Answer to Carl Lumma on min7

Carl

You wrote

<< I have been following your posts to the tuning digest, and have visited
your web page, with great interest. Could you explain "invariant math
index", and how it can be used to evaluate the different inversions of
the min7 chord? >>

I mentionned that it could be irrelevant abstract standpoint. I don't want
presume equivalence in perception of harmonic and subharmonic series. I'm
not qualified for that and wait to observe consensus before presuming of
reality. Since I use for calculation dual notions, I want to precise that
it's only an observation about spectral stability of differential tones and
"reciprocal differentials", in different inversions of chords. I wait for
interpretation if relevant.

Differential tone for (a,b) dyad of tones is (b-a). Pertinence of this
notion is well-established.

Reciprocal notion is abstraction. "Inverse differential" for dyad (a,b)
is 1/(1/a - 1/b). I use it for reciprocal calculation on aligned harmonics
and subharmonics with tones.

I underline after a pertinent remark of Paul H. Erlich that alignement of
differential tones alone is not a good characterization : (300 400 500 600
800) and (337 437 537 637 837) have same alignement, but alignement fit
with real tones only in first case.

I use matrix for simple calculation of both differentials. I put tones on
diagonal. In inferior cases I note differential d = b-a of what is seen on
diagonal horizontaly (a) and verticaly (b). In superior symetric case I
note ab/d which is inverse differential. Here is an example of major triad
(20 25 30), minor triad (20 24 30) and process :

60 150 30 60 120 30

100 25 5 120 24 6 ab/d b

20 5 10 20 4 10 a d (=b-a)

Good alignement of differential tones 5 (with correspondence in alignement
of first three abstract subharmonics on 5) characterize major harmony
(harmonic series).

Good alignement of "inverse differentials" 120 (with correspondence in
first alignement of three real harmonics on 120) characterize generalized
minor harmony (subharmonic series).

Now, min7 (in 5-limit) is a mixed chord in sense of representation as
harmonic or subharmonic series are equivalent : 45/(5 3 15 9) = (9 15 3 5).
I use it for comparaison on form changes to keep focus on spectral
stability. I would like to say that general concept of chord implies
"circular intervals" (what I name in French "ton" well distinguished, as
class, of "note" and "intervalle") and relative invariance about a certain
class of form changing. I give matrix for normal (10 12 15 18) only.

22.5 36 90 18

30 60 15 3

60 12 3 6

10 2 5 8

Spectral alignement of (both) differentials are presented on following
figure (hoping keeped by posting). Columns are indicated by "circularly"
ordered harmonics and lines corresponds to octaves. Numbers indicate played
tones and then form of chord :

1 9 5 45 3 15 1 9 5 45 3 15 1 9 5 45 3 15
-------------------------------------------------------------------------
*

* * * * * * * * * *
9 * * * 15 *9 * * 15 *9 *10 *15
* 5 ** * * 5 *6 * * * *6
* 3 * * *
* *
-------------------------------------------------------------------------
*
* * *
* ** * ** * **
* * 18 * * 18 20 * *
9 10 12 15 * 10 12 15 * 12 15
* * * * * *
* ** * ** **
*
-------------------------------------------------------------------------

In the 6 forms both differentials spectrum have 2 elements aligned on
"circular interval" (modulo 2) of fundamental basis 1 and "high" 45. All
other elements are aligned on 9-5-3-15 with successive quantities :
2244 3333 3333 2244 2244 2244.

It's much less evident that it could seem.

With less compact forms spectrum is spread out. With Dominanth chords
(harmonic series), non aligned inverse differentials are spread out.
Situation is completly dual with subharmonic series : non-aligned
differentials are spread out.

But it's just an observation for the moment.

Pierre

Paul

C'était probablement une erreur de répondre à la question de Carl Lumma,
malgré plein de précautions qui affirment clairement que ce n'est qu'une
façon simple de calculer des alignements. Je sais que les tons
différentiels ça s'entend : j'ai fait un cours de physique des ondes. Et
j'ai indiqué clairement que la notion réciproque est une abstraction. Y
a-t-il derrière ta promptitude à dénoncer sa "pure fantaisie" une crainte
que j'embrouille les esprits avec de dangereuses notions abstraites ?

Les acousticiens ont présentement le haut du pavé et ça ne me gêne pas pour
affirmer que le besoin d'un soupçon sémiotique se fair sentir.

Je ne sais si tu te doutes de la difficulté que je peux avoir à écrire des
messages en anglais, moi qui aurais envoyé se faire voir quiconque aurait
dit que je pouvais écrire en anglais, il n'y a pas de cela un mois. Je vais
continuer à écrire en anglais pour discuter d'idées mais je n'ai pas
tellement le goût de perdre mon temps à mettre des points sur les "i" avec
un handicap pareil. Je peux très bien défendre des opinions dans ma langue
dans des débats à côté des questions. Je t'y invite.

Quant à la question de l'écoute, je n'ai présentement aucun moyen d'écouter
quoi que ce soit. J'aimerais bien, mais je ne peux pas. Je dois patienter.

Tout de même sans rancune.

Pierre

Paul

What I have in mind when I speak of language barrier is mostly huge time
consumed. It's why I can't appreciate, with the pleasure I could have, if I
was fluent in English, your attempt to tease out substance. I don't want to
criticize your manner as such. I want only emphasize that it's costly to
follow.

Besides, I have speakers without driver. I will try to find that.

Pierre