5th-octave temperaments: Difference between revisions

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Quint: move to very low accuracy temperaments
 
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Supporting ETs: {{Optimal ET sequence|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}
Supporting ETs: {{Optimal ET sequence|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}
== Quint ==
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 16/15, 27/25
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
: Mapping generators: ~9/8, ~7
[[Optimal tuning]]s:
* [[POTE]]: ~9/8 = 1\5, ~7/4 = 1017.903 (~21/20 = 57.903)
* [[CTE]]: ~9/8 = 1\5, ~7/4 = 968.8259 (~63/64 = 8.8259)
{{Optimal ET sequence|legend=1| 5, 15ccd }}
[[Badness]]: 0.048312


== Obscenity ==
== Obscenity ==