7th-octave temperaments: Difference between revisions
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{{ | {{Technical data page}} | ||
7th octave temperament | {{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. | |||
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]] | ||
==Akjaysmic (rank-3)== | |||
{{Main|Akjaysma}} | == 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 {{multival| 0 0 7 … }} (in fact, it is {{Multival|legend=| 0 0 7 0 11 16 }}) | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 25/24, 81/80 | |||
{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }} | |||
[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~7/4 = 941.861 | |||
{{Optimal ET sequence|legend=1| 7, 14c }} | |||
[[Badness]]: 0.041714 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 25/24, 33/32, 45/44 | |||
Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }} | |||
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.090 | |||
{{Optimal ET sequence|legend=1| 7, 14c }} | |||
Badness: 0.023524 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 25/24, 27/26, 33/32, 40/39 | |||
Mapping: {{mapping| 7 11 16 0 24 26 | 0 0 0 1 0 0 }} | |||
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 949.236 | |||
{{Optimal ET sequence|legend=1| 7, 14c }} | |||
Badness: 0.023003 | |||
==== Austinpowers ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 25/24, 33/32, 45/44, 65/63 | |||
Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }} | |||
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 952.555 | |||
{{Optimal ET sequence|legend=1| 7, 14cf }} | |||
Badness: 0.022569 | |||
== Akjaysmic (rank-3) == | |||
{{Main| Akjaysma }} | |||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 15: | Line 72: | ||
[[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 | ||
{{Optimal ET sequence|legend=1| 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802 }} | {{Optimal ET sequence|legend=1| 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802 }} | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 28: | Line 85: | ||
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 | ||
{{Optimal ET sequence|legend=1| 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197 }} | {{Optimal ET sequence|legend=1| 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. | == 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 | Subgroup: 2.3.5.7 | ||
| Line 47: | Line 105: | ||
{{Optimal ET sequence|legend=1|140, 1407, 1547}}, ... | {{Optimal ET sequence|legend=1|140, 1407, 1547}}, ... | ||
== Jackpot == | |||
Jackpot identifies [[29/16]] with 6\7. | |||
[[Subgroup]]: 2.3.29 | |||
[[Comma list]]: 17249876309/17179869184 | |||
{{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) | |||
[[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]] | ||