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7th-octave temperaments: Difference between revisions

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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.
{{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.


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-chromatic equivalence continuum#Absurdity|Syntonic chromatic equivalence continuum]]
* ''Absurdity'' ''→'' [[Syntonic–chromatic equivalence continuum#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 {{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 }}


==Nitrogen==
Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 952.555
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.
 
{{Optimal ET sequence|legend=1| 7, 14cf }}
 
Badness: 0.022569
 
== 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 ==
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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Comma list: 3955078125/3954653486, 140737488355328/140710042265625
Comma list: 3955078125/3954653486, 140737488355328/140710042265625


Mapping: 7 10 17 20, 0 22 -15 -7
Mapping: [{{val|7 10 17 20}}, {{val|0 22 -15 -7}}]


Mapping generators: ~1157625/1048576, ~1029/1024
Mapping generators: ~1157625/1048576, ~1029/1024
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Optimal tuning (CTE): ~1157625/1048576 = 1\7, ~1029/1024 = 8.531
Optimal tuning (CTE): ~1157625/1048576 = 1\7, ~1029/1024 = 8.531


Vals: 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]]
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