8th-octave temperaments: Difference between revisions
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{{ | {{Infobox fractional-octave|8}} | ||
Discussed elsewhere are: | Discussed elsewhere are: | ||
* ''Octant'', → [[Schismatic family#Octant|Schismatic family]] | * ''Octant'', → [[Schismatic family#Octant|Schismatic family]] | ||
* ''Octoid'', → [[Ragismic microtemperaments#Octoid|Ragismic microtemperaments]] | * ''Octoid'', → [[Ragismic microtemperaments#Octoid|Ragismic microtemperaments]] | ||
== 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]]. | ||
Revision as of 03:59, 14 February 2025
Discussed elsewhere are:
- Octant, → Schismatic family
- Octoid, → Ragismic microtemperaments
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 = 1\8, ~3
Supporting ETs: 16, 24, 32, 48, 56, 80, 72, 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 = 1\8, ~24/23 = 76.284
Optimal tuning (CTE): ~24/23 = 76.284
Supporting ETs: 472, 1400, 1872, 3272, 5144
Octoid (5-limit)
For the 7-limit temperament, see Ragismic microtemperaments#Octoid.
Subgroup: 2.3.5
Comma list: 59604644775390625/59296646043258912
Subgroup-val mapping: [⟨8 1 3], ⟨0 3 4]]
Optimal tuning (CWE): ~2125764/1953125 = 1\8, ~4374/3125 = 584.025
Supporting ETs: 8, 152, 224, 72, 376, 80, 600, 528, 296, 64c, 232, 88bc, 824, 680