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). Template:ErrorTemplate:Category handler Each step represents a frequency ratio of Template:Radic, or 35.29 cents Template:ErrorTemplate:Category handler.

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, 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

File:34ed2.svg

Just intonation intervals approximated in 34-ET

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

• J. Murray Barbour, Tuning and Temperament, Michigan State College Press, 1951.

External links

• 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.