59th-octave temperaments

Fractional-octave temperaments

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← 58th-octave • 59th-octave • 60th-octave →

59edo features a 5/4 is nearly pure (stretched only 0.127 ¢), as the denominator of a convergent to log₂5. Hence is natural to consider temperaments that temper out the 59-5-comma, [137 0 -59⟩. Additionally, any edo join between equal temperaments whose greatest common divisor is 59 can be used to produce a 59th-octave temperament.

Additionally, 59edo tunes some harmonics well and others poorly, being an excellent tuning for the 2.9.5.21.11 subgroup but representing harmonics like 3, 7, and 15 poorly, thus there is room for improvement on these harmonics through 59th-octave temperaments.

59-5-commatic

59-5-commatic tempers out the comma which equates 5/4 with 19 steps of 59edo, and leaves the harmonic 3 as an independent generator.

Subgroup: 2.3.5

Comma list: [137 0 -59⟩

Mapping: [⟨59 0 137], ⟨0 1 0]]

mapping generators: ~[-65 0 28⟩ = 1\59, ~3

Optimal tuning (CTE): ~3/2 = 701.955

Supporting ETs: 59N, N = 1 to 80, largest patent val: 4720

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