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 2^1/34, or 35.29 cents

Cent (music)

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

 .

History

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

Diesis

In classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents...

 and the syntonic comma

Syntonic comma

In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...

), division into 34 steps did not arise 'naturally' out of older music theory, although Cyriac Schneegass proposed a meantone system with 34 divisions based in effect on half a chromatic semitone

Semitone

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....

 (the difference between a major third

Major third

In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

 and a minor third

Minor third

In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

, 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 well-known 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

Modulation (music)

In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest...

 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

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

The following table outlines some of the intervals of this tuning system and their match to various ratios in the harmonic series

Harmonic series (music)

Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

.

interval name	size (steps)	size (cents)	just ratio	just (cents)	error
perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...	20	705.88	3:2	701.95	+3.93
lesser septimal tritone Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7...	17	600	7:5	582.51	+17.49
11:8 wide fourth	16	564.71	11:8	551.32	+13.39
undecimal wide fourth	15	529.41	15:11	536.95	−7.54
perfect fourth Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...	14	494.12	4:3	498.04	−3.93
septimal major third Septimal major third In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal...	12	423.53	9:7	435.08	−11.55
undecimal major third	12	423.53	14:11	417.51	+6.02
major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...	11	388.24	5:4	386.31	+1.92
undecimal neutral third Neutral third A neutral third is a musical interval wider than a minor third but narrower than a major third . Three distinct intervals may be termed neutral thirds:...	10	352.94	11:9	347.41	+5.53
minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...	9	317.65	6:5	315.64	+2.01
tridecimal minor third	8	282.35	13:11	289.21	−6.86
septimal minor third Septimal minor third In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...	8	282.35	7:6	266.87	+15.48
tridecimal semimajor second	7	247.06	15:13	247.74	−0.68
septimal whole tone Septimal whole tone In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to a 8/7 ratio of frequencies. It is about 231 cents wide in just intonation. Although 24 equal temperament does not match this interval particularly well, its...	7	247.06	8:7	231.17	+15.88
whole tone, major tone	6	211.76	9:8	203.91	+7.85
whole tone, minor tone	5	176.47	10:9	182.40	−5.93
neutral second Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:... , greater undecimal	5	176.47	11:10	165.00	+11.47
neutral second, lesser undecimal	4	141.18	12:11	150.64	−9.46
15:14 semitone	3	105.88	15:14	119.44	−13.56
diatonic semitone	3	105.88	16:15	111.73	−5.85
21:20 semitone	2	70.59	21:20	84.47	−13.88
chromatic semitone	2	70.59	25:24	70.67	−0.08
28:27 semitone	2	70.59	28:27	62.96	+7.63
septimal sixth-tone Septimal sixth-tone In music, septimal sixth-tone is a septimal sixth-tone, an interval with the ratio of 50:49 , about 34.98 cents, which in just intonation is the difference between the lesser septimal tritone, and its inversion, the greater septimal tritone ....	1	35.29	50:49	34.98	+0.31

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.