41st-octave temperaments
Countercomp temperaments are a notable subset of 41st-octave temperaments which have a page of their own.
Therefore, this page collects different 41st-octave temperaments.
Niobium
Niobium can be viewed as the 41st-octave counterpart of atomic, although their badnesses diverge sharply. Much like Kirnberger's atom (12 & 612) is a difference between a stack of 12 fifths flattened by a comma closed against 12edo, niobium (41 & 2460) fulfills a similar function for 41edo. Hence it is also not a countercomp temperament.
It is the difference between a circle of 41 3/2 fifths flattened by [215 -121 -10⟩ and the appropriate octave. The temperament also maps 2657205/2097152 into 14\41 and in higher limits, 3645/3584 to 1\41.
Subgroup: 2.3.5
Comma list: [-875 492 41⟩
Mapping: [⟨41 0 875], ⟨0 1 -12]]
- mapping generators: ~[-64 36 3⟩, ~3
Optimal tuning (CTE): ~[-64 36 3⟩ = 1\41, ~3/2 = 701.9536
Optimal ET sequence: 41, …, 2378, 2419, 2460, 4961, 7421, 17302, 24723, 66748, 91471, 116194
Badness: 5.57
7-limit
Subgroup: 2.3.5.7
Comma list: [-5 -12 -9 16⟩, [-55 30 2 1⟩
Mapping: [⟨41 0 875 259], ⟨0 1 -12 -6]]
- mapping generators: ~ 3645/3584, ~3
Optimal tuning (CTE): ~ 3645/3584 = 1\41, ~3/2 = 701.953
Optimal ET sequence: 41, ..., 2460
Badness: 10.676
11-limit
Subgroup: 2.3.5.7.11
Comma list: 151263/151250, 6576668672/6576582375, 1722499009/1721868840
Mapping: [⟨41 0 875 259 -98], ⟨0 1 -12 -6 10]]
- mapping generators: ~ 3645/3584, ~3
Optimal tuning (CTE): ~ 3645/3584 = 1\41, ~3/2 = 701.955
Optimal ET sequence: 41, ..., 2460
Badness: 3.308
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 10648/10647, 196625/196608, 105644/105625, 14350336/14348907
Mapping: [⟨41 0 875 259 -98 -256], ⟨0 1 -12 -6 10 17]]
- mapping generators: ~ 3645/3584, ~3
Optimal tuning (CTE): ~ 3645/3584 = 1\41, ~3/2 = 701.954
Optimal ET sequence: 41, ..., 2460
Badness: 2.099