7th-octave temperaments: Difference between revisions
Cmloegcmluin (talk | contribs) →Nitrogen: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
higher-limit akjaysmic goes here, the comma page should only describe the temp associated with the comma itself. |
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* ''Brahmagupta →'' [[Ragismic microtemperaments#Brahmagupta|Ragismic microtemperaments]] | * ''Brahmagupta →'' [[Ragismic microtemperaments#Brahmagupta|Ragismic microtemperaments]] | ||
* ''Absurdity'' ''→'' [[Syntonic-chromatic equivalence continuum#Absurdity|Syntonic chromatic equivalence continuum]] | * ''Absurdity'' ''→'' [[Syntonic-chromatic equivalence continuum#Absurdity|Syntonic chromatic equivalence continuum]] | ||
==Akjaysmic (rank-3)== | |||
{{Main|Akjaysma}} | |||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: {{monzo| 47 -7 -7 -7 }} | |||
[[Mapping]]: [{{val| 7 0 0 47 }}, {{val| 0 1 0 -1 }}, {{val| 0 0 1 -1 }}] | |||
Mapping generators: ~1157625/1048576, ~3, ~5 | |||
[[POTE generator]]s: ~3/2 = 701.965, ~5/4 = 386.330 | |||
{{Optimal ET sequence|legend=1| 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: [{{val| 7 0 0 47 -168 }}, {{val| 0 1 0 -1 10 }}, {{val| 0 0 1 -1 5 }}] | |||
Mapping generators: ~29160/26411, ~3, ~5 | |||
POTE generators: ~3/2 = 701.968, ~5/4 = 386.332 | |||
{{Optimal ET sequence|legend=1| 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197 }} | |||
==Nitrogen== | ==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. | 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. | ||
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[[Category:7edo]] | [[Category:7edo]] | ||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
Revision as of 00:13, 14 June 2023
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, ...