CPS's and magic chord

>>Only for CPS's where k:n is 1:2 are otonal and utonal resources equal. Ah-
>>actually, for all k of any n (power set of n), they are equal, that's
>>true...
>
>I meant generally. Clearly some CPSs favor otonality, while others favor
>utonality. What does your last sentence mean?

If n is the set of factors and k fixes the size of the subsets, you'll get the power set of n if k is allowed to be anything. The power set will contain an equal number of otonal and utonal chords, and not only for the original n! As Wilson points out in D'Allesandro, the power set also contains every CPS above it in Pascal's triangle --- in fact a different triangle top for each of the combinations of (n-1) out of n, recursively until n hits 1!

>>Actually, Kraig, I am beginning to believe that the 12tET minor triad
>>approximates 16:19:24 more often than 10:12:15. You and Wilson both told >>me about the 19-limit approximations in 12-tone... I think you guys were >>right.
>
>I guess my comment to that effect came too late.

Too late for what? It's good to know your view. Incidentally, I don't think it's only JI enthusiasts that hear the chord this way (I certainly don't have a very good ear at the 19-limit!). While the 19-limit version has a higher harmonic series representation than the 5-limit utonal one, as you once said, both sets of numbers are too high to rule out the effects of their constituent intervals. The intervals of the utonal triad do not all evoke the same VF, whereas the intervals of the 19-limit otonal chord do. Might this give a boost to the 19-limit triad's probability of being heard in the case of 0 300 700? Was our earlier observation of 'something funny with fractions with a base 2 under' a manifestation of some sort of VF rally effect?

>>Don't you mean at least one interval more dissonant than the tempered 8/5?
>
>I was thinking octave equivalence, which seemed appropriate since you did
>not list any inversions.

For the record, I twice disclum my use of only root position just versions.

>>I hear the first three of the above as clearly more consonant than the
>>12tET augmented triad.
>
>Well, you may be listening in particular for the effects of otonal-in-JI
>chords, which one might call consonance.

I think I must be -- lower roughness and higher tonalness.

>there are other ways of perceiving consonance, and these ways are important >in music.

Certainly in 12tET music, none of the just versions I list could replace the augmented triad (although I feel strongly that the same need not be true of music written in 12tET and transcribed for JI). Your point is well taken, and in recent conversation with a friend of mine, I was unable to persuade that the 7/5 is 'roughly as consonant' as the 8/5 when the 5's are fixed. My friend is a trained classical musician, and his ear locates 8/5's in triads quicker than you can randomly change their roots while he listens.

-C.

>>I meant generally. Clearly some CPSs favor otonality, while others favor
>>utonality. What does your last sentence mean?

>If n is the set of factors and k fixes the size of the subsets, you'll get
the power set of n if k is allowed to be >anything. The power set will
contain an equal number of otonal and utonal chords, and not only for the
original n! >As Wilson points out in D'Allesandro, the power set also
contains every CPS above it in Pascal's triangle --- in >fact a different
triangle top for each of the combinations of (n-1) out of n, recursively
until n hits 1!

Yes, I talked about this here several years ago. The "power set" can be
interpreted as an Euler genus, the unit "cube" in n-dimensional space, and
the geometric interpretation is that the beautiful figures representing the
CPSs stack together to fill n-dimensional space.

>Too late for what? It's good to know your view. Incidentally, I don't
think it's only JI enthusiasts that hear the >chord this way (I certainly
don't have a very good ear at the 19-limit!). While the 19-limit version
has a higher >harmonic series representation than the 5-limit utonal one, as
you once said, both sets of numbers are too high to >rule out the effects of
their constituent intervals. The intervals of the utonal triad do not all
evoke the same VF, >whereas the intervals of the 19-limit otonal chord do.
Might this give a boost to the 19-limit triad's probability of >being heard
in the case of 0 300 700? Was our earlier observation of 'something funny
with fractions with a base >2 under' a manifestation of some sort of VF
rally effect?

Yep, I've said before that I believe the modern effect of the minor triad
requires that it have the "right" root, therefore 16:19:24 is the "right"
tuning (given that one has already learned to live with 12-tET-type
deviations from 5-limit JI in the major triad). These observations go back
at least to Helmholtz; they certainly did not originate with Tom Stone!

>>>Don't you mean at least one interval more dissonant than the tempered
8/5?
>
>>I was thinking octave equivalence, which seemed appropriate since you did
>>not list any inversions.

>For the record, I twice disclum my use of only root position just versions.

What?

>Certainly in 12tET music, none of the just versions I list could replace
the augmented triad (although I feel strongly >that the same need not be
true of music written in 12tET and transcribed for JI). Your point is well
taken, and in >recent conversation with a friend of mine, I was unable to
persuade that the 7/5 is 'roughly as consonant' as the
>8/5 when the 5's are fixed.

How does that have anything to do with it? By the way, my roommate in
college asked me why, when played as diads, the 12-tET diminished fifth and
minor sixth sounded equally dissonant. I told him (and I think I was right)
that the first is essentially heard as a 7:5 17 cents out-of-tune, and the
second is heard as an 8:5 14 cents out-of-tune. Since the numbers are about
the same size, and are fairly small, a rough equivalence in consonance can
be expected.

>My friend is a trained classical musician, and his ear locates 8/5's in
triads quicker than you can randomly >change their roots while he listens.

What does that mean?

"Paul H. Erlich" wrote:

> Yep, I've said before that I believe the modern effect of the minor triad
> requires that it have the "right" root, therefore 16:19:24 is the "right"
> tuning (given that one has already learned to live with 12-tET-type
> deviations from 5-limit JI in the major triad). These observations go back
> at least to Helmholtz; they certainly did not originate with Tom Stone!

Paul!
I believe you, but where!
-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

>> Yep, I've said before that I believe the modern effect of the minor triad
>> requires that it have the "right" root, therefore 16:19:24 is the "right"
>> tuning (given that one has already learned to live with 12-tET-type
>> deviations from 5-limit JI in the major triad). These observations go
back
>> at least to Helmholtz; they certainly did not originate with Tom Stone!

>Paul!
> I believe you, but where!
>-- Kraig Grady

Helmholtz shows how lowering the minor third brings the difference tone in
tune with the minor triad, which he considers a good thing. The earliest
explicit mention of 16:19:24 I've seen was in a pamphlet from the '40s, I
think, called "Voicing with Acoustics" or something like that (from the NY
Public Library). It recommends this tuning in the context of voicings like
1:2:3:4:6:8:12:16:19:24, but for first-inversion minor triads, the 5-limit
version is preferred since the major third at the bottom confers stability
to the bass note.