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 , 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,Tuning and Temperament, Michigan State College Press, 1951 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 namesize (steps)size (cents)midijust ratiojust (cents)midierroroctave3412002:112000ultramajor seventh331164.7149:251165.02 -0.31diatonic (septimal) major seventh321129.4127:141137.04 -7.63just major seventh311094.1215:81088.27 +5.85neutral seventh301058.8211:61049.36 +9.46classical minor seventh291023.539:51017.6 +5.93diatonic minor seventh28988.2416:9996.09 -7.85harmonic seventh27952.947:4968.83 -15.89diatonic major sixth26917.6512:7933.13 -15.48just major sixth25882.355:3884.36 -2.01diatonic augmented fifth24847.0613:8840.53 +6.53just minor sixth23811.768:5813.69 -1.93diatonic minor sixth22776.4711:7782.49 -6.02tridecimal inframinor sixth21741.1820:13745.79 -4.61perfect fifth20705.883:2701.95+3.93mavila fifth19670.5928:19671.31 -0.72diatonic augmented fourth18635.2913:9636.62 -1.33septendecimal tritone17600.0017:12603.00−3.00lesser septimal tritone17600.007:5582.51+17.49tridecimal narrow tritone16564.7118:13563.38+1.3211:8 wide fourth16564.7111:8551.32+13.39undecimal wide fourth15529.4115:11536.95−7.54perfect fourth14494.124:3498.04−3.93tridecimal major third13458.8213:10454.21+4.61septimal major third12423.539:7435.08−11.55undecimal major third12423.5314:11417.51+6.02major third11388.245:4386.31+1.92tridecimal neutral third10352.9416:13359.47−6.53undecimal neutral third10352.9411:9347.41+5.53minor third9317.656:5315.64+2.01tridecimal minor third8282.3513:11289.21−6.86septimal minor third8282.357:6266.87+15.48tridecimal semimajor second7247.0615:13247.74−0.68septimal whole tone7247.068:7231.17+15.88whole tone, major tone6211.769:8203.91+7.85whole tone, minor tone5176.4710:9182.40−5.93neutral second, greater undecimal5176.4711:10165.00+11.47neutral second, lesser undecimal4141.1812:11150.64−9.46greater tridecimal -tone4141.1813:12138.57+2.60lesser tridecimal -tone4141.1814:13128.30+12.8815:14 semitone3105.8815:14119.44−13.56diatonic semitone3105.8816:15111.73−5.8517th harmonic3105.8817:16104.96+0.9321:20 semitone270.5921:2084.47−13.88chromatic semitone270.5925:2470.67−0.0828:27 semitone270.5928:2762.96+7.63septimal sixth-tone135.2950:4934.98+0.31
References J. Murray Barbour, Tuning and Temperament, Michigan State College Press, 1951.
External links 34edo on Xenharmonic Wiki Dirk de Klerk. "Equal Temperament", Acta Musicologica, Vol. 51, Fasc. 1 (Jan. - Jun., 1979), pp. 140-150. Stickman: Neil Haverstick - Neil Haverstick is a composer and guitarist who uses microtonal tunings, especially 19, 31 and 34 tone equal temperament.