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No-twos subgroup temperaments: Difference between revisions

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Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].


* Arcturus, Aldebaran and Polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.
* Arcturus, antares, aldebaran, delta centauri and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals.
* BPS has a ~9/7 generator, two of which give the ~5/3.
* BPS has a ~9/7 generator, two of which give the ~5/3.
* Sirius has a ~25/21 generator, three of which give the ~5/3.
* Sirius has a ~25/21 generator, three of which give the ~5/3.
* Deneb has a ~11/9 generator, three of which give the ~9/5.
* Deneb has a ~11/9 generator, three of which give the ~9/5.
* Canopus has a ~7/5 generator, five of which give the ~27/5 (9/5 up a tritave).
* Canopus has a ~15/7 generator, five of which give the ~45/1 (5/1 up two tritaves).
* Alnilam has a ~81/55 generator, ten of which give the ~243/5 (9/5 up three tritaves).
* Miaplacidus has a ~7/5 generator, eight of which give the ~15/1 (5/1 up one tritave).  
* Izar has a ~16807/10125 generator, twelve of which give the ~2187/5 (9/5 up five tritaves).
* Alnilam has a ~55/27 generator, ten of which give the ~1215/1 (5/1 up five tritaves).
* Nekkar has a ~16807/10935 generator, sixteen of which give the ~6561/5 (9/5 up six tritaves).
* Izar has a ~30375/16807 generator, twelve of which give the ~1215/1 (5/1 up five tritaves).
* Nekkar has a ~32805/16807 generator, sixteen of which give the ~32805/1 (5/1 up eight tritaves).
* Mintaka does not include the 5th harmonic, and has an ~11/7 generator, two of which give the ~27/11, and three of which give the ~27/7 (9/7 and a tritave).
* Mintaka does not include the 5th harmonic, and has an ~11/7 generator, two of which give the ~27/11, and three of which give the ~27/7 (9/7 and a tritave).
* Antipyth uses 5/1 as a period, and has a ~7/5 generator. There is one-to-one correspondence between the 5.7 subgroup and mapped intervals.
* Antipyth uses 5/1 as a period, and has a ~7/5 generator. There is one-to-one correspondence between the 5.7 subgroup and mapped intervals.
* Juggernaut uses half-pentave(~11/5) as a period, and has a ~7/5 generator.
* Juggernaut uses half-pentave (~11/5) as a period, and has a ~7/5 generator.


= 3.5.7-subgroup temperaments =
= 3.5.7-subgroup temperaments =
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{{Main| Arcturus }}
{{Main| Arcturus }}


For extensions of this temperament that include the prime 2, see [[trienstonic clan #Opossum|opossum]], [[jubilismic clan #Crepuscular|crepuscular]], [[kleismic family #Catalan|catalan]], [[tetracot family #Bunya|bunya]], [[sensamagic clan #Bohpier|bohpier]], and [[gamelismic clan #Superkleismic|superkleismic]].
[[Strong extension]]s of this temperament that include the octave are [[trienstonic clan #Opossum|opossum]] and [[jubilismic clan #Crepuscular|crepuscular]]. [[Weak extension]]s are [[kleismic family #Catalan|catalan]], [[tetracot family #Bunya|bunya]], [[gamelismic clan #Superkleismic|superkleismic]], and [[sensamagic clan #Bohpier|bohpier]]. Non-octave extensions are documented below.


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7
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[[Comma list]]: 15625/15309
[[Comma list]]: 15625/15309


{{Mapping|legend=2| 1 0 -7 | 0 1 6 }}
{{Mapping|legend=3| 1 0 -7 | 0 1 6 }}
: mapping generators: ~3, ~5
: mapping generators: ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1903.863{{c}}, ~5/3 = 878.923{{c}}
* [[WE]]: ~3 = 1903.8634{{c}}, ~5/3 = 878.9230{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 878.291{{c}}
: [[error map]]: {{val| +1.908 -3.527 +0.849 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 878.2907{{c}}
: error map: {{val| 0.000 -6.068 -1.037 }}


[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]
[[Optimal ET sequence]]: [[2edt|b2]], [[9edt|b9d]], [[11edt|b11]], [[13edt|b13]], [[340edt|b340cc]]


[[Badness]] (Sintel): 0.535
[[Badness]] (Sintel): 0.535


=== Polturus ===
=== Polturus ===
This extension of arcturus adds [[polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
This extension of arcturus adds [[#Polaris|polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.


Subgroup: 3.5.7.11
Subgroup: 3.5.7.11
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Comma list: 15625/15309, 177147/171875
Comma list: 15625/15309, 177147/171875


Subgroup-val mapping: 1 1 -1 5 | 0 1 6 -6 }}
{{Mapping|legend=2| 1 1 -1 5 | 0 1 6 -6 }}
: gencom: [3/1 5/3; 15625/15309 177147/171875]


Optimal tuning (POTE): ~5/3 = 884.268{{c}}
Optimal tunings:  
* WE: ~3 = 1890.5039{{c}}, ~5/3 = 879.7029{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~5/3 = 884.8724{{c}}


Optimal ET sequence: [[13edt|b13e]], [[15edt|b15]], [[28edt|b28e]], [[43edt|b43dee]]
Optimal ET sequence: [[13edt|b13e]], [[15edt|b15]], [[28edt|b28e]], [[43edt|b43dee]]


Badness (Sintel): 2.507
Badness (Sintel): 2.51
 
== Antares ==
Antares tempers out [[21875/19683]], the amount by which, assuming [[tritave equivalence]], five [[5/3]]'s exceed [[9/7]], or equivalently the amount by which five [[9/5]]'s fall short of [[7/3]]. It is basically to [[#Arcturus|arcturus]] what [[mavila family #Mavila|mavila]] is to [[meantone family #Meantone|meantone]], and is even more inaccurate than mavila. It falls into [[exotemperament]] range with the "comma" it tempers out (~182.8{{cent}}) being greater than a [[Bohlen–Pierce]] step. In terms of diatonic interval regions, the ~[[9/5]] is extremely sharp and becomes a [[neutral seventh]], and the ~[[7/5]] becomes a [[supermajor third]].
 
Antares was named by {{u|CompactStar}} after the star {{w|Antares}}, following a series of non-octave temperaments that were named after stars.
 
[[Subgroup]]: 3.5.7
 
[[Comma list]]: 21875/19683
 
{{Mapping|legend=3| 1 0 9 | 0 1 -5 }}
: mapping generators: ~3, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1913.7665{{c}}, ~5/3 = 858.1255{{c}}
: [[error map]]: {{val| +11.811 -14.422 -4.388 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 851.8677{{c}}
: error map: {{val| 0.000 -32.391 -20.844 }}
 
[[Optimal ET sequence]]: [[9edt|b9]], [[20edt|b20]], [[29edt|b29]]
 
[[Badness]] (Sintel): 2.95


== BPS ==
== BPS ==
{{Main| BPS }}
{{Main| BPS }}


For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions will be documented below.
For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions are documented below.  


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7
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[[Comma list]]: 245/243
[[Comma list]]: 245/243


{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}
{{Mapping|legend=3| 1 1 2 | 0 2 -1 }}
: mapping generators: ~3, ~9/7
: mapping generators: ~3, ~9/7


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=== Alhena ===
=== Alhena ===
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11/2). }}
Alhena is a strong extension of BPS to the 3.5.7.11/2.13/4 subgroup that equates the "semitone" of [[27/25]]~[[49/45]] to [[13/12]], and then three of these intervals to [[14/11]].
 
Subgroup: 3.5.7.11/2
 
Comma list: 245/243 ({{monzo| -5 1 2 0 }}), 3773/3750 ({{monzo| -1 -4 3 1 }})
 
{{Mapping|legend=2| 1 1 2 -1 | 0 2 -1 11 }}
 
Optimal tunings:
* Subgroup WE: ~3 = 1903.6644{{c}}, ~9/7 = 441.3233{{c}}
* Subgroup CWE: ~3 = 1901.9550{{c}}, ~9/7 = 440.9989{{c}}


This is a strong extension to BPS in the subgroup 3.5.7.11/2.13/4 that equates the "semitone" of [[27/25]]~[[49/45]] to [[13/12]], and then three of these intervals to [[14/11]].
Optimal ET sequence: [[13edt|b13]], [[43edt|b43]], [[56edt|b56]], [[69edt|b69]], [[151edt|b151]], [[220edt|b220d]], [[289edt|b289dé]] *


: <nowiki/>* é is used as the wart for 11/2.
==== 3.5.7.11/2.13/4 subgroup ====
Subgroup: 3.5.7.11/2.13/4
Subgroup: 3.5.7.11/2.13/4


Comma list: 196/195, 325/324, 1001/1000
Comma list: 196/195 ({{monzo| -1 -1 2 0 -1 }}), 325/324 ({{monzo| -4 2 0 0 1 }}), 1001/1000 ({{monzo| 0 -3 1 1 1 }})


Subgroup-val mapping: {{mapping| 1 1 2 -1 2 | 0 2 -1 11 -4 }}
{{Mapping|legend=2| 1 1 2 -1 2 | 0 2 -1 11 -4 }}
: mapping generators: ~3, ~7/3


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~9/7 = 441.025{{c}}
Optimal tunings:
* Subgroup WE: ~3 = 1903.5584{{c}}, ~9/7 = 441.3534{{c}}
* Subgroup CWE: ~3 = 1901.9550{{c}}, ~9/7 = 441.0249{{c}}


Supporting ETs: [[13edt|b13]], [[69edt|b69]], [[56edt|b56]], [[82edt|b82]], [[43edt|b43]], [[125edt|b125]], [[30edt|b30]], [[151edt|b151]], [[95edt|b95]], [[17edt|b17é]], [[194edt|b194d]], [[99edt|b99]], [[181edt|b181d]], [[108edt|b108é]] *
Optimal ET sequence: [[13edt|b13]], [[43edt|b43]], [[56edt|b56]], [[69edt|b69]], [[289edt|b289d]], [[358edt|b358ddé]], [[427edt|b427cddé]] *


<nowiki/>* é is used as the wart for 11/2.
: <nowiki/>* é is used as the wart for 11/2.


Badness (Sintel): 0.187
Badness (Sintel): 0.187


=== Mintra ===
=== Mintra ===
: ''See also: [[No-twos subgroup temperaments #Mintaka|Mintaka]] and [[No-twos subgroup temperaments #Deneb|Deneb]].''
This temperament splits 27/7 (the BPS generator up a tritave) into three by means of [[11/7]] or, equivalently, [[7/1]] in three by means of [[21/11]], and is the intersection of BPS, [[#Deneb|deneb]], and [[#Mintaka|mintaka]] temperaments as well as the most natural temperament satisfied in the 3.5.7.11 subgroup in [[39edt]].


This temperament splits 27/7 (the BPS generator up a tritave) into three by means of [[11/7]] or, equivalently, [[7/1]] in three by means of [[21/11]], and is the intersection of BPS, Deneb, and Mintaka temperaments as well as the most natural temperament satisfied in the 3.5.7.11 subgroup in [[39edt]].
The 13-limit extension uses the canonical extension for prime 13 described at [[#Tridecimal mintaka]].


Subgroup: 3.5.7.11
Subgroup: 3.5.7.11
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Comma list: 245/243, 1331/1323
Comma list: 245/243, 1331/1323


Subgroup-val mapping: {{mapping| 1 5 0 1 | 0 -6 3 2 }}
{{Mapping|legend=2| 1 -1 3 3 | 0 6 -3 -2 }}
mapping generators: ~3, ~21/11
: mapping generators: ~3, ~11/7


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 780.752{{c}}
Optimal tunings:
* WE: ~3 = 1904.1393{{c}}, ~11/7 = 781.6168{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/7 = 780.7525{{c}}


Supporting ETs: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90
Optimal ET sequence: [[17edt|b17]], [[39edt|b39]], [[95edt|b95]], [[134edt|b134]], [[229edt|b229de]]


Badness (Sintel): 0.302
Badness (Sintel): 0.302


==== Tridecimal mintra ====
==== Tridecimal mintra ====
This temperament uses the canonical extension for prime 13 described at [[#Tridecimal mintaka]].
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


Comma list: 245/243, 275/273, 1575/1573
Comma list: 245/243, 275/273, 1331/1323


Subgroup-val mapping: {{mapping| 1 5 0 1 10 | 0 -6 3 2 -13 }}
{{Mapping|legend=2| 1 5 0 1 10 | 0 -6 3 2 -13 }}
: mapping generators: ~3, ~21/11


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 780.428{{c}}
Optimal tunings:
* WE: ~3 = 1903.9326{{c}}, ~11/7 = 781.1856{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/7 = 780.4280{{c}}


Supporting ETs: 39, 17, 22, 56, 5f, 61, 95, 100, 134, 73f, 139cf, 83cf, 173e, 178cef
Optimal ET sequence: [[17edt|b17]], [[22edt|b22]], [[39edt|b39]], [[290edt|b229cde]]


Badness (Sintel): 0.373
Badness (Sintel): 0.373
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{{Main| Dubhe }}
{{Main| Dubhe }}


This temperament is a simple 3.5.7.17 weak extension of BPS that splits the generator of 9/7 into two intervals of [[17/15]]. The name was suggested by [[MidnightBlue]] after dubhe, a bright double star (the ninth brightest) and similarities to the word ''double''.
This temperament is a simple 3.5.7.17 weak extension of BPS that splits the generator of 9/7 into two intervals of [[17/15]], tempering out [[2025/2023]]. The name was suggested by [[MidnightBlue]] after Dubhe, a bright double star (the ninth brightest) and similarities to the word ''double''.


Subgroup: 3.5.7.17
Subgroup: 3.5.7.17


Comma list: 245/243, 2025/2023
Comma list: 245/243, 875/867


Subgroup-val mapping: {{mapping| 1 1 2 2 | 0 4 -2 5 }}
{{Mapping|legend=2| 1 1 2 2 | 0 4 -2 5 }}
: mapping generators: ~3, ~17/15


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~17/15 = 220.142{{c}}
Optimal tunings:
* WE: ~3 = 1903.4591{{c}}, ~17/15 = 220.2094{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~17/15 = 220.1420{{c}}


Supporting ETs: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44
Optimal ET sequence: [[8edt|b8]], [[9edt|b9]], [[17edt|b17]], [[26edt|b26]], [[69edt|b69]], [[95edt|b95]], [[121edt|b121]]


Badness (Sintel): 0.177
Badness (Sintel): 0.177


== Canopus ==
== Vega ==
{{Main| Canopus }}
[[Subgroup]]: 3.5.7
 
[[Comma list]]: 6561/6125
 
{{Mapping|legend=3| 1 0 4 | 0 2 -3 }}
: mapping generators: ~3, ~81/35
 
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1891.9431{{c}}, ~81/35 = 1397.0975{{c}}
: [[error map]]: {{val| -10.012 +7.881 +7.654 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~81/35 = 1404.6664{{c}}
: error map: {{val| 0.000 +23.019 +24.995 }}
 
[[Optimal ET sequence]]: [[4edt|b4]], [[11edt|b11d]], [[15edt|b15]], [[19edt|b19]], [[23edt|b23]], [[42edt|b42d]], [[65edt|b65cd]]
 
[[Badness]] (Sintel): 1.34
 
== Sirius ==
{{Main| Sirius }}
 
This tempers out the [[Don Page comma]] between [[5/3]] and [[7/5]], allowing an accurate representation of the 3:5:7 chord, similar to the 5:7:9 in [[#Procyon|procyon]].


For extensions to this temperament that include the prime 2, see [[Canopic clan]]. No-2's extensions will be documented below.
For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7


[[Comma list]]: 16875/16807
[[Comma list]]: 3125/3087


{{Mapping|legend=2| 1 -2 -1 | 0 5 4 }}
{{Mapping|legend=3| 1 1 1 | 0 3 5 }}
: mapping generators: ~3, ~15/7
: mapping generators: ~3, ~25/21


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
* [[WE]]: ~3 = 1902.4455{{c}}, ~25/21 = 293.7393{{c}}
: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
: [[error map]]: {{val| +0.490 -2.650 +2.316 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~25/21 = 293.7594{{c}}
: error map: {{val| 0.000 -0.381 +1.093 }}
: error map: {{val| 0.000 -3.080 +1.926 }}
 
[[Optimal ET sequence]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]],  [[123edt|b123]], [[136edt|b136]]
 
[[Badness]] (Sintel): 0.213
 
=== Mizar ===
Mizar exploits the sirius tuning of the 25/21 generator being close to [[13/11]] (in order to split 7/5 evenly); additionally this tempers out [[459/455]], equating [[17/13]] to [[35/27]].
 
The mapping for prime 17 is similar to what [[dubhe]] does: tempering out [[2025/2023]] to split the 7-limit generator in half; in this case, 25/7 is split into two intervals of [[17/9]], which turns out to occupy the position of a [[macrodiatonic]] fifth, specifically a macro-[[flattone]] fifth.
 
Subgroup: 3.5.7.11
 
Comma list: 3125/3087, 295245/290521


[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
{{Mapping|legend=2| 1 -2 -4 12 | 0 6 10 -17 }}
: mapping generators: ~3, ~729/385


[[Badness]] (Sintel): 0.0996
Optimal tunings:  
* WE: ~3 = 1901.0672{{c}}, ~729/385 = 1097.6627{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.1722{{c}}


=== Suhail ===
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]], [[168edt|b168]]
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}


Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].
Badness (Sintel): 1.82


==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


Comma list: 1575/1573, 1625/1617, 4459/4455
Comma list: 275/273, 847/845, 59049/57967
 
{{Mapping|legend=2| 1 -2 -4 12 11 | 0 6 10 -17 -15 }}
 
Optimal tunings:
* WE: ~3 = 1900.6176{{c}}, ~729/385 = 1097.5046{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.2837{{c}}
 
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]]
 
Badness (Sintel): 1.12
 
==== 3.5.7.11.13.17 subgroup ====
Subgroup: 3.5.7.11.13.17
 
Comma list: 275/273, 459/455, 847/845, 2025/2023
 
{{Mapping|legend=2| 1 -2 -4 12 11 2 | 0 6 10 -17 -15 1 }}
 
Optimal tunings:
* WE: ~3 = 1901.0269{{c}}, ~17/9 = 1097.7583{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~17/9 = 1098.2979{{c}}
 
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]]
 
Badness (Sintel): 0.841
 
=== Remus ===
By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with [[mos scale]]s of 18, 25, 32, or 39 steps.
 
This is essentially [[#Electra|electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.
 
Subgroup: 3.5.7.11
 
Comma list: 1331/1323, 3125/3087


Subgroup-val mapping: {{mapping| 3 4 5 6 7 | 0 5 4 7 0 }}
{{Mapping|legend=2| 1 -5 -9 -5 | 0 9 15 10 }}
: mapping generators: ~13/9, ~65/63
: mapping generators: ~3, ~11/5


Optimal tunings:  
Optimal tunings:  
* WE: ~13/9 = 634.144{{c}}, ~65/63 = 49.695{{c}}
* WE: ~3 = 1902.6279{{c}}, ~11/5 = 1366.3374{{c}}
* CWE: ~13/9 = 633.985{{c}}, ~65/63 = 49.733{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8996{{c}}


Optimal ET sequence: [[39edt|b39]], [[114edt|b114]], [[153edt|b153]], [[498edt|b498cf]], [[651edt|b651cf]]
Optimal ET sequence: [[7edt|b7]], [[25edt|b25d]], [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[149edt|b149]]


Badness (Sintel): 0.330
Badness (Sintel): 0.538
 
==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13


== Izar ==
Comma list: 275/273, 847/845, 1331/1323
[[Subgroup]]: 3.5.7


[[Comma list]]: 13841287201/13839609375
{{Mapping|legend=2| 1 -5 -9 -5 -7 | 0 9 15 10 13 }}


{{Mapping|legend=2| 1 7 5 | 0 -12 -7 }}
Optimal tunings:
: mapping generators: ~3, ~16807/10125
* WE: ~3 = 1902.4456{{c}}, ~11/5 = 1366.1830{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8649{{c}}


[[Optimal tuning]]s:
Optimal ET sequence: [[7edt|b7]], [[25edt|b25df]], [[32edt|b32]], [[39edt|b39]], [[110edt|b110]], [[149edt|b149]]
* [[WE]]: ~3 = 1901.958{{c}}, ~16807/10125 = 877.283{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~16807/10125 = 877.281{{c}}


[[Optimal ET sequence]]: [[13edt|b13]], [[141edt|b141]], [[154edt|b154]], …, [[258edt|b258]], [[271edt|b271]], [[800edt|b800]], [[1071edt|b1071]], [[1342edt|b1342]], [[1613edt|b1613]], [[4568edt|b4568]], [[6181edt|b6181]]
Badness (Sintel): 0.286


[[Badness]] (Sintel): 0.017
== Canopus ==
{{Main| Canopus }}


== Nekkar ==
For extensions to this temperament that include the prime 2, see [[Canopic clan]]. No-2's extensions will be documented below.
This temperament is the no-2's [[restriction]] of [[squares]], and as such is named after a star that belonged to the obsolete constellation of Quadrans Muralis, whose name has to do with squares. However, seeing the sheer complexity and size of the commas, nekkar is much more naturally thought of as 3.5.7.11 than 3.5.7, whereupon it becomes a strong extension of [[mintaka]].


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7


[[Comma list]]: 35303692060125/33232930569601
[[Comma list]]: 16875/16807


{{Mapping|legend=2|1 8 3|0 -16 -3}}
{{Mapping|legend=3| 1 -2 -1 | 0 5 4 }}
: mapping generators: ~3, ~16807/10935
: mapping generators: ~3, ~15/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1900.155{{c}}, ~16807/10935 = 775.963{{c}}
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~16807/10935 = 776.767{{c}}
: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}
: error map: {{val| 0.000 -0.381 +1.093 }}


[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]]
[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]


[[Badness]] (Sintel): 17.120
[[Badness]] (Sintel): 0.0996


=== 3.5.7.11 subgroup ===
=== Suhail ===
{{See also| Mintaka }}
Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].
 
This continues the canonical 11-limit extension of squares.


Subgroup: 3.5.7.11
Subgroup: 3.5.7.11


Comma list: 1331/1323, 120285/117649
Comma list: 9375/9317, 12005/11979


Subgroup-val mapping: {{mapping| 1 8 3 3 | 0 -16 -3 -2 }}
{{Mapping|legend=2| 3 4 5 6 | 0 5 4 7 }}
: mapping generators: ~3, ~11/7
: mapping generators: ~539/375, ~77/75


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 776.781{{c}}
Optimal tunings:
 
* WE: ~539/375 = 634.9453{{c}}, ~77/75 = 49.8753{{c}}
Supporting ETs: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd
* CWE: ~539/375 = 633.9850{{c}}, ~77/75 = 49.8596{{c}}


Badness (Sintel): 1.375
Optimal ET sequence: [[36edt|b36]], [[39edt|b39]], [[75edt|b75]], [[114edt|b114]], [[153edt|b153]], [[267edt|b267]]


=== 3.5.7.11.13 subgroup ===
Badness (Sintel): 0.421
This uses the [[no-twos subgroup temperaments #Minalzidar|minalzidar]] mapping of 13.


==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


Comma list: 169/165, 351/343, 11011/10935
Comma list: 1575/1573, 1625/1617, 4459/4455


Subgroup-val mapping: {{mapping| 1 8 3 3 6 | 0 -16 -3 -2 -9 }}
{{Mapping|legend=2| 3 4 5 6 7 | 0 5 4 7 0 }}
: mapping generators: ~3, ~11/7


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 776.678{{c}}
Optimal tunings:
* WE: ~13/9 = 634.1444{{c}}, ~65/63 = 49.6946{{c}}
* CWE: ~13/9 = 633.9850{{c}}, ~65/63 = 49.7330{{c}}


Supporting ETs: 22, 5c, 27, 49, 71f, 17cf
Optimal ET sequence: [[36edt|b36]], [[39edt|b39]], [[114edt|b114]], [[153edt|b153]], [[498edt|b498cf]], [[651edt|b651cf]]


Badness (Sintel): 1.723
Badness (Sintel): 0.330


== Procyon ==
== Procyon ==
This tempers out the [[Don Page comma]] between [[7/5]] and [[9/7]], allowing an accurate representation of the 5:7:9 chord, similar to the 3:5:7 in Sirius.
This tempers out the [[Don Page comma]] between [[7/5]] and [[9/7]], allowing an accurate representation of the 5:7:9 chord, similar to the 3:5:7 in [[#Sirius|sirius]].


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7
Line 266: Line 382:
[[Comma list]]: 823543/820125
[[Comma list]]: 823543/820125


{{Mapping|legend=2| 1 2 2 | 0 -7 -3 }}
{{Mapping|legend=3| 1 2 2 | 0 -7 -3 }}
: mapping generators: ~3, ~17/9
: mapping generators: ~3, ~49/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1902.198{{c}}, ~49/45 = 145.412{{c}}
* [[WE]]: ~3 = 1902.1979{{c}}, ~49/45 = 145.4124{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~49/45 = 145.368{{c}}
: [[error map]]: {{val| +0.243 +0.195 -0.667 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~49/45 = 145.3680{{c}}
: error map: {{val| 0.000 +0.020 -1.020 }}


[[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79
[[Optimal ET sequence]]: [[13edt|b13]], [[92edt|b92]], [[105edt|b105]], [[118edt|b118]], [[131edt|b131]], [[144edt|b144]], [[157edt|b157]], [[327edt|b327]], [[484edt|b484]], [[641edt|b641]], [[1125edt|b1125d]]


[[Badness]] (Sintel): 0.200
[[Badness]] (Sintel): 0.200


=== Erigone ===
=== Erigone ===
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}
Erigone splits the 9/1-complement generator of procyon into three, allowing for an accurate representation of 11/9 at 19 generators and 13/9 at 13 generators.
 
Subgroup: 3.5.7.11
 
Comma list: 12005/11979, 456533/455625
 
{{Mapping|legend=2| 1 -12 -4 -10 | 0 21 9 19 }}
: mapping generators: ~3, ~99/49
 
Optimal tunings:
* WE: ~3 = 1902.1806{{c}}, ~99/49 = 1219.6482{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~99/49 = 1219.5121{{c}}


Erigone splits the (tritave-augmented) generator of [[no-twos subgroup temperaments #Procyon|procyon]] into three, allowing for an accurate representation of 11/9 at -19 generators and 13/9 at -13 generators.
Optimal ET sequence: [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[471edt|b471]], [[641edt|b641]], [[811edt|b811]], [[1452edt|b1452d]]


Badness (Sintel): 0.312
==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


Comma list: 847/845, 1575/1573, 4459/4455
Comma list: 847/845, 1575/1573, 4459/4455


Subgroup-val mapping: {{mapping| 1 9 5 9 7 | 0 -21 -9 -19 -13 }}
{{Mapping|legend=2| 1 -12 -4 -10 -6 | 0 21 9 19 13 }}
: mapping generators: ~3, ~49/33


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1901.9699{{c}}, ~49/33 = 682.4486{{c}}
* WE: ~3 = 1901.9695{{c}}, ~99/49 = 1219.5210{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~49/33 = 682.4427{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~99/49 = 1219.5123{{c}}
 
Optimal ET sequence: [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]]
 
Badness (Sintel): 0.214


Optimal ET sequence: [[25edt|b25ce]], [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]]
===== Hemigone =====
By tempering out [[3971/3969]], erigone's generator ([[~]][[99/49]]) is split into two [[27/19]]'s. Meanwhile, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]). [[2277/2275]] may be used in the same way to extend erigone for prime 23.


Badness (Sintel): 0.21396
Subgroup: 3.5.7.11.13.17


==== Hemigone ====
Comma list: 847/845, 1575/1573, 3185/3179, 4459/4455
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11.13.17). }}


By tempering out [[3971/3969]], erigone's tritave-augmented generator ([[49/11]]) is split into two [[19/9]]s. Then, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]).
{{Mapping|legend=2| 1 -12 -4 -10 -6 -8 | 0 42 18 38 26 33 }}
: mapping generators: ~3, ~121/85


Optimal tunings:
* WE: ~3 = 1901.8706{{c}}, ~121/85 = 609.7295{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~121/85 = 609.7549{{c}}
Optimal ET sequence: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]
Badness (Sintel): 0.453
====== 3.5.7.11.13.17.19 subgroup ======
Subgroup: 3.5.7.11.13.17.19
Subgroup: 3.5.7.11.13.17.19


Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 4459/4455
Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 4459/4455


Subgroup-val mapping: {{mapping| 1 30 14 28 20 25 2 | 0 -42 -18 -38 -26 -33 1 }}
{{Mapping|legend=2| 1 -12 -4 -10 -6 -8 3 | 0 42 18 38 26 33 -1 }}
: mapping generators: ~3, ~19/9


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1902.0918{{c}}, ~19/9 = 1292.3032{{c}}
* WE: ~3 = 1902.0918{{c}}, ~27/19 = 609.7886{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~19/9 = 1292.2083{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~27/19 = 609.7467{{c}}


Optimal ET sequence: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]]
Optimal ET sequence: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]]


Badness (Sintel): 0.45479
Badness (Sintel): 0.455
 
==== 3.5.7.11.13.17.19.23 subgroup ====
[[2277/2275]] may be used in the same way to extend the simpler [[#Erigone|erigone]] to the 3.5.7.11.13.23 subgroup.


====== 3.5.7.11.13.17.19.23 subgroup ======
Subgroup: 3.5.7.11.13.17.19.23
Subgroup: 3.5.7.11.13.17.19.23


Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 2277/2275, 4459/4455
Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 2277/2275, 4459/4455


Subgroup-val mapping: {{mapping| 1 30 14 28 20 25 2 64 | 0 -42 -18 -38 -26 -33 1 -90 }}
{{Mapping|legend=2| 1 -12 -4 -10 -6 -8 3 -26 | 0 42 18 38 26 33 -1 90 }}
: mapping generators: ~3, ~19/9


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1902.0149{{c}}, ~19/9 = 1292.2401{{c}}
* WE: ~3 = 1902.0149{{c}}, ~27/19 = 609.7748{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~19/9 = 1292.1988{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~27/19 = 609.7562{{c}}


Optimal ET sequence: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]
Optimal ET sequence: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]


Badness (Sintel): 0.54174
Badness (Sintel): 0.542
 
== Sirius ==
{{Main| Sirius }}
 
This tempers out the [[Don Page comma]] between [[5/3]] and [[7/5]], allowing an accurate representation of the 3:5:7 chord, similar to the 5:7:9 in Procyon.


For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].
== Miaplacidus ==
Miaplacidus tempers out [[5859375/5764801]], the amount by which eight [[7/5]]'s fall short of [[15/1]]. It thus tunes the chord 7:9:15 accurately. Having a generator of ~7/5, it possesses [[mos scale]]s of the families {{mos scalesig|link=1|3L 4s<3/1>}}, {{mos scalesig|link=1|3L 7s<3/1>}}, {{mos scalesig|link=1|3L 10s<3/1>}}, and {{mos scalesig|link=1|13L 3s<3/1>}}. Good tunings for this temperament are [[13edt]], [[16edt]], and [[29edt]].


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7


[[Comma list]]: 3125/3087
[[Comma list]]: 5859375/5764801


{{Mapping|legend=2| 1 1 1 | 0 3 5 }}
{{Mapping|legend=3| 1 -1 -1 | 0 8 9 }}
: mapping generators: ~3, ~25/21
: mapping generators: ~3, ~7/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1902.4455{{c}}, ~25/21 = 293.7393{{c}}
* [[WE]]: ~3 = 1901.8789{{c}}, ~7/5 = 585.8428{{c}}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~25/21 = 293.7594{{c}}
: [[error map]]: {{val| -0.076 -1.450 +1.880 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~7/5 = 585.8590{{c}}
: error map: {{val| 0.000 -1.396 +1.950 }}
 
[[Optimal ET sequence]]: [[13edt|b13]], [[120edt|b120]], [[133edt|b133]], …, [[211edt|b211]], [[224edt|b224]]


[[Optimal ET sequence]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]],  [[123edt|b123]], [[136edt|b136]]
[[Badness]] (Sintel): 1.24


[[Badness]] (Sintel): 0.213
== Izar ==
[[Subgroup]]: 3.5.7


=== Remus ===
[[Comma list]]: 13841287201/13839609375
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}


By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with mos scales of 18, 25, 32, or 39 steps.
{{Mapping|legend=3| 1 -5 -2 | 0 12 7 }}
: mapping generators: ~3, ~30375/16807


This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-sevens [[restriction]] of this temperament.
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.9584{{c}}, ~30375/16807 = 1024.6759{{c}}
: [[error map]]: {{val| +0.003 +0.005 -0.012 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~30375/16807 = 1024.6743{{c}}
: error map: {{val| 0.000 +0.002 -0.016 }}


Subgroup: 3.5.7.11.13
[[Optimal ET sequence]]: [[13edt|b13]], [[141edt|b141]], [[154edt|b154]], …, [[258edt|b258]], [[271edt|b271]], [[800edt|b800]], [[1071edt|b1071]], [[1342edt|b1342]], [[1613edt|b1613]], [[4568edt|b4568]], [[6181edt|b6181]]


Comma list: 275/273, 1625/1617, 1575/1573
[[Badness]] (Sintel): 0.0166


Subgroup-val mapping: {{mapping| 1 4 6 5 6 | 0 -9 -15 -10 -13 }}
== Nekkar ==
: mapping generators: ~3, ~15/11
This temperament is the no-2's [[restriction]] of [[squares]], and as such is named after a star that belonged to the obsolete constellation of Quadrans Muralis, whose name has to do with squares. However, seeing the sheer complexity and size of the commas, nekkar is much more naturally thought of as a temperament of 3.5.7.11 than 3.5.7, whereupon it is a restriction of undecimal squares, and a strong extension of [[#Mintaka|mintaka]]. However, the 13-limit extension restricts to [[#Minalzidar|minalzidar]] rather than tridecimal mintaka.


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~15/11 = 536.090{{c}}
[[Subgroup]]: 3.5.7


Supporting ETs: 39, 7, 32, 71, 110, 46, 149, 188, 181
[[Comma list]]: {{monzo| -24 -3 16 }}


Badness (Sintel): 0.286
{{Mapping|legend=3| 1 -8 0 | 0 16 3 }}
: mapping generators: ~3, ~32805/16807


=== Mizar ===
[[Optimal tuning]]s:
{{Todo|inline=1|complete section|comment= Catalog the intermediate extensions (3.5.7.11, 3.5.7.11.13) instead. }}
* [[WE]]: ~3 = 1900.1550{{c}}, ~32805/16807 = 1124.1916{{c}}
: [[error map]]: {{val| -1.800 -0.488 +3.749 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~32805/16807 = 1125.1876{{c}}
: error map: {{val| 0.000 +1.047 +6.737 }}


This temperament uses a weak extension to the 3.5.7.17 subgroup similar to what [[dubhe]] does: tempering out [[2025/2023]] to split the 7-limit generator in half; in this case, 25/7 is split into two intervals of [[17/9]], which turns out to occupy the position of a [[macrodiatonic]] fifth, specifically a macro-[[flattone]] fifth.
[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[311edt|b311dd]]


Subgroup: 3.5.7.17
[[Badness]] (Sintel): 17.1


Comma list: 2025/2023, 3125/3087
=== 3.5.7.11 subgroup ===
Subgroup: 3.5.7.11


Subgroup-val mapping: {{mapping| 1 -2 -4 2 | 0 6 10 1 }}
Comma list: 1331/1323, 120285/117649
: mapping generators: ~3, ~17/9


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~17/9 = 1097.800{{c}}
{{Mapping|legend=2| 1 -8 0 1 | 0 16 3 2 }}


Supporting ETs: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd
Optimal tunings:  
* WE: ~3 = 1900.6084{{c}}, ~21/11 = 1124.4193{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.1738{{c}}


Badness (Sintel): 0.383
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[262edt|b262d]]


==== 3.5.7.11.13.17 subgroup ====
Badness (Sintel): 1.37
This exploits the sirius tuning of the 25/21 generator being close to [[13/11]] (in order to split 7/5 evenly); additionally this tempers out [[459/455]], equating [[17/13]] to [[35/27]].


Subgroup: 3.5.7.11.13.17
=== 3.5.7.11.13 subgroup ===
Subgroup: 3.5.7.11.13


Comma list: 275/273, 459/455, 1625/1617, 2025/2023
Comma list: 169/165, 351/343, 11011/10935


Subgroup-val mapping: {{mapping| 1 -2 -4 12 11 2 | 0 6 10 -17 -15 1 }}
{{Mapping|legend=2| 1 -8 0 1 -3 | 0 16 3 2 9 }}
: mapping generators: ~3, ~17/9


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~17/9 = 1098.298
Optimal tunings:
* WE: ~3 = 1902.3248{{c}}, ~21/11 = 1125.4837{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.2772{{c}}


Supporting ETs: 26, 71, 45, 19, 97f, 116d
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71f]]


Badness (Sintel): 0.841
Badness (Sintel): 1.72


== Bohlenic ==
== Bohlenic ==
This temperament is identical to [[13edt]] (equal-tempered [[Bohlen–Pierce scale]]), but has an independent generator for 11.
This temperament is identical to [[13edt]] (equal-tempered [[Bohlen–Pierce scale]]), but has an independent generator for 11.


Subgroup: 3.5.7.11
[[Subgroup]]: 3.5.7.11


Comma list: 245/243, 3125/3087
[[Comma list]]: 245/243, 3125/3087


Subgroup-val mapping: ((mapping| 13 19 23 0 | 0 0 0 1 }}
{{Mapping|legend=3| 13 19 23 0 | 0 0 0 1 }}
: mapping generators: ~27/25, ~11
: mapping generators: ~27/25, ~11


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~27/25 = 146.304{{c}}, ~11 = 4151.318{{c}}
* [[WE]]: ~27/25 = 146.4737{{c}}, ~11/9 = 343.9912{{c}}
* CWE: ~27/25 = 146.304{{c}}, ~11 = 4147.705{{c}}
* [[CWE]]: ~27/25 = 146.3042{{c}}, ~11/9 = 343.7950{{c}}


Optimal ET sequence: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]
[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]], [[299edt|b299ccde]], [[338edt|b338ccde]]


Badness (Sintel): 0.499
[[Badness]] (Sintel): 0.499


=== 3.5.7.11.13 subgroup ===
=== 3.5.7.11.13 subgroup ===
Line 434: Line 587:
Comma list: 245/243, 275/273, 847/845
Comma list: 245/243, 275/273, 847/845


Subgroup-val mapping: ((mapping| 13 19 23 0 2 | 0 0 0 1 1 }}
Subgroup-val mapping: {{mapping| 13 19 23 0 2 | 0 0 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 146.304{{c}}, ~11 = 4149.733{{c}}
* WE: ~27/25 = 146.4722{{c}}, ~11/9 = 341.2865{{c}}
* CWE: ~27/25 = 146.304{{c}}, ~11 = 4146.033{{c}}
* CWE: ~27/25 = 146.3042{{c}}, ~11/9 = 342.1231{{c}}


[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]
Optimal ET sequence: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]


[[Badness]] (Sintel): 0.365
Badness (Sintel): 0.365


== Tuning diagrams ==
== Tuning diagrams ==
Line 499: Line 652:
= 3.5.11-subgroup temperaments =
= 3.5.11-subgroup temperaments =
== Polaris ==
== Polaris ==
{{Main| Polaris }}
Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]].  It contains [[mos scale]]s of the families {{mos scalesig|link=1|2L 5s<3/1>}}, {{mos scalesig|link=1|2L 7s<3/1>}}, and {{mos scalesig|link=1|2L 9s<3/1>}}. The hendecatonic scale is generally the most usable of these, with the heptatonic and enneatonic scales usually being lopsided. Good tunings for polaris include [[15edt]], [[17edt]] and [[32edt]].


Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]].  
[[Subgroup]]: 3.5.11
 
[[Comma list]]: 177147/171875
 
{{Mapping|legend=3| 1 0 11 | 0 1 -6 }}
: mapping generators: ~3, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1899.1140{{c}}, ~5/3 = 890.5067{{c}}
: [[error map]]: {{val| -2.841 +3.307 +1.212 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 892.8502{{c}}
: error map: {{val| 0.000 +7.691 +6.156 }}
 
[[Optimal ET sequence]]: [[15edt|b15]], [[17edt|b17]], [[32edt|b32]], [[113edt|b113]], [[145edt|b145ce]], [[177edt|b177ce]], [[209edt|b209ce]]
 
[[Badness]] (Sintel): 1.01
 
; Scales
* [[Polaris9]]
* [[Polaris11]]
 
== Delta centauri ==
{{Main| Delta centauri }}
 
Delta centauri is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The slightly less accurate extension to the 3.5.11.17 subgroup equates twelve generators with [[17/9]].


[[Subgroup]]: 3.5.11
[[Subgroup]]: 3.5.11


[[Comma list]]: 177147/171875
[[Comma list]]: 1953125/1948617
 
{{Mapping|legend=3| 1 0 -11 | 0 1 9 }}
: mapping generators: ~3, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.1019{{c}}, ~5/3 = 883.9539{{c}}
: [[error map]]: {{val| +0.147 -0.258 +0.064 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 883.9008{{c}}
: error map: {{val| 0.000 -0.458 -0.121 }}
 
[[Optimal ET sequence]]: [[13edt|b13]], [[15edt|b15]], [[28edt|b28]], [[43edt|b43]], [[71edt|b71]], [[170edt|b170]], [[241edt|b241]], [[411edt|b411]], [[652edt|b652]], [[893edt|b893]], [[1545edt|b1545]], [[3742edt|b3742ce]]
 
[[Badness]] (Sintel): 0.114
 
=== 3.5.11.17 subgroup ===
[[Subgroup]]: 3.5.11.17


{{Mapping|legend=2| 1 2 1 | 0 1 -6 }}
[[Comma list]]: 1377/1375, 265625/264627
: [[gencom]]: [3/1 5/3; 177147/171875]


[[Optimal tuning]] ([[POTE]])): ~5/3 = 892.6{{c}}
{{Mapping|legend=3| 1 0 -11 -15 | 0 1 9 12 }}


[[Support]]ing [[EDT]]s: 17, 15, 32, 49, 13[+11], 47, 19, 11[+11], 81, 66, 79[+11], 62[+11], 28[+11], 21[-11]
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.9930{{c}}, ~5/3 = 884.1054{{c}}
: [[error map]]: {{val| +0.038 -0.215 +1.645 -1.669 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 884.0911{{c}}
: error map: {{val| 0.000 -0.268 +1.592 -1.727 }}
 
[[Optimal ET sequence]]: [[13edt|b13g]], [[15edt|b15]], [[28edt|b28]], [[43edt|b43]], [[71edt|b71]], [[185edt|b185]], [[256edt|b256]], [[327edt|b327]]
 
[[Badness]] (Sintel): 0.146


== Deneb ==
== Deneb ==
{{Main| Deneb }}
{{Main| Deneb }}
Deneb tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]].


[[Subgroup]]: 3.5.11
[[Subgroup]]: 3.5.11
Line 521: Line 722:
[[Comma list]]: 6655/6561
[[Comma list]]: 6655/6561


{{Mapping|legend=2| 1 2 2 | 0 -3 1 }}
{{Mapping|legend=3| 1 2 2 | 0 -3 1 }}
: [[gencom]]: [3/1 11/9; 6655/6561]
: mapping generators: ~3, ~11/9
 
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1903.7592{{c}}, ~11/9 = 340.5646{{c}}
: [[error map]]: {{val| +1.804 -0.489 -3.235 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~11/9 = 340.0519{{c}}
: error map: {{val| 0.000 -2.559 -7.356 }}


[[Optimal tuning]] ([[POTE]])): ~11/9 = 340.242{{c}}
[[Optimal ET sequence]]: [[5edt|b5]], [[6edt|b6]], [[11edt|b11]], [[17edt|b17]], [[28edt|b28]], [[67edt|b67]], [[95edt|b95]], [[123edt|b123]], [[218edt|b218e]], [[341edt|b341cee]]


[[Support]]ing [[EDT]]s: 28, 11, 17, 6, 39, 5, 67, 45, 50, 16, 23, 73, 61, 62
[[Badness]] (Sintel): 0.255


=== Fomalhaut ===
=== Fomalhaut ===
Line 533: Line 740:
Fomalhaut was considered in the 23-limit from the start, as an attempt to approximate the no-2's, no-7's 23-limit as accurately as possible using 25 to 35 notes per equave, defined as the b28 & b33 temperament in this limit.  
Fomalhaut was considered in the 23-limit from the start, as an attempt to approximate the no-2's, no-7's 23-limit as accurately as possible using 25 to 35 notes per equave, defined as the b28 & b33 temperament in this limit.  


Fomalhaut follows the convention of naming no-twos temperaments after stars.
Fomalhaut follows the convention of naming no-2's temperaments after stars.


Subgroup: 3.5.11.13
Subgroup: 3.5.11.13
Line 539: Line 746:
Comma list: 6655/6561, 274625/264627
Comma list: 6655/6561, 274625/264627


Subgroup-val mapping: {{mapping| 1 5 1 -2 | 0 -9 3 11 }}
{{Mapping|legend=2| 1 -4 4 9 | 0 9 -3 -11 }}
: gencom: [3/1 99/65; 6655/6561 274625/264627]
: mapping generators: ~3, ~65/33


Optimal tuning (POTE): ~99/65 = 748.0156{{c}}
Optimal tunings:  
* WE: ~3 = 1905.7441{{c}}, ~65/33 = 1156.2566{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~65/33 = 1153.9402{{c}}


Supporting EDTs: {{EDs| b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ff |equave=t}}
Optimal ET sequence [[5edt|b5]], [[23edt|b23f]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89f]]
 
Badness (Sintel): 2.69


==== 3.5.11.13.17 subgroup ====
==== 3.5.11.13.17 subgroup ====
Line 551: Line 762:
Comma list: 1105/1089, 4225/4131, 6655/6561
Comma list: 1105/1089, 4225/4131, 6655/6561


Subgroup-val mapping: {{mapping| 1 5 1 -2 1 | 0 -9 3 11 4 }}
{{Mapping|legend=2| 1 -4 4 9 5 | 0 9 -3 -11 -4 }}
: gencom: [3/1 99/65; 1105/1089 4225/4131 6655/6561]
 
Optimal tunings:  
* WE: ~3 = 1905.8547{{c}}, ~33/17 = 1156.2973{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.9162{{c}}


Optimal tuning (POTE): ~17/11 = 748.0236{{c}}
Optimal ET sequence: [[5edt|b5]], [[23edt|b23f]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fg]]


Supporting EDTs: {{EDs| b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ffg |equave=t}}
Badness (Sintel): 1.15


==== 3.5.11.13.17.19 subgroup ====
==== 3.5.11.13.17.19 subgroup ====
Line 563: Line 777:
Comma list: 247/243, 325/323, 1105/1089, 4675/4617
Comma list: 247/243, 325/323, 1105/1089, 4675/4617


Subgroup-val mapping: {{mapping| 1 5 1 -2 1 7 | 0 -9 3 11 4 -11 }}
{{Mapping|legend=2| 1 -4 4 9 5 -4 | 0 9 -3 -11 -4 11 }}
: gencom: [3/1 99/65; 247/243 325/323 1105/1089 4675/4617]
 
Optimal tunings:  
* WE: ~3 = 1905.9433{{c}}, ~33/17 = 1156.3787{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.9775{{c}}


Optimal tuning (POTE): ~17/11 = 747.9960{{c}}
Optimal ET sequence: [[5edt|b5]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fgh]]


Supporting EDTs: {{EDs| b28, b33, b5, b61, b56f, b23f, b38ch, b66ch, b89fgh, b10cfh |equave=t}}
Badness (Sintel): 0.882


==== 3.5.11.13.17.19.23 subgroup ====
==== 3.5.11.13.17.19.23 subgroup ====
Line 575: Line 792:
Comma list: 209/207, 247/243, 255/253, 325/323, 4675/4617
Comma list: 209/207, 247/243, 255/253, 325/323, 4675/4617


Subgroup-val mapping: {{mapping| 1 5 1 -2 1 7 6 | 0 -9 3 11 4 -11 -8 }}
{{Mapping|legend=2| 1 -4 4 9 5 -4 -2 | 0 9 -3 -11 -4 11 8 }}
: gencom: [3/1 99/65; 209/207 247/243 255/253 325/323 4675/4617]
 
Optimal tunings:
* WE: ~3 = 1905.4597{{c}}, ~33/17 = 1155.9938{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.8954{{c}}
 
Optimal ET sequence: [[5edt|b5]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fgh]]
 
Badness (Sintel): 0.838
 
== Electra ==
[[File:Electra chords.wav|thumb|Just 11:13:15 and 143:165:195 triads on A]]
 
Electra tempers out [[1875/1859]] and [[14641/14625]] in the 3.5.11.13 subgroup, with a generator near [[15/11]] or its tritave complement, [[11/5]]. It possesses [[mos scale]]s of the families {{mos scalesig|link=1|4L 3s<3/1>}}, {{mos scalesig|link=1|7L 4s<3/1>}}, and {{mos scalesig|link=1|7L 11s<3/1>}}.
 
The 11:13:15 triad is a characteristic feature of many electra scales, and has been suggeested by {{u|CompactStar}} as a fundamental sonority for no-2's music, due to being relatively consonant compared to other no-2's chords. There is also a "minor" version of this triad, 143:165:195. If 11:13:15 is treated as the fundamental sonority of no-2's music, then electra can be viewed as the analogue to [[meantone family #Meantone|meantone]].
 
The name was given by CompactStar in 2023 after the star {{w|Electra (star)|Electra}}, following a series of non-octave temperaments that were named after stars.
 
[[Subgroup]]: 3.5.11
 
[[Comma list]]: 2373046875/2357947691
 
{{Mapping|legend=3| 1 -5 -5 | 0 9 10 }}
: mapping generators: ~3, ~11/5
 
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.8666{{c}}, ~11/5 = 1366.1408{{c}}
: [[error map]]: {{val| -0.088 -0.379 +0.757 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~11/5 = 1366.1964{{c}}
: error map: {{val| 0.000 -0.321 +0.8712 }}
 
[[Optimal ET sequence]]: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[1239edt|b1239e]], [[1310edt|b1310e]], …, [[1665edt|b1665e]], [[1736edt|b1736e]]
 
[[Badness]] (Sintel): 0.656
 
=== 3.5.11.13 subgroup ===
Subgroup: 3.5.11.13
 
Comma list: 1875/1859, 14641/14625
 
{{Mapping|legend=2| 1 -5 -5 -7 | 0 9 10 13 }}
 
Optimal tunings:
* WE: ~3 = 1901.8899{{c}}, ~11/5 = 1365.9871{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1366.0281{{c}}
 
Optimal ET sequence: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[181edt|b181]]


Optimal tuning (POTE): ~17/11 = 748.0874{{c}}
Badness (Sintel): 0.405


Supporting EDTs: {{EDs| b28, b5, b33, b23f, b61, b56f, b38ch, b10cfhi, b66ch, b51ffg |equave=t}}
; Scales
* [[Electra7]]


== Alnilam ==
== Alnilam ==
Effectively a [[microtemperament]], Alnilam takes a generator of an 81/55 flat fifth and equates 9 of them with [[11/9]]. The name was given by {{u|CompactStar}} to continue with the theme of naming no-twos temperaments after proper star names, but also to indirectly reference [[mavila]].
Nearly a [[microtemperament]], alnilam takes a generator of [[~]][[55/27]] and equates nine of them tritave reduced with [[27/11]]. The name was given by {{u|CompactStar}} to continue with the theme of naming no-2's temperaments after proper star names, but also to indirectly reference [[mavila]] as its tritave-complement generator is ~[[81/55]], a flat fifth.


[[Subgroup]]: 3.5.11
[[Subgroup]]: 3.5.11
Line 589: Line 853:
[[Comma list]]: {{monzo| -35 9 10 }}
[[Comma list]]: {{monzo| -35 9 10 }}


{{Mapping|legend=2| 1 5 -1 | 0 -10 9 }}
{{Mapping|legend=3| 1 -5 8 | 0 10 -9 }}
: [[gencom]]: [3/1 81/55; {{monzo| 0 -35 9 0 10 }}]
: mapping generators: ~3, ~55/27


[[Optimal tuning]] ([[CTE]]): ~81/55 = 672.410{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.3576{{c}}, ~55/27 = 1229.7879{{c}}
: [[error map]]: {{val| +0.403 -0.222 -0.548 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~55/27 = 1229.5321{{c}}
: error map: {{val| 0.000 -0.768 -1.467 }}


[[Support]]ing EDTs: {{EDs| 99, 17, 82, 116, 181, 65, 14[-5], 280, 48, 215, 31, 133, 314, 263
[[Optimal ET sequence]]: [[17edt|b17]], [[65edt|b65]], [[82edt|b82]], [[99edt|b99]], [[577edt|b577]], [[676edt|b676]], [[775edt|b775e]], [[874edt|b874e]], [[973edt|b973e]], [[2045edt|b2045ceee]]


= 3.7.11 subgroup temperaments =
[[Badness]] (Sintel): 3.84
 
= 3.7.11-subgroup temperaments =
== Mintaka ==
== Mintaka ==
{{Main| Mintaka }}
{{Main| Mintaka }}


Extensions to prime 5 are covered at [[#Mintra]] and [[#Nekkar]].
Mintaka tempers out [[1331/1323]] in the 3.7.11 subgroup. It is the common [[restriction]] of [[#Mintra|mintra]] and [[#Nekkar|nekkar]].


[[Subgroup]]: 3.7.11
[[Subgroup]]: 3.7.11
Line 606: Line 876:
[[Comma list]]: 1331/1323
[[Comma list]]: 1331/1323


{{Mapping|legend=2| 1 0 1 | 0 3 2 }}
{{Mapping|legend=3| 1 0 1 | 0 3 2 }}
: mapping generators: ~3, ~21/11
: mapping generators: ~3, ~21/11


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[POTE]]: ~3 = 1901.955{{c}}, ~11/7 = 778.961{{c}}
* [[WE]]: ~3 = 1902.4401{{c}}, ~21/11 = 1123.2803{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~11/7 = 778.803{{c}}
: [[error map]]: {{val| +0.485 +1.015 -2.317 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~21/11 = 1123.1517{{c}}
: error map: {{val| 0.000 +0.629 -3.060 }}


[[Support]]ing [[ET]]s: {{EDs| b22, b5, b17, b39, b12, b61, b27, b7, b83, b49, b56, b32, b29, b100 |equave=t}}
[[Optimal ET sequence]]: [[5edt|b5]], [[17edt|b17]], [[22edt|b22]], [[61edt|b61]], [[83edt|b83]], [[105edt|b105]], [[188edt|b188]], [[293edt|b293e]]
 
[[Badness]] (Sintel): 0.0528


=== Tridecimal mintaka ===
=== Tridecimal mintaka ===
This extension to prime 13 works in the sharper half of the mintaka tuning range, where the most important add-5 extension is [[#Mintra|mintra]].
This extension for prime 13 works in the sharper half of the mintaka tuning range. It is the no-5 restriction of [[#Mintra|tridecimal mintra]].


Subgroup: 3.7.11.13
Subgroup: 3.7.11.13
Line 622: Line 896:
Comma list: 1331/1323, 218491/216513
Comma list: 1331/1323, 218491/216513


Subgroup-val mapping: {{mapping| 1 0 1 10 | 0 3 2 -13 }}
{{Mapping|legend=2| 1 0 1 10 | 0 3 2 -13 }}
: mapping generators: ~3, ~21/11


Optimal tunings:
Optimal tunings:
* POTE: ~3 = 1901.955{{c}}, ~11/7 = 780.155{{c}}
* WE: ~3 = 1903.5668{{c}}, ~11/7 = 1122.7513{{c}}
* CWE: ~3 = 1901.955{{c}}, ~11/7 = 780.183{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/7 = 1121.7718{{c}}
 
Optimal ET sequence: [[17edt|b17]], [[22edt|b22]], [[39edt|b39]], [[217edt|b217ef]], [[256edt|b256ef]], [[295edt|b295def]], [[334edt|b334deef]], [[373edt|b373deef]]


Supporting ETs: {{EDs| b39, b22, b17, b5f, b61, b56, b100, b139f, b95, b178ef, b83f, b134, b73f, b217ef |equave=t}}
Badness (Sintel): 0.604


=== Minalzidar ===
=== Minalzidar ===
This extension for prime 13 works in the flatter half of the mintaka tuning range, where the most important add-5 extension is [[#Nekkar|Nekkar]].
This extension for prime 13 works in the flatter half of the mintaka tuning range. It is the no-5 restriction of [[#Nekkar|tridecimal nekkar]].


Subgroup: 3.7.11.13
Subgroup: 3.7.11.13
Line 638: Line 913:
Comma list: 351/343, 1331/1323
Comma list: 351/343, 1331/1323


Subgroup-val mapping: {{mapping| 1 0 1 -3 | 0 3 2 9 }}
{{Mapping|legend=2| 1 0 1 -3 | 0 3 2 9 }}
: mapping generators: ~3, ~21/11
: mapping generators: ~3, ~21/11


Optimal tunings:
Optimal tunings:
* POTE: ~3 = 1901.955{{c}}, ~11/7 = 774.432{{c}}
* WE: ~3 = 1899.2045{{c}}, ~21/11 = 1125.8915{{c}}
* CWE: ~3 = 1901.955{{c}}, ~11/7 = 774.782{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1127.1732{{c}}


Supporting ETs: {{EDs| b5, b27, b22, b32, b17f, b37f, b12ff, b49, b59, b42df, b76, b39ff, b86d, b71f |equave=t}}
Optimal ET sequence: [[22edt|b22]], [[27edt|b27]], [[86edt|b86d]], [[113edt|b113d]]
 
Badness (Sintel): 0.472


== Mebsuta ==
== Mebsuta ==
Mebsuta is a microtemperament in the 3.7.11 subgroup that sets the relative sizes of [[9/7]] and [[11/9]] to be in the ratio of 5:4; its generator is identifiable as the ratio between these intervals, 81/77. It produces a 21L 1s [[mos scale]] against the tritave, which serves as a well-temperament of [[22edt]]; that scale's chroma is identified with [[1331/1323]].
Mebsuta is a microtemperament in the 3.7.11 subgroup that sets the relative sizes of [[9/7]] and [[11/9]] to be in the ratio of 5:4; its generator is identifiable as the ratio between these intervals, 81/77. It produces a 21L 1s [[mos scale]] against the tritave, which serves as a well-temperament of [[22edt]]; that scale's chroma is identified with [[1331/1323]].
Mebsuta naturally extends itself with prime 19, identifying the two-generator interval as [[21/19]], since its square differs from [[11/9]] (the four-generator interval) by the small comma [[3971/3969]].


It is also possible to set the chroma 1331/1323 equal to [[245/243]], producing an accurate if complex mapping for prime 5 at 32 generators up; it is notable that this sets eight [[11/9]]'s equal to [[5/1]], which is the 3.5.11-subgroup [[restriction]] of [[mohaha]].
It is also possible to set the chroma 1331/1323 equal to [[245/243]], producing an accurate if complex mapping for prime 5 at 32 generators up; it is notable that this sets eight [[11/9]]'s equal to [[5/1]], which is the 3.5.11-subgroup [[restriction]] of [[mohaha]].
Line 656: Line 935:
[[Comma list]]: 387420489/386683451
[[Comma list]]: 387420489/386683451


{{Mapping|legend=2| 1 2 2 | 0 -5 4 }}
{{Mapping|legend=3| 1 2 2 | 0 -5 4 }}
: mapping generators: ~3, ~81/77
: mapping generators: ~3, ~81/77


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[POTE]]: ~3 = 1901.955{{c}}, ~81/77 = 86.957{{c}}
* [[WE]]: ~3 = 1901.8347{{c}}, ~81/77 = 86.9519{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~81/77 = 86.957{{c}}
: [[error map]]: {{val| -0.120 +0.084 +0.159 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~81/77 = 86.8601{{c}}
: error map: {{val| 0.000 +0.284 +0.433 }}
 
[[Optimal ET sequence]]: [[21edt|b21]], [[22edt|b22]], [[87edt|b87]], [[109edt|b109]], [[131edt|b131]], [[153edt|b153]], [[175edt|b175]], [[503edt|b503]], [[678edt|b678]], [[853edt|b853]], [[1028edt|b1028]], [[2909edt|b2909e]], [[3937edt|b3937de]], [[4965edt|b4965dee]]


[[Support]]ing [[ET]]s: {{EDs| b22, b175, b153, b197, b131, b328, b109, b21, b219, b87, b43, b372, b65, b23 |equave=t}}
[[Badness]] (Sintel): 0.128


=== 3.7.11.19 subgroup ===
=== 3.7.11.19 subgroup ===
Mebsuta naturally extends itself with prime 19, identifying the two-generator interval as [[21/19]], since its square differs from [[11/9]] (the four-generator interval) by the small comma [[3971/3969]].
Subgroup: 3.7.11.19
Subgroup: 3.7.11.19


Comma list: 3971/3969, 41553/41503
Comma list: 3971/3969, 41553/41503


Subgroup-val mapping: {{mapping| 1 2 2 3 | 0 -5 4 -7 }}
{{Mapping|legend=2| 1 2 2 3 | 0 -5 4 -7 }}
: mapping generators: ~3, ~81/77


Optimal tunings:
Optimal tunings:
* POTE: ~3 = 1901.955{{c}}, ~[[81/77]] = 86.929{{c}}
* WE: ~3 = 1901.8715{{c}}, ~81/77 = 86.9248{{c}}
* CWE: ~3 = 1901.955{{c}}, ~[[81/77]] = 86.932{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~81/77 = 86.9325{{c}}


Supporting ETs: {{EDs| b22, b175, b197, b153, b131, b219, b372, b109, b328, b241, b87, b21, b65, b43 |equave=t}}
Optimal ET sequence: [[21edt|b21]], [[22edt|b22]], [[109edt|b109]], [[131edt|b131]], [[153edt|b153]], [[175edt|b175]], [[372edt|b372]], [[547edt|b547]], [[1269edt|b1269]], [[1816edt|b1816]]
 
Badness (Sintel): 0.128


=== Adhara ===
=== Adhara ===
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.7.11.13). }}
Adhara cleaves the step of mebsuta in three to produce a remarkable Don Page temperament for the chord 7:9:11:13:17 (that is, setting [[13/11]] to two-thirds of 9/7, and [[17/13]] to four-thirds of 11/9). It can be extended to even higher subgroups fairly naturally, and encompasses several prominent tunings within its structure (such as [[65edt]], [[131edt]], and [[197edt]]).
Adhara cleaves the step of mebsuta in three to produce a remarkable Don Page temperament for the chord 7:9:11:13:17 (that is, setting [[13/11]] to two-thirds of 9/7, and [[17/13]] to four-thirds of 11/9). It can be extended to even higher subgroups fairly naturally, and encompasses several prominent tunings within its structure (such as [[65edt]], [[131edt]], and [[197edt]]).


It is also possible to set two-thirds of 11/9 to [[8/7]], giving rise to an add-8 extension.  
It is also possible to set two-thirds of 11/9 to [[8/7]], giving rise to an add-8 extension.  


[[Subgroup]]: 3.7.11.13.17
Subgroup: 3.7.11.13
 
Comma list: 107811/107653, 387420489/386683451


[[Comma list]]: 14161/14157, 107811/107653, 1108809/1108723
{{Mapping|legend=2| 1 2 2 2 | 0 -15 12 22 }}
: mapping generators: ~3, ~121/119
 
Optimal tunings:
* WE: ~3 = 1901.5487{{c}}, ~121/119 = 29.0313{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~121/119 = 29.0392{{c}}
 
Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]]
 
Badness (Sintel): 1.26
 
==== 3.7.11.13.17 subgroup ====
Subgroup: 3.7.11.13.17
 
Comma list: 14161/14157, 107811/107653, 1108809/1108723


{{Mapping|legend=2| 1 2 2 2 2 | 0 -15 12 22 38 }}
{{Mapping|legend=2| 1 2 2 2 2 | 0 -15 12 22 38 }}
: mapping generators: ~3, ~119/117


[[Optimal tuning]]s:
Optimal tunings:
* [[POTE]]: ~3 = 1901.955{{c}}, ~119/117 = 28.979{{c}}
* WE: ~3 = 1901.7879{{c}}, ~119/117 = 28.9764{{c}}
* [[CTE]]: ~3 = 1901.955{{c}}, ~119/117 = 28.970{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~119/117 = 28.9772{{c}}
 
Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247f]]


[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247f]], [[3216edt|b3216defff]]
Badness (Sintel): 0.274


==== 3.7.11.13.17.19 subgroup ====
==== 3.7.11.13.17.19 subgroup ====
This includes the natural extension of mebsuta to prime 19.
Subgroup: 3.7.11.13.17.19
Subgroup: 3.7.11.13.17.19


Comma list: 3213/3211, 3971/3969, 14161/14157, 41553/41503
Comma list: 3213/3211, 3971/3969, 14161/14157, 41553/41503


Subgroup-val mapping: {{mapping| 1 2 2 2 2 3 | 0 -15 12 22 38 -21 }}
{{Mapping|legend=2| 1 2 2 2 2 3 | 0 -15 12 22 38 -21 }}
: mapping generators: ~3, ~119/117
: mapping generators: ~3, ~119/117


Optimal tunings:
Optimal tunings:
* POTE: ~3 = 1901.955{{c}}, ~119/117 = 28.973{{c}}
* WE: ~3 = 1901.8410{{c}}, ~119/117 = 28.9716{{c}}
* CTE: ~3 = 1901.955{{c}}, ~119/117 = 28.970{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~119/117 = 28.9729{{c}}


Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[525edt|b525]], [[722edt|b722]], [[919edt|b919]], [[2035edt|b2035df]]
Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[525edt|b525]], [[722edt|b722]], [[919edt|b919]], [[2035edt|b2035df]]
Badness (Sintel): 0.225


= Other tritave-based subgroups =
= Other tritave-based subgroups =
== Aldebaran ==
== Aldebaran ==
{{main|Aldebaran}}
{{Main| Aldebaran }}


[[Subgroup]]: 3.5.13
[[Subgroup]]: 3.5.13
Line 725: Line 1,022:
[[Comma list]]: 3159/3125
[[Comma list]]: 3159/3125


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


[[Support]]ing [[ET]]s: 15, 17, 13, 32, 47, 28, 11[-13], 19[+13], 43, 9[-13], 7[-13], 49[+13], 21[+13], 41[-13]
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1900.8391{{c}}, ~5/3 = 887.8617{{c}}
: [[error map]]: {{val| -1.116 +2.387 -1.219 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 888.1251{{c}}
: error map: {{val| 0.000 +3.766 +0.098 }}


[[CTE tuning|CTE generator]]: ~[[5/3]] = 887.76
[[Optimal ET sequence]]: [[15edt|b15]], [[92edt|b92]], [[107edt|b107]], [[122edt|b122]], [[137edt|b137]], [[152edt|b152]], [[319edt|b319c]], [[471edt|b471c]]
 
[[Badness]] (Sintel): 0.140


== Keladic ==
== Keladic ==
Line 736: Line 1,040:
[[Comma list]]: 351/343
[[Comma list]]: 351/343


[[Sval]] [[mapping]]: [{{val| 1 1 0 }}, {{val| 0 1 3 }}]
{{Mapping|legend=3| 1 0 -3 | 0 1 3 }}
: mapping generators: ~3, ~7


Sval mapping generators: ~3, ~7/3
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1899.1302{{c}}, ~7/3 = 1478.4616{{c}}
: [[error map]]: {{val| -2.825 +8.766 -5.143 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~7/3 = 1479.4872{{c}}
: error map: {{val| 0.000 +12.616 -2.066 }}


[[Optimal tuning]]s:
[[Optimal ET sequence]]: [[4edt|b4]], [[5edt|b5]], [[9edt|b9]], [[86edt|b86d]], [[95edt|b95d]], [[104edt|b104d]], [[113edt|b113d]], [[122edt|b122d]], [[131edt|b131d]]
* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[7/3]] = 1480.661
* [[CWE]]: ~3 = 1\1ed3, ~[[7/3]] = 1479.487


[[Support]]ing [[ET]]s: {{EDs|b9, b5, b14, b13, b23, b22, b32, b6f, b31, b19f, b17f, b41, b7ff, b40|equave=t}}
[[Badness]] (Sintel): 0.125


== Sadalmelik ==
== Sadalmelik ==
Line 751: Line 1,058:
[[Comma list]]: 85293/83521
[[Comma list]]: 85293/83521


[[Sval]] [[mapping]]: [{{val| 1 0 2 }}, {{val| 0 4 1 }}]
{{Mapping|legend=3| 1 0 2 | 0 4 1 }}
: mapping generators: ~3, ~17/9
 
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1900.2977{{c}}, ~17/9 = 1109.8480{{c}}
: [[error map]]: {{val| -1.657 -1.136 +5.488 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~17/9 = 1110.3763{{c}}
: error map: {{val| 0.000 +0.978 +9.331 }}
 
[[Optimal ET sequence]]: [[5edt|b5]], [[7edt|b7]], [[12edt|b12]], [[65edt|b65]], [[77edt|b77]], [[89edt|b89]], [[101edt|b101g]], [[113edt|b113g]], [[238edt|b238gg]]
 
[[Badness]] (Sintel): 0.322
 
= No-twos no-threes subgroup temperaments =
{{See also|Ed5}}
 
No-twos temperaments exclude the second harmonic, thus meaning the next lowest prime, 3, is the best remaining option for an equave. If you then also remove the 3rd harmonic, the best remaining option for the equave would be [[5/1]] (also known as the pentane). Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical.
 
== Pentadacus ==
{{Main| Pentadacus }}
 
Discovered and named by {{u|CompactStar}} in 2026, Pentadacus is an efficient 5.7.11 subgroup temperament generated by a [[meantone]]-esque small whole tone interval that represents [[55/49]]. Stacking 3 of these tones gives [[7/5]] and 7 of them give [[11/5]], basically making it a compressed [[14ed5]] generator chain. It is a no-twos version of 2.5.7.11 [[didacus]], and is named as a play on "didacus", with the "di-" part being rebracketed as the Greek prefix for 2 and changed to "penta-" (5).
 
[[Subgroup]]: 5.7.11
 
[[Comma list]]: 831875/823543
 
{{Mapping|legend=3| 1 1 1 | 0 3 7 }}
: mapping generators: ~5, ~55/49
 
[[Optimal tuning]]s:
* [[WE]]: ~5 = 2785.665{{c}}, ~55/49 = 194.939{{c}}
: [[error map]]: {{val| -0.649 +1.656 -1.080 }}
* [[CWE]]: ~5 = 2786.314{{c}}, ~55/49 = 194.908{{c}}
: error map: {{val| 0.000 +2.213 -0.645 }}
 
[[Optimal ET sequence]]: TBA
 
[[Badness]] (Sintel): 0.216
 
=== Pentapassion ===
Named by [[User:2^67-1|Cole]] in 2026, Pentapassion is a [[weak extension]] of Pentadacus to the 5.7.11.13 subgroup that splits the generator in half, making the new generator a semitone. [[13/5]] (the pentave-reduced 13th harmonic) is found at 17 of these new generators up. It is named in reference to the octave-equivalent [[Passion]] temperament, which has a similar generator, and in both temperaments [[7/5]] is mapped to 6 generators up.
 
[[Subgroup]]: 5.7.11.13
 
[[Comma list]]: [[847/845]], 831875/823543


Sval mapping generators: ~3, ~17/9
{{Mapping|legend=3| 1 1 1 1 | 0 6 14 17 }}
: mapping generators: ~5, ~343/325


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: ~3 = 1\1ed3, ~[[17/9]] = 1109.689
* [[WE]]: ~5 = 2785.855{{c}}, ~343/325 = 97.393{{c}}
* [[CWE]]: ~3 = 1\1ed3, ~[[17/9]] = 1110.376
: [[error map]]: {{val| -0.459 +1.384 -1.968 +1.000 }}
* [[CWE]]: ~5 = 2786.314{{c}}, ~343/325 = 97.386{{c}}
: error map: {{val| 0.000 +1.803 -1.601 +1.347 }}
 
[[Optimal ET sequence]]: TBA


[[Support]]ing [[ET]]s: {{EDs|b12, b5, b7, b17, b29, b19, b41, b53, b31, b65, b22f, b9ff, b77, b43|equave=t}}
[[Badness]] (Sintel): 0.277


= No-twos-or-threes subgroup temperaments =
== Antipyth ==
== Antipyth ==
{{main|Antipyth}}
{{Todo|inline=1|rename}}
{{Main| Antipyth }}


[[Subgroup]]: 5.7.11
[[Subgroup]]: 5.7.11
Line 769: Line 1,126:
[[Comma list]]: 859375/823543
[[Comma list]]: 859375/823543


{{Mapping|legend=2| 1 2 7 | 0 1 7 }}
{{Mapping|legend=3| 1 0 -7 | 0 1 7 }}
: mapping generators: ~5, ~7


Mapping generators: ~5, ~7/25
[[Optimal tuning]]s:
* [[WE]]: ~5 = 2782.0966{{c}}, ~7/5 = 592.8517{{c}}
: [[error map]]: {{val| -4.217 +6.122 -1.356 }}
* [[CWE]]: ~5 = 2786.3137{{c}}, ~7/5 = 593.3647{{c}}
: error map: {{val| 0.000 +10.852 +2.235 }}


[[Optimal tuning]] ([[CTE]]): ~5 = 1\1ed5, ~[[7/5]] = 592.728
[[Optimal ET sequence]]: [[5ed5|c5]], [[9ed5|c9e]], [[14ed5|c14]], [[33ed5|c33]], [[47ed5|c47]], [[61ed5|c61]]


[[Support]]ing [[ET]]s: {{EDs|c14, c5, c19, c33, c47, c9e, c61, c75, c23e, c24e, c52e, c80e, c89e, c37e|equave=5}}
[[Badness]] (Sintel): 1.04


== Juggernaut ==
== Juggernaut ==
{{main|Juggernaut}}
{{Todo|inline=1|rename|comment=CompactStar: possibly rename juggernaut (Cole suggested the name "Macrodec", not sure about it yet though)}}
{{Main| Juggernaut }}


[[Subgroup]]: 5.7.11
[[Subgroup]]: 5.7.11
Line 784: Line 1,147:
[[Comma list]]: 125/121
[[Comma list]]: 125/121


{{Mapping|legend=2| 2 4 3 | 0 1 0 }}
{{Mapping|legend=3| 2 0 3 | 0 1 0 }}
: mapping generators: ~11/5, ~7


Mapping generators: ~11/5, ~7/25
[[Optimal tuning]]s:
* [[WE]]: ~11/5 = 1388.4013{{c}}, ~7/5 = 591.9463{{c}}
: [[error map]]: {{val| -9.511 -0.077 +13.886 }}
* [[CWE]]: ~11/5 = 1393.1569{{c}}, ~7/5 = 590.1275{{c}}
: error map: {{val| 0.000 +7.615 +28.153 }}


[[Optimal tuning]] ([[CTE]]): ~11/5 = 1\2ed5, ~[[7/5]] = 582.512
[[Optimal ET sequence]]: [[2ed5|c2]], [[4ed5|c4]], [[10ed5|c10]], [[14ed5|c14]], [[66ed5|c66e]], [[80ed5|c80e]], [[94ed5|c94e]], [[108ed5|c108ee]], [[122ed5|c122ee]]


[[Support]]ing [[ET]]s: {{EDs|c14, c10, c6, c18, c24, c22, c32, c16, c38, c8d, c34, c26d, c46, c52e|equave=5}}
[[Badness]] (Sintel): 0.116


=== Tridecimal juggernaut ===
=== Tridecimal juggernaut ===
[[Subgroup]]: 5.7.11.13
Subgroup: 5.7.11.13
 
Comma list: 125/121, 637/605
 
{{Mapping|legend=2| 2 0 3 8 | 0 1 0 -2 }}
 
Optimal tunings:
* WE: ~11/5 = 1392.6466{{c}}, ~7/5 = 570.6655{{c}}
* CWE: ~11/5 = 1393.1569{{c}}, ~7/5 = 570.9139{{c}}
 
Optimal ET sequence: [[4ed5|c4]], [[6ed5|c6]], [[10ed5|c10]], [[24ed5|c24]], [[34d5|c34]], [[44ed5|c44]]
 
Badness (Sintel): 0.116
 
== Catfish ==
[[File:Salmon7.wav|thumb|Catfish[7] scale and 11:13:17 chord on A in 20ed11/5 tuning]]
 
Named by {{u|CompactStar}} in 2023, catfish tempers out 54925/54043 and 1795625/1771561 in the 5.11.13.17 subgroup.
 
CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{mos scalesig|2L 3s<11/5>}}, {{mos scalesig|2L 5s<11/5>}}, {{mos scalesig|2L 7s<11/5>}}, and {{mos scalesig|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord.
 
[[Subgroup]]: 5.11.13
 
[[Comma list]]: {{monzo| 14 -19 9 }}
 
{{Mapping|legend=3| 1 -4 -10 | 0 9 19 }}
: mapping generators: ~5, ~325/121
 
[[Optimal tuning]]s:
* [[WE]]: ~5 = 2785.1693{{c}}, ~325/121 = 1699.4922{{c}}
: [[error map]]: {{val| -1.144 +3.435 -1.869 }}
* [[CWE]]: ~5 = 2786.3137{{c}}, ~325/121 = 1700.1381{{c}}
: error map: {{val| 0.000 +4.670 -1.042 }}
 
[[Optimal ET sequence]]: [[18ed5|c18]], [[41ed5|c41]], [[59ed5|c59]], [[372ed5|c372e]], [[431ed5|c431e]], [[490ed5|c490e]]
 
Badness (Sintel): 10.3


[[Comma list]]: 125/121, 637/625
=== 5.11.13.17 subgroup ===
Subgroup: 5.11.13.17


{{Mapping|legend=2| 2 4 3 0 | 0 1 0 -2 }}
Comma list: 54925/54043, 1795625/1771561


Mapping generators: ~11/5, ~7/25
{{Mapping|legend=2| 1 -4 -10 -8 | 0 9 19 16 }}
: mapping generators: ~5, ~325/121


[[Optimal tuning]] ([[CTE]]): ~11/5 = 1\2ed5, ~[[7/5]] = 582.512
Optimal tunings:  
* WE: ~5 = 2785.0096{{c}}, ~325/121 = 1699.3010{{c}}
* CWE: ~5 = 2786.3137{{c}}, ~325/121 = 1700.0352{{c}}


[[Support]]ing [[ET]]s: {{EDs|c10, c14, c6, c24, c34, c16f, c44, c18f, c38, c26f, c54, c64|equave=5}}
Optimal ET sequence: [[18ed5|c18]], [[41ed5|c41]], [[59ed5|c59]]


= Graphs =
Badness (Sintel): 0.905
See: [[Catalog of 3.5.7 subgroup rank two temperaments#Graphs]]


== Projective tuning space diagrams ==
; Scales
See: [[Catalog of 3.5.7 subgroup rank two temperaments#Projective tuning space diagrams]]
* [[Catfish7]]
* [[Catfish9]]


[[Category:Temperament collections]]
[[Category:Catalogs of subgroup temperaments]]
[[Category:Non-octave temperaments]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/No-twos_subgroup_temperaments"