Английская Википедия: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). Шаблон:Audio Each step represents a frequency ratio of Шаблон:Radic, or 35.29 cents Шаблон:Audio.

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).Шаблон: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, the first recognition of its potential importance appears to be in an article published in 1979 by the Dutch theorist Dirk de Klerk.Шаблон: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.Шаблон: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.

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	Шаблон:Audio	3:2	701.95	Шаблон:Audio	+Шаблон:03.93
septendecimal tritone	17	600.00	Шаблон:Audio	17:12	603.00		−Шаблон:03.00
lesser septimal tritone	17	600.00		7:5	582.51	Шаблон:Audio	+17.49
tridecimal narrow tritone	16	564.71	Шаблон:Audio	18:13	563.38	Шаблон:Audio	+Шаблон:01.32
11:8 wide fourth	16	564.71		11:8Шаблон:0	551.32	Шаблон:Audio	+13.39
undecimal wide fourth	15	529.41	Шаблон:Audio	15:11	536.95	Шаблон:Audio	−Шаблон:07.54
perfect fourth	14	494.12	Шаблон:Audio	4:3	498.04	Шаблон:Audio	−Шаблон:03.93
tridecimal major third	13	458.82		13:10	454.21	Шаблон:Audio	+Шаблон:04.61
septimal major third	12	423.53	Шаблон:Audio	9:7	435.08	Шаблон:Audio	−11.55
undecimal major third	12	423.53		14:11	417.51	Шаблон:Audio	+Шаблон:06.02
major third	11	388.24	Шаблон:Audio	5:4	386.31	Шаблон:Audio	+Шаблон:01.92
tridecimal neutral third	10	352.94	Шаблон:Audio	16:13	359.47	Шаблон:Audio	−Шаблон:06.53
undecimal neutral third	10	352.94		11:9Шаблон:0	347.41	Шаблон:Audio	+Шаблон:05.53
minor third	Шаблон:09	317.65	Шаблон:Audio	6:5	315.64	Шаблон:Audio	+Шаблон:02.01
tridecimal minor third	Шаблон:08	282.35	Шаблон:Audio	13:11	289.21	Шаблон:Audio	−Шаблон:06.86
septimal minor third	Шаблон:08	282.35		7:6	266.87	Шаблон:Audio	+15.48
tridecimal semimajor second	Шаблон:07	247.06	Шаблон:Audio	15:13	247.74	Шаблон:Audio	−Шаблон:00.68
septimal whole tone	Шаблон:07	247.06		8:7	231.17	Шаблон:Audio	+15.88
whole tone, major tone	Шаблон:06	211.76	Шаблон:Audio	9:8	203.91	Шаблон:Audio	+Шаблон:07.85
whole tone, minor tone	Шаблон:05	176.47	Шаблон:Audio	10:9Шаблон:0	182.40	Шаблон:Audio	−Шаблон:05.93
neutral second, greater undecimal	Шаблон:05	176.47		11:10	165.00	Шаблон:Audio	+11.47
neutral second, lesser undecimal	Шаблон:04	141.18	Шаблон:Audio	12:11	150.64	Шаблон:Audio	−Шаблон:09.46
greater tridecimal Шаблон:2/3-tone	Шаблон:04	141.18		13:12	138.57	Шаблон:Audio	+Шаблон:02.60
lesser tridecimal Шаблон:2/3-tone	Шаблон:04	141.18		14:13	128.30	Шаблон:Audio	+12.88
15:14 semitone	Шаблон:03	105.88	Шаблон:Audio	15:14	119.44	Шаблон:Audio	−13.56
diatonic semitone	Шаблон:03	105.88		16:15	111.73	Шаблон:Audio	−Шаблон:05.85
17th harmonic	Шаблон:03	105.88		17:16	104.96	Шаблон:Audio	+Шаблон:00.93
21:20 semitone	Шаблон:02	Шаблон:070.59	Шаблон:Audio	21:20	Шаблон:084.47	Шаблон:Audio	−13.88
chromatic semitone	Шаблон:02	Шаблон:070.59		25:24	Шаблон:070.67	Шаблон:Audio	−Шаблон:00.08
28:27 semitone	Шаблон:02	Шаблон:070.59		28:27	Шаблон:062.96	Шаблон:Audio	+Шаблон:07.63
septimal sixth-tone	Шаблон:01	Шаблон:035.29	Шаблон:Audio	50:49	Шаблон:034.98	Шаблон:Audio	+Шаблон: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		CШаблон:Music/DШаблон:Music		D		DШаблон:Music		EШаблон:Music		E		F		FШаблон:Music/GШаблон:Music		G		GШаблон:Music		AШаблон:Music		A		AШаблон:Music/BШаблон:Music		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.