source file: mills2.txt
Date: Sat, 12 Jul 1997 17:35:47 +0200

Subject: Basic dimensionality, and related issues

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

Looking back at my notes on matrices, I see that, if you define
the logarithms of prime numbers to be orthonormal vectors, scales
can be written as a vector space using these vectors.  Then, the
dimension of the vector space is the same as my definition of the
dimension of the scale.  I can talk about the "basic dimension of
the scale" if this will avoid confusion with other concepts of
scale dimension.  The logic being that the dimension is the
number of vectors that form a basis.

This works until transcendental numbers come into it.  Then, the
same basis can be defined using different primes:

     (x) (x/log(2) 0)(log(2)) (x/log(2) 0      )(log(2))
     (y)   (y/log(2) 0)(log(3))   (0        y/log(3)(log(3))

To avoid this ambiguity, the matrix defining a scale from its
basis must only contain integers -- or, at a stretch, rationals.
I think this covers all musically meaningful situations.  Any
counter examples?

Are people understanding this matrix stuff?  There was one
error a few posts back that John Chalmers spotted (I wrote a
chromatic semitone instead of a diesis) which may have
confused some of you.


A discussion on the interval 9/1 has resurfaced.  Partly, this
relates to scale dimensionality and partly to definitions of
consonances.  I will concentrate on the former.

To take Daniel Wolf's example:

>(1) I compose a piece consisting only of chromatically planed (parallel)
>Major ninth
>chords without fifths in just intonation. If the orchestration is such thlog(9) can be taken as a basis here.  It doesn't affect the
basic dimensionality.

and another:

>(4) Using a TX81Z (with a resolution of 1.56 cent deviations from 12tet),> I
>perform
>a melody based upon random walks over a lattice with two dimensions: 3s a>nd
>9s. The
>best approximation of 9 does not coincide with the best approximation of
>3^2. For all practical purposes, 3 and 9, in this temperament, are
>relatively prime.

3*3is a universal theorem.  If you are using an interval that
is not 2*log(3), that interval is not log(9).

The TX81Z uses 768TET tuning tables.  Any such table will be 1-D.
To what extent it approximates a higher dimensional scale is a
matter of interpretation.  In this case, though, I strongly
disagree with you.  On a TX81Z, 3/1 is 768*ln(3)/ln(2)17.2512
steps, approximated to 1217/768 oct.  9/1 is 768*ln(9)/ln(2)2434.5024.  The closest approximation is 2435/768 oct, but 3*3
gives 1217*2/768$34/768.  The latter is less accurate by
0.0024/768 oct 3.1 mu-oct (0.0038 c*nts) --  surely negligible
in comparison with the inaccuracy of both intervals.  I see no
problem in making the two consistent.

Daniel again:

>It>is interesting to hear how certain structures survive radically different>mappings intact as well as to hear what features are projected differentl>y
>under alternative intonations. Basic dimensionality is a precisely defined mathematical idea.
Music being a subjective thing, it makes more sense to define an
n-D scale in the abstract, and then approximate it for
performance, than to take a real life scale and try to assign a
dimension to it.

Paul Erlich wrote:

>Also, if 9/5 and 9/7 are to be used as distinctive harmonic entities (i.e.,
>consonances), and not only arising from 3*3/5 and 3*3/7, then it is useful to-

>include a 9-axis into the triangular lattice.

If the dissonance of a composite number is independent of its
factors, why use lattices at all?  If you do, though, this
would introduce a concept of harmonic dimensionality different
from scale dimensionality.  What I previously called "n-D
harmony" can be renamed "n-D JI" to avoid confusion here.

After introducing a term, I should define it.  A scale containing
only integer frequency ratios is a JI.  An n-D JI is then a JI
with a basic dimension of n.  This means a scale involving 11, but
not 7, is still a 4-D JI although it is not 7-limit, so the two
concepts are not equivalent.


On the concept of the prime limit: this is useful because it
states the resources available for scale construction.  Unlike
the Partch limit, 5 prime limit intervals can never take you out
of the 5 prime limit.  However, the prime limit is not a useful
measure of dissonance or even scale complexity.  Any interval can
be written as 3 prime limit, within the limits of human hearing.
It may be a very complex 3 prime limit interval, but it can
always be done.


On dissonance of composites in general:  it is much easier to form
consonant chords using composite numbers than prime numbers of
about the same size.  As an octave invariant example, 1:3:5:15
includes the interval 15/1.  Try finding an equally consonant 4
note octave invariant chord including 13/1.  If the consonance of
an interval reflects its propensity to form consonant chords, then
composites are more consonant than primes and, in some
circumstances, the metric should handicap high primes accordingly.
This brings us one step closer to octave invariance.

Paul also wrote:

>The assumption of exact octave equivalence, though it has some
>psychoacoustic basis, should be taken more as a conveniece than an exact
>truth.

Hey, he agrees with me!


An unrelated (or is it) point.  Recently, Gary Morrison wrote:

>   By fibonacci, what I'm referring to is J. Yasser's fibonacci-like
>sequence of tunings (5 7 12 19 31 50 81...).  (By fibonacci-like I mean
>that the next number in the sequence is the sum of the previous two.)  12

Firstly, wasn't it Kornerup who first identified this sequence?

Secondly, there are lots of Fibonacci-like scale sequences.
Fibonacci+3, for a start.  The special thing about this
sequence is that the number of steps in a tone and semitone, or
tone and minor third, or minor third and perfect fourth, are
neighbouring steps in the Fibonacci series.  Another meantone
type ET is 43TET.  12+31C.  Hey ...

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