Re: Complex numbers and tuning parameters

Hello, there, and here's one quick reaction to the question of complex
numbers as elements in tuning.

One interpretation of the two-dimensional "a + bi" formula which
occurs to me from synthesizing tuning tables is to let "a" represent
the choice of notes or intervals within a set of "standard" pitches,
and to let "b" equal a deviation (for example in cents, or in decimal
values of some scale interval such as a diatonic semitone or diesis)
from a given standard pitch or interval.

A purist might argue that both the real and the imaginary portions of
this complex formula actually pertain to the same "dimension," namely
pitch, but I might argue that conceptually and musically, the choice
of _which_ basic tone or interval and _how far_ from that tone could
be taken as a two-dimensional process.

For example, taking a 31-note version of 1/4-comma meantone as the
basis of our "a" dimension, with a diesis (~1/5-tone) as the basic
unit, and letting cents be the "b" units of our "i" dimension, we
might specify a two-voice cadence using vertical intervals such as the
following:

10 + 21.51i (a Pythagorean major third:
10 = 10 dieses, a pure 5:4 major third
21.51i = 21.51 cents, the syntonic comma of 81:80)

18 + 5.38i (a pure 3:2 fifth:
18 = 18 dieses, a tempered fifth around 696.57 cents
5.38i = ~5.38 cents, making the tempered fifth pure)

In other words, this cadence is identical to a standard Pythagorean
progression from 81:64 major third to 3:2 fifth -- but mapped onto a
system where 1/4-comma meantone is the norm.

In fact, one kind of two-dimensional "x-y" keyboard scheme discussed
in a recent thread is to have two or more keyboards with the same
note patterns or tunings at different pitch levels.

In such a scheme, the "a" dimension of "a + bi" might tell us which
note to press, while the "i" dimension might tell us on which keyboard
to press it.

While such a keyboard array, with fixed pitches for each note and
keyboard, would present a few discrete values for the "i" dimension
(e.g. three keyboards in the same Pythagorean tuning, each tuned a
Pythagorean comma apart), in a conceptual "a + bi" scheme we could
vary the note by any desired quantity.

It seems to me that this kind of scheme could apply both to
traditional Pythagorean and meantone tunings, and to various n-tet
tunings. In this case the "real-number" system would be the standard
notes of the tuning, and the "complex" system these notes plus or
minus any desired deviation -- in other words, any intermediate
values.

From this point of view, for example, we might interpret 12-tet as a
variation on a 17-note Pythagorean scale (Gb-A#); each 12-tet
accidental might have two possible mappings, for example the C#/Db key
as a deviation from Pythagorean C# or Db.

For example, 12-tet E might be notated as "6 - 7.82i" -- the seventh note
of a 17-note Pythagorean series 0-16

(C,Db,C#,D,Eb,D#,E,F,Gb,F#,G,Ab,G#,A,Bb,A#,B)

with a deviation of -7.82 cents from an 81:64 major third in relation to
C.

Most respectfully,

Margo Schulter
mschulter@value.net