8th-octave temperaments: Difference between revisions
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{{ | {{Technical data page}} | ||
{{Infobox fractional-octave|8}} | |||
An 8th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 8. | |||
Temperaments discussed elsewhere include: | |||
* ''Octant'' → [[Schismatic family#Octant|Schismatic family]] | |||
== Octatonic == | |||
[[12/11]] is very close to 1 step of 8edo, and hence this temperament tempers out the octatonic comma, the difference between a stack of 8 12/11's and the octave. The octatonic temperament makes a [[consistent circle]]. | |||
[[Subgroup]]: 2.3.11 | |||
[[Comma list]]: {{monzo|15 8 0 0 -8}} | |||
{{Mapping|legend=1|8 0 15|0 1 1}} | |||
: Mapping generators: ~12/11, ~3 | |||
[[Support]]ing [[ET]]s: {{EDOs|16, 24, 32, 48, 56, 72, 80, 120, 128, 152}}, ... | |||
== Octium == | == Octium == | ||
Octium temperament is named after an "extended provisional" name for oxygen, the 8th element. The term [[oxygen]] itself is already the name of an exotemperament. | Octium temperament is named after an "extended provisional" name for oxygen, the 8th element. The term [[oxygen]] itself is already the name of an exotemperament. It reaches [[24/23]] in one generator and [[19/16]] in two. | ||
Subgroup: 2.3.19.23.29.31 | Subgroup: 2.3.19.23.29.31 | ||
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{{Mapping|legend=2| 8 2 35 25 47 2 | 0 21 -2 22 -16 74 }} | {{Mapping|legend=2| 8 2 35 25 47 2 | 0 21 -2 22 -16 74 }} | ||
: | : Mapping generators: ~10051/9216, ~24/23 | ||
[[Optimal tuning]] ([[CTE]]): ~ | [[Optimal tuning]] ([[CTE]]): ~24/23 = 76.284 | ||
[[Support]]ing [[ET]]s: {{EDOs|472, 1400, 1872, 3272, 5144}} | [[Support]]ing [[ET]]s: {{EDOs|472, 1400, 1872, 3272, 5144}} | ||
== Octoid == | |||
: ''For the 7-limit temperament, see [[Ragismic microtemperaments #Octoid]].'' | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: 59604644775390625/59296646043258912 | |||
{{Mapping|legend=1| 8 1 3 | 0 3 4 }} | |||
: Mapping generators: ~2125764/1953125, ~4374/3125 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2125764/1953125 = 150.001{{c}}, ~4374/3125 = 584.027{{c}} | |||
* [[CWE]]: ~2125764/1953125 = 150.000{{c}}, ~4374/3125 = 584.025{{c}} | |||
{{Optimal ET sequence|legend=1| 8, 56bcc, 64c, 72, 152, 224, 376, 976 }} | |||
[[Badness]] (Sintel): 6.687 | |||
{{Navbox fractional-octave}} | |||