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No-sevens subgroup temperaments: Difference between revisions

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{{Technical data page}}
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 7. Many of these have been removed from this page and placed on their appropriate family pages in an effort to include extensions other than incremental-prime-limit; sections for them remain on this page due to extensive linkage.
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 7. Many of these have been removed from this page and placed on their appropriate family pages in an effort to include extensions other than incremental-prime-limit; sections for them remain on this page due to extensive linkage.


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[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents
=== Superpine ===
{{see also|Meantone family #Superpine}}
{{see also|Meantone family #Trimean}}
Subgroup: 2.3.5.11
Comma list: 81/80, 1350/1331
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 167.712
* CWE: ~2 = 1200.000, ~11/10 = 167.882
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13
Comma list: 81/80, 144/143, 975/968
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 5 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 167.729
* CWE: ~2 = 1200.000, ~11/10 = 167.846
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}


=== Dicot ===
=== Dicot ===
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== 2.3.5.13 temperaments ==
== 2.3.5.13 temperaments ==
=== Majvam ===
=== Majvam ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Majvam]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Majvam]].''
: ''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''
: ''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''


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==== 2.3.5.13.17 ====
==== 2.3.5.13.17 ====
The comma list shown here is interestingly a possible minimal S-expression-based comma list for majvam: {[[676/675|S13/S15 = S26]], [[24576/24565|S16/S17]], [[2601/2600|S51]]} (though the order of the last two commas is swapped to make the commas appear in ascending prime limit).
The comma list shown here is interestingly a possible minimal [[S-expression]]-based comma list for majvam: {[[676/675|S13/S15 = S26]], [[24576/24565|S16/S17]], [[2601/2600|S51]]} (though the order of the last two commas is swapped to make the commas appear in ascending prime limit).


Subgroup: 2.3.5.13.17
Subgroup: 2.3.5.13.17
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=== Tricot ===
=== Tricot ===
{{main|Tricot family #2.3.5.13 subgroup }}
{{Main| Tricot family #2.3.5.13 subgroup }}


=== Taylor ===
=== Taylor ===
{{main| Schismatic family #Taylor (2.3.5.13) }}
{{Main| Schismatic family #Taylor (2.3.5.13) }}
 
=== Vulture ===
: ''For the 5-limit restriction and full 13-limit extension, see [[Vulture family]].''
 
This entry is interesting because [[vulture]] and [[buzzard]] unite to the same temperament on the 2.3.5.13.19 subgroup. It results in a surprising decrease in Dirichlet badness, and up to [[octave equivalence]] finds [[13/8]] at 27 generators and [[19/16]] at 41 generators. In this temperament, the [[schisma]] is equated with [[325/324]], [[361/360]], [[513/512]], and [[625/624]]. [[270edo]] is especially ideal, whose step size being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Smaller edos such as [[58edo|58]] (58h val), [[111edo|111]], and [[164edo|164]] are also possible.
 
[[Subgroup]]: 2.3.5.13
 
[[Comma list]]: 676/675, 256000/255879
 
{{Mapping|legend=2| 1 0 -6 -7 | 0 4 21 27 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~320/243 = 475.5502
: [[error map]]: {{val| 0.000 +0.2457 +0.2401 -0.6728 }}
* [[CWE]]: ~2 = 1200.0000, ~320/243 = 475.5543
: [[error map]]: {{val| 0.000 +0.2622 +0.3266 -0.5616 }}
 
{{Optimal ET sequence|legend=1| 53, 164, 217, 270 }}
 
[[Badness]] (Sintel): 0.301
 
==== 2.3.5.13.19 ====
Subgroup: 2.3.5.13.19
 
Comma list: 676/675, 1216/1215, 20007/20000
 
Sval mapping: {{mapping| 1 0 -6 -7 -12 | 0 4 21 27 41 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~320/243 = 475.5498
* CWE: ~2 = 1200.0000, ~320/243 = 475.5533
 
{{Optimal ET sequence|legend=0| 53, 164, 217, 270 }}
 
Badness (Sintel): 0.190


== 2.3.5.17 temperaments ==
== 2.3.5.17 temperaments ==
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=== Stützel ===
=== Stützel ===
{{Main| Meantone family #Stützel (2.3.5.19) }}
{{Main| Meantone family #Stützel (2.3.5.19) }}
== Higher-limit subgroup temperaments ==
=== Reversed marvel ===
As the [[Marvel|marvel temperament]] is based on the marvel comma, [[225/224]], identifying the [[7/4|harmonic seventh (7/4)]] as a stack of two [[15/8|classical major sevenths (15/8)]] [[Octave reduction|octave-reduced]] and tempering 3rd and 5th harmonics slightly flat, tempering [[226/225]] instead results in sharper 3rd and 5th harmonics, and a stack of two classical major sevenths equivalent to the 113th harmonic instead of the 7th, so it might as well be called reversed marvel.
Subgroup: 2.3.5.113
[[Comma list]]: 226/225
{{Mapping|legend=2|1 0 0 -1|0 1 0 2|0 0 1 2}}
: Sval mapping generators: ~2, ~3, ~5
[[Optimal tuning]]s:
* CTE: ~3/2 = 702.449, ~5/4 = 387.373
* CWE: ~3/2 = 702.522, ~5/4 = 387.479
{{Optimal ET sequence|legend=1| 12, 22, 31, 34, 46, 53 }}


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]
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