41st-octave temperaments

Fractional-octave temperaments

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← 40th-octave • 41st-octave • 42nd-octave →

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