In column A are the frequency ratios needed for harmonious chords. Column B is obtained by taking the products of the ratios in column A with one another, and column C is obtained by taking the products of the ratios of column A with those of column B. In this way, 37 frequency ratios are obtained, which map fairly precisely into the 34 values in the 34-note octave, as shown.
This mapping makes the 34-tone octave more "natural" than any other equal-temperament scheme with fewer than 53 notes to the octave. This fact seems to have been overlooked, at least through 1980 -- the year in which The New Grove Dictionary of Music was published. (See "interval" in that dictionary.)
| . | A | B | C | . | 34-Step Octave | 12-Step Octave | ||||
| # | . | . | . | = | n | 2n/34 | % error | t | 2t/12 | % error |
| 0 | 1/1 | . | . | 1.0000 | 0 | 1.0000 | 0 | 0 | 1.0000 | 0 |
| 1 | . | . | 128/125 | 1.0240 | 1 | 1.0206 | -0.33 | . | . | . |
| 2 | . | 25/24 | . | 1.0416 | 2 | 1.0416 | +0.00 | . | . | . |
| 3 | . | 16/15 | . | 1.0667 | 3 | 1.0631 | -0.34 | 1 | 1.0595 | -0.68. |
| 4 | . | . | 27/25 | 1.0800 | 4 | 1.0850 | +0.46 | . | . | . |
| 5 | . | 10/9 | . | 1.1111 | 5 | 1.1073 | -0.34 | . | . | . |
| 6 | . | 9/8 | . | 1.1250 | 6 | 1.1301 | +0.46 | 2 | 1.1225 | -0.22 |
| 7 | . | . | 144/125 | 1.1520 | 7 | 1.1534 | +0.12 | . | . | . |
| 8 | . | . | 125/108 | 1.1574 | -0.35 | . | . | . | ||
| 9 | . | . | 75/64 | 1.1719 | 8 | 1.1771 | +0.45 | . | . | . |
| 10 | . | . | 32/27 | 1.1852 | -0.68 | . | . | . | ||
| 11 | 6/5 | . | . | 1.2000 | 9 | 1.2014 | +0.12 | 3 | 1.1892 | -0.91 |
| . | . | . | . | . | 10 | 1.2261 | . | . | . | . |
| 12 | 5/4 | . | . | 1.2500 | 11 | 1.2514 | +0.11 | 4 | 1.2599 | +0.79 |
| 13 | . | 32/25 | . | 1.2800 | 12 | 1.2772 | -0.22 | . | . | . |
| 14 | . | . | 125/96 | 1.3021 | 13 | 1.3035 | +0.11 | . | . | . |
| 15 | 4/3 | . | . | 1.3333 | 14 | 1.3303 | -0.23 | 5 | 1.3348 | +0.11 |
| 16 | . | . | 27/20 | 1.3500 | 15 | 1.3577 | +0.57 | . | . | . |
| 17 | . | 25/18 | . | 1.3889 | 16 | 1.3857 | -0.23 | . | . | . |
| 18 | . | . | 45/32 | 1.4063 | 17 | 1.4142 | +0.57 | 6 | 1.4142 | +0.57 |
| 19 | . | . | 64/45 | 1.4222 | -0.57 | -0.57 | ||||
| 20 | . | 36/25 | . | 1.4400 | 18 | 1.4433 | +0.23 | . | . | . |
| 21 | . | . | 40/27 | 1.4815 | 19 | 1.4731 | -0.57 | . | . | . |
| 22 | 3/2 | . | . | 1.5000 | 20 | 1.5034 | +0.23 | 7 | 1.4983 | -0.11 |
| 23 | . | . | 192/125 | 1.5360 | 21 | 1.5344 | -0.11 | . | . | . |
| 24 | . | 25/16 | . | 1.5625 | 22 | 1.5660 | +0.22 | . | . | . |
| 25 | 8/5 | . | . | 1.6000 | 23 | 1.5982 | -0.11 | 8 | 1.5874 | -0.79 |
| . | . | . | . | . | 24 | 1.6311 | . | . | . | . |
| 26 | 5/3 | . | . | 1.6667 | 25 | 1.6647 | -0.12 | 9 | 1.6818 | +0.91 |
| 27 | . | . | 27/16 | 1.6875 | 26 | 1.6990 | +0.68 | . | . | . |
| 28 | . | . | 128/75 | 1.7067 | -0.45 | . | . | . | ||
| 29 | . | . | 216/125 | 1.7280 | 27 | 1.7340 | +0.35 | . | . | . |
| 30 | . | . | 125/72 | 1.7361 | -0.12 | . | . | . | ||
| 31 | . | 16/9 | . | 1.7778 | 28 | 1.7697 | -0.46 | 10 | 1.7818 | +0.22 |
| 32 | . | 9/5 | . | 1.8000 | 29 | 1.8062 | +0.34 | . | . | . |
| 33 | . | . | 50/27 | 1.8519 | 30 | 1.8434 | -0.46 | . | . | . |
| 34 | . | 15/8 | . | 1.8750 | 31 | 1.8813 | +0.34 | 11 | 1.8877 | +0.68. |
| 35 | . | 48/25 | . | 1.9200 | 32 | 1.9201 | +0.01 | . | . | . |
| 36 | . | . | 125/64 | 1.9531 | 33 | 1.9596 | +0.33 | . | . | . |
| 37 | 2/1 | . | . | 2.0000 | 34 | 2.0000 | 0 | 12 | 2.0000 | 0 |
For some current references to harmony theory in literature, see Harmony, Schoenberg, and the Last Samurai.
For more information on actual use of the 34-tone scale, see Neil Haverstick's Virtual Chautauqua page.
Page created March 12, 2001. Return to Journal.
Copyright © 2001 by Steven H. Cullinane. All rights reserved.