Indian 5-limit, Symmetry

I think we are in agreement that because of the skhisma, a
sufficiently long Pythagorean series approximates 5 limit
intervals. While Greek music is a lost art at the present
time, it was not during the Hellenistic and Roman periods.
Islamic music theory is heavily dependent upon Greek theory,
even if some writers admitted that certain scales and tetrachords
weren't played any more and proceeded to discuss the 14 and 17
tone Pythagorean cycles.

As for "broken symmetry, one might consider a static analog,
the symmetrical division of an octave into repeating modules
as in Messiaen's modes of limited transposition (2 1 1 2 1 1
2 1 1, etc.) and certain types of scales generated by cycles
of intervals relatively prime to the octave. These "well-formed"
or MOS (moment of symmetry) scales have 2 or more instances of a
repeated block of intervals and a truncated series. For example,
the diatonic scale in 12-tet is 2 2 1 2 2 1 2 (as the Greek
Hypophrygian mode) where the block is 2 2 1 and the broken set is 2.
In 19 tet, the homologous 12 tone mode is 2 2 1 2 1 2 2 1 2 1 2 1
where the repeating block is 2 2 1 2 1.

In 13-tet, 5 or 8 degrees generates 2 2 1 2 2 1 2 1, and there are
many more examples in other ET's where instead of a symmetrical
division of the octave, there is an almost symmetrical division.
(The symmetrical divisions are in general of a type called "maximally
even" by John Clough and his students and collaborators, who have also
developed the theory of well-formed scales.)

--John



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Subject: Re: 5-limit in India?
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