41st-octave temperaments: Difference between revisions
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== Niobium == | == Niobium == | ||
Niobium can be viewed as the 41st-octave counterpart of atomic. The temperament also maps 2657205/2097152 into 14\41. | Niobium can be viewed as the 41st-octave counterpart of [[atomic]], although their badnesses diverge sharply. Much like [[atomic|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 {{monzo|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 | [[Subgroup]]: 2.3.5 | ||
| Line 20: | Line 22: | ||
[[Badness]]: 5.57 | [[Badness]]: 5.57 | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: {{monzo| -5 -12 -9 16 }}, {{monzo| -55 30 2 1 }} | |||
{{Mapping|legend=1| 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|legend=1| 41, ..., 2460 }} | |||
[[Badness]]: 10.676 | |||
=== 11-limit === | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 151263/151250, 6576668672/6576582375, 1722499009/1721868840 | |||
{{Mapping|legend=1| 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|legend=1| 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|legend=1| 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|legend=1| 41, ..., 2460 }} | |||
[[Badness]]: 2.099 | |||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||
[[Category:41edo]] | [[Category:41edo]] | ||
Latest revision as of 18:15, 17 August 2026
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