7th-octave temperaments: Difference between revisions
→Jackpot: explain what jackpot is |
→Septimal: Renamed temperament from "septimal" to "austinpowers" to avoid potential confusion with "septimal" the term for the 7-limit. Asked on Discord and Facebook first and commenters were all in favour. Note that if jamesbond is ever renamed in the future, then you are free to rename austinpowers along with it. I am not attached to the name :) |
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Badness: 0.023003 | Badness: 0.023003 | ||
==== | ==== Austinpowers ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 07:04, 21 October 2024
Template:Fractional-octave navigation a 7th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 7. The most notable 7th-octave family is the whitewood family – tempering out 2187/2048 and associating 4\7 to 3/2.
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
Jamesbond
This temperament uses exactly the same 5-limit as 7et, but the harmonic 7 is mapped to an independent generator. It is so named because its wedgie starts with ⟨⟨ 0 0 7 … ]].
Subgroup: 2.3.5.7
Comma list: 25/24, 81/80
Mapping: [⟨7 11 16 0], ⟨0 0 0 1]]
Wedgie: ⟨⟨ 0 0 7 0 11 16 ]]
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.861
Badness: 0.041714
11-limit
Subgroup: 2.3.5.7.11
Comma list: 25/24, 33/32, 45/44
Mapping: [⟨7 11 16 0 24], ⟨0 0 0 1 0]]
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.090
Badness: 0.023524
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 25/24, 27/26, 33/32, 40/39
Mapping: [⟨7 11 16 0 24 26], ⟨0 0 0 1 0 0]]
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 949.236
Badness: 0.023003
Austinpowers
Subgroup: 2.3.5.7.11.13
Comma list: 25/24, 33/32, 45/44, 65/63
Mapping: [⟨7 11 16 0 24 6], ⟨0 0 0 1 0 1]]
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 952.555
Badness: 0.022569
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, ...
Jackpot
Jackpot identifies 29/16 with 6\7.
Subgroup: 2.3.29
Comma list: 17249876309/17179869184
Mapping: [⟨7 0 34], ⟨0 1 0]]
- mapping generators: ~32/29, ~3
Optimal tuning (CTE): ~32/29 = 1\7, ~3/2 = 701.955 (~24576/24389 = 16.239)
Supporting ETs: 7, 77, 70, 147, 224, 84, 63, 301, 217, 371, 56, 161, 91, 378