34 equal temperament

In musical theory, 34 equal temperament, also referred to as 34-TET, 34-EDO or 34-ET, is the tempered tuning derived by dividing the octave into 34 equal-sized steps (equal frequency ratios). Each step represents a frequency ratio of 342, or 35.29 cents .

History and use

Unlike divisions of the octave into 19, 31 or 53 steps, which can be considered as being derived from ancient Greek intervals (the greater and lesser diesis and the syntonic comma), division into 34 steps did not arise 'naturally' out of older music theory, although Cyriakus Schneegass proposed a meantone system with 34 divisions based in effect on half a chromatic semitone (the difference between a major third and a minor third, 25:24 or 70.67 cents). Wider interest in the tuning was not seen until modern times, when the computer made possible a systematic search of all possible equal temperaments. While Barbour discusses it,[1] the first recognition of its potential importance appears to be in an article published in 1979 by the Dutch theorist Dirk de Klerk. The luthier Larry Hanson had an electric guitar refretted from 12 to 34 and persuaded American guitarist Neil Haverstick to take it up.

As compared with 31-et, 34-et reduces the combined mistuning from the theoretically ideal just thirds, fifths and sixths from 11.9 to 7.9 cents. Its fifths and sixths are markedly better, and its thirds only slightly further from the theoretical ideal of the 5:4 ratio. Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B, thus making a distinction between major tones, ratio 9:8 and minor tones, ratio 10:9. This can be regarded either as a resource or as a problem, making modulation in the contemporary Western sense more complex. As the number of divisions of the octave is even, the exact halving of the octave (600 cents) appears, as in 12-et. Unlike 31-et, 34 does not give an approximation to the harmonic seventh, ratio 7:4.

Interval size

The following table outlines some of the intervals of this tuning system and their match to various ratios in the harmonic series.

 interval name size (steps) size (cents) midi just ratio just (cents) midi error octave 34 1200 2:1 1200 0 perfect fifth 20 705.88 3:2 701.95 +03.93 septendecimal tritone 17 600.00 17:12 603.00 −03.00 lesser septimal tritone 17 600.00 7:5 582.51 +17.49 tridecimal narrow tritone 16 564.71 18:13 563.38 +01.32 11:8 wide fourth 16 564.71 11:80 551.32 +13.39 undecimal wide fourth 15 529.41 15:11 536.95 −07.54 perfect fourth 14 494.12 4:3 498.04 −03.93 tridecimal major third 12 458.82 13:10 454.21 +04.61 septimal major third 12 423.53 9:7 435.08 −11.55 undecimal major third 12 423.53 14:11 417.51 +06.02 major third 11 388.24 5:4 386.31 +01.92 tridecimal neutral third 10 352.94 16:13 359.47 −06.53 undecimal neutral third 10 352.94 11:90 347.41 +05.53 minor third 09 317.65 6:5 315.64 +02.01 tridecimal minor third 08 282.35 13:11 289.21 −06.86 septimal minor third 08 282.35 7:6 266.87 +15.48 tridecimal semimajor second 07 247.06 15:13 247.74 −00.68 septimal whole tone 07 247.06 8:7 231.17 +15.88 whole tone, major tone 06 211.76 9:8 203.91 +07.85 whole tone, minor tone 05 176.47 10:90 182.40 −05.93 neutral second, greater undecimal 05 176.47 11:10 165.00 +11.47 neutral second, lesser undecimal 04 141.18 12:11 150.64 −09.46 greater tridecimal ​2⁄3-tone 04 141.18 13:12 138.57 +02.60 lesser tridecimal ​2⁄3-tone 04 141.18 14:13 128.30 +12.88 15:14 semitone 03 105.88 15:14 119.44 −13.56 diatonic semitone 03 105.88 16:15 111.73 −05.85 17th harmonic 03 105.88 17:16 104.96 +00.93 21:20 semitone 02 070.59 21:20 084.47 −13.88 chromatic semitone 02 070.59 25:24 070.67 −00.08 28:27 semitone 02 070.59 28:27 062.96 +07.63 septimal sixth-tone 01 035.29 50:49 034.98 +00.31

Scale diagram

The following are 15 of the 34 notes in the scale:

 Interval (cents) 106 106 70 35 70 106 106 106 70 35 70 106 106 106 Note name C C♯/D♭ D D♯ E♭ E F F♯/G♭ G G♯ A♭ A A♯/B♭ B C Note (cents) 0 106 212 282 318 388 494 600 706 776 812 882 988 1094 1200

The remaining notes can easily be added.

References

1. Tuning and Temperament, Michigan State College Press, 1951