7th-octave temperaments
7th octave temperament occurs through temperament merging any two edos whose greatest common divisor is 7. A comma that frequently appears in 7th-octave temps is akjaysma, which sets 105/64 to be equal to 5\7.
Temperaments discussed elsewhere include:
- Septant → Schismatic family
- Brahmagupta → Ragismic microtemperaments
- Absurdity → Syntonic chromatic equivalence continuum
Akjaysmic (rank-3)
Subgroup: 2.3.5.7
Comma list: [47 -7 -7 -7⟩
Mapping: [⟨7 0 0 47], ⟨0 1 0 -1], ⟨0 0 1 -1]]
Mapping generators: ~1157625/1048576, ~3, ~5
POTE generators: ~3/2 = 701.965, ~5/4 = 386.330
Optimal ET sequence: 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802
11-limit
Subgroup: 2.3.5.7.11
Comma list: 184549376/184528125, 199297406/199290375
Mapping: [⟨7 0 0 47 -168], ⟨0 1 0 -1 10], ⟨0 0 1 -1 5]]
Mapping generators: ~29160/26411, ~3, ~5
POTE generators: ~3/2 = 701.968, ~5/4 = 386.332
Optimal ET sequence: 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197
Nitrogen
Described as 140 & 1407 temperament in the 7-limit, named after the 7th element for being period-7 and also because 140 and 1407 only contain numbers 7 and 14, atomic number and atomic weight of nitrogen respectively. On top of this connection to the number 7, it also reaches 7th harmonic 7 generators down.
Subgroup: 2.3.5.7
Comma list: 3955078125/3954653486, 140737488355328/140710042265625
Mapping: [⟨7 10 17 20], ⟨0 22 -15 -7]]
Mapping generators: ~1157625/1048576, ~1029/1024
Optimal tuning (CTE): ~1157625/1048576 = 1\7, ~1029/1024 = 8.531
Optimal ET sequence: 140, 1407, 1547, ...