7th-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|7}}
{{Fractional-octave navigation|7}}
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.
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]]
* ''Brahmagupta →'' [[Ragismic microtemperaments #Brahmagupta|Ragismic microtemperaments]]
* ''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]] starts with {{multival| 0 0 7 … }}.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 25/24, 81/80
{{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


* ''Septant →'' [[Schismatic family#Septant|Schismatic family]]
{{Optimal ET sequence|legend=1| 7, 14c }}
* ''Brahmagupta →'' [[Ragismic microtemperaments#Brahmagupta|Ragismic microtemperaments]]
 
* ''Absurdity'' ''→'' [[Syntonic-chromatic equivalence continuum#Absurdity|Syntonic chromatic equivalence continuum]]
Badness: 0.023524
==Akjaysmic (rank-3)==
 
{{Main|Akjaysma}}
==== 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
 
==== Septimal ====
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


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{{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 ===


Line 33: Line 92:


{{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==
 
== 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.


Revision as of 07:55, 23 January 2024

Template:Fractional-octave navigation 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 starts with ⟨⟨ 0 0 7 … ]].

Subgroup: 2.3.5.7

Comma list: 25/24, 81/80

Mapping: [⟨7 11 16 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨ 0 0 7 0 11 16 ]]

Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.861

Optimal ET sequence: 7, 14c

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

Optimal ET sequence: 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: [⟨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: 7, 14c

Badness: 0.023003

Septimal

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

Optimal ET sequence: 7, 14cf

Badness: 0.022569

Akjaysmic (rank-3)

Main article: 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

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, ...

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