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No-fives subgroup temperaments: Difference between revisions

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=== Archy ===
=== Archy ===
See [[Archytas clan #Archy]].  
See [[Archytas clan #Archy]].  
==== Supra ====
See [[Archytas clan #Supra]].
===== Supraphon =====
See [[Archytas clan #Supraphon]].


==== Suhajira ====
==== Suhajira ====
Line 55: Line 49:
=== Slendroschismic ===
=== Slendroschismic ===
See [[5th-octave temperaments #Slendroschismic]].  
See [[5th-octave temperaments #Slendroschismic]].  
==== Pentadecoid ====
See [[15th-octave temperaments #Pentadecoid]].


=== Navy ===
=== Navy ===
Line 598: Line 589:


=== Purpleheart ===
=== Purpleheart ===
{{See also| Whitewood }}
{{Main| Whitewood }}


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
Line 866: Line 857:
=== Neutral ===
=== Neutral ===
See [[Rastmic clan #Neutral]].  
See [[Rastmic clan #Neutral]].  
==== Namo ====
See [[Rastmic clan #Namo]].


=== Io ===
=== Io ===
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.


The name ''io'' was coined by [[User:CompactStar|CompactStar]] in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.
The name ''io'' was coined by {{u|CompactStar}} in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.


[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11
Line 891: Line 879:


[[Badness]] (Sintel): 0.185
[[Badness]] (Sintel): 0.185
=== Alphaxenean ===
Alphaxenean tempers out the [[Alpharabian comma]], and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It has a [[semi-octave]] period, and the generator can be taken to be the aforementioned undecimal quartertone or that plus a period, which represents ~[[16/11]]. The interval class of a perfect fifth is found by two generator steps and a period.
[[Subgroup]]: 2.3.11
[[Comma list]]: 131769/131072
{{Mapping|legend=2| 2 1 8 | 0 2 -1 }}
: mapping generators: ~363/256, ~16/11
[[Optimal tuning]]s:
* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}
: [[error map]]: {{val| +0.318 -0.094 -0.897 }}
* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}
: error map: {{val| 0.000 -0.491 -2.050 }}
{{Optimal ET sequence|legend=1| 22, 24, 94, 118, 142, 450e, 592e, 1326beeee }}
[[Badness]] (Sintel): 0.395
=== Octatonic ===
[[12/11]] is very close to one step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.
[[Subgroup]]: 2.3.11
[[Comma list]]: 214990848/214358881
{{Mapping|legend=2| 8 0 15 | 0 1 1 }}
: mapping generators: ~12/11, ~3
[[Optimal tuning]]s:
* [[WE]]: ~12/11 = 149.9917{{c}}, ~3/2 = 701.9324{{c}}
: [[error map]]: {{val| -0.066 -0.089 +0.424 }}
* [[CWE]]: ~12/11 = 150.0000{{c}}, ~3/2 = 701.9147{{c}}
: error map: {{val| 0.000 -0.040 +0.597 }}
{{Optimal ET sequence|legend=1| 8, 16, 24, 104, 128, 152, 176, 200, 1024e, 1224e, 1424e, 1624e }}
[[Badness]] (Sintel): 0.515
=== Betaxenean ===
Betaxenean tempers out the [[Betarabian comma]]. It splits a [[3/1|perfect twelfth]] into two halves of [[512/297]].
[[Subgroup]]: 2.3.11
[[Comma list]]: 264627/262144
{{Mapping|legend=2| 1 0 9 | 0 2 -7 }}
: mapping generators: ~2, ~512/297
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5931{{c}}, ~512/297 = 950.6872{{c}}
: [[error map]]: {{val| +0.593 -0.581 -0.791 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~512/297 = 950.1972{{c}}
: error map: {{val| 0.000 -1.561 -2.699 }}
{{Optimal ET sequence|legend=1| 5, 14e, 19, 24, 125, 149, 173, 197e, 221e, 466bee, 687beee }}
[[Badness]] (Sintel): 0.709


=== Infraug ===
=== Infraug ===
Line 910: Line 958:
[[Badness]] (Sintel): 0.734
[[Badness]] (Sintel): 0.734


==== 2.3.11.13 ====
==== 2.3.11.13 subgroup ====
Subgroup: 2.3.11.13
Subgroup: 2.3.11.13


Comma list: 144/143, 729/704
Comma list: 144/143, 729/704


Subgrouo-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}
Subgroup-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 930: Line 978:
Comma list: 363/361, 729/704
Comma list: 363/361, 729/704


Subgrouo-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
Subgroup-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
: mapping generators: ~2, ~19/11
: mapping generators: ~2, ~19/11


Line 937: Line 985:
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}


{{Optimal ET sequence|legend=1| 14, 19, 33 }}
{{Optimal ET sequence|legend=0| 14, 19, 33 }}


Badness (Sintel): 1.59
Badness (Sintel): 1.59
Line 1,013: Line 1,061:


Badness (Sintel): 0.415
Badness (Sintel): 0.415
=== Profanity ===
Profanity identifies [[11/9]] with 2\7.
[[Subgroup]]: 2.3.11
[[Comma list]]: 19487171/19131876
{{Mapping|legend=2| 7 0 2 | 0 1 2 }}
: mapping generators: ~1458/1331, ~3
[[Optimal tuning]]s:
* [[WE]]: ~1458/1331 = 171.4369{{c}}, ~3/2 = 702.9304{{c}}
: [[error map]]: {{val| +0.058 +1.033 -2.467 }}
* [[CWE]]: ~1458/1331 = 171.4286{{c}}, ~3/2 = 702.9442{{c}}
: error map: {{val| 0.000 -0.989 -2.572 }}
{{Optimal ET sequence|legend=1| 7, … 49, 56, 63, 70 }}
[[Badness]] (Sintel): 3.03


== Temperaments with a 2.3.13 gene ==
== Temperaments with a 2.3.13 gene ==
Line 1,052: Line 1,120:


[[Badness]] (Sintel): 0.200
[[Badness]] (Sintel): 0.200
=== Tridecapyth ===
Tridecapyth sets a stack of ten [[9/8]]'s equal to [[13/4]]. It shows a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping – for 118edo we get the 118f [[val]] that takes the second-closest, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, [[41edo]] uses this mapping, and so does [[94edo]], the val sum of 41 and 53. Thus it is of interest to flatter tunings of [[garibaldi]]/[[cassandra]] with fifths tending close to pure. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].
[[Subgroup]]: 2.3.13
{{mapping|legend=2| 1 0 -28 | 0 1 20 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9778{{c}}, ~3/2 = 702.0170{{c}}
: [[error map]]: {{val| -0.022 +0.040 -0.011 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0283{{c}}
: error map: {{val| 0.000 +0.070 -0.019 }}
{{Optimal ET sequence|legend=1| 12, 29f, 41, 53, 94, 147, 494, 641, 788, 2217, 3005 }}
[[Badness]] (Sintel): 0.211


=== Superflat ===
=== Superflat ===
Line 1,069: Line 1,155:
: [[error map]]: {{val| +3.129 -3.177 -4.349 }}
: [[error map]]: {{val| +3.129 -3.177 -4.349 }}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
: error map: {{val| 0.000 -8.347 +14.960 }}
: error map: {{val| 0.000 -8.347 -14.960 }}


{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}
{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}
Line 1,095: Line 1,181:
Badness (Sintel): 0.135
Badness (Sintel): 0.135


==== 2.3.13.23 ====
==== 2.3.13.23 subgroup ====
Subgroup: 2.3.13.23
Subgroup: 2.3.13.23


Line 1,113: Line 1,199:
* [[Shoal17]]
* [[Shoal17]]


Music:
; Music
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)


=== Glacier ===
=== Glacier ===
This 2.3.13-subgroup gene is not nearly as good as shoal, but it can extend extremely well to other no-5 subgroups. It is the common [[restriction]] of [[#Bleu|bleu]] and [[#Navy|navy]]. It is very well represented in [[26edo]], where a pure 13/12 can serve as the generator, but [[94edo]] provides a much better tuning.
This 2.3.13-subgroup gene is not nearly as good as shoal, but it can extend extremely well to other no-5 subgroups. It is the common [[restriction]] of [[#Bleu|bleu]] and [[#Navy|navy]]. It is very well represented in [[26edo]], where a nearly pure 13/12 can serve as the generator, but [[94edo]] provides a much better tuning.


[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13
Line 1,143: Line 1,229:


{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
: mapping generators: ~17/12, ~3


: sval mapping generators: ~17/12, ~3
[[Optimal tuning]]s:  
* [[WE]]: ~17/12 = 600.1471{{c}}, ~3/2 = 701.9563{{c}} (~17/16 = 101.8091{{c}})
: [[error map]]: {{val| +0.294 +0.295 -1.969 }}
* [[CWE]]: ~17/12 = 600.0000{{c}}, ~3/2 = 702.0260{{c}} (~17/16 = 102.0260{{c}})
: error map: {{val| 0.000 +0.071 -2.929 }}


: [[gencom]]: [17/12 3; 289/288]
{{Optimal ET sequence|legend=1| 10, 12, 58, 70, 82, 94, 106, 118, 224g }}


[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
[[Badness]] (Sintel): 0.0454


{{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}
=== Boethian ===
Boethian is a [[5L 2s|diatonic-based]] temperament that makes [[513/512]] vanish, so that the major third (C–E) is ~[[24/19]] and the minor third (C–E♭) is ~[[19/16]]. As such, it functions as a 2.3.19-subgroup analogue of [[meantone]], though the small size of the comma puts it at [[schismic]] level of accuracy. In particular, the equal temperaments in the tuning spectrum up to 1/2-comma (flattened) boethian temperament (very close to [[12edo]]) are included in the schismic tuning spectrum in the 5-limit, so boethian intersects with schismic in the prime-5 infill extension thereof, called [[nestoria]], which also tempers out [[361/360]], the difference between 19/18 and 20/19 or between 19/15 and 24/19.


[[Tp tuning #T2 tuning|RMS error]]: 0.2247 cents
[[Subgroup]]: 2.3.19


=== Gigapyth ===
[[Comma list]]: 513/512
[[Subgroup]]: 2.3.85


[[Comma list]]: 2.3.85 {{val|-40 1 6 }}
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~3


{{mapping|legend=2| 1 4 6 | 0 -6 1 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.2498{{c}}, ~3/2 = 701.4958{{c}}
: mapping generators: ~2, ~85/64
: [[error map]]: {{val| +0.250 -0.209 -0.501 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.3445{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.034
: error map: {{val| 0.000 -0.610 -1.547 }}
 
[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**
 
<nowiki>*</nowiki>Wart for 85
 
==== 2.3.7.85 subgroup ====
[[Subgroup]]: 2.3.7.85
 
[[Comma list]]: 1029/1024, 7225/7203
 
{{mapping|legend=2| 1 4 2 6 | 0 -6 2 1}}
 
: mapping generators: ~2, ~85/64
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.031
 
[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**
 
<nowiki>*</nowiki>Wart for 85
 
=== Dog ===
The dog temperament is based by [[2L 5s]] or [[7L 2s]] scale that makes [[81/76]] vanish, so [[19/16]] is a major third. It can be viewed as a 2.3.19 subgroup analogue of [[Pelogic family|mavila]].
 
Subgroup: 2.3.19
 
[[Comma list]]: 81/76
 
[[Gencom]]: [2 4/3; 81/76]
 
[[Mapping|Sval mapping]]: [{{val|1 2 6}}, {{val|0 -1 -4}}]
 
[[Tp tuning|POL2 generator]]: ~4/3 = 521.403


{{Optimal ET sequence|legend=1| 5h, 7, 16, 23 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}


[[Tp tuning #T2 tuning|RMS error]]: 4.943 cents
[[Badness]] (Sintel): 0.0294


=== Boethian ===
=== Deviaug ===
Boethian is a [[5L 2s|diatonic-based]] temperament that makes [[513/512]] vanish, so that the major third (C–E) is ~[[24/19]] and the minor third (C–E♭) is ~[[19/16]]. As such, it functions as a 2.3.19-subgroup analogue of [[meantone]], though the small size of the comma puts it at [[schismic]] level of accuracy. In particular, the equal temperaments in the tuning spectrum up to 1/2-comma (flattened) boethian temperament (very close to [[12edo]]) are included in the schismic tuning spectrum in the 5-limit, so boethian intersects with schismic in the prime-5 infill extension thereof, called [[nestoria]], which also tempers out [[361/360]], the difference between 19/18 and 20/19 or between 19/15 and 24/19.  
Deviaug tempers out [[6912/6859]] in the 2.3.19 subgroup, setting [[24/19]] to 1/3 of an octave.  


[[Subgroup]]: 2.3.19
[[Subgroup]]: 2.3.19


[[Comma list]]: [[513/512]]
[[Comma list]]: 6912/6859


{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
{{Mapping|legend=2| 3 0 8 | 0 1 1 }}
: mapping generators: ~2, ~3
: mapping generators: ~24/19, ~3


[[Optimal tuning]] (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.3288{{c}}
[[Optimal tuning]]s:  
* [[WE]]: ~24/19 = 399.8571{{c}}, ~3/2 = 701.9795{{c}}
: [[error map]]: {{val| -0.429 -0.404 +2.894 }}
* [[CWE]]: ~24/19 = 400.0000{{c}}, ~3/2 = 701.8810{{c}}
: error map: {{val| 0.000 -0.074 +4.368 }}


{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296 }}
{{Optimal ET sequence|legend=1| 9, 12, 75, 87, 99, 111, 123h, 135h }}


[[Badness]] (Smith): 0.000374
[[Badness]] (Sintel): 0.237


=== Lipsett ===
=== Lipsett ===
Lipsett temperament is a pleasantly melodic little temperament with a highly useable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director off the Canadian short film ’21-87’. Leia’s prison cell in Star Wars is numbered ‘2187’, as a nod to the influence the film had on George Lucas.
Lipsett is a pleasantly melodic little temperament with a highly usable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director of the Canadian short film ''21-87''. Leia's prison cell in ''Star Wars'' is numbered 2187, as a nod to the influence the film had on George Lucas.


[[Subgroup]]: 2.3.23
[[Subgroup]]: 2.3.23


[[Comma list]]: [[2187/2116]]
[[Comma list]]: 2187/2116


{{Mapping|legend=2| 1 0 -1 | 0 2 7}}
{{Mapping|legend=2| 1 0 -1 | 0 2 7 }}
: mapping generators: ~2, ~46/27


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~46/27 = 948.526
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5339{{c}}, ~46/27 = 948.5629{{c}}
: [[error map]]: {{val| +0.534 -4.829 +11.132 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~46/27 = 948.3272{{c}}
: error map: {{val| 0.000 -5.301 +10.016 }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}


[[Badness]] (Smith): 8.998 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.801


=== Porpoise ===
=== Porpoise ===
Line 1,237: Line 1,306:
[[Comma list]]: 24576/24389
[[Comma list]]: 24576/24389


[[Mapping]]: [{{val|1 2 5}}, {{val|0 3 -1}}]
{{Mapping|legend=2| 1 2 5 | 0 -3 -1 }}
: mapping generators: ~2, ~32/29


[[CTE tuning|CTE generator]]: ~32/29 = 166.067
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5519{{c}}, ~32/29 = 165.7453{{c}}
: [[error map]]: {{val| -0.448 -0.087 +2.437 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/29 = 165.9004{{c}}
: error map: {{val| 0.000 +0.344 +4.522 }}


{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}
{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}


=== Sematology ===
[[Badness]] (Sintel): 0.317
This temperament tempers out 4107/4096 and thus equates 2 [[37/32]]'s with [[4/3]].


[[Subgroup]]: 2.3.37
=== Jackpot ===
Named by {{u|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1|3rd harmonic]]. The name refers to the [[Temperament merging|EDO join]] of 7 & 77.


[[Comma list]]: 4107/4096
[[Subgroup]]: 2.3.29


[[Gencom]]: [2 37/32; 4107/4096]
[[Comma list]]: 17249876309/17179869184


[[Mapping]]: [{{val|1 1 5}}, {{val|0 -2 1}}]
{{Mapping|legend=2| 7 0 34 | 0 1 0 }}
: mapping generators: ~32/29, ~3


[[POTE generator]]: ~[[37/32]] = 249.075
[[Optimal tuning]]s:
* [[WE]]: ~32/29 = 171.4434{{c}}, ~3/2 = 701.8515{{c}}
: [[error map]]: {{val| +0.103 -0.000 -0.503 }}
* [[CWE]]: ~32/29 = 171.4286{{c}}, ~3/2 = 701.8456{{c}}
: error map: {{val| 0.000 -0.109 -1.006 }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130 }}
{{Optimal ET sequence|legend=1| 7, 42, 49, 56, 63, 70, 77, 147, 224 }}


==== 2.3.7.37 subgroup ====
[[Badness]] (Sintel): 1.02
[[Subgroup]]: 2.3.7.37


[[Comma list]]: 4107/4096, 259/256
=== Sematology ===
This temperament tempers out [[4107/4096]] and thus equates a stack of two [[37/32]]'s with [[4/3]].


[[Gencom]]: [2 37/32; 4107/4096 259/256]
[[Subgroup]]: 2.3.37


[[Mapping]]: [{{val|1 1 1 5}}, {{val|0 -2 -1 1}}]
[[Comma list]]: 4107/4096


[[POTE generator]]: ~[[37/32]] = 247.782
{{Mapping|legend=2| 1 0 6 | 0 2 -1 }}
: mapping generators: ~2, ~64/37


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53d }}
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.2184{{c}}, ~64/37 = 950.9546{{c}}
==== 2.3.5.37 subgroup ====
: [[error map]]: {{val| +0.218 -0.046 -0.988 }}
It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/37 = 950.8250{{c}}
 
: error map: {{val| 0.000 -0.305 -2.169 }}
[[Subgroup]]: 2.3.5.37
 
[[Comma list]]: 4107/4096, 17592186044416/17562397269605
 
[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605]
 
[[Mapping]]: [{{val|1 1 4 5}}, {{val|0 -2 -8 1}}]
 
[[POTE generator]]: ~[[37/32]] = 251.393


{{Optimal ET sequence|legend=1| 5, 14c, 19, 43, 62 }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130, 443l, 573ll, 703ll, 1536blllll }}


===== 2.3.5.7.37 subgroup =====
[[Badness]] (Sintel): 0.0690
[[Subgroup]]: 2.3.5.7.37
 
[[Comma list]]: 4107/4096, 17592186044416/17562397269605, 259/256
 
[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]
 
[[Mapping]]: [{{val|1 1 4 1 5}}, {{val|0 -2 -8 -1 1}}]
 
[[POTE generator]]: ~[[37/32]] = 251.204
 
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}


=== Reversed mavila ===
=== Reversed mavila ===
Line 1,304: Line 1,364:
[[Comma list]]: 81/74
[[Comma list]]: 81/74


[[Gencom]]: [2 4/3; 81/74]
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3


[[Mapping]]: [{{val|1 1 0}}, {{val|0 -1 12}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.9908{{c}}, ~3/2 = 676.4865{{c}}
: [[error map]]: {{val| +1.991 -23.478 +60.575 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 676.7603{{c}}
: error map: {{val| 0.000 -25.195 +55.697 }}


[[POTE generator]]: ~4/3 = 521.397
{{Optimal ET sequence|legend=1| 2, 5l, 7l, 9, 16l }}


{{Optimal ET sequence|legend=1| 5l, 7l, 9, 16l }}
[[Badness]] (Sintel): 0.623


=== Reversed meantone ===
=== Reversed meantone ===
{{Main| Reversed meantone }}
{{Main| Reversed meantone }}


Subgroup: 2.3.41
[[Subgroup]]: 2.3.41


[[Comma list]]: 82/81
[[Comma list]]: 82/81


[[Gencom]]: [2 4/3; 82/81]
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3


[[Mapping|Sval mapping]]: [{{val|1 2 7}}, {{val|0 -1 -4}}]
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.6907{{c}}, ~3/2 = 705.3096{{c}}
[[Tp tuning|POL2 generator]]: ~4/3 = 494.509
: [[error map]]: {{val| -0.309 +3.045 -8.752 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2699{{c}}
{{Optimal ET sequence|legend=1| 5, 12, 17 }}
: error map: {{val| 0.000 +3.315 -7.983 }}
 
==== 2.3.7.41 subgroup ====
Subgroup: 2.3.7.41
 
[[Comma list]]: 64/63, 82/81
 
[[Gencom]]: [2 4/3; 64/63 82/81]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]
 
[[POTE generator]]: ~4/3 = 490.0323
 
[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}
 
==== 2.3.7.11.41 subgroup ====
Subgroup: 2.3.7.11.41
 
[[Comma list]]: 64/63, 82/81, 99/98
 
[[Gencom]]: [2 4/3; 64/63 82/81 99/98]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]
 
[[POTE generator]]: ~4/3 = 492.1787


[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
{{Optimal ET sequence|legend=1| 5, 12, 17, 97m, 114m, 131m }}


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
[[Badness]] (Sintel): 0.0841


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]
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