34 equal temperament

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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). About this sound Play  Each step represents a frequency ratio of 342, or 35.29 cents About this sound Play .

History[edit]

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).[citation needed] 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.[citation needed] The luthier Larry Hanson had an electric guitar refretted from 12 to 34 and persuaded American guitarist Neil Haverstick to take it up.[citation needed]

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.

Scale diagram[edit]

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.

Interval size[edit]

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
perfect fifth 20 705.88 About this sound Play 3:2 701.95 About this sound Play +03.93
septendecimal tritone 17 600.00 About this sound Play 17:12 603.00 03.00
lesser septimal tritone 17 600.00 7:5 582.51 About this sound Play +17.49
tridecimal narrow tritone 16 564.71 About this sound Play 18:13 563.38 About this sound Play +01.32
11:8 wide fourth 16 564.71 11:80 551.32 About this sound Play +13.39
undecimal wide fourth 15 529.41 About this sound Play 15:11 536.95 About this sound Play 07.54
perfect fourth 14 494.12 About this sound Play 4:3 498.04 About this sound Play 03.93
tridecimal major third 12 458.82 About this sound Play 13:10 454.21 About this sound Play +04.61
septimal major third 12 423.53 9:7 435.08 About this sound Play −11.55
undecimal major third 12 423.53 14:11 417.51 About this sound Play +06.02
major third 11 388.24 About this sound Play 5:4 386.31 About this sound Play +01.92
tridecimal neutral third 10 352.94 About this sound Play 16:13 359.47 About this sound Play 06.53
undecimal neutral third 10 352.94 11:90 347.41 About this sound Play +05.53
minor third 09 317.65 About this sound Play 6:5 315.64 About this sound Play +02.01
tridecimal minor third 08 282.35 About this sound Play 13:11 289.21 About this sound Play 06.86
septimal minor third 08 282.35 7:6 266.87 About this sound Play +15.48
tridecimal semimajor second 07 247.06 About this sound Play 15:13 247.74 About this sound Play 00.68
septimal whole tone 07 247.06 8:7 231.17 About this sound Play +15.88
whole tone, major tone 06 211.76 About this sound Play 9:8 203.91 About this sound Play +07.85
whole tone, minor tone 05 176.47 About this sound Play 10:90 182.40 About this sound Play 05.93
neutral second, greater undecimal 05 176.47 11:10 165.00 About this sound Play +11.47
neutral second, lesser undecimal 04 141.18 About this sound Play 12:11 150.64 About this sound Play 09.46
greater tridecimal 23-tone 04 141.18 13:12 138.57 About this sound Play +02.60
lesser tridecimal 23-tone 04 141.18 14:13 128.30 About this sound Play +12.88
15:14 semitone 03 105.88 About this sound Play 15:14 119.44 About this sound Play −13.56
diatonic semitone 03 105.88 16:15 111.73 About this sound Play 05.85
17th harmonic 03 105.88 17:16 104.96 About this sound Play +00.93
21:20 semitone 02 070.59 About this sound Play 21:20 084.47 About this sound Play −13.88
chromatic semitone 02 070.59 25:24 070.67 About this sound Play 00.08
28:27 semitone 02 070.59 28:27 062.96 About this sound Play +07.63
septimal sixth-tone 01 035.29 About this sound Play 50:49 034.98 About this sound Play +00.31

References[edit]

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

External links[edit]