29th-octave temperaments: Difference between revisions
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Move 5-limit mystery to here as it's the most logical place for it to be |
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[[29edo]] is notable for being the first equal division to have a more precise [[3/2]] than [[12edo]], and the first tuning to be consistent in the [[15-odd-limit]]. 29th-octave temperaments occur naturally when temperament-merging edos whose greatest common divisor is 29. | [[29edo]] is notable for being the first equal division to have a more precise [[3/2]] than [[12edo]], and the first tuning to be consistent in the [[15-odd-limit]]. 29th-octave temperaments occur naturally when temperament-merging edos whose greatest common divisor is 29. | ||
== Mystery (5-limit) == | |||
: ''Main article: [[Mystery]] and for higher-limit versions see [[Hemifamity temperaments #Mystery]]'' | |||
The mystery temperament in the 5-limit is described by tempering out the comma {{monzo| 46 -29 }}, where a circle of 29 fifths closes on 17 octaves, and it is supported by small multiples of 29edo. | |||
Subgroup: 2.3.5 | |||
[[Comma]]: {{monzo| 46 -29 }} | |||
[[Mapping]]: [{{val| 29 46 0 }}, {{val| 0 0 1 }}] | |||
Mapping generators: ~531441/524288, ~5 | |||
[[POTE generator]]: ~5/4 = 387.408 | |||
{{Optimal ET sequence|legend=1| 29, 58, 87, 232, 319 }} | |||
[[Badness]]: 1.020556 | |||
== Copper == | == Copper == | ||
Copper temperament is derived from a 5-limit comma called [[copper comma]], because it is constructed the same way towards 29edo as [[Kirnberger's atom]] is towards 12edo. A fifth of each of these tunings is modified by a tiny amount, then a circle of these fifths is set to close eventually at the octave. | Copper temperament is derived from a 5-limit comma called [[copper comma]], because it is constructed the same way towards 29edo as [[Kirnberger's atom]] is towards 12edo. A fifth of each of these tunings is modified by a tiny amount, then a circle of these fifths is set to close eventually at the octave. | ||