58 Equal Temperament
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In
music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...
, 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, and the first that distinguishes between all the elements of the 11-limit
tonality diamond In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality.Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", ''Harry Partch: An Anthol ...
. The next-smallest equal temperament to do both these things is
72 equal temperament In music, 72 equal temperament, called twelfth-tone, 72-TET, 72-EDO, or 72-ET, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps (equal frequency ratios). Each step represents a frequency r ...
. 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 In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three adj ...
(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 Marchetto da Padova (Marchettus of Padua; fl. 1305 – 1319) was an Italian music theorist and composer of the late medieval era. His innovations in notation of time-values were fundamental to the music of the Italian ars nova, as was his w ...
proposed a system that is approximately 29-EDO, which is a subset of 58-EDO, in 1318.


Interval size


See also

*
Harry Partch's 43-tone scale The 43-tone scale is a just intonation scale with 43 pitches in each octave. It is based on an eleven-limit tonality diamond, similar to the seven-limit diamond previously devised by Max Friedrich Meyer and refined by Harry Partch. The first of ...
; 58-EDO is the smallest equal temperament that can reasonably approximate this scale


References


External links


Xenharmonic Wiki article on 58edo
{{Musical tuning Equal temperaments Microtonality