Re: Pythagorean in action: a quick example

Hello, there, and I'm very pleased to see recent articles by Brett Barbaro
and Kraig Grady emphasizing that Pythagorean tuning is indeed an important
integer-ratio tuning in various world musics.

At the same time, this discussion suggests to me that one way to make
the practical aspects of Pythagorean tuning more familiar is to
present a quick example which might demonstrate the qualities of this
tuning on any keyboard instrument that can support it, for example a
microtuneable synthesizer.

Here's an example of a three-voice cadence from 13th-century France
that may sum up some of these pleasant qualities in only three
sonorities. Note that following MIDI conventions, I use C4 to show
middle C, with higher numbers showing higher octaves:

1 2 3 | 1 2 3 |
D4 E4 F4
C4 B3 C4
G3 F4

Here the first mildly unstable sonority, G3-C4-D4, has two richly
stable medieval intervals (the fifth and fourth) plus a relatively
tense major second. The overall effect is one of an energetic but
relatively "concordant" or blending sound, in part because all three
intervals are in their pure ratios: 6:8:9 (frequency-ratio), or 12:9:8
(string-ratio, with the largest string length representing the lowest
note, the medieval approach).

In the second cadential sonority G3-B3-E4, Pythagorean tuning makes
both the relatively concordant major third and relatively tense major
sixth (ranking about on par with the major second) more acoustically
"active" or tense. These intervals have rather complex ratios of 81:64
(M3) and 27:16 (M6), with the sonority as a whole at 64:81:108
(frequency-ratio), or 81:64:48 (string-ratio). This somewhat tense or
"vibrant" quality accentuates the directed cadential resolution about
to follow.

In this directed progression, the major third between the lower two
voices expands to a stable fifth, while the major sixth between the
outer voices expands to a stable octave. Note that both upper voices
move by characteristically narrow and "efficient" Pythagorean
semitones of 256:243 (B3-C4, E4-F4), while the lowest voice descends
by a generous whole-tone of 9:8 (G3-F3). These melodic motions can be
expressive not only in polyphonic music like this, but also in
Gregorian chant and other forms of monophonic music.

This progression takes us to the final sonority, F3-C4-F4, with three
pure intervals: the octave (2:1), fifth (3:2), and fourth (4:3), or
together 2:3:4 (frequency-ratio) or 6:4:3 (string-ratio). Pythagorean
tuning optimizes all three intervals of this favorite medieval
three-voice sonority, the standard of complete and "most perfect"
concord.

Of course, medieval European polyphony is just one of the many world
musics favoring Pythagorean tuning. As these styles become more
familiar, the tuning itself may be better understood for its beauty in
practice as well as its intriguing theoretical aspects.

Most respectfully,

Margo Schulter
mschulter@value.net