7th-octave temperaments: Difference between revisions
→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 :) |
Tristanbay (talk | contribs) →Profanity: Added mapping generators Tags: Mobile edit Mobile web edit |
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{{ | {{Technical data page}} | ||
{{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 comma that frequently appears in 7th-octave temps is [[akjaysma]], which sets [[105/64]] to be equal to 5\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: | Temperaments discussed elsewhere include: | ||
* ''Septant →'' [[Schismatic family #Septant|Schismatic family]] | * ''Septant →'' [[Schismatic family#Septant|Schismatic family]] | ||
* ''Brahmagupta →'' [[Ragismic microtemperaments #Brahmagupta|Ragismic microtemperaments]] | * ''Brahmagupta →'' [[Ragismic microtemperaments#Brahmagupta|Ragismic microtemperaments]] | ||
* ''Absurdity'' ''→'' [[Syntonic | * ''Absurdity'' ''→'' [[Syntonic–chromatic equivalence continuum#Absurdity|Syntonic–chromatic equivalence continuum]] | ||
== Jamesbond == | == 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 {{multival| 0 0 7 … }} | 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]]" (a kind of mathematical object representing the temperament) starts with {{multival| 0 0 7 … }} (in fact, it is {{Multival|legend=| 0 0 7 0 11 16 }}) | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }} | {{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~7/4 = 941.861 | [[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~7/4 = 941.861 | ||
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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) | ||
[[Support]]ing [[ET]]s: {{EDOs|7, 77, 70, 147, 224, 84, 63, 301, 217, 371, 56, 161, 91, 378}} | [[Support]]ing [[ET]]s: {{EDOs|7, 77, 70, 147, 224, 84, 63, 301, 217, 371, 56, 161, 91, 378}} | ||
== Profanity == | |||
Profanity identifies [[11/9]] with 2\7. | |||
[[Subgroup]]: 2.3.11 | |||
[[Comma list]]: 19487171/19131876 | |||
{{mapping|legend=1| 7 0 2 | 0 1 2 }} | |||
: Mapping generators: ~1458/1331, ~3 | |||
[[Support]]ing [[ET]]s: {{EDOs|7, 49, 56, 63, 70, 77, 133}} | |||
{{Navbox fractional-octave}} | |||
[[Category:7edo]] | [[Category:7edo]] | ||
Latest revision as of 09:04, 10 July 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" (a kind of mathematical object representing the temperament) starts with ⟨⟨ 0 0 7 … ]] (in fact, it is ⟨⟨ 0 0 7 0 11 16 ]])
Subgroup: 2.3.5.7
Comma list: 25/24, 81/80
Mapping: [⟨7 11 16 0], ⟨0 0 0 1]]
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
Profanity
Profanity identifies 11/9 with 2\7.
Subgroup: 2.3.11
Comma list: 19487171/19131876
Mapping: [⟨7 0 2], ⟨0 1 2]]
- Mapping generators: ~1458/1331, ~3
Supporting ETs: 7, 49, 56, 63, 70, 77, 133