VIRTUAL PITCH CONTINUUM
72 TONE EQUAL TEMPERAMENT AS A PITCH CONTINUUM
. . . [Regarding] 72ET as representing the harmonic series, from a practical standpoint, you could consider 72ET to be a pitch continuum, where ANY interval is approximated well (with a maximum error of 8 1/3 cents), and this viewpoint is particularly appropriate to strings and such instruments, where the variable tuning is all by ear anyway. For fixed pitch instruments, it may be more reasonable to think of the 72 tones as only representing intervals that are "nearby" (the max error leads to ambiguity); in that view, the harmonic series is represented very well up to harmonic 12 (worst error 3.9c), but with 13 the error jumps to 7.2c, so from there on, the approximations are not clear representations of the harmonic relations... IMHO anyway. So one may need to be content with "only" pure octaves, fifths, thirds, sevenths, ninths, and elevenths! Not bad! From: Canright, David, Date: Sat, 21 Apr 2001 09:39:02 -0700
72-tET Approximations of Just Intervals
Below is a table of [the first 23]. . . intervals found in the harmonic series . . ., and their closest approximations by 72-tET. These are ordered by first appearance in the harmonic series (by numerator, then by denominator; . . .); Not all of the "octave complements" appear; of course, their tempered approximations have equal but opposite errors. Each entry includes the ratio, cents error relative to nearest tempered (so, for example, 3/2 is +2.0c relative to note #42), tempered note # (0-71), keyboard # (0-5), key # (0-11), where keyboards 1-5 are assumed progressively sharper relative to keyboard 0.
| ratio | cents error | note | key |
|---|---|---|---|
| 1/1 | +0.0¢ | 0 | 0:0 |
| 3/2 | +2.0¢ | 42 | 0:7 |
| 4/3 | -2.0¢ | 30 | 0:5 |
| 5/3 | +1.0¢ | 53 | 5:8 |
| 5/4 | +3.0¢ | 23 | 5:3 |
| 6/5 | -1.0¢ | 19 | 1:3 |
| 7/4 | +2.2¢ | 58 | 4:9 |
| 7/5 | -0.8¢ | 35 | 5:5 |
| 7/6 | +0.2¢ | 16 | 4:2 |
| 8/5 | -3.0¢ | 49 | 1:8 |
| 8/7 | -2.2¢ | 14 | 2:2 |
| 9/5 | +0.9¢ | 61 | 1:10 |
| 9/7 | +1.8¢ | 26 | 2:4 |
| 9/8 | +3.9¢ | 12 | 0:2 |
| 10/7 | +0.8¢ | 37 | 1:6 |
| 10/9 | -0.9¢ | 11 | 5:1 |
| 11/6 | -0.6¢ | 63 | 3:10 |
| 11/7 | -0.8¢ | 47 | 5:7 |
| 11/8 | +1.3¢ | 33 | 3:5 |
| 11/9 | -2.6¢ | 21 | 3:3 |
| 11/10 | -1.7¢ | 10 | 4:1 |
| 12/7 | -0.2¢ | 56 | 2:9 |
| 12/11 | +0.6¢ | 9 | 3:1 |
Subject: table ordered by harmonic limit/Date: Wed, 2 May 2001 11:56:53 -0700/From: Canright, David
VIRTUAL PITCH CONTINUUM KEYBOARD VERSION WITH ERRORS <4.9¢
72-tET Approximation of Just Intervals Ordered by First Appearance
72-tET Approximation of Just Intervals Ordered by Interval Size
Our friend Scott Van Duyne from CCRMA, Stanford has kindly provided the following information: Preliminary Studies in the Virtual Pitch Continuum by Joe Maneri and Scott Van Duyne (Published by Accentuate Music, 42 Cornell Dr., Plainview, NY 11803) is an entire book on composition in the 72 note scale. It runs in a sort of work book style, but there are many short musical examples and a variety of practical pitch organization approaches. It doesn't really deal with just intonation. For that, try Ezra Sim's article in Computer Music Journal from about 1988 or so, which arranges the 72 notes into scales of 18 tones, and transposes around among them. Date: Fri, 10 Jun 94 21:48:21 -0700 From: eig@ccrma.Stanford.EDU (Enrique Moreno)
The latest information regarding
Joe Maneri's book "Preliminary Studies in the Virtual Pitch Continuum"
is that it is undergoing revision. Information about the book may be obtained
from Julia Werntz, Boston Microtonal Society. 27 Valentine St. Cambridge,
MA 02139. Phone 617-491-6057. Fax 617-491-6730. As of May 5, 2001 the
cost is $20.
Alois
Haba Recordings from MUSICA BONA Czech Classical Music
December 4, 2003