Do you like clean numbers ?

(You can skip text and look at numbers)

Je cite plus bas un extrait d'un message privé en me demandant si ces
nombres parlent d'eux-mêmes, si ça permet d'entrer aisément dans la théorie
qui les produit. Je dis seulement que ça concerne la base shrutale d'un
gammoïde et le vecteur de coordonnées des tons (ici l'octave). D'ici
quelques jours je vais rendre disponible le texte que je prépare sur le
système indien où les définitions de shruti, matrice S et base shrutale
canonique permettent de tout comprendre, si bien sûr, l'intérêt se fait
sentir.

Pierre

P.S the numbers ...

[A B C](x,y,z)= Ax + By + Cz

Pythagore ... (diatonisme car 114 proche de 90)

[114 90](5,7) = 1200 (cents)
[5 4](5,7) = 53 (excellent T53)
[4 3](5,7) = 41 (très bon T41)
[1 1](5,7) = 12 (demi-tons T12)

[0 1](5,7) = 7 (degrés)

Indien ...

[22 70 90](10,5,7) = 1200 (cents)

[1 1 1](10,5,7) = 22 (shrutis)
[0 1 1](10,5,7) = 12 (demi-tons)
[0 0 1](10,5,7) = 7 (degrés)

Slendro ... (monotonisme car 27 et 36 petits devant 204)

[27 36 204](4,2,5) = 1200 (cents)

[1 1 1](4,2,5) = 11 (shrutis)
[0 0 1](4,2,5) = 5 (très bon T5)

Zarlino ... (demi-tons diatonique 112 et chromatique 70)

[22 70 112](3,5,7) = 1200 (cents)

[1 1 1](3,5,7) = 15 (shrutis)
[0 1 1](3,5,7) = 12 (demi-tons)
[0 0 1](3,5,7) = 7 (degrés)

Wim

I'll post you privately in french for more details. Now I would like
dissipate a recurent misinterpretation of my (apparent exclusive) attention
to just ratios (in last point). I'll take citations in reverse order.

[1]

You wrote

<< So, are just ratios really enough to explain Indian music, or is there some
more? What about the melodic quality of all the steps? >>

I'm not qualified to speak about music in general and again much less about
Indian music. My general opinion is : musical phenomenon is much more rich
and complex than what it is written about and what it will be written in
future.

And by analogy I would say :

There is much more in phonology than in acoustic.
There is much more in language than in phonology.
There is much more in litterature than in language.

So my attention to ratios don't mean I'm attempting to reduce reality. It's
only a specialized attack of "algorithmic compression" on data restricted
to intervals.

[2]

You wrote (on just srutis)

<< Many theorists do that as well: Alain Daniélou for instance. >>

I don't have nothing like knowledge that could be compared to those of
great specialists of music. But my weakness is my strenght. Indian system
of 22 srutis is just, for me, an excellent illustration for derivative
power of a universal axiomatic theory. I did'nt develop concepts to fit
reality. It's only after, by chance, that I've found this almost perfect
case of macrotonal coherence such as I'd soon theorized.

[3]

You wrote

<< In your research you seem to consider the shrutis all the time as just
ratios. >>

There are three aspects about this remark I would like to comment.

----- First aspect

My clean numbers show almost only tempered srutis and common rational
relations connecting just and tempered structures.

Traditional mathematical approach of "tempered/just" conflict in attempt to
balance "coherence in combinatory of width" with "good qualities of
sonance", use irrational numbers. Reason is clear. None finite simple
algebra structure like group structure exists which allies more than one
independant rational factor and algebra closure. Then, groups of tempered
tones are usually common frame of work. Finding modes with good melodic and
harmonic properties in such frame concern much more musical judgement than
mathematical possibilities.

The tempered representation of intervals is truncated. It contains only
information on width. In number 200 or 204 (cents) we can't seen which is
more consonant. But the problem is not at this level, for a simple
logarithm calculation can resolve it. In a set of tempered tones, the best
fit for consonance on individual basis can't constitute a coherent
algebraic set nor an harmonic coherent choice. Complex macrotonal analysis
of all the set, with non easy ponderation of two "fudge" factors, are
necessary on mathematical basis to propose pertinent structure that may
retain musical attention.

I have wanted to solve the problem at fundamental basis. Which simple
algebraic structure is possible, like group (which is very important for
mind), if closure is relaxed (for obvious goal to process directly
rationals) ? By a suprizing paradox, I've found solution of that pure
mathematical problem in musical concept of chord. What remains on chord if
we do abstraction of rationality or sonance qualities ? Answer : there
exist a unique solution of linear equation (ax=b) where a and b are on the
chord. This "nothing" was a key allowing macrotonality.

Decoupling macrotonality of microtonality, it has been possible to
construct, on pure mathematical basis, the appropriate structure in which
rationals and tempered correponding can be commonly process. Obviously, I
can't develop here this theory of chordoid.

With the tools derived in this frame of chordoid structure, we can now
manage bidimensional information on intervals and express relations that
are invariant in tempered approximation.

My concern is not tones (just or tempered) with which we can do
calculation, in other words, it's not arithmetic but algebra. Algebra
manages universal relations on a structured set of elements, independently
of individuality of elements.

The most important step from there (after first problem is resolved) is to
explicit how simplicity of group is possible. Which axioms give isomorphism
of tones' class on cyclical group of degrees like it's in use since
millenaries. I'm astonished that Z/12 (frame) group were remarked while Z/7
(musical) group were neglected. (Probably fishes don't remark water. :)
It's just a break to change aspect.)

----- Second aspect

I don't have a particular interest on JI tones derived from opinion on best
intonation. I wrote in "Fétichisme du numérique et rationalité musicale"
what I think about obsessive precision on tuning. Global context perception
seems to me much more important in musical flow than acoustical phenomenon
tracked only with attention.

In my opinion, what is important in Indian Srutis System is global impact
on the music (how such raga has to be played) and not the possible number
of cents out-of-tune. I would think that much good music are made with
instuments lightly out-of-tune and much insignificant music with
instruments perfectly tuned. I would say also that lattitude of expressive
variations by good musicians is large compared with the few cents of
mistuning, and that the referential allowing variations' significance is in
the brain and not in the instrument.

I recall an experience (reported in La Recherche 229, feb 1991, Un
synthétiseur dans la brousse, S. Aroum) on very large possible gap between
instrument tuning (xylophone) and musical perception. A synthesizer has
first been precisely tuned by autochtones. Analysis of tuning was unable to
reveal scale structure. Returning with voice recorder reveals well-known
pentatonic modes and an accomodation (with large tolerance) of a unique
xylophone's tuning for use with all modes.

----- Third aspect

The thought scheme explaining my propension to isolate intervals (and
numbers about them) is what I would call the relatively independance of
tonal segment among various segments of intelligibility. Central concept is
parametric invariance in acoustic channel.

Timbre recognizing stands on complex invariance perception. This highly
sophisticated ability of mind to track source of sound (I don't mean as
position but as significance) was important for survivance. Music inherit
of this ability and use it largely in a proper segment of intelligibility.

Global ordering structure of intensities is also an invariant which opens
on a new segment of intelligibility. Timbres don't have this type of
invariance. Intensity and tempo continue variations are probably the main
vector of emotion.

(( Another segments may be found also in relation with our natural
insertion. There exist an instinctive comprehension of sounds expressed by
animals. I like cats and I express relatively well sounds they like. But I
don't theorize on that. ))

Important thing is the following. There exist a UNIQUE ELEMENTARY INVARIANT
in acoustic channel (if relative source position is fixed). It's time. It's
why duration intervals and frequency intervals may alone support algebraic
structure which necessitate elementary invariance for conservation of
composition law under width or intensity translation.

Intervals are not forcely what is most important in music. Acousmatic don't
use them. But they defines alone tonal intelligibility segment. Simple
algebraic structures on intervals act as paradigm to construct (in mind)
similitude class of forms spread out on syntagmatic axis by compositors.
Reason is that fast decoding of evolutive (varying) forms in flow context
necessitate categorial perception of intervals as positions in a discrete
structure. This fast schematic apprehension is completed, obviouly, by
other sensitive perceptions.

----------

I stop here. And I take a break on the List. I did'nt write a line of my
text on Hindu System since 3 days.

Pierre

CORRECTION

<< which necessitate elementary invariance for conservation of
composition law under width or intensity translation >>
^^^^^

It's not width translation but height translation (of notes). Width is an
invariant.

Excuse me!

--- In tuning@egroups.com, Pierre Lamothe <plamothe@a...> wrote:
[snip]
Pierre, this was a fascinating post. Please, please, Pierre, is there
any way you could get your theory into a form I could understand? I'm
really dying to understand it! So far I see just ratios, but you say
that you're actually proposing temperament to increase the number of
possible relations (I think) -- I'd like to understand how that is
implied in your diagrams, or however you do imply it . . . (???)