7th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|7}} | {{Infobox fractional-octave|7}} | ||
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 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]]. | ||
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[[Mapping]]: [{{val| 7 0 0 47 }}, {{val| 0 1 0 -1 }}, {{val| 0 0 1 -1 }}] | [[Mapping]]: [{{val| 7 0 0 47 }}, {{val| 0 1 0 -1 }}, {{val| 0 0 1 -1 }}] | ||
Mapping generators: ~1157625/1048576, ~3, ~5 | : Mapping generators: ~1157625/1048576, ~3, ~5 | ||
[[POTE generator]]s: ~3/2 = 701.965, ~5/4 = 386.330 | [[POTE generator]]s: ~3/2 = 701.965, ~5/4 = 386.330 | ||
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=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Mapping: [{{val| 7 0 0 47 -168 }}, {{val| 0 1 0 -1 10 }}, {{val| 0 0 1 -1 5 }}] | 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 | : Mapping generators: ~29160/26411, ~3, ~5 | ||
POTE generators: ~3/2 = 701.968, ~5/4 = 386.332 | POTE generators: ~3/2 = 701.968, ~5/4 = 386.332 | ||
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== 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. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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{{mapping|legend=1| 7 0 34 | 0 1 0 }} | {{mapping|legend=1| 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) | [[Optimal tuning]] ([[CTE]]): ~32/29 = 1\7, ~3/2 = 701.955 (~24576/24389 = 16.239) | ||
Revision as of 13:32, 13 April 2025
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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