Tuning Lattices

Hello,

I haven't read the entire archive, so forgive me if some of this is redundant. But I wanted to contribute a few words about the relation between my orbifolds and traditional tuning lattices.

1) Take a very simple log-frequency tuning lattice, with acoustically pure fifths on the X axis and octaves on the Y axis: moving one unit to the right raises the pitch by perfect fifth, while moving one unit upward raises the pitch by one octave.
Now let's imagine that the axes are continuous -- if we move fractionally along the positive X axis, we glide slowly upward. In this continuous version of our tuning lattice, every pitch will now represented by a line: for any distance that we move along the X axis, we can compensate with a move along the Y axis. Thus points arbitrarily far away in the XY plane can represent the same pitch. Consequently, distance in this space isn't going to tell you about voice leading.
This generalizes to higher dimensions. It's typically impossible to embed familiar tuning lattices in continuous spaces where every point represents a *distinct* pitch. Moreover, "distance" in the resulting spaces measures something other than the distance relevant for thinking about voice leading.
(Note, by the way, that if you "glue together" all points representing the same pitch class, you end up with a log-frequency line corresponding to one of the axes.)

2) However, as someone on this list pointed out, it is possible to represent alternative tunings in my orbifolds. Basically, any octave repeating scale (including an octave-repeating chromatic or diatonic scale representing a tuning system) gives you a way to measure distance in circular pitch class space. Musicians typically refer to this unit of distance as a "scale step." Only in an equal temperament will all (chromatic) scale steps have the same log-frequency size.

As this happens, my "ChordGeometries" program depicts this nicely. Start the program, then do the following:
a) Select "Master Parameters" from the "Commands" menu.
b) Type in some (equal tempered) [sorry!] pitch classes into the "Display These PCs" menu.
c) Press "return"
d) Select "Dyadic Space" from the "Geometries" menu
e) Select "Parameters" from the "Commands" menu
f) Use the pop up menu that says "No Lines" to check "Oblique Steps," "Parallel Steps," and "Contrary Steps"
g) Look at the resulting figure

What you see represents the metrical grid that a tuning system imposes on the continuous orbifolds. It sort of looks like the terrain of a hilly city, viewed from above. It's worth looking at a bunch of scales (using the "Master Parameters" window) to see how they generate different metrics.

Hope that helps,
DT
--
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Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri