RE: For all strict-JI fans . . . (SRUTIS)

Paul,

I don't know exactly what << out of >> means in << shouldn't both
correspond to #4 out of 24 >> so I cannot understand how this notion is
linked to the srutis counting. Does it refer to sruti or temperament?

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All is in definitions!

Either we share the same sruti definition and then you have simply to count
or else we have distinct definitions and then we have to compare them.

Why are you astonished that two distinct tones may have a same count of
srutis while it seems you would not be if they would have a same count of
steps?

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If you would share my sruti definition, then you would rattach, at any
system having <21/20 16/15 15/14 10/9> as steps, this set

<21/20 64/63 225/224 28/27>

as unique possibility for srutis. In that case, it would remain only to
count :

10/9 = (21/20) (64/63) (225/224) (28/27)

1 2 3 4

9/8 = (21/20) (64/63) (225/224) (21/20)

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Since you seem to see a problem in that equality and the count 24 for the
octave, you have to give first to me your sruti definition (if it is not
something varying from case to case according to contextual fudge factors).

What can I say without that? It could maybe permit to see why I distinguish
macrotonality from microtonality.

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Now I give to you my sruti definition.

It is not derived from matrices using. It is only 7 years after I defined
them I found that srutis are easily manipulated by matrices : what might
talk about its consistency level.

I used already since many years a Turbo Pascal program finding structures
like gammoids and gammiers when I discover, in one of seldom visit at music
library of Laval University, the Donald Lentz Hindu Classical System book.
I saw immediately how this system would be a great illustration of the
gammier theory.

Using my program I found in few minutes the near corresponding harmonic
generator and in few hours the explanation of all anomalies shown by Donald
Lentz. It is there, about this problem, I defined srutis.

What I mean here by defining srutis? The srutis had already a rich
polysemic sense in Hindu music. I mean I distinguished in sruti concept
between a universal aspect, that may be applied to all gammoids, and
particular aspect, that might be applied to only few gammoids sharing a
specific property existing in the Hindu Classical system.

In that Hindu system we find something very specific : the interval between
the maximal tone of a degree (SA RI GA ...) and the minimal tone of the
next degree is always the minimal step (256/243). There is no exception in
the Hindi region where the sruti 12 is 729/512 but one exception where the
sruti 12 is 64/45.

If a similar particularity is included in a sruti definition then you
cannot have a universal algebraic tool. However I have not invented a new
concept. I took simply what was universal while already there in Hindu system.

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In the Hindu Classical system the first degree has four tones being in order

<256/243 16/15 10/9 9/8>

Do you know their Hindu names? I translate only :

<one-sruti two-srutis three-srutis four-srutis>

So one of the srutis, in the precise sense of increments, is forcely the
minimal step 256/243 for starting with unison 1

one-sruti = (zero-sruti)(sruti increment)
256/243 = (1)(256/243)

Since 16/15 is two-srutis (meaning composed of 2 srutis) another sruti
increment is forcely 81/80 for

two-srutis = (one-sruti)(sruti increment)
16/15 = (256/243)(81/80)

Continuing, since 10/9 is three-srutis (meaning composed of 3 srutis)
another sruti increment is forcely 25/24 for

three-srutis = (two-srutis)(sruti increment)
10/9 = (16/15)(25/24)

Finally since 9/8 is four-srutis (meaning composed of 4 srutis) the next
increment is forcely the sruti already used 81/80 for

four-srutis = (three-srutis)(sruti increment)
9/8 = (10/9)(81/80)

That gives the three srutis

<256/243 81/80 25/24>

having in cents approximately

[90 22 70]

permitting to obtain the very particular 22-tone Hindu system with the
particular sequence of these well-defined three increments

(90 22 70 22)(90 22 70 22)(90 22 70 22)(90)(90 22 70 22)(90 22 70 22)(90)

That construction is very particular but the manner we have obtained the
three sruti increments is not only universal but very useful in musical
system algebra.

It is exactly that definition I wrote recently in condensed manner

sB = e'(tG)

where

the tonal generator tG means the steps (degree 1) and

the srutal basis sB means the srutis

and where as seen, the srutis are composed of the minimal step and the
unison vectors necessary to construct the steps starting with the minimal.

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Maybe there exist another definition having its proper virtues. I need to
know it if I have to say something new.

Pierre