Dimensions, etc.

Paul E. is correct (as usual) that the best fifths of 24- and 34-tets
is not cyclic through the whole gamut. However, there are other intervals
that are, and in this respect, ET's are 1-D. In fact, in all ET's with
more than 6 tones, there is at least one pair of intervals other than
1 and N-1 that is a cyclic generator. (For 6-tet, the pair 1-5 is the only
cyclic generator).

Paul is also correct that meantone rolls the 3x5 plane into an infinite
cylinder. This is true for any non-cyclic temperament which defines one
harmonic dimension in terms of a cycle of another generating interval.
Thus a meantone-like tuning which defined the 7/4 by 10 ascending fifths
would roll up (compactify?) the 3x7 plane. A temperament which tweaked the
third to obtain the just 7th would roll up the 5 x 7 plane (disregarding
the Fifth for the moment).

I imagine 1/3 comma meantone, with its just 6/5-5/3, would roll up a plane
lying obliquely to the coordinate axes. Paul?

In summary, the dimensionality of a scale or tonal space depends somewhat
on how we conceive or use it. Gerry Balzano described tunings of 12, 20,
30, n(n-1) tones in terms of triads whose component "thirds" spanned n and
n-1 degrees and scales of 2*n-1 tones. Fokker showed how regions of tonal
spaces
correspond various to ET's and these regions induce different
dimensionalities on ET's of the same number of tones. Thus dimensionality
is in the mind of the theorist.

My other major point is, as I think Graham first mentioned, that one can
restrict oneself to various subspaces of these tonal spaces. Fokker in
New Music with 31 Notes (translated by Leigh Gerdine from Fokker's
German original, published by Orpheus), discusses Diamond-like struct
�
in the 3x5, 3x7 and 5x7 planes generated by mirroring triads such as 1.3.5,
1.3.7, and 1.5.7

A skew Diamond may be generated by mirroring the 3.5.7 triad. These
scales are equivalent to portions of the 7 limit Partch Diamond and
are perhaps visualized most easily by constructing cross-sets (to use
E.W.'s term) of the 1.3.7 triad by its inversion, the sub-1.3.7., etc.
For the 3.5.7, multiply 1/1 5/3 7/6 by 1/1 6/5 12/7. Alternatively,
one may construct a 7-prime-limit Diamond and delete the notes containing
the omitted factors.

Other scales are possible in subspace (that has a nice SF'nal sound to
it). A very nice pseudo-diatonic scale in the (2)x3x7 subspace is the
septimal or subminor scale 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1. Five-less
chord progressions can sound very consonant and I recommend trying them.

A final and minor point: I believe Erv's 22-tone constant structure was
based on the 1.3.7.9.11.15 Eikosany with 2 additional tones to complete
the gamut. In CPS, the 1 (2, 4, etc.) factor axis may be an essential
conceptual dimension, even if it can be omitted (compactified?) when
plotting notes on paper.

It is the presence of small "kommata" that inevitably occur when one uses
complex intervals or moves very far from the origin and which are less
than single ET degrees that makes it impossible to do 1-to-1 mappings of
some scales into some ET's.

--John

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