7th-octave temperaments: Difference between revisions

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== 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]]" (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 }})
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| 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 }}
: mapping generators: ~10/9, ~7


[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~7/4 = 941.861
[[Optimal tuning]]s:
* [[WE]]: ~10/9 = 172.790{{c}}, ~7/4 = 949.343{{c}}
: [[error map]]: {{val| +9.533 -1.261 -21.668 -0.418 }}
* [[CWE]]: ~10/9 = 171.429{{c}}, ~7/4 = 948.499{{c}}
: error map: {{val| -0.000 -16.241 -43.457 -20.327 }}


{{Optimal ET sequence|legend=1| 7, 14c }}
{{Optimal ET sequence|legend=1| 7(d), 14c }}


[[Badness]]: 0.041714
[[Badness]] (Sintel): 1.06


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}
Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}


Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.090
Optimal tunings:
* WE: ~10/9 = 172.830{{c}}, ~7/4 = 948.784{{c}}
* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 946.554{{c}}


{{Optimal ET sequence|legend=1| 7, 14c }}
{{Optimal ET sequence|legend=1| 7(d), 14c }}


Badness: 0.023524
Badness (Sintel): 0.778


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 7 11 16 0 24 26 | 0 0 0 1 0 0 }}
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 tunings:
* WE: ~10/9 = 172.390{{c}}, ~7/4 = 954.559{{c}}
* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 952.367{{c}}


{{Optimal ET sequence|legend=1| 7, 14c }}
{{Optimal ET sequence|legend=1| 7(d), 14c }}


Badness: 0.023003
Badness (Sintel): 0.951


==== Austinpowers ====
==== Austinpowers ====
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Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }}
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 tunings:
* WE: ~10/9 = 172.873{{c}}, ~7/4 = 960.581{{c}}
* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 958.793{{c}}


{{Optimal ET sequence|legend=1| 7, 14cf }}
{{Optimal ET sequence|legend=1| 7(df), 14cf }}


Badness: 0.022569
Badness (Sintel): 0.933


== Akjaysmic (rank-3) ==
== Akjaysmic (rank-3) ==
{{Main| Akjaysma }}
{{See also| Akjaysma }}
Subgroup: 2.3.5.7
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| 47 -7 -7 -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|legend=1| 7 0 0 47 | 0 1 0 -1 | 0 0 1 -1 }}
: mapping generators: ~1157625/1048576, ~3, ~5


: Mapping generators: ~1157625/1048576, ~3, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~1157625/1048576 = 171.427811{{c}}, ~3/2 = 701.962313{{c}}, ~5/4 = 386.328628{{c}}
: [[error map]]: {{val| -0.0053 +0.0020 +0.0043 +0.0062 }}
* [[CWE]]: ~1157625/1048576 = 171.428571{{c}}, ~3/2 = 701.964859{{c}}, ~5/4 = 386.330310{{c}}
: error map: {{val| 0.0000 +0.0099 +0.0166 +0.0218 }}


[[POTE generator]]s: ~3/2 = 701.965, ~5/4 = 386.330
{{Optimal ET sequence|legend=1| 56, 63, 77, 84, 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}


{{Optimal ET sequence|legend=1| 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802 }}
[[Badness]] (Sintel): 2.22


=== 11-limit ===
=== 11-limit ===
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Comma list: 184549376/184528125, 199297406/199290375
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: {{mapping| 7 0 0 47 -168 | 0 1 0 -1 10 | 0 0 1 -1 5 }}
: mapping generators: ~29160/26411, ~3, ~5


: Mapping generators: ~29160/26411, ~3, ~5
Optimal tunings:  
* WE: ~29160/26411 = 171.427802{{c}}, ~3/2 = 701.964561{{c}}, ~5/4 = 386.329837{{c}}
* CWE: ~29160/26411 = 171.428571{{c}}, ~3/2 = 701.967291{{c}}, ~5/4 = 386.331624{{c}}


POTE generators: ~3/2 = 701.968, ~5/4 = 386.332
{{Optimal ET sequence|legend=0| 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee }}


{{Optimal ET sequence|legend=1| 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197 }}
Badness (Sintel): 1.32


== 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.
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: 3955078125/3954653486, 140737488355328/140710042265625
[[Comma list]]: 3955078125/3954653486, {{monzo| 47 -7 -7 -7 }}


Mapping: [{{val|7 10 17 20}}, {{val|0 22 -15 -7}}]
{{Mapping|legend=1| 7 10 17 20 | 0 22 -15 -7 }}
: mapping generators: ~1157625/1048576, ~1029/1024


Mapping generators: ~1157625/1048576, ~1029/1024
[[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 }}


Optimal tuning (CTE): ~1157625/1048576 = 1\7, ~1029/1024 = 8.531
{{Optimal ET sequence|legend=1| 140, 847, 987, 1127, 1267, 1407, 1547, 2954 }}


{{Optimal ET sequence|legend=1|140, 1407, 1547}}, ...
[[Badness]] (Sintel): 1.50


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}


[[Category:7edo]]
[[Category:7edo]]

Revision as of 08:56, 30 May 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:

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]]

mapping generators: ~10/9, ~7

Optimal tunings:

  • WE: ~10/9 = 172.790 ¢, ~7/4 = 949.343 ¢
error map: ⟨+9.533 -1.261 -21.668 -0.418]
  • CWE: ~10/9 = 171.429 ¢, ~7/4 = 948.499 ¢
error map: ⟨-0.000 -16.241 -43.457 -20.327]

Optimal ET sequence: 7(d), 14c

Badness (Sintel): 1.06

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 tunings:

  • WE: ~10/9 = 172.830 ¢, ~7/4 = 948.784 ¢
  • CWE: ~10/9 = 171.429 ¢, ~7/4 = 946.554 ¢

Optimal ET sequence: 7(d), 14c

Badness (Sintel): 0.778

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 tunings:

  • WE: ~10/9 = 172.390 ¢, ~7/4 = 954.559 ¢
  • CWE: ~10/9 = 171.429 ¢, ~7/4 = 952.367 ¢

Optimal ET sequence: 7(d), 14c

Badness (Sintel): 0.951

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 tunings:

  • WE: ~10/9 = 172.873 ¢, ~7/4 = 960.581 ¢
  • CWE: ~10/9 = 171.429 ¢, ~7/4 = 958.793 ¢

Optimal ET sequence: 7(df), 14cf

Badness (Sintel): 0.933

Akjaysmic (rank-3)

See also: Akjaysma

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

Optimal tunings:

  • WE: ~1157625/1048576 = 171.427811 ¢, ~3/2 = 701.962313 ¢, ~5/4 = 386.328628 ¢
error map: ⟨-0.0053 +0.0020 +0.0043 +0.0062]
  • CWE: ~1157625/1048576 = 171.428571 ¢, ~3/2 = 701.964859 ¢, ~5/4 = 386.330310 ¢
error map: ⟨0.0000 +0.0099 +0.0166 +0.0218]

Optimal ET sequence: 56, 63, 77, 84, 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454

Badness (Sintel): 2.22

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

Optimal tunings:

  • WE: ~29160/26411 = 171.427802 ¢, ~3/2 = 701.964561 ¢, ~5/4 = 386.329837 ¢
  • CWE: ~29160/26411 = 171.428571 ¢, ~3/2 = 701.967291 ¢, ~5/4 = 386.331624 ¢

Optimal ET sequence: 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee

Badness (Sintel): 1.32

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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