58 equal temperament

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In music, 58 equal temperament (also called 58-ET or 58-EDO) divides the octave into 58 equal parts of approximately 20.69 cents each. It is notable as the simplest equal division of the octave to faithfully represent the 17-limit, [1] and the first that distinguishes between all the elements of the 11-limit tonality diamond. The next-smallest equal temperament to do both these things is 72 equal temperament.

Contents

Compared to 72-EDO, which is also consistent in the 17-limit, 58-EDO's approximations of most intervals are not quite as good (although still workable). One obvious exception is the perfect fifth (slightly better in 58-EDO), and another is the tridecimal minor third (11:13), which is significantly better in 58-EDO than in 72-EDO. The two systems temper out different commas; 72-EDO tempers out the comma 169:168, thus equating the 14:13 and 13:12 intervals. On the other hand, 58-EDO tempers out 144:143 instead of 169:168, so 14:13 and 13:12 are left distinct, but 13:12 and 12:11 are equated.

58-EDO, unlike 72-EDO, is not a multiple of 12, so the only interval (up to octave equivalency) that it shares with 12-EDO is the 600-cent tritone (which functions as both 17:12 and 24:17). On the other hand, 58-EDO has fewer pitches than 72-EDO and is therefore simpler.

History and use

The medieval Italian music theorist Marchetto da Padova proposed a system that is approximately 29-EDO, which is a subset of 58-EDO, in 1318. [2]

Interval size

interval namesize
(steps)
size
(cents)
just
ratio
just
(cents)
error
(cents)
octave 5812002:112000
perfect fifth 34703.453:2701.96+1.49
greater septendecimal tritone 2960017:12603.003.00
lesser septendecimal tritone 24:17597.00+3.00
septimal tritone 28579.317:5582.513.20
eleventh harmonic 27558.6211:8551.32+7.30
15:11 wide fourth26537.9315:11536.95+0.98
perfect fourth 24496.554:3498.041.49
septimal narrow fourth23475.8621:16470.78+5.08
tridecimal major third22455.1713:10454.21+0.96
septimal major third 21434.489:7435.080.60
undecimal major third20413.7914:11417.513.72
major third 19393.105:4386.31+6.79
tridecimal neutral third 17351.7216:13359.477.75
undecimal neutral third 11:9347.41+4.31
minor third 15310.346:5315.645.30
tridecimal minor third14289.6613:11289.21+0.45
septimal minor third 13268.977:6266.87+2.10
tridecimal semifourth12248.2815:13247.74+0.54
septimal whole tone 11227.598:7231.173.58
whole tone, major tone 10206.909:8203.91+2.99
whole tone, minor tone 9186.2110:9182.40+3.81
greater undecimal neutral second 8165.5211:10165.00+0.52
lesser undecimal neutral second 7144.8312:11150.645.81
septimal diatonic semitone 6124.1415:14119.44+4.70
septendecimal semitone; 17th harmonic5103.4517:16104.961.51
diatonic semitone 16:15111.738.28
septimal chromatic semitone 482.7621:2084.471.71
chromatic semitone 362.0725:2470.678.60
septimal third tone 28:2762.960.89
septimal quarter tone 241.3836:3548.777.39
septimal diesis 49:4835.70+5.68
septimal comma 120.6964:6327.266.57
syntonic comma 81:8021.510.82

See also

References

  1. "consistency / consistent".
  2. "Marchettus, the cadential diesis, and neo-Gothic tunings".