COMPUTER ASSISTED COMPOSITION IN EQUAL TUNINGS: TONAL
COGNITION AND THE THIRTEEN TONE MARCH
John Chow Seymour
University of North Texas
College of Music
ABSTRACT
This essay first describes the technological method I used
to create a working environment for creating music with
equal tunings in divisions of the octave other than twelve.
It then makes some general comments on the tuning of
thirteen equal tones, and presents a partial analysis of my
own composition in this tuning. This composition, simply
entitled "Thirteen Tone March" was created as part of a
project to attempt the composition of tonal music in tuning
systems with numbers of equal tones higher than twelve.
The essay concludes with an aesthetic discussion of the
reasons for such a project, and its results.
1. A WORKING ENVIRONMENT FOR
COMPOSITION IN EQUAL TUNINGS
In standard equal tuning, the octave is divided into twelve
equal steps. Given any frequency F, one can determine
the frequency f of any note x that is y of these steps away
from F by the following formula:
f(x)
F. 2
Using the formula above, I created a retuning patch in
the Max/MSP environment that lets the user assign values to d, F, and M; it then accepts MIDI input and outputs frequency in Hz (and passes velocity through unchanged). Its output can be sent to any synthesis module
that accepts frequency and velocity. To ensure problemfree polyphony, a simple but safely redundant mechanism
is put into place.
As an example, the common reference pitch "A 440"
would let M equal 69 (on some keyboard controllers) and
F equal 440.000 Hz. Playing a note on a MIDI controller
12 physical keys above the given A would let x equal 81,
x - M would be 12, and if d = 12 then f(x) would
be 880.000 Hz, one octave above the reference note. If
d were 13, however, one would need to play a note 13
physical keys higher then the reference note to produce an
octave. It would look like an A-sharp on the keyboard,
but would sound an A. Similarly, playing a C key three
physical keys above the reference A while d = 12 would
obviously result in a C (523.251 Hz) or 3/12 of an octave.
While d = 13, playing the same key would sound only
3/13 of an octave above A (516.323 Hz), while d = 18 it
would sound 3/18 of an octave above, and so on.
The physical key corresponding to the reference note
M will be the one key that is never retuned, and all of the
other keys on the controller will be retuned around it. For
this reason, I refer to the reference note as the "pole." F
and M can be set together by picking a pole; in this case
it is assumed that this note will have the same frequency
that it does in standard tuning. In my scores written for
this environment, the tuning is indicated at the top of the
page by giving values for d and for the pole.
2. TUNING IN THIRTEEN EQUAL TONES
I greatly enjoy Easley Blackwood's microtonal compositions and am impressed with his essay from around fifteen
years ago that proposes some useful theories for dealing
with nineteen, seventeen, sixteen and fifteen note equal
tunings, especially in that his approach to these tuning systems is well adapted to the idiosyncrasies of each system
[1]. My approach to the tuning of 13 equal tones per octave is to take advantage of the ambiguity that can arise
from different ways of approximating tonal elements of
standard 12-note tuning. After working in this tuning for
some time, I composed the Thirteen Tone March, each
Intuitively enough, the resulting frequency is y steps
above F if y is positive, and below F is y is negative.
To divide the octave into equal steps numbering other
than 12, we can replace 12 with a variable, d (for "division" of the octave.)
To implement this in a MIDI interface,
* Let M be the MIDI note number that corresponds
to the known frequency F.
* Let x be the MIDI note number of a key that's been
played.
* The distance y can now be understood as the difference in the MIDI note numbers of the played and
known notes, or x - M. (This order preserves the
positive-negative relationship described above).
Thus, given a known note M with frequency F, the
frequency of any note x can be expressed as below:
f(x) = F. 2 (2)
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