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). {{audio|34-tet scale on C.mid|Play}} Each step represents a frequency ratio of {{radic|2|34}}, or 35.29 cents {{audio|1 step in 34-et on C.mid|Play}}.
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).{{citation needed|date=August 2015}} 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.{{citation needed|date=August 2015}} 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|date=August 2015}}
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.
| class="wikitable"
|align=center bgcolor="#ffffb4"|interval name |align=center bgcolor="#ffffb4"|size (steps) |align=center bgcolor="#ffffb4"|size (cents) |align=center bgcolor="#ffffb4"|midi |align=center bgcolor="#ffffb4"|just ratio |align=center bgcolor="#ffffb4"|just (cents) |align=center bgcolor="#ffffb4"|midi |align=center bgcolor="#ffffb4"|error |
| align=center|octave
|align=center|34 |align=center|1200 |align=center| |align=center|2:1 |align=center|1200 |align=center| |align=center|0 |
| align=center|perfect fifth
|align=center|20 |align=center|705.88 |align=center|{{audio|help=no|10 steps in 17-et on C.mid|Play}} |align=center|3:2 |align=center|701.95 |align=center|{{audio|help=no|Just perfect fifth on C.mid|Play}} |align=center|+{{0}}3.93 |
| align=center|septendecimal tritone
|align=center|17 |align=center|600.00 |align=center|{{audio|help=no|Tritone on C.mid|Play}} |align=center|17:12 |align=center|603.00 |align=center| |align=center|−{{0}}3.00 |
| align=center bgcolor="#D4D4D4"|lesser septimal tritone
|align=center bgcolor="#D4D4D4"|17 |align=center bgcolor="#D4D4D4"|600.00 |align=center bgcolor="#D4D4D4"| |align=center bgcolor="#D4D4D4"|7:5 |align=center bgcolor="#D4D4D4"|582.51 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Lesser septimal tritone on C.mid|Play}} |align=center bgcolor="#D4D4D4"|+17.49 |
| align=center|tridecimal narrow tritone
|align=center|16 |align=center|564.71 |align=center|{{audio|help=no|8 steps in 17-et on C.mid|Play}} |align=center|18:13 |align=center|563.38 |align=center|{{audio|help=no|Tridecimal narrow tritone on C.mid|Play}} |align=center|+{{0}}1.32 |
| align=center bgcolor="#D4D4D4"|11:8 wide fourth
|align=center bgcolor="#D4D4D4"|16 |align=center bgcolor="#D4D4D4"|564.71 |align=center bgcolor="#D4D4D4"| |align=center bgcolor="#D4D4D4"|11:8{{0}} |align=center bgcolor="#D4D4D4"|551.32 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Eleventh harmonic on C.mid|Play}} |align=center bgcolor="#D4D4D4"|+13.39 |
| align=center|undecimal wide fourth
|align=center|15 |align=center|529.41 |align=center|{{audio|help=no|15 steps in 34-et on C.mid|Play}} |align=center|15:11 |align=center|536.95 |align=center|{{audio|help=no|Undecimal augmented fourth on C.mid|Play}} |align=center|−{{0}}7.54 |
| align=center|perfect fourth
|align=center|14 |align=center|494.12 |align=center|{{audio|help=no|7 steps in 17-et on C.mid|Play}} |align=center|4:3 |align=center|498.04 |align=center|{{audio|help=no|Just perfect fourth on C.mid|Play}} |align=center|−{{0}}3.93 |
| align=center|tridecimal major third
|align=center|13 |align=center|458.82 |align=center| |align=center|13:10 |align=center|454.21 |align=center|{{audio|help=no|Tridecimal major third on C.mid|Play}} |align=center|+{{0}}4.61 |
| align=center bgcolor="#D4D4D4"|septimal major third
|align=center bgcolor="#D4D4D4"|12 |align=center bgcolor="#D4D4D4"|423.53 |align=center bgcolor="#D4D4D4"|{{audio|help=no|6 steps in 17-et on C.mid|Play}} |align=center bgcolor="#D4D4D4"|9:7 |align=center bgcolor="#D4D4D4"|435.08 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Septimal major third on C.mid|Play}} |align=center bgcolor="#D4D4D4"|−11.55 |
| align=center|undecimal major third
|align=center|12 |align=center|423.53 |align=center| |align=center|14:11 |align=center|417.51 |align=center|{{audio|help=no|Undecimal major third on C.mid|Play}} |align=center|+{{0}}6.02 |
| align=center|major third
|align=center|11 |align=center|388.24 |align=center|{{audio|help=no|11 steps in 34-et on C.mid|Play}} |align=center|5:4 |align=center|386.31 |align=center|{{audio|help=no|Just major third on C.mid|Play}} |align=center|+{{0}}1.92 |
| align=center|tridecimal neutral third
|align=center|10 |align=center|352.94 |align=center|{{audio|help=no|5 steps in 17-et on C.mid|Play}} |align=center|16:13 |align=center|359.47 |align=center|{{audio|help=no|Tridecimal neutral third on C.mid|Play}} |align=center|−{{0}}6.53 |
| align=center|undecimal neutral third
|align=center|10 |align=center|352.94 |align=center| |align=center|11:9{{0}} |align=center|347.41 |align=center|{{audio|help=no|Undecimal neutral third on C.mid|Play}} |align=center|+{{0}}5.53 |
| align=center|minor third
|align=center|{{0}}9 |align=center|317.65 |align=center|{{audio|help=no|9 steps in 34-et on C.mid|Play}} |align=center|6:5 |align=center|315.64 |align=center|{{audio|help=no|Just minor third on C.mid|Play}} |align=center|+{{0}}2.01 |
| align=center|tridecimal minor third
|align=center|{{0}}8 |align=center|282.35 |align=center|{{audio|help=no|4 steps in 17-et on C.mid|Play}} |align=center|13:11 |align=center|289.21 |align=center|{{audio|help=no|Tridecimal minor third on C.mid|Play}} |align=center|−{{0}}6.86 |
| align=center bgcolor="#D4D4D4"|septimal minor third
|align=center bgcolor="#D4D4D4"|{{0}}8 |align=center bgcolor="#D4D4D4"|282.35 |align=center bgcolor="#D4D4D4"| |align=center bgcolor="#D4D4D4"|7:6 |align=center bgcolor="#D4D4D4"|266.87 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Septimal minor third on C.mid|Play}} |align=center bgcolor="#D4D4D4"|+15.48 |
| align=center|tridecimal semimajor second
|align=center|{{0}}7 |align=center|247.06 |align=center|{{audio|help=no|7 steps in 34-et on C.mid|Play}} |align=center|15:13 |align=center|247.74 |align=center|{{audio|help=no|Tridecimal five-quarter tone on C.mid|Play}} |align=center|−{{0}}0.68 |
| align=center bgcolor="#D4D4D4"|septimal whole tone
|align=center bgcolor="#D4D4D4"|{{0}}7 |align=center bgcolor="#D4D4D4"|247.06 |align=center bgcolor="#D4D4D4"| |align=center bgcolor="#D4D4D4"|8:7 |align=center bgcolor="#D4D4D4"|231.17 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Septimal major second on C.mid|Play}} |align=center bgcolor="#D4D4D4"|+15.88 |
| align=center|whole tone, major tone
|align=center|{{0}}6 |align=center|211.76 |align=center|{{audio|help=no|3 steps in 17-et on C.mid|Play}} |align=center|9:8 |align=center|203.91 |align=center|{{audio|help=no|Major tone on C.mid|Play}} |align=center|+{{0}}7.85 |
| align=center|whole tone, minor tone
|align=center|{{0}}5 |align=center|176.47 |align=center|{{audio|help=no|5 steps in 34-et on C.mid|Play}} |align=center|10:9{{0}} |align=center|182.40 |align=center|{{audio|help=no|Minor tone on C.mid|Play}} |align=center|−{{0}}5.93 |
| align=center bgcolor="#D4D4D4"|neutral second, greater undecimal
|align=center bgcolor="#D4D4D4"|{{0}}5 |align=center bgcolor="#D4D4D4"|176.47 |align=center bgcolor="#D4D4D4"| |align=center bgcolor="#D4D4D4"|11:10 |align=center bgcolor="#D4D4D4"|165.00 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Greater undecimal neutral second on C.mid|Play}} |align=center bgcolor="#D4D4D4"|+11.47 |
| align=center bgcolor="#D4D4D4"|neutral second, lesser undecimal
|align=center bgcolor="#D4D4D4"|{{0}}4 |align=center bgcolor="#D4D4D4"|141.18 |align=center bgcolor="#D4D4D4"|{{audio|help=no|2 steps in 17-et on C.mid|Play}} |align=center bgcolor="#D4D4D4"|12:11 |align=center bgcolor="#D4D4D4"|150.64 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Lesser undecimal neutral second on C.mid|Play}} |align=center bgcolor="#D4D4D4"|−{{0}}9.46 |
| align=center|greater tridecimal {{2/3}}-tone
|align=center|{{0}}4 |align=center|141.18 |align=center| |align=center|13:12 |align=center|138.57 |align=center|{{audio|help=no|Greater tridecimal two-third tone on C.mid|Play}} |align=center|+{{0}}2.60 |
| align=center bgcolor="#D4D4D4"|lesser tridecimal {{2/3}}-tone
|align=center bgcolor="#D4D4D4"|{{0}}4 |align=center bgcolor="#D4D4D4"|141.18 |align=center bgcolor="#D4D4D4"| |align=center bgcolor="#D4D4D4"|14:13 |align=center bgcolor="#D4D4D4"|128.30 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Lesser tridecimal two-third tone on C.mid|Play}} |align=center bgcolor="#D4D4D4"|+12.88 |
| align=center bgcolor="#D4D4D4"|15:14 semitone
|align=center bgcolor="#D4D4D4"|{{0}}3 |align=center bgcolor="#D4D4D4"|105.88 |align=center bgcolor="#D4D4D4"|{{audio|help=no|3 steps in 34-et on C.mid|Play}} |align=center bgcolor="#D4D4D4"|15:14 |align=center bgcolor="#D4D4D4"|119.44 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Septimal diatonic semitone on C.mid|Play}} |align=center bgcolor="#D4D4D4"|−13.56 |
| align=center|diatonic semitone
|align=center|{{0}}3 |align=center|105.88 |align=center| |align=center|16:15 |align=center|111.73 |align=center|{{audio|help=no|Just diatonic semitone on C.mid|Play}} |align=center|−{{0}}5.85 |
| align=center|17th harmonic
|align=center|{{0}}3 |align=center|105.88 |align=center| |align=center|17:16 |align=center|104.96 |align=center|{{audio|help=no|Just major semitone on C.mid|Play}} |align=center|+{{0}}0.93 |
| align=center bgcolor="#D4D4D4"|21:20 semitone
|align=center bgcolor="#D4D4D4"|{{0}}2 |align=center bgcolor="#D4D4D4"|{{0}}70.59 |align=center bgcolor="#D4D4D4"|{{audio|help=no|1 step in 17-et on C.mid|Play}} |align=center bgcolor="#D4D4D4"|21:20 |align=center bgcolor="#D4D4D4"|{{0}}84.47 |align=center bgcolor="#D4D4D4"|{{audio|help=no|Septimal chromatic semitone on C.mid|Play}} |align=center bgcolor="#D4D4D4"|−13.88 |
| align=center|chromatic semitone
|align=center|{{0}}2 |align=center|{{0}}70.59 |align=center| |align=center|25:24 |align=center|{{0}}70.67 |align=center|{{audio|help=no|Just chromatic semitone on C.mid|Play}} |align=center|−{{0}}0.08 |
| align=center|28:27 semitone
|align=center|{{0}}2 |align=center|{{0}}70.59 |align=center| |align=center|28:27 |align=center|{{0}}62.96 |align=center|{{audio|help=no|Septimal minor second on C.mid|Play}} |align=center|+{{0}}7.63 |
| align=center|septimal sixth-tone
|align=center|{{0}}1 |align=center|{{0}}35.29 |align=center|{{audio|help=no|1 step in 34-et on C.mid|Play}} |align=center|50:49 |align=center|{{0}}34.98 |align=center|{{audio|help=no|Septimal sixth-tone on C.mid|Play}} |align=center|+{{0}}0.31 |
Scale diagram
The following are 15 of the 34 notes in the scale:
| class="wikitable"
|align=center bgcolor="#ffeeee"|Interval (cents) | |colspan=2 align=center bgcolor="#ffeeee"|106 |colspan=2 align=center bgcolor="#ffeeee"|106 |colspan=2 align=center bgcolor="#ffeeee"|70 |colspan=2 align=center bgcolor="#ffeeee"|35 |colspan=2 align=center bgcolor="#ffeeee"|70 |colspan=2 align=center bgcolor="#ffeeee"|106 |colspan=2 align=center bgcolor="#ffeeee"|106 |colspan=2 align=center bgcolor="#ffeeee"|106 |colspan=2 align=center bgcolor="#ffeeee"|70 |colspan=2 align=center bgcolor="#ffeeee"|35 |colspan=2 align=center bgcolor="#ffeeee"|70 |colspan=2 align=center bgcolor="#ffeeee"|106 |colspan=2 align=center bgcolor="#ffeeee"|106 |colspan=2 align=center bgcolor="#ffeeee"|106 | |
| align=center bgcolor="#fffbee"|Note name
|colspan=2 align=center bgcolor="#fffbee"|C |colspan=2 align=center bgcolor="#fffbee"|C{{music|♯}}/D{{music|♭}} |colspan=2 align=center bgcolor="#fffbee"|D |colspan=2 align=center bgcolor="#fffbee"|D{{music|♯}} |colspan=2 align=center bgcolor="#fffbee"|E{{music|♭}} |colspan=2 align=center bgcolor="#fffbee"|E |colspan=2 align=center bgcolor="#fffbee"|F |colspan=2 align=center bgcolor="#fffbee"|F{{music|♯}}/G{{music|♭}} |colspan=2 align=center bgcolor="#fffbee"|G |colspan=2 align=center bgcolor="#fffbee"|G{{music|♯}} |colspan=2 align=center bgcolor="#fffbee"|A{{music|♭}} |colspan=2 align=center bgcolor="#fffbee"|A |colspan=2 align=center bgcolor="#fffbee"|A{{music|♯}}/B{{music|♭}} |colspan=2 align=center bgcolor="#fffbee"|B |colspan=2 align=center bgcolor="#fffbee"|C |
| align=center bgcolor="#eeeeff"|Note (cents)
|colspan=2 align=center bgcolor="#eeeeff"| 0 |colspan=2 align=center bgcolor="#eeeeff"|106 |colspan=2 align=center bgcolor="#eeeeff"|212 |colspan=2 align=center bgcolor="#eeeeff"|282 |colspan=2 align=center bgcolor="#eeeeff"|318 |colspan=2 align=center bgcolor="#eeeeff"|388 |colspan=2 align=center bgcolor="#eeeeff"|494 |colspan=2 align=center bgcolor="#eeeeff"|600 |colspan=2 align=center bgcolor="#eeeeff"|706 |colspan=2 align=center bgcolor="#eeeeff"|776 |colspan=2 align=center bgcolor="#eeeeff"|812 |colspan=2 align=center bgcolor="#eeeeff"|882 |colspan=2 align=center bgcolor="#eeeeff"|988 |colspan=2 align=center bgcolor="#eeeeff"|1094 |colspan=2 align=center bgcolor="#eeeeff"|1200 |
The remaining notes can easily be added.
References
- J. Murray Barbour, Tuning and Temperament, Michigan State College Press, 1951.
{{Reflist}}
External links
- [https://www.jstor.org/stable/932181 Dirk de Klerk. "Equal Temperament"], Acta Musicologica, Vol. 51, Fasc. 1 (Jan. - Jun., 1979), pp. 140-150.
- [http://www.microstick.net/ Stickman: Neil Haverstick] - Neil Haverstick is a composer and guitarist who uses microtonal tunings, especially 19, 31 and 34 tone equal temperament.
{{Microtonal music}}
{{Musical tuning}}