8th-octave temperaments: Difference between revisions
No edit summary |
→Octium: that is also a valid mapping and it's simpler, so minimal gen = 24/23 |
||
| Line 24: | Line 24: | ||
{{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 = 1\8, ~ | : mapping generators: ~10051/9216 = 1\8, ~24/23 = 76.284 | ||
[[Optimal tuning]] ([[CTE]]): ~231173/221184 = 76.284 | [[Optimal tuning]] ([[CTE]]): ~231173/221184 = 76.284 | ||
[[Support]]ing [[ET]]s: {{EDOs|472, 1400, 1872, 3272, 5144}} | [[Support]]ing [[ET]]s: {{EDOs|472, 1400, 1872, 3272, 5144}} | ||
Revision as of 17:53, 21 October 2023
Template:Fractional-octave navigation 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.
Subgroup: 2.3.5.7.11
Comma list: [15 8 0 0 -8⟩
Mapping: [⟨8 0 0 0 15], ⟨0 1 0 0 1], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]
- mapping generators: ~12/11 = 1\8, ~3, ~5, ~7
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.
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): ~231173/221184 = 76.284