8th-octave temperaments: Difference between revisions
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An 8th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 8. | An 8th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 8. | ||
Temperaments discussed elsewhere include: | |||
* ''Octant'' | * ''Octant'' → [[Schismatic family#Octant|Schismatic family]] | ||
== Octatonic == | == 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]]. | [[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 | [[Subgroup]]: 2.3.11 | ||
Comma list: {{monzo|15 8 0 0 -8}} | [[Comma list]]: {{monzo|15 8 0 0 -8}} | ||
{{Mapping|legend=1|8 0 15|0 1 1}} | {{Mapping|legend=1|8 0 15|0 1 1}} | ||
: Mapping generators: ~12/11 | : Mapping generators: ~12/11, ~3 | ||
[[Support]]ing [[ET]]s: {{EDOs|16, 24, 32, 48, 56, 80 | [[Support]]ing [[ET]]s: {{EDOs|16, 24, 32, 48, 56, 72, 80, 120, 128, 152}}, ... | ||
== Octium == | == Octium == | ||
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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 | : Mapping generators: ~10051/9216, ~24/23 | ||
[[Optimal tuning]] ([[CTE]]): ~24/23 = 76.284 | [[Optimal tuning]] ([[CTE]]): ~24/23 = 76.284 | ||
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[[Support]]ing [[ET]]s: {{EDOs|472, 1400, 1872, 3272, 5144}} | [[Support]]ing [[ET]]s: {{EDOs|472, 1400, 1872, 3272, 5144}} | ||
== Octoid | == Octoid == | ||
''For the 7-limit temperament, see [[Ragismic microtemperaments#Octoid]].'' | : ''For the 7-limit temperament, see [[Ragismic microtemperaments #Octoid]].'' | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Comma list: 59604644775390625/59296646043258912 | [[Comma list]]: 59604644775390625/59296646043258912 | ||
{{Mapping|legend= | {{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}} | {{Navbox fractional-octave}} | ||
Latest revision as of 23:32, 20 February 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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
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: [15 8 0 0 -8⟩
Mapping: [⟨8 0 15], ⟨0 1 1]]
- Mapping generators: ~12/11, ~3
Supporting ETs: 16, 24, 32, 48, 56, 72, 80, 120, 128, 152, ...
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. It reaches 24/23 in one generator and 19/16 in two.
Subgroup: 2.3.19.23.29.31
Comma list: 94221/94208, 419957/419904, 219501/219488, 7997934975003/7996960669696
Subgroup-val mapping: [⟨8 2 35 25 47 2], ⟨0 21 -2 22 -16 74]]
- Mapping generators: ~10051/9216, ~24/23
Optimal tuning (CTE): ~24/23 = 76.284
Supporting ETs: 472, 1400, 1872, 3272, 5144
Octoid
- For the 7-limit temperament, see Ragismic microtemperaments #Octoid.
Subgroup: 2.3.5
Comma list: 59604644775390625/59296646043258912
Mapping: [⟨8 1 3], ⟨0 3 4]]
- Mapping generators: ~2125764/1953125, ~4374/3125
- WE: ~2125764/1953125 = 150.001 ¢, ~4374/3125 = 584.027 ¢
- CWE: ~2125764/1953125 = 150.000 ¢, ~4374/3125 = 584.025 ¢
Optimal ET sequence: 8, 56bcc, 64c, 72, 152, 224, 376, 976
Badness (Sintel): 6.687