29th-octave temperaments: Difference between revisions
No edit summary |
Move 5-limit mystery to here as it's the most logical place for it to be |
||
| Line 2: | Line 2: | ||
[[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. | ||
Revision as of 23:57, 3 August 2023
Template:Fractional-octave navigation 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 [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: [46 -29⟩
Mapping: [⟨29 46 0], ⟨0 0 1]]
Mapping generators: ~531441/524288, ~5
POTE generator: ~5/4 = 387.408
Optimal ET sequence: 29, 58, 87, 232, 319
Badness: 1.020556
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.
Surprisingly, despite 29edo's fifth being closer to 3/2 than 12edo's, copper has a higher TE error than atomic and is not a very high accuracy temperament.
Subgroup: 2.3.5
Comma list: [-481 261 29⟩
Mapping: [⟨29 0 481], ⟨0 1 -9]]
Mapping generators: ~[-199 12 108⟩ = 1\29, ~3/2 = 701.905
Optimal tuning (CTE): ~3/2 = 701.905
Supporting ETs: 29, 754, 783, 812, 1566, 1537, 2320, 3103, 3132, ...