RE: Meantone, Partch

John Chalmers wrote,

>I agree that in general Meantone tone systems are 2-D,
>but under the
>common assumption of octave equivalence, the dimensionality may be reduced
>to 1. In which case, both ET's and meantone tunings are generated by one
>interval, the tempered fifth.

Some ET's cannot be generated by their best fifth, e.g. 24 and 34. I
like to see meantone systems as 2-D even if octave equivalence is
assumed, since meantone systems are meant to approximate 5-limit JI.
Topologically, meantone tunings wrap the 5-limit plane onto an infinite
cylinder. I like to see ETs as having as many dimensions as odd factors
with which they are consistent. ETs consistent with the 5-limit wrap
said plane onto a torus. 22tET can be considered 5-dimensional because
it consistently represents all ratios of 3, 5, 7, 9, and 11. If we allow
skipping of 13, we can add 15 and 17 and treat 22tET as a 7-dimensional
tuning! Of course, one doesn't need a very large portion of the lattice
to cover all 22 notes . . .

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On Wed, 9 Jul 1997, Graham Breed wrote:
> So, it looks like I was thinking along the same lines
> as Fokker. Did he come up with any other stuff like this? Is
> there a book I can read?

These two books probably best represent Fokker's thinking along these
lines. They can both be requested through Interlibrary Loan from the
library I work at, among others.

Title: Selections from the harmonic lattice of perfect fifths and
major thirds containing 12, 19, 22, 31, 41 or 53 notes.
Author: Fokker, A. D. (Adriaan Daniel), 1887-
Published: Amsterdam, Koninkl. Nederl. Akademie van Wetenschappen, 1968.

Title: Unison vectors and periodicity blocks in the three-dimensional
(3-5-7) harmonic lattice of notes.
Author: Fokker, A. D. (Adriaan Daniel), 1887-
Published: Amsterdam, Koninkl. Nederl. Akademie van Wetenschappen, 1968.

--pH http://library.wustl.edu/~manynote <*>
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

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