RE: Comment on Kami's post

John Chalmers wrote,

>Whenever 1 or 2 appears as a factor, the dimensionality may be
>lowered by the assumption of octave equivalence.

I was under the impression that having 1 (or 2) as a factor was common,
based on some material John sent me, which led me to the discovery I
mentioned in my comment on Kami's post. It certainly is not a trivial factor
and seems to aid in the geometrical interpretation of these things. Without
a 1 (or 2), the CPS is just a "border" between these tiled hyperpolyhedra.
Whether or not you assume octave equivalence doesn't alter the
dimensionality of a given CPS, as far as I can see, since no
octave-equivalent pairs of tones are present. An exception would have to be
made in the case of 2 appearing as a factor along with the repetition of
other factors, but John said,

>Erv Wilson [. . .] seldom used repeated factors.

On another point:


>to measure distances between intervals use the Minkowski metric
>(Tenney's harmonic distance)

I would dispute this, since it takes the axes as orthogonal, contrary to my
view of Euler geni (?) as oblique. Tenney would view 6/5 as spanning the
same harmonic distance as 16/15, that distance being log(3)+log(5)g(15),
while I would argue that the former (and its octave equivalence class) is
simpler and so should be represented by a shorter distance.

Here's how I would measure distance. I take each axis as representing a new
prime number, starting with 3. In two dimensions (5-limit), the solution is
to make major triads or 1)3 [1.3.5] CPS, and minor triads or 2)3 [1.3.5]
CPS, isosceles triangles (which fill the plane) where the three edges 3/1,
5/1, and 5/3 have length log(3), log(5), and log(5), respectively (i.e., the
length is the log of the limit). By contrast, Tenney would take these
lengths as log(3), log(5), and log(15), respectively, so that the consonant
triads are no more compact in space than a chord like 12:15:16.

In three dimensions (7-limit or 9-limit), one additionally represents 7/1,
7/3, and 7/5 as edges of length log(7), and space is littered with
transpositions of 1)4 [1.3.5.7] and 3)4 [1.3.5.7] tetrads (otonal and
utonal, in Partch language), which have one of each of the six edge types
(are these called isosceles tetrahedra?), and the "holes" are filled with
transpositions of the 2)4 [1.3.5.7] hexany (isosceles octahedra?), which
have two of each of the six edge types. 9-limit constructs are not as pretty
geometrically, but the math works out: assuming you travel along edges only
(is this what is meant by the Minkowski metric? My relativity teacher didn't
think so!), an interval is consonant with respect to the n-limit if the
shortest distance needed to traverse the interval is less than or equal to
log(n), since log(3)+log(3)g(9), etc.

The practical problem with these metrics is that they don't take "punning"
into account. Eventually a really distant point will sound like a really
close one. A simple, accurate, and consistent temperament can remedy this
situation, but only with more (euclidean) dimensions around which to "wrap"
the lattices. For example, the set of meantone temperaments makes three
steps of 3/1 the same as one step of 5/3. So the plane becomes a tube. This
tube may still be infinitely long, though, and so will have punning
problems. But meantone equal temperaments like 12 and 19 cut it off and
attach the two ends together, making a torus. The geometric distance between
any two tones is then a good indication of how consonant the interval
between them is.


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I wrote:

>(b) A similar flip-flop may obtain in Just Intonation between chords that
>are ambiguously harmonic or subharmonic.

Paul wrote:

''Do you really believe this phenomenon exists? If so, can you suggest an
experiment that could demonstrate its existence?''

1. Sure. All chords built on repeated intervals will have this character.

2. I assume that you mean some kind of perceptual experiment. Short of an
EEG, all tests I can imagine involve some kind of identification or
labeling, which has a lot to do with training or conditioning - and the
phenomena here is already quite technical for trained musicians. So, with
trained musicians, I would suggest testing chords aside from Major or minor
triads, due to our overconditioning with these particular animals (although
some voicings of four-tone chords containing both Major and minor triads
also have this ambiguity: try 10:12:15:16).

One totally subjective example:

The triad: 330 - 440 - 495 (Hz).

I hear it sometimes as /8 : /9 : /12 and sometimes as 6 : 8 : 9. In
particular, if it is arpeggiated descending I tend to hear it as
subharmonic, ascending, harmonic. As a single gestalt, I flip-flop with a
slight preference towards the subharmonic interpretation - possibly because
of the smaller interval on top.

As for a test with untrained listeners, I have no idea of how to do it
without running into the problems of finding adjectives to characterize
chords.

Perhaps you can be more specific about what kind of experimental
information you seek.



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