665th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|665}} | {{Infobox fractional-octave|665}} | ||
[[665edo]] is a notable tuning system, | [[665edo]] is a notable tuning system, most prominently notable because of its extremely precise [[3/2|perfect fifth]], as well as strong representations of higher harmonics. As such it gives rise to some fractional-octave temperaments. Satanic temperaments (referencing the 3/2 approximation) are a notable subset of 665th-octave temperaments which have a page of their own. | ||
== Satanic == | == Satanic == | ||
Latest revision as of 18:13, 23 October 2026
665edo is a notable tuning system, most prominently notable because of its extremely precise perfect fifth, as well as strong representations of higher harmonics. As such it gives rise to some fractional-octave temperaments. Satanic temperaments (referencing the 3/2 approximation) are a notable subset of 665th-octave temperaments which have a page of their own.
Satanic
Subgroup: 2.3.5
Comma list: [-1054 665⟩ (satanic comma)
Mapping: [⟨665 1054 0], ⟨0 0 1]]
- Mapping generators: ~[485 -306⟩, ~5
Optimal tuning (CTE): ~[485 -306⟩ = 1＼665, ~5/4 = 386.31371386
Optimal ET sequence: 665, …, 4655, 5320, 5985, 6650, 7315, 7980, 24605, 32585
Badness: 0.935807
Monzosatanic
Subgroup: 2.3.5.7
Comma list: 19073486328125/19042491875328, 450359962737049600/450283905890997363
Mapping: [⟨665 1054 1544 0], ⟨0 0 0 1]]
- Mapping generators: ~[485 -306⟩, ~7
Optimal tuning (CTE): ~[485 -306⟩ = 1＼665, ~7/4 = 968.826
Optimal ET sequence: 665, 3325, 3990, 4655c, 5320c, 5985c
Badness: 1.492380
Neosatanic
Subgroup: 2.3.5.7.13.17.19
Comma list: 4200/4199, 6860/6859, 285719/285610, 369664/369603, 781250/780759
Mapping: [⟨665 1054 1544 0 594 4585 958], ⟨0 0 0 1 1 -1 1]]
- Mapping generators: ~[485 -306⟩, ~7
Optimal tuning (POTE): ~[485 -306⟩ = 1＼665, ~7/4 = 968.7358