58 equal temperament
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, 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.
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.
|interval name||size (steps)||size (cents)||just ratio||just (cents)||error|
|greater septendecimal tritone||29||600||17:12||603.00||−3.00|
|lesser septendecimal tritone||24:17||597.00||+3.00|
|15:11 wide fourth||26||537.93||15:11||536.95||+0.98|
|septimal narrow fourth||23||475.86||21:16||470.78||+5.08|
|tridecimal major third||22||455.17||13:10||454.21||+0.96|
|septimal major third||21||434.48||9:7||435.08||−0.60|
|undecimal major third||20||413.79||14:11||417.51||−3.72|
|tridecimal neutral third||17||351.72||16:13||359.47||−7.75|
|undecimal neutral third||11:9||347.41||+4.31|
|tridecimal minor third||14||289.66||13:11||289.21||+0.45|
|septimal minor third||13||268.97||7:6||266.87||+2.10|
|septimal whole tone||11||227.59||8:7||231.17||−3.58|
|whole tone, major tone||10||206.90||9:8||203.91||+2.99|
|whole tone, minor tone||9||186.21||10:9||182.40||+3.81|
|greater undecimal neutral second||8||165.52||11:10||165.00||+0.52|
|lesser undecimal neutral second||7||144.83||12:11||150.64||−5.81|
|septimal diatonic semitone||6||124.14||15:14||119.44||+4.70|
|septendecimal semitone; 17th harmonic||5||103.45||17:16||104.96||−1.51|
|septimal chromatic semitone||4||82.76||21:20||84.47||−1.71|
|septimal third tone||3||62.07||28:27||62.96||−0.89|
|septimal quarter tone||2||41.38||36:35||48.77||−7.39|
- Harry Partch's 43-tone scale; 58-EDO is the smallest equal temperament that can reasonably approximate this scale