Multi-Dimensionality - reply to Paul E.

Paul writes:

>from a purely psychoacoustic perspective, it is unlikely
>that making exceptions to the prevailing odd-limit by allowing large
>powers of a small prime makes much sense. Using Kameoka & Kuriagawa, for
>example, there is no sense in which 27/16 will be more consonant that
>5/3 for any listener.

Certainly, the interval 5/3 sounds sweeter than 27/16, when each is sounded
in isolation. I seem to have omitted a necessary part of the context -
sorry. I was referring, not to the tuning of intervals themselves, but to
the tuning of JI scale members, used for tonal music. If you built a minor
triad on the second scale member, and it was tuned to 9/8, the chord would
obviously be more consonant with a 27/16-sixth-scale-member
fifth-of-the-chord than a 5/3 one.

To restate, then, if all of the members of your scale are made up of ratios
involving 3^x, and possibly other primes and their powers, but not 5^x, then
a single note on the 6th of the scale, tuned to 5/3 or any ratio involving 5,
would sound out of place. It could be used, though, as an intersection, a
corner around which to turn into a passage involving more scale members using
5^x in their ratios. I've experienced this. I'ts like the first time in a
piece that has been 5-limit (prime) up to that point, that you use a true 7/4
- zow! But if you continue to use more ratios involving 7, it looses its
shock value and comes to sound quite pleasant. I doubt that a 9/5 or 16/9 in
the same situation, if all 9s had been avoided up to that point, would come
as such a shock, even though 9>7. I believe that that is because 9 _is_ 3 3s
- it contains 3s, for all rational purposes. I have yet to see a convincing
example of the use of a ratio involving 9 where it did not function and sound
like it was part of the prime dimension based on 3.

David



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> In 12tET, there are 2 patterns available : hW hW hW hW and Wh Wh Wh Wh. In
>8tET, there is only one diminished scale. Blackwood said that a possible
>16tET diminished scale would be 0 1 4 5 8 9 12 13 16. All these scales have
>a two note pattern that fills a minor third. The pattern is repeated 4
>times in an octave.

How specifically does the term "symmetric mode" apply to this? I
presume that a diminished scale is a special case of a symmetric mode, in
particular where the two-step repeating pattern spans a minor third?



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Larry Polansky has noted that Tenney's use of the octotonic is an
approximation of the first eight primes, reduced to an octave: 1, 17, 19,=

5, 11, 3, 13, 14. Others have noted this in Stravinsky and work of the
Skryabinists.



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