source file: mills2.txt
Date: Sun, 6 Jul 1997 12:05:03 +0200

Subject: Scale Dimensionality

From: gbreed@cix.compulink.co.uk (Graham Breed)

David Finnamore wrote:
>BTW, I like a concept that Graham Breed introduced me to: harmonic
>dimensionality using prime limits.  As I understand it, using all octaves
>(like P. D. Q. Bach's Don Octave) would be 1-dimensional, Pythagorean scales
>(3-limit in practice) would be 2-dimensional, 5-limit would be 3-dimensional.

This is a correct exposition of my idea of n-D harmony.  I thought
of it before I knew anything about prime limits, and I suggest
the terminology be adopted to avoid confusion with odd limits.

I have a related idea of scale dimensionality: this is the number
of generative intervals for a scale.  An n-D scale can be
represented on an n-D rectangular lattice.  Just n-D harmony
requires an n-D scale.  Meantone temperaments are 2-D scales,
usually approximating 3- or 4-D harmony.  ETs are 1-D scales.
This is inconsistent with meantone temperaments being "linear"
temperaments.

The idea of scale dimensionality is closely linked to my matrix
representation.  A meantone scale can be written:

          H(k) H + (k2)p(k)
                     (k3)

Where H is the metric of 2-D harmonic space, H(k) is an approx-
imation to H in the tempered scale, k2 is the amount by which an
octave is sharpened, k3 is the amount by which a pefect 12th is
sharpened and p(k) is the comma as a proportion of which this
sharpening occurs.  Note that k2 and k3 form a vector.

1/4-comma meantone is: k2:0, k3/4, p(k)4 4 -1)H.

H(k) is a tempered Pythagorean scale.  To get a tone and a
semitone out of this:

                 (t) (-3  2)N(k)
                 (s)k  ( 8 -5)

The power of matrix representation is that a temperament can be
derived from the intervals you wish to remove.  However, that
gets a bit complicated so I won't illustrate it here.

                 Graham

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