meant 'minor diesis', or whatever.

'Major diesis' -- it's big.

> >In any case, the
> >accordion timbre obscures tuning quite well, but the second midi
> >sounded 'bad' on first listening.
>
> Now wait just a minute, that's a reed organ I'll have you know, one
> of the best GM patches for tuning comparisons.

Yes, but strangely, it sounded unlike the reed organ you used for
your other examples! "Tuning affects timbre"!

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> perhaps using sinusoids is best. Not using
> sinusoids presents other problems I'd rather not deal with anyway.
If you
> are inclined to agree then I'd suggest starting with a
few "favorite" 4-note
> or larger chords like 4:5:6:7 or 8:9:10:11 as a base, use a
sinusoid for
> each note, and then add a variable value to each element of the
ratio, as
> in:
>
> 4+v : 5+v : 6+v : 7+v
>
> and initially just provide a way of semi-continuously varying v
over a range
> like -.5 to +.5. And just listen to the results. How does the
implied
> "fundamental" vary, if at all, and does this depend on the range
over which
> v is varied?

"Whereas periodic, harmonic tone complexes generally evoke
unambiguous low pitches,
inharmonic complexes can evoke ambiguous, multiple low pitches and
small pitch shifts. A halfcentury
ago, Schouten and deBoer (deBoer, 1976) conducted a classic set of
experiments to
determine whether pitch perception relies on spacings between
adjacent frequency components
(or equivalently on waveform envelope periods) rather than on
harmonic relationships between
components (or equivalently on the waveform fine structure). An AM
tone consists of a complex
comprising three successive harmonics that evokes a clear,
unambiguous pitch at its (missing)
fundamental frequency. When all three harmonics were shifted either
upward or downward in
frequency by the same amount, while keeping their frequency-spacings
constant, the low pitch of
the complex first shifted slightly by a much smaller amount than this
frequency difference, an
amount that was related to harmonic structure. When the frequencies
were further shifted,
listeners could hear one of two ambiguous pitches in the vicinity of
the original pitch."

http://homepage.mac.com/cariani/CarianiWebsite/CarianiNP99.pdf

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >It's the *overall fundamental* which represents the frequency
of
> >> >periodicity, not the fundamentals of the individual pitches,
let
> >> >along the harmonics.
> >>
> >> Yes, but periodicity detection is based on all stimuli (harmonics
> >> and all) *below about 4KHz.*.
> >
> >No, it's the periodicity that needs to be below 4KHz, not the
> >frequencies of the components.
>
> You have a source on this?
>
> -Carl

What's the source for what you were saying? This article:

http://homepage.mac.com/cariani/CarianiWebsite/CarianiNP99.pdf

makes it clear that the *waveform* is what our neural systems phase-
lock to, not the decomposition into sinusoidal components.
Periodicity pitch, if it exists at all, is practically by definition
not a function of the Fourier-analysed frequencies -- otherwise it
would be called *spectral pitch*. When he says,

"In the auditory system, as in many other sensory systems, receptor
cells
depolarize when stereocilia are deflected in a particular direction,
such that the timings of spikes
predominantly occur during one phase of the stimulus waveform as it
presents itself to the
individual receptor (for example, after having been mechanically
filtered by the cochlea). This
form of stimulus-locking is known as phase-locking. In the auditory
system, depending upon the
species, strong phase-locking can exist up to several kHz,
dramatically declining as pro-gressively
higher frequencies are reached."

he is talking about the *waveform* frequency, not that of any
sinusoidal components.

"Periodicities related to the fundamental F0, and
hence, to the pitch period, are distributed across the entire array
in the responses of fibers, with
CFs [characteristic frequencies] ranging from 200 Hz to over 10 kHz.
Given that the stimulus has relatively little power
above 1 kHz (Fig. 3C), this result is perhaps even more remarkable.
To a greater or a lesser
degree, all temporal discharge patterns follow the stimulus waveform,
reflecting the relation
between the respective fiber CFs and the stimulus spectrum."

>>>>>>>>[Carl]
>>>>>>>>There are certainly irrational ("tempered") chords that
>>>>>>>>are targets in their own right.
>>>>>>>
>>>>>>>[Paul]
>>>>>>>so why aren't these "magic"?
>>>>>>
>>>>>>[Carl]
>>>>>>Because they don't sound in any way more concordant than
>>>>>>similar JI chords.
>>>>>
>>>>>[Paul]
>>>>>So why are they "targets"?
>>>>
>>>>[Carl]
>>>>I was thinking along more along the 'puns are natural' line
>>>>than the 'magic' line.
>>>
>>>[Paul]
>>>What's the difference, pray tell?
>>
>>[Carl]
>>"If you listen to held, isolated chords like this, you're
>>closer to the 'minimalist' aesthetic where JI otonalities are
>>quite popular. In a context with movement, counterpoint, and
>>voice leading, the comparison might strike you in a very
>>different way."
>
>[Paul]
>Carl, what does this quote from me have to do with what I was
>asking you above?? What's the difference between the 'puns are
>natural' line and the 'magic' line? The two seem completely
>identical to me.

I thought the idea of magic chords is that tempering can improve
the concordance of certain isolated chords, without pitch-space
effects. By 'puns are natural' I mean the idea that tempering
can improve pitch-space stuff. It's the first idea I'm questioning.

>"Tuning affects timbre"!

Def. I played a 7-limit John deLaubenfels' version of my piece
Retrofit for my parents and they thought I had switched the timbre!
In fact, the overarching thing I hear from people about tuning,
from barbershop to boys choirs to what they like about their piano
tuner to what Michael Harrison's piano sounds like, is a comment
on timbre!

In fact, I remember asking myself why barbershop quartets and
harmonicas (diatonic ones, anyway) sounded 'that way' for most
of my life. It wasn't until I started getting into tuning theory
that I began to hear the difference as a quality in the intervals.
One day in particular I was listening to barbershop and all of a
sudden I recognized some 7-limit intervals I had been playing on
my returned keyboard.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I thought the idea of magic chords is that tempering can improve
> the concordance of certain isolated chords, without pitch-space
> effects. By 'puns are natural' I mean the idea that tempering
> can improve pitch-space stuff. It's the first idea I'm questioning.

What do you mean by "pitch-space stuff"? I have no idea what the
difference is between the "magic" chords I've mentioned before and
the chords that you and Gene are bringing up now. Please clue me in?

>> >> Yes, but periodicity detection is based on all stimuli (harmonics
>> >> and all) *below about 4KHz.*.
>> >
>> >No, it's the periodicity that needs to be below 4KHz, not the
>> >frequencies of the components.
>>
>> You have a source on this?
>>
>> -Carl
>
>What's the source for what you were saying?

Personal communication. But deBoer and Cariani are sources we
have discussed.

>This article:
>
>http://homepage.mac.com/cariani/CarianiWebsite/CarianiNP99.pdf
>
>makes it clear that the *waveform* is what our neural systems phase-
>lock to, not the decomposition into sinusoidal components.
>Periodicity pitch, if it exists at all, is practically by definition
>not a function of the Fourier-analysed frequencies -- otherwise it
>would be called *spectral pitch*. When he says,
>
>"In the auditory system, as in many other sensory systems, receptor
>cells depolarize when stereocilia are deflected in a particular
>direction, such that the timings of spikes predominantly occur
>during one phase of the stimulus waveform as it presents itself to
>the individual receptor (for example, after having been mechanically
>filtered by the cochlea). This form of stimulus-locking is known
>as phase-locking. In the auditory system, depending upon the species,
>strong phase-locking can exist up to several kHz, dramatically
>declining as pro-gressively higher frequencies are reached."
>
>he is talking about the *waveform* frequency, not that of any
>sinusoidal components.

Then why does he say "after having been mechanically filtered by
the cochlea"? Periodicity pitch works on the whole waveform, but
only after it has been decomposed and resynthesized. And if
the resynthesis is blind to components above 4K (no CFs above 4K)...

"
>To a greater or a lesser
>degree, all temporal discharge patterns follow the stimulus waveform,
>reflecting the relation between the respective fiber CFs and the
>stimulus spectrum."

This seem to exactly back up what I said, but I'll have to read
the entire paper to be sure.

-Carl

>> I thought the idea of magic chords is that tempering can improve
>> the concordance of certain isolated chords, without pitch-space
>> effects. By 'puns are natural' I mean the idea that tempering
>> can improve pitch-space stuff. It's the first idea I'm questioning.
>
>What do you mean by "pitch-space stuff"? I have no idea what the
>difference is between the "magic" chords I've mentioned before and
>the chords that you and Gene are bringing up now. Please clue me in?

Sorry to be confusulating. By pitch space stuff I mean comma
shifts/drift/puns. I put both magic chords and this pitch space
stuff under the heading of 'things Paul likes to point out where
temperament gives a more 'natural' result than JI'. This is a
notion that is very foreign to much of the JI crowd, and apparently
it was to Kurt.

Anyway, nobody disagrees that the pitch space stuff is audible and
musically important. I'm trying to look into whether the magic
effect really exists. Since I am denying (at least for the sake of
argument) the magic effect, you asked how I could say that some
tempered chords could be targets in their own right. And I replied
that they are targets in a pitch-spacey sense. And you seemed to
agree (in the quote I quoted) that pitch-space effects can bear on
our perception of concordance. Even just inverting the dim7 chords
as I do, the 3-tET seems to take on an identity by virtue of its
inversion symmetry.

There is no difference, to my knowledge, between the magic chords
Gene and I are bringing up and the ones you've brought up. Except
that the planar versions are maybe working a little better for me.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >he is talking about the *waveform* frequency, not that of any
> >sinusoidal components.
>
> Then why does he say "after having been mechanically filtered by
> the cochlea"?

Probably because there's no other way to hear anything! Our cochlea
is there in the pathway. Anwway, where does he say this?

> Periodicity pitch works on the whole waveform, but
> only after it has been decomposed and resynthesized.

The cochlear fibers, though, are *each* transmitting an awful lot in
common with the original waveform, even those that are tuned to 10KHz!
So it sure looks, at least here, as if frequency components that high
are aiding in the periodicity detection mechanism.

> And if
> the resynthesis

What resynthesis?

> is blind to components above 4K (no CFs above 4K)...

Yes?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Anyway, nobody disagrees that the pitch space stuff is audible and
> musically important. I'm trying to look into whether the magic
> effect really exists. Since I am denying (at least for the sake of
> argument) the magic effect, you asked how I could say that some
> tempered chords could be targets in their own right. And I replied
> that they are targets in a pitch-spacey sense.

I see no distinction!

> And you seemed to
> agree (in the quote I quoted) that pitch-space effects can bear on
> our perception of concordance. Even just inverting the dim7 chords
> as I do, the 3-tET seems to take on an identity by virtue of its
> inversion symmetry.

Not sure what you mean. In any case, shouldn't you have written 4-tET?

> There is no difference, to my knowledge, between the magic chords
> Gene and I are bringing up and the ones you've brought up.

So there is no distinction? Whew.

>> >he is talking about the *waveform* frequency, not that of any
>> >sinusoidal components.
>>
>> Then why does he say "after having been mechanically filtered by
>> the cochlea"?
>
>Probably because there's no other way to hear anything! Our cochlea
>is there in the pathway. Anwway, where does he say this?

Right above where you deleted there, in a block you and then I quoted.

>> And if
>> the resynthesis
>
>What resynthesis?

It's my understanding that in the midbrain and/or on the way to
the midbrain the frequency components are 'recombined' (in the
loosest sense) by combination-sensitive neurons, a process which
I speculate (emphasis on speculate) is the origin of virtual pitch.

>> is blind to components above 4K (no CFs above 4K)...
>
>Yes?

If it is blind above 4K (I'll have to look at the paper), then
the more energy you put up there the less of it will contribute
to a virtual pitch. Hence my statement on chords that are below
4K but higher than other chords. Which was admittedly speculative
but I don't think it's the VF that must be above 4K as you said
but rather the components that are used to calculate a VF (and
thus, obviously also the VF itself).

-Carl

>> And you seemed to
>> agree (in the quote I quoted) that pitch-space effects can bear on
>> our perception of concordance. Even just inverting the dim7 chords
>> as I do, the 3-tET seems to take on an identity by virtue of its
>> inversion symmetry.
>
>Not sure what you mean. In any case, shouldn't you have written 4-tET?

Yes.

>> There is no difference, to my knowledge, between the magic chords
>> Gene and I are bringing up and the ones you've brought up.
>
>So there is no distinction? Whew.

Correct.

>> Anyway, nobody disagrees that the pitch space stuff is audible and
>> musically important. I'm trying to look into whether the magic
>> effect really exists. Since I am denying (at least for the sake of
>> argument) the magic effect, you asked how I could say that some
>> tempered chords could be targets in their own right. And I replied
>> that they are targets in a pitch-spacey sense.
>
>I see no distinction!

One only has to do with isolated concordance judgements, the other
has to do with melodic and ... judgements, which can in turn
influence our perception of concordance.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> Anyway, nobody disagrees that the pitch space stuff is audible
and
> >> musically important. I'm trying to look into whether the magic
> >> effect really exists. Since I am denying (at least for the sake
of
> >> argument) the magic effect, you asked how I could say that some
> >> tempered chords could be targets in their own right. And I
replied
> >> that they are targets in a pitch-spacey sense.
> >
> >I see no distinction!
>
> One only has to do with isolated concordance judgements, the other
> has to do with melodic and ... judgements, which can in turn
> influence our perception of concordance.

But there isn't a set of chords that's an example of one, and a set
of chords that's an example of the other, is there? You seemed to
agree in your last message. So the distinction is fading into
mist . . .

>
> -Carl

>> One only has to do with isolated concordance judgements, the other
>> has to do with melodic and ... judgements, which can in turn
>> influence our perception of concordance.
>
>But there isn't a set of chords that's an example of one, and a set
>of chords that's an example of the other, is there? You seemed to
>agree in your last message. So the distinction is fading into
>mist . . .

You lost me here. All I can think to say is that melody and
harmony are to some extent different perceptual processes, and
hence the magic effect is different from the pun effect, and
that this bit summed up perfectly what I was trying to say:

> "If you listen to held, isolated chords like this, you're
> closer to the 'minimalist' aesthetic where JI otonalities are quite
> popular. In a context with movement, counterpoint, and voice
> leading, the comparison might strike you in a very different way."

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> You lost me here. All I can think to say is that melody and
> harmony are to some extent different perceptual processes, and
> hence the magic effect is different from the pun effect, and
> that this bit summed up perfectly what I was trying to say:
>
> > "If you listen to held, isolated chords like this, you're
> > closer to the 'minimalist' aesthetic where JI otonalities are
quite
> > popular. In a context with movement, counterpoint, and voice
> > leading, the comparison might strike you in a very different way."
>
> -Carl

Oh! I didn't think of inverting a single chord as having counterpoint
or voice leading. So maybe you could concoct a similar set of
examples for the stack-of-fourth examples, as you did for the
diminished tetrads? Let me suggest an example where you "invert" the
quartal tetrad (this is guitar notation so transpose down an octave):

B,-E-A-d
B,-D-A-e
D-A-B-e
E-A-d-b
E-B-d-a
A-B-d-e
A-d-e-b
A-e-b-d'
B-e-a-d'

etc.

>So maybe you could concoct a similar set of
>examples for the stack-of-fourth examples, as you did for the
>diminished tetrads? Let me suggest an example where you "invert" the
>quartal tetrad (this is guitar notation so transpose down an octave):
>
>B,-E-A-d
>B,-D-A-e
>D-A-B-e
>E-A-d-b
>E-B-d-a
>A-B-d-e
>A-d-e-b
>A-e-b-d'
>B-e-a-d'

What do , and ' mean, and what does case mean?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >So maybe you could concoct a similar set of
> >examples for the stack-of-fourth examples, as you did for the
> >diminished tetrads? Let me suggest an example where you "invert"
the
> >quartal tetrad (this is guitar notation so transpose down an
octave):
> >
> >B,-E-A-d
> >B,-D-A-e
> >D-A-B-e
> >E-A-d-b
> >E-B-d-a
> >A-B-d-e
> >A-d-e-b
> >A-e-b-d'
> >B-e-a-d'
>
> What do , and ' mean, and what does case mean?
>
> -Carl

http://www.music.vt.edu/musicdictionary/appendix/octaveregisters/octav
eregisters.html

on 12/3/03 12:41 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>>>>> There are certainly irrational ("tempered") chords that are
>>>>>> targets in their own right. The diminished 7ths and augmented
>>>>>> triads of 12-equal being classic examples.
>>>>>
>>>>> so why aren't these "magic"?
>>>>
>>>> Because they don't sound in any way more concordant than similar
>>>> JI chords.
>>>
>>> So why are they "targets"?
>>
>> I was thinking along more along the 'puns are natural' line than
>> the 'magic' line.
>
> What's the difference, pray tell?
>
>> Anywho, here's a dim7 chord (in 12-equal) with added octave,
>> being inverted...
>>
>> http://lumma.org/tuning/300.mid
>>
>> ...and here's the same thing with a chain of 6:5s and added octave
>> (and thus a kleisma-fudged 6:5)...
>>
>> http://lumma.org/tuning/6-5.mid
>
> I'm confused. A kleisma is 15635:15552, about 8 cents, but four 6:5s
> differ from an octave by 648:625, about 63 cents. In any case, the
> accordion timbre obscures tuning quite well, but the second midi
> sounded 'bad' on first listening.

For me the 2nd midi sounds bad in the *first* chord on *every* listening.
The next 3 chords sound fine. Contextual shift? Or a pitch issue?

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> For me the 2nd midi sounds bad in the *first* chord on *every*
listening.
> The next 3 chords sound fine. Contextual shift? Or a pitch issue?

The outer interval is 216/125, or 946.92 cents, in the first chord,
but it's 5/3 in the next 3 chords.

on 12/3/03 1:00 PM, Carl Lumma <ekin@lumma.org> wrote:

>>>>>>> There are certainly irrational ("tempered") chords that are
>>>>>>> targets in their own right. The diminished 7ths and augmented
>>>>>>> triads of 12-equal being classic examples.
>>>>>>
>>>>>> so why aren't these "magic"?
>>>>>
>>>>> Because they don't sound in any way more concordant than similar
>>>>> JI chords.
>>>>
>>>> So why are they "targets"?
>>>
>>> I was thinking along more along the 'puns are natural' line than
>>> the 'magic' line.
>>
>> What's the difference, pray tell?
>
> "If you listen to held, isolated chords like this, you're
> closer to the 'minimalist' aesthetic where JI otonalities are quite
> popular. In a context with movement, counterpoint, and voice leading,
> the comparison might strike you in a very different way."
>
>>> Anywho, here's a dim7 chord (in 12-equal) with added octave,
>>> being inverted...
>>>
>>> http://lumma.org/tuning/300.mid
>>>
>>> ...and here's the same thing with a chain of 6:5s and added octave
>>> (and thus a kleisma-fudged 6:5)...
>>>
>>> http://lumma.org/tuning/6-5.mid
>>
>> I'm confused. A kleisma is 15635:15552, about 8 cents, but four 6:5s
>> differ from an octave by 648:625, about 63 cents.
>
> Sorry, wrong comma name. I meant 'minor diesis', or whatever.
>
>> In any case, the
>> accordion timbre obscures tuning quite well, but the second midi
>> sounded 'bad' on first listening.
>
> Now wait just a minute, that's a reed organ I'll have you know, one
> of the best GM patches for tuning comparisons. Because they're midi
> files I can't predict how they'll sound on your hardware, but the
> patch is easy enough to change with any sequencer software and the
> benefits of distributing midi files (as opposed to mp3s) for quick
> tests like this are many.
>
> If you're listening with quicktime in your browser, I recommend
> against that. At least on my system, it applies reverb and other
> effects inconsistently (ie, the first time you play a file after
> opening it, it'll have the effects, and subsequently it won't).
>
> -Carl

I have no effects on the Mac in my browser using quicktime, and rendering is
consistent each time. My quicktime is a couple of years old here, though -
and on the Mac.

-Kurt

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48659.html#48986

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > Yeah but to me it just sounds like a restless version of
12:16:21:28
> > or 9:12:16:21...
> >
> > http://lumma.org/tuning/0_491_982_1473.mid
> > http://lumma.org/tuning/12-16-21-28.mid
> > http://lumma.org/tuning/9-12-16-21.mid
> >
> > -Carl
>
> Thanks, If you listen to held, isolated chords like this, you're
> closer to the 'minimalist' aesthetic where JI otonalities are quite
> popular. In a context with movement, counterpoint, and voice
leading,
> the comparison might strike you in a very different way.
>
> Regardless, could we try a lower register (like where your left
hand
> might be)?

***These seem a little short, too. I kinda wish they would go on for
about twice the length...

J. Pehrson

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_48659.html#48990

> >> Yeah but to me it just sounds like a restless version of
12:16:21:28
> >> or 9:12:16:21...
> >>
> >> http://lumma.org/tuning/0_491_982_1473.mid
> >> http://lumma.org/tuning/12-16-21-28.mid
> >> http://lumma.org/tuning/9-12-16-21.mid
> >>
> >> -Carl
> >
> >Thanks, If you listen to held, isolated chords like this, you're
> >closer to the 'minimalist' aesthetic where JI otonalities are
quite
> >popular. In a context with movement, counterpoint, and voice
leading,
> >the comparison might strike you in a very different way.
>
> Agreed.
>
> >Regardless, could we try a lower register (like where your left
hand
> >might be)?
>
> It's all done with smoke and mirrors (Scala). Look ma, no hands!
>
> http://lumma.org/tuning/0_491_982_1473-lower.mid
> http://lumma.org/tuning/12-16-21-28-lower.mid
> http://lumma.org/tuning/9-12-16-21-lower.mid
>
> -Carl

***The "middle" one seems to have slower beating. Is that what I'm
supposed to be looking for??

J. Pehrson

>> > http://lumma.org/tuning/0_491_982_1473.mid
>> > http://lumma.org/tuning/12-16-21-28.mid
>> > http://lumma.org/tuning/9-12-16-21.mid
>> >
//
>***These seem a little short, too. I kinda wish they would go on for
>about twice the length...
>
>J. Pehrson

I liked the length I chose, but one can slow them down with the tempo
adjustment on any MIDI sequencer (including SONAR).

-Carl

>> It's all done with smoke and mirrors (Scala). Look ma, no hands!
>>
>> http://lumma.org/tuning/0_491_982_1473-lower.mid
>> http://lumma.org/tuning/12-16-21-28-lower.mid
>> http://lumma.org/tuning/9-12-16-21-lower.mid
>>
>> -Carl
>
>
>***The "middle" one seems to have slower beating. Is that what I'm
>supposed to be looking for??
>
>J. Pehrson

What we're trying to test here is which chord sounds the most
"just", whatever that means.

-Carl

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48659.html#49020

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > >> Anyway, nobody disagrees that the pitch space stuff is audible
> and
> > >> musically important. I'm trying to look into whether the magic
> > >> effect really exists. Since I am denying (at least for the
sake
> of
> > >> argument) the magic effect, you asked how I could say that some
> > >> tempered chords could be targets in their own right. And I
> replied
> > >> that they are targets in a pitch-spacey sense.
> > >
> > >I see no distinction!
> >
> > One only has to do with isolated concordance judgements, the other
> > has to do with melodic and ... judgements, which can in turn
> > influence our perception of concordance.
>
> But there isn't a set of chords that's an example of one, and a set
> of chords that's an example of the other, is there? You seemed to
> agree in your last message. So the distinction is fading into
> mist . . .
>
> >
> > -Carl

***Ok... just so I get a bit more out of this discussion. What
exactly *is* a planar temperament?? And, I'm assuming from the
context, that it has a certain *something* that both JI and 12-tET
lack. What is this *something* we are looking for??

The ones I've been hearing have a particularly *smooth* sound without
being *locked* as much as JI... is that the idea??

Thanks!

JP

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_48659.html#49030

> >> One only has to do with isolated concordance judgements, the
other
> >> has to do with melodic and ... judgements, which can in turn
> >> influence our perception of concordance.
> >
> >But there isn't a set of chords that's an example of one, and a
set
> >of chords that's an example of the other, is there? You seemed to
> >agree in your last message. So the distinction is fading into
> >mist . . .
>
> You lost me here. All I can think to say is that melody and
> harmony are to some extent different perceptual processes, and
> hence the magic effect is different from the pun effect, and
> that this bit summed up perfectly what I was trying to say:
>
> > "If you listen to held, isolated chords like this, you're
> > closer to the 'minimalist' aesthetic where JI otonalities are
quite
> > popular. In a context with movement, counterpoint, and voice
> > leading, the comparison might strike you in a very different way."
>
> -Carl

***Regarding the "magic" chords... Paul, are these similar to the
important "magic" chord that you found for me in Blackjack? I'm
assuming they are...

JP

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> ***Ok... just so I get a bit more out of this discussion. What
> exactly *is* a planar temperament??

It's a regular temperament with three generators, which means two
generators if you are ignoring octaves. If we are working in the 7-
limit, a planar temperament is determined by only one comma.

> The ones I've been hearing have a particularly *smooth* sound
without
> being *locked* as much as JI... is that the idea??

That's what I would expect to hear just from theory.

>> The ones I've been hearing have a particularly *smooth* sound
>> without being *locked* as much as JI... is that the idea??
>
>That's what I would expect to hear just from theory.

You are aware that JP is referring here to the non-tempered
versions in both cases?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> It's all done with smoke and mirrors (Scala). Look ma, no hands!
> >>
> >> http://lumma.org/tuning/0_491_982_1473-lower.mid
> >> http://lumma.org/tuning/12-16-21-28-lower.mid
> >> http://lumma.org/tuning/9-12-16-21-lower.mid
> >>
> >> -Carl
> >
> >
> >***The "middle" one seems to have slower beating. Is that what
I'm
> >supposed to be looking for??
> >
> >J. Pehrson
>
> What we're

?

> trying to test here is which chord sounds the most
> "just", whatever that means.
>
> -Carl

I definitely think the latter two sound more "just", since they are!

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> /tuning/topicId_48659.html#49030
>
> > >> One only has to do with isolated concordance judgements, the
> other
> > >> has to do with melodic and ... judgements, which can in turn
> > >> influence our perception of concordance.
> > >
> > >But there isn't a set of chords that's an example of one, and a
> set
> > >of chords that's an example of the other, is there? You seemed
to
> > >agree in your last message. So the distinction is fading into
> > >mist . . .
> >
> > You lost me here. All I can think to say is that melody and
> > harmony are to some extent different perceptual processes, and
> > hence the magic effect is different from the pun effect, and
> > that this bit summed up perfectly what I was trying to say:
> >
> > > "If you listen to held, isolated chords like this, you're
> > > closer to the 'minimalist' aesthetic where JI otonalities are
> quite
> > > popular. In a context with movement, counterpoint, and voice
> > > leading, the comparison might strike you in a very different
way."
> >
> > -Carl
>
>
> ***Regarding the "magic" chords... Paul, are these similar to the
> important "magic" chord that you found for me in Blackjack? I'm
> assuming they are...
>
> JP

Yes, and also to the "augmagic" chords on the Blackjack keyboard
chord chart I made for you, etc.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_48659.html#49251

> >> The ones I've been hearing have a particularly *smooth* sound
> >> without being *locked* as much as JI... is that the idea??
> >
> >That's what I would expect to hear just from theory.
>
> You are aware that JP is referring here to the non-tempered
> versions in both cases?
>
> -Carl

***Oh, no. I'm not quite *that* oblivious. The ones I was
referring to were the tempered ones, that had that particular effect
(This was just a general observation...)

JP

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > ***Ok... just so I get a bit more out of this discussion. What
> > exactly *is* a planar temperament??
>
> It's a regular temperament with three generators, which means two
> generators if you are ignoring octaves. If we are working in the 7-
> limit, a planar temperament is determined by only one comma.
>
> > The ones I've been hearing have a particularly *smooth* sound
> without
> > being *locked* as much as JI... is that the idea??
>
> That's what I would expect to hear just from theory.

Yes. To clarify why this is, Joseph, the temperament makes the
various consonant intervals comprising the chord "fit together" which
they wouldn't in JI -- the chord in JI would have to have some "wolf".

>> What we're
>
>?

?

>> trying to test here is which chord sounds the most
>> "just", whatever that means.
>>
>> -Carl
>
>I definitely think the latter two sound more "just", since they are!

Then what in your words, are we listening for?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What we're
> >
> >?
>
> ?
>
> >> trying to test here is which chord sounds the most
> >> "just", whatever that means.
> >>
> >> -Carl
> >
> >I definitely think the latter two sound more "just", since they
are!
>
> Then what in your words, are we listening for?
>
> -Carl

Some abstract concept loosely associated with concordance. But Joseph
and Kurt at least seem to be hearing it anyway!

>>> trying to test here is which chord sounds the most
>>> "just", whatever that means.
>>
>>I definitely think the latter two sound more "just", since they are!
>
>Then what in your words, are we listening for?

I put quotes on just to mean Right or True, or whatever sense it
originally was applied for JI.

Maybe I should have said "magic". Which of these chords, if any,
sound "magic" to you, JP?

-Carl

> > Then what in your words, are we listening for?
> >
> > -Carl
>
> Some abstract concept loosely associated with concordance. But
> Joseph and Kurt at least seem to be hearing it anyway!

They do? Joseph picked JI chords as beating less, and Kurt
said his results depended on the order in which he listened
(though I didn't fully understand your response, Kurt).

-Carl

on 12/8/03 12:07 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> Then what in your words, are we listening for?
>>>
>>> -Carl
>>
>> Some abstract concept loosely associated with concordance. But
>> Joseph and Kurt at least seem to be hearing it anyway!
>
> They do? Joseph picked JI chords as beating less, and Kurt
> said his results depended on the order in which he listened

No, that was Paul, and as I recall it was only on the first listening.

> (though I didn't fully understand your response, Kurt).

I was just saying that the first chord sounded "bad" in one of the samples,
and the subsequent (2 or 3?) chords did not. Paul or someone explained to
me that the first chord was actually different, so I was not just imagining
it and it was not just *contextual*.

Unless you are talking about a different set of samples. The original
context is now gone from the quoting, and I'm not going to go looking for
it. Too many messages with the same subject!

>
> -Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> I was just saying that the first chord sounded "bad" in one of the
samples,
> and the subsequent (2 or 3?) chords did not.

That was when you listened to the various inversions of the stack-of-
6/5s-diminished-seventh-chord.

> Paul or someone explained to
> me that the first chord was actually different, so I was not just
imagining
> it and it was not just *contextual*.

Well, I just pointed out that your perception could be explained if
the *outer* interval was the most important one, since this was a
discordant interval in the first chord but a pure 5/3 in the
inversions. However, all the chords are equally "different" or "the
same" if you don't pay extra attention to the outer interval -- it's
just a series of inversions of "the same" chord.

> Unless you are talking about a different set of samples.

I was -- and Joseph was too.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48659.html#49315

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
> wrote:
> >
> > > ***Ok... just so I get a bit more out of this discussion. What
> > > exactly *is* a planar temperament??
> >
> > It's a regular temperament with three generators, which means two
> > generators if you are ignoring octaves. If we are working in the
7-
> > limit, a planar temperament is determined by only one comma.
> >
> > > The ones I've been hearing have a particularly *smooth* sound
> > without
> > > being *locked* as much as JI... is that the idea??
> >
> > That's what I would expect to hear just from theory.
>
> Yes. To clarify why this is, Joseph, the temperament makes the
> various consonant intervals comprising the chord "fit together"
which
> they wouldn't in JI -- the chord in JI would have to have
some "wolf".

***So, it's another situation where the "aggregate justness" is
greater when tempered than a "truly just" situation. Our "born
again" JI fans don't seem to dig this approach, but I'm very happy
with this kind of thinking, myself (rather in the Blackjack vein,
also...)

JP

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_48659.html#49341

> >>> trying to test here is which chord sounds the most
> >>> "just", whatever that means.
> >>
> >>I definitely think the latter two sound more "just", since they
are!
> >
> >Then what in your words, are we listening for?
>
> I put quotes on just to mean Right or True, or whatever sense it
> originally was applied for JI.
>
> Maybe I should have said "magic". Which of these chords, if any,
> sound "magic" to you, JP?
>
> -Carl

***Well, the "tempered" ones... but I may be prejudiced, since I
believe those are the ones that are "supposed" to be "magic..."

JP

>***So, it's another situation where the "aggregate justness" is
>greater when tempered than a "truly just" situation. Our "born
>again" JI fans don't seem to dig this approach, but I'm very happy
>with this kind of thinking, myself (rather in the Blackjack vein,
>also...)

So now you *hear* it that way, JP?

-Carl

>***Well, the "tempered" ones... but I may be prejudiced, since I
>believe those are the ones that are "supposed" to be "magic..."

That certainly would make you prejudiced.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_48659.html#49492

> >***So, it's another situation where the "aggregate justness" is
> >greater when tempered than a "truly just" situation. Our "born
> >again" JI fans don't seem to dig this approach, but I'm very happy
> >with this kind of thinking, myself (rather in the Blackjack vein,
> >also...)
>
> So now you *hear* it that way, JP?
>
> -Carl

***Actually, I don't think so... :) It doesn't "lock..." But, I'd
rather compose in a situation where I can notate everything easily
(such as 72-tET) than work up all the plusses and minuses of, let's
say, Ben Johnston's notation...

(Casting no aspersions on Ben Johnston, who is, of course, a
brilliant composer and musical thinker...)

JP

>/tuning/topicId_48659.html#49492
>
>> >***So, it's another situation where the "aggregate justness" is
>> >greater when tempered than a "truly just" situation. Our "born
>> >again" JI fans don't seem to dig this approach, but I'm very happy
>> >with this kind of thinking, myself (rather in the Blackjack vein,
>> >also...)
>>
>> So now you *hear* it that way, JP?
>>
>> -Carl
>
>***Actually, I don't think so... :) It doesn't "lock..." But, I'd
>rather compose in a situation where I can notate everything easily
>(such as 72-tET) than work up all the plusses and minuses of, let's
>say, Ben Johnston's notation...
>
>(Casting no aspersions on Ben Johnston, who is, of course, a
>brilliant composer and musical thinker...)

Thanks for reporting back. I agree about the notation issue, too.
Live performers can easily bend from 72 to JI, at least as easily as
they can play the exact 72-tET intervals. And if it's strict JI
you want to write in, the harmonic meaning of the notation will be
a lot less ambiguous than with 12-tET notation at the same harmonic
limit, which means that it would be possible to write automatic
tuning software that would escape many of the problems faced by John
deLaubenfels' software, Hermode Tuning software, or the Groven piano
software (to name three).

-Carl