7th-octave temperaments: Difference between revisions

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→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 :)
- jamesbond (more properly addressed in whitewood family)
 
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{{Fractional-octave navigation|7}}
{{Technical data page}}
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]].  
{{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-chromatic equivalence continuum #Absurdity|Syntonic chromatic equivalence continuum]]
* ''[[Absurdity]]'' → [[Porwell temperaments #Absurdity|Porwell temperaments]]


== Jamesbond ==
== Nitrogen ==
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 … }}.
Nitrogen may be described as the {{nowrap| 140 & 1407 }} temperament in the 7-limit. It was named after the 7th element for having a 7th-octave period 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 the 7th harmonic 7 generators down.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 25/24, 81/80
[[Comma list]]: 3955078125/3954653486, {{monzo| 47 -7 -7 -7 }}
 
{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }}
 
{{Multival|legend=1| 0 0 7 0 11 16 }}
 
[[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
 
[[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
 
Comma list: 3955078125/3954653486, 140737488355328/140710042265625
 
Mapping: [{{val|7 10 17 20}}, {{val|0 22 -15 -7}}]
 
Mapping generators: ~1157625/1048576, ~1029/1024
 
Optimal tuning (CTE): ~1157625/1048576 = 1\7, ~1029/1024 = 8.531
 
{{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 10 17 20 | 0 22 -15 -7 }}
: mapping generators: ~1157625/1048576, ~1029/1024


{{mapping|legend=1| 7 0 34 | 0 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~1157625/1048576 = 171.4278{{c}}, ~1029/1024 = 8.5308{{c}}
: [[error map]]: {{val| -0.005 +0.001 -0.002 +0.015 }}
* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~1029/1024 = 8.5308{{c}}
: error map: {{val| 0.000 +0.008 +0.010 +0.030 }}


: mapping generators: ~32/29, ~3
{{Optimal ET sequence|legend=1| 140, 847, 987, 1127, 1267, 1407, 1547, 2954 }}


[[Optimal tuning]] ([[CTE]]): ~32/29 = 1\7, ~3/2 = 701.955 (~24576/24389 = 16.239)
[[Badness]] (Sintel): 1.50


[[Support]]ing [[ET]]s: {{EDOs|7, 77, 70, 147, 224, 84, 63, 301, 217, 371, 56, 161, 91, 378}}
{{Navbox fractional-octave}}


[[Category:7edo]]
[[Category:7edo]]
[[Category:Temperament collections]]

Latest revision as of 11:47, 8 June 2026

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.

← 6th-octave • 7th-octave • 8th-octave →

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:

Nitrogen

Nitrogen may be described as the 140 & 1407 temperament in the 7-limit. It was named after the 7th element for having a 7th-octave period 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 the 7th harmonic 7 generators down.

Subgroup: 2.3.5.7

Comma list: 3955078125/3954653486, [47 -7 -7 -7⟩

Mapping: [⟨7 10 17 20], ⟨0 22 -15 -7]]

mapping generators: ~1157625/1048576, ~1029/1024

Optimal tunings:

  • WE: ~1157625/1048576 = 171.4278 ¢, ~1029/1024 = 8.5308 ¢
error map: ⟨-0.005 +0.001 -0.002 +0.015]
  • CWE: ~1157625/1048576 = 171.4286 ¢, ~1029/1024 = 8.5308 ¢
error map: ⟨0.000 +0.008 +0.010 +0.030]

Optimal ET sequence: 140, 847, 987, 1127, 1267, 1407, 1547, 2954

Badness (Sintel): 1.50

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