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Limits (Paul E)

🔗Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul)

3/22/1997 10:31:37 AM
From: PAULE

For a reference on Euler-Fokker genera, see Fokker, Adriaan D. 1975. _New
Music with 31 Notes._ Verlag fur systematische Musikwissenschaft GmbH, Bonn.

The prime limit definition should NOT be the default, despite its utility in
describing a tuning system. The odd limit definition is not intended to
describe a tuning system; it is intended to describe musical practice.
Despite what Jonathan Walker said, one does not consider music in
Pythagorean tuning to be a 531441-limit system under the odd definition of
limit. Even under Partch's own (odd) definition, Partch's music was
11-limit, not 33-limit nor some product of powers of numbers through 11.
Partch listened carefully and came up with the most useful definition for
the ear and for musical practice, not for studying the mathematics of tuning
systems.

The definition is: A piece of music is N-limit if all intervals whose
(heard) rational approximations use odd numbers up to N, and all chords
containing only such intervals, are considered consonant, while intervals
which cannot be so approximated, or any chord containg at least one such
interval, are considered dissonant. In other words, it's how high up the
harmonic series (or how far down the subharmonic series) the ear is asked to
go in a particular piece.

So with a few caveats (such as second-inversion triads) common-practice
music is 5-limit. The difference between the prime and odd definitions
really shows up at the first composite odd number, 9. As Dan Wolf and I have
pointed out before, there can be a very real difference between 7-limit
music, where 9 is only possible as 3*3, and 9-limit music, where 9 is
independently available as a harmonic building block.

I'm sure I could have said some things better here, I look forward to
discussing more because this is an important issue!

-Paul E.

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