59th-octave temperaments: Difference between revisions
Created page with "{{Fractional-octave navigation|59}} == 59-5-commatic == 59edo is a denominator to convergent of log<sub>2</sub>5. Subgroup: 2.3.5 Comma list: {{monzo|137 0 -59}} {{Mapp..." |
|||
| (3 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{ | {{Infobox fractional-octave|59}} | ||
[[59edo]] features a [[5/4]] is nearly pure (stretched only 0.127{{c}}), as the denominator of a convergent to log<sub>2</sub>5. Hence is natural to consider temperaments that temper out the 59-5-comma, {{monzo|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 == | ||
[[ | |||
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 | Subgroup: 2.3.5 | ||
| Line 14: | Line 19: | ||
[[Support]]ing [[ET]]s: 59N, N = 1 to 80, largest patent val: [[4720edo|4720]] | [[Support]]ing [[ET]]s: 59N, N = 1 to 80, largest patent val: [[4720edo|4720]] | ||
{{Navbox fractional-octave}} | |||
{{stub}} | |||
Latest revision as of 16:44, 8 August 2026
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
|
This page is a stub. You can help the Xenharmonic Wiki by expanding it. |