Re: Heuristic approach of coherene

Paul,

I appreciate you take time to give detailed comments at this moment of
febrile activity in harmonic entropy. I'll be able, starting with that, to
clarify my ideas. Before to comment, I'll take time to analyse your
response and see what are the main points on which I could give better
explanations. For the moment, I'll note only few details at the beginning
of your response.

You wrote :

>Those look correct, but . . .
>> ( 3/2 4/3 20/9) == (108:135:160) == (20,864,1080)
>> ( 3/2 4/3 9/4) == (4:5:6) == (20,30,6)
>I believe you mean 15/8 instead of 4/3 in the two chords above.

Oh! yes. It's (G B D). Thank you for correction.

Then you wrote :

>Those would be the four best concordances
>against the 1/1 in the octave above it.
>Or if you also went into the octave below,
>the four best concordances against the 1/1
>would be an octave below 6/5, 4/3, 3/2, and 8/5.
>If you then allowed octave-
>equivalence, you have a total of seven tones
>per octave (this seems to help lead to what
>you do below -- let me know if it is a misinterpretation).

Yes, it's not what I wanted to say. It is totally equivalent to start with
the four ratios below unison (3/5, 2/3, 3/4, 4/5), but free use principle
is applied only to 5 tones (unison + the 4 chosen tones, above or below)
and not applied to 7 tones.

[ For this brief comment, I'll use the term "tone" as equivalent to French
"ton". This sense is maybe not valid in English. Let me know if yes or no
and how I could translate if it's not valid. Tone, in that sense, don't
correspond to pitch or frequency but only to interval modulo 2.]

The heuristic process try (arbitrary) to build a coherent set of tones
beginning with unison (1) and 4 other tones chosen here for exceptional
concordance qualities. (I could start with different tones). After that,
there is no subsequent direct choice of tones but only a principle choice :
free use of these 5 tones.

In N-tET context it's almost insignificant for there is no possibility to
exceed N tones. In JI context, you have to restrict free use to few tones
only for you go toward infinity using freely intervals of intervals of
intervals...

What means free use ? Harmonicaly, it means you can use all these tones as
a chord. Melodicaly, it means you can use all possibilities in succession
for these tones. Then the set of all tones (13 dyads including
unison-unison) spanned by all pentads (and subchords) of this chord and the
set of all possible steps (13 steps modulo 2 including repetition) in all
sequences (using frequencies having with a tonic these 5 tones) are
strictly equivalent.

So the choice to use freely the 5 starting tones (1 5/4 4/3 3/2 5/3)
implies surging (verticaly and horizontaly) of 8 other tones. These 13
tones are not equivalent. It would be aberrant to use freely (16/15 10/9
9/8). The intervals between these tones, in our chosen context, are not
tones as such but modal sruties with which is excluded building of chord
and making of melodic steps.

Until now, nothing has been said about particular coherence of these 13
tones (or much other coherent choice). It's toward what I want to progress
once ambiguities will be eliminated.

I announce here what is in view. There are two coherence levels. The first
level is strictly mathematical (how algebra is possible without closure
axiom). The second level refer to tonal categorization (how congruence
class of tones may arised and how it is, in same time, similar and
different of linguistic structures).

I'll give later more structured comments starting with your questions.

Pierre Lamothe