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No-twos subgroup temperaments: Difference between revisions

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''(WIP, further entries in the [[catalog of 3.5.7 subgroup rank two temperaments]] will eventually be documented here)''
{{Technical data page}}
{{Todo|WIP|inline=1|text=Further entries in the [[catalog of 3.5.7 subgroup rank two temperaments]] will eventually be documented here.}}


This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 2. Because of the absence of octaves, these are all [[nonoctave]] scales using a period of a [[tritave]], or if harmonic 3 is also excluded, [[5/1]].
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 2. Because of the absence of octaves, these are all [[nonoctave]] scales using a period of a [[tritave]], or if harmonic 3 is also excluded, [[5/1]].
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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 =
== Arcturus ==
== Arcturus ==
{{main|Arcturus}}
{{Main| Arcturus }}


As 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 [[Shibboleth family #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


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


[[Sval]] [[mapping]]: [{{val| 1 0 -7 }}, {{val| 0 1 6 }}]
{{Mapping|legend=3| 1 0 -7 | 0 1 6 }}
: mapping generators: ~3, ~5


Sval mapping generators: ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1903.8634{{c}}, ~5/3 = 878.9230{{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 }}


[[POTE generator]]: ~5/3 = 878.042
[[Optimal ET sequence]]: [[2edt|b2]], [[9edt|b9d]], [[11edt|b11]], [[13edt|b13]], [[340edt|b340cc]]


[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]
[[Badness]] (Sintel): 0.535


=== Polturus ===
=== Polturus ===
This extension of arcturus adds [[#Polaris|polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
Subgroup: 3.5.7.11
Comma list: 15625/15309, 177147/171875
{{Mapping|legend=2| 1 1 -1 5 | 0 1 6 -6 }}
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]]
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]].


This extension of Arcturus adds [[Polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
Antares was named by {{u|CompactStar}} after the star {{w|Antares}}, following a series of non-octave temperaments that were named after stars.


[[Subgroup]]: 3.5.7.11
[[Subgroup]]: 3.5.7


[[Comma list]]: 15625/15309, 177147/171875
[[Comma list]]: 21875/19683


[[Gencom]]: [3/1 5/3; 15625/15309 177147/171875]
{{Mapping|legend=3| 1 0 9 | 0 1 -5 }}
: mapping generators: ~3, ~5


[[Mapping]]: [{{val|1 1 -1 5}}, {{val|0 1 6 -6}}]
[[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 }}


[[POTE generator]]: ~[[5/3]] = 884.268
[[Optimal ET sequence]]: [[9edt|b9]], [[20edt|b20]], [[29edt|b29]]


[[EDT]]s: 15, 13e, 28e, 43dee
[[Badness]] (Sintel): 2.95


== BPS ==
== BPS ==
{{main|Bohlen-Pierce-Stearns}}
{{Main| BPS }}


For extensions to this temperament that include the prime 2, see [[Sensamagic clan]]. No-twos 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


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


Sval mapping generators: ~3, ~9/7
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1903.7398{{c}}, ~9/7 = 440.9014{{c}}
: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~9/7 = 440.6646{{c}}
: error map: {{val| 0.000 -3.030 -5.580 }}


[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~9/7 = 440.4881
[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]], [[367edt|b367cdd]], [[462edt|b462cdd]]


[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
[[Badness]] (Sintel): 0.0659


Badness (Sintel): 0.066
=== Alhena ===
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]].


=== Alhena ===
Subgroup: 3.5.7.11/2
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]].


[[Subgroup]]: 3.5.7.11/2.13/4
Comma list: 245/243 ({{monzo| -5 1 2 0 }}), 3773/3750 ({{monzo| -1 -4 3 1 }})


[[Comma list]]: 196/195, 325/324, 1001/1000
{{Mapping|legend=2| 1 1 2 -1 | 0 2 -1 11 }}


[[Sval]] [[mapping]]: [{{val| 1 1 2 -1 2}}, {{val| 0 2 -1 11 -4}}]
Optimal tunings:  
* Subgroup WE: ~3 = 1903.6644{{c}}, ~9/7 = 441.3233{{c}}
* Subgroup CWE: ~3 = 1901.9550{{c}}, ~9/7 = 440.9989{{c}}


Sval mapping generators: ~3, ~7/3
Optimal ET sequence: [[13edt|b13]], [[43edt|b43]], [[56edt|b56]], [[69edt|b69]], [[151edt|b151]], [[220edt|b220d]], [[289edt|b289dé]] *


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~9/7 = 441.025
: <nowiki/>* é is used as the wart for 11/2.


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]]
==== 3.5.7.11/2.13/4 subgroup ====
(Note that é is used as the wart for 11/2.)
Subgroup: 3.5.7.11/2.13/4


Badness (Sintel): 0.187
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 }})


=== Mintra ===
{{Mapping|legend=2| 1 1 2 -1 2 | 0 2 -1 11 -4 }}


''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]] and [[No-twos subgroup temperaments#Deneb|Deneb]].''
Optimal tunings:
* Subgroup WE: ~3 = 1903.5584{{c}}, ~9/7 = 441.3534{{c}}
* Subgroup CWE: ~3 = 1901.9550{{c}}, ~9/7 = 441.0249{{c}}


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]].
Optimal ET sequence: [[13edt|b13]], [[43edt|b43]], [[56edt|b56]], [[69edt|b69]], [[289edt|b289d]], [[358edt|b358ddé]], [[427edt|b427cddé]] *


[[Subgroup]]: 3.5.7.11
: <nowiki/>* é is used as the wart for 11/2.


[[Comma list]]: 245/243, 1331/1323
Badness (Sintel): 0.187


{{Mapping|legend=2|1 5 0 1|0 -6 3 2}}
=== Mintra ===
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]].


Sval mapping generators: ~3, ~21/11
The 13-limit extension uses the canonical extension for prime 13 described at [[#Tridecimal mintaka]].


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~11/7 = 780.752
Subgroup: 3.5.7.11


[[Support]]ing [[ET]]s: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90
Comma list: 245/243, 1331/1323


Badness (Sintel): 0.302
{{Mapping|legend=2| 1 -1 3 3 | 0 6 -3 -2 }}
: mapping generators: ~3, ~11/7


==== Tridecimal Mintra ====
Optimal tunings:
* WE: ~3 = 1904.1393{{c}}, ~11/7 = 781.6168{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/7 = 780.7525{{c}}


This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments#Tridecimal Mintaka|Tridecimal Mintaka]].
Optimal ET sequence: [[17edt|b17]], [[39edt|b39]], [[95edt|b95]], [[134edt|b134]], [[229edt|b229de]]


[[Subgroup]]: 3.5.7.11.13
Badness (Sintel): 0.302


[[Comma list]]: 245/243, 275/273, 1575/1573
==== Tridecimal mintra ====
Subgroup: 3.5.7.11.13


{{Mapping|legend=2|1 5 0 1 10|0 -6 3 2 -13}}
Comma list: 245/243, 275/273, 1331/1323


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


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~11/7 = 780.428
Optimal tunings:
* WE: ~3 = 1903.9326{{c}}, ~11/7 = 781.1856{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/7 = 780.4280{{c}}


[[Support]]ing [[ET]]s: 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


=== Dubhe ===
=== Dubhe ===
{{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


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


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~17/15 = 220.142
Optimal tunings:
* WE: ~3 = 1903.4591{{c}}, ~17/15 = 220.2094{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~17/15 = 220.1420{{c}}


[[Support]]ing [[ET]]s: 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
== Vega ==
[[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 an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].
[[Subgroup]]: 3.5.7
[[Comma list]]: 3125/3087
{{Mapping|legend=3| 1 1 1 | 0 3 5 }}
: mapping generators: ~3, ~25/21
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.4455{{c}}, ~25/21 = 293.7393{{c}}
: [[error map]]: {{val| +0.490 -2.650 +2.316 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~25/21 = 293.7594{{c}}
: 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
{{Mapping|legend=2| 1 -2 -4 12 | 0 6 10 -17 }}
: mapping generators: ~3, ~729/385
Optimal tunings:
* WE: ~3 = 1901.0672{{c}}, ~729/385 = 1097.6627{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.1722{{c}}
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]], [[168edt|b168]]
Badness (Sintel): 1.82
==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13
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
{{Mapping|legend=2| 1 -5 -9 -5 | 0 9 15 10 }}
: mapping generators: ~3, ~11/5
Optimal tunings:
* WE: ~3 = 1902.6279{{c}}, ~11/5 = 1366.3374{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8996{{c}}
Optimal ET sequence: [[7edt|b7]], [[25edt|b25d]], [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[149edt|b149]]
Badness (Sintel): 0.538
==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13
Comma list: 275/273, 847/845, 1331/1323
{{Mapping|legend=2| 1 -5 -9 -5 -7 | 0 9 15 10 13 }}
Optimal tunings:
* WE: ~3 = 1902.4456{{c}}, ~11/5 = 1366.1830{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8649{{c}}
Optimal ET sequence: [[7edt|b7]], [[25edt|b25df]], [[32edt|b32]], [[39edt|b39]], [[110edt|b110]], [[149edt|b149]]
Badness (Sintel): 0.286


== Canopus ==
== Canopus ==
{{main|Canopus}}
{{Main| Canopus }}


For extensions to this temperament that include the prime 2, see [[Mirkwai clan]]. No-twos extensions will be documented below.
For extensions to this temperament that include the prime 2, see [[Canopic clan]]. No-2's extensions will be documented below.


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7
Line 154: Line 329:
[[Comma list]]: 16875/16807
[[Comma list]]: 16875/16807


{{Mapping|legend=2|1 3 3|0 -5 -4}}
{{Mapping|legend=3| 1 -2 -1 | 0 5 4 }}
: mapping generators: ~3, ~15/7
 
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{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]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
 
[[Badness]] (Sintel): 0.0996


Sval mapping generators: ~3, ~7/5
=== Suhail ===
Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].


[[Optimal tuning]]s:
Subgroup: 3.5.7.11
* [[CTE]]: ~3 = 1\1edt, ~7/5 = 584.017
 
* [[POTE|PETE]]: ~3 = 1\1edt, ~7/5 = 583.9584
Comma list: 9375/9317, 12005/11979
 
{{Mapping|legend=2| 3 4 5 6 | 0 5 4 7 }}
: mapping generators: ~539/375, ~77/75
 
Optimal tunings:
* WE: ~539/375 = 634.9453{{c}}, ~77/75 = 49.8753{{c}}
* CWE: ~539/375 = 633.9850{{c}}, ~77/75 = 49.8596{{c}}
 
Optimal ET sequence: [[36edt|b36]], [[39edt|b39]], [[75edt|b75]], [[114edt|b114]], [[153edt|b153]], [[267edt|b267]]
 
Badness (Sintel): 0.421
 
==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13
 
Comma list: 1575/1573, 1625/1617, 4459/4455
 
{{Mapping|legend=2| 3 4 5 6 7 | 0 5 4 7 0 }}
 
Optimal tunings:
* WE: ~13/9 = 634.1444{{c}}, ~65/63 = 49.6946{{c}}
* CWE: ~13/9 = 633.9850{{c}}, ~65/63 = 49.7330{{c}}
 
Optimal ET sequence: [[36edt|b36]], [[39edt|b39]], [[114edt|b114]], [[153edt|b153]], [[498edt|b498cf]], [[651edt|b651cf]]
 
Badness (Sintel): 0.330
 
== 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|sirius]].
 
[[Subgroup]]: 3.5.7
 
[[Comma list]]: 823543/820125
 
{{Mapping|legend=3| 1 2 2 | 0 -7 -3 }}
: mapping generators: ~3, ~49/45
 
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1902.1979{{c}}, ~49/45 = 145.4124{{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 }}
 
[[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
 
=== Erigone ===
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}}
 
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
 
Comma list: 847/845, 1575/1573, 4459/4455
 
{{Mapping|legend=2| 1 -12 -4 -10 -6 | 0 21 9 19 13 }}
 
Optimal tunings:
* WE: ~3 = 1901.9695{{c}}, ~99/49 = 1219.5210{{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
 
===== 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.
 
Subgroup: 3.5.7.11.13.17
 
Comma list: 847/845, 1575/1573, 3185/3179, 4459/4455
 
{{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
 
Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 4459/4455
 
{{Mapping|legend=2| 1 -12 -4 -10 -6 -8 3 | 0 42 18 38 26 33 -1 }}
 
Optimal tunings:
* WE: ~3 = 1902.0918{{c}}, ~27/19 = 609.7886{{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]]
 
Badness (Sintel): 0.455
 
====== 3.5.7.11.13.17.19.23 subgroup ======
Subgroup: 3.5.7.11.13.17.19.23
 
Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 2277/2275, 4459/4455
 
{{Mapping|legend=2| 1 -12 -4 -10 -6 -8 3 -26 | 0 42 18 38 26 33 -1 90 }}
 
Optimal tunings:
* WE: ~3 = 1902.0149{{c}}, ~27/19 = 609.7748{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~27/19 = 609.7562{{c}}
 
Optimal ET sequence: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]
 
Badness (Sintel): 0.542
 
== 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
 
[[Comma list]]: 5859375/5764801
 
{{Mapping|legend=3| 1 -1 -1 | 0 8 9 }}
: mapping generators: ~3, ~7/5
 
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.8789{{c}}, ~7/5 = 585.8428{{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]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
[[Badness]] (Sintel): 1.24


== Izar ==
== Izar ==
Line 169: Line 501:
[[Comma list]]: 13841287201/13839609375
[[Comma list]]: 13841287201/13839609375


{{Mapping|legend=2|1 7 5|0 -12 -7}}
{{Mapping|legend=3| 1 -5 -2 | 0 12 7 }}
: mapping generators: ~3, ~30375/16807


Sval mapping generators: ~3, ~16807/10125
[[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 }}


[[Optimal tuning]] (CTE): ~3 = 1\1edt, ~16807/10125 = 877.280
[[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]]


[[Support]]ing [[ET]]s: {{EDs|b13, b11cd, b193, b15cd, b180, b24c, b167, b37c, b154, 141, b50c, b28cd, b128, b63c|equave=t}}
[[Badness]] (Sintel): 0.0166


== Nekkar ==
== Nekkar ==
This temperament is the no-twos 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]].
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.


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


[[Comma list]]: 35303692060125/33232930569601
[[Comma list]]: {{monzo| -24 -3 16 }}


{{Mapping|legend=2|1 8 3|0 -16 -3}}
{{Mapping|legend=3| 1 -8 0 | 0 16 3 }}
: mapping generators: ~3, ~32805/16807


Sval mapping generators: ~3, ~16807/10935
[[Optimal tuning]]s:
* [[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 }}


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~16807/10935 = 776.767
[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[311edt|b311dd]]


[[Support]]ing [[ET]]s: 22, 49, 5c, 71, 27, 17c, 120, 93, 76c, 32cc, 169d, 115, 191d, 164d
[[Badness]] (Sintel): 17.1


=== 3.5.7.11 subgroup ===
=== 3.5.7.11 subgroup ===
''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]].''
Subgroup: 3.5.7.11
 
Comma list: 1331/1323, 120285/117649
 
{{Mapping|legend=2| 1 -8 0 1 | 0 16 3 2 }}
 
Optimal tunings:
* WE: ~3 = 1900.6084{{c}}, ~21/11 = 1124.4193{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.1738{{c}}
 
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[262edt|b262d]]
 
Badness (Sintel): 1.37
 
=== 3.5.7.11.13 subgroup ===
Subgroup: 3.5.7.11.13
 
Comma list: 169/165, 351/343, 11011/10935


This continues the canonical 11-limit extension of squares.
{{Mapping|legend=2| 1 -8 0 1 -3 | 0 16 3 2 9 }}
 
Optimal tunings:
* WE: ~3 = 1902.3248{{c}}, ~21/11 = 1125.4837{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.2772{{c}}
 
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71f]]
 
Badness (Sintel): 1.72
 
== Bohlenic ==
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]]: 1331/1323, 120285/117649
[[Comma list]]: 245/243, 3125/3087


{{Mapping|legend=2|1 8 3 3|0 -16 -3 -2}}
{{Mapping|legend=3| 13 19 23 0 | 0 0 0 1 }}
: mapping generators: ~27/25, ~11


Sval mapping generators: ~3, ~11/7
[[Optimal tuning]]s:
* [[WE]]: ~27/25 = 146.4737{{c}}, ~11/9 = 343.9912{{c}}
* [[CWE]]: ~27/25 = 146.3042{{c}}, ~11/9 = 343.7950{{c}}


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~11/7 = 776.781
[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]], [[299edt|b299ccde]], [[338edt|b338ccde]]


[[Support]]ing [[ET]]s: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd
[[Badness]] (Sintel): 0.499


=== 3.5.7.11.13 subgroup ===
=== 3.5.7.11.13 subgroup ===
This uses the [[No-twos subgroup temperaments#Minalzidar|Minalzidar]] mapping of 13.
Subgroup: 3.5.7.11.13


[[Subgroup]]: 3.5.7.11.13
Comma list: 245/243, 275/273, 847/845


[[Comma list]]: 169/165, 351/343, 11011/10935
Subgroup-val mapping: {{mapping| 13 19 23 0 2 | 0 0 0 1 1 }}


{{Mapping|legend=2|1 8 3 3 6|0 -16 -3 -2 -9}}
Optimal tunings:
* WE: ~27/25 = 146.4722{{c}}, ~11/9 = 341.2865{{c}}
* CWE: ~27/25 = 146.3042{{c}}, ~11/9 = 342.1231{{c}}


Sval mapping generators: ~3, ~11/7
Optimal ET sequence: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~11/7 = 776.678
Badness (Sintel): 0.365


[[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf
== Tuning diagrams ==
{| class="wikitable" style="margin: auto auto auto auto;"
|-
| [[File:357plot_cplx_damage.png|alt=357plot_cplx_damage.png|357plot_cplx_damage.png]]
|-
| Complexity vs. damage plot. {{nowrap| ''z'' &lt; 1 }} corresponds to the "Middle Path" inclusion criterion.
|}


== Sirius ==
{{center|<div style{{=}}"display: inline-grid; margin-right: 25px;">
{{main|Sirius}}
{{(!}} class{{=}}"wikitable"
{{!-}}
{{!}} [[File:357ptslines1n.png|320px]]
{{!-}}
{{!}} Temperaments supported by 13edt, labelled by name
{{!)}}
</div><div style{{=}}"display: inline-grid; margin-right: 25px;">
{{(!}} class{{=}}"wikitable"
{{!-}}
{{!}} [[File:357ptslines2n.png|320px]]
{{!-}}
{{!}} Temperaments not supported by 13edt, labelled by name
{{!)}}
</div><div style{{=}}"display: inline-grid;">
{{(!}} class{{=}}"wikitable"
{{!-}}
{{!}} [[File:357ptslines12n.png|320px]]
{{!-}}
{{!}} Both sets, labelled by name
{{!)}}
</div>}}
{{center|<div style{{=}}"display: inline-grid; margin-right: 25px;">
{{(!}} class{{=}}"wikitable"
{{!-}}
{{!}} [[File:357ptslines1c.png|320px]]
{{!-}}
{{!}} Temperaments supported by 13edt, labelled by comma
{{!)}}
</div><div style{{=}}"display: inline-grid; margin-right: 25px;">
{{(!}} class{{=}}"wikitable"
{{!-}}
{{!}} [[File:357ptslines2c.png|320px]]
{{!-}}
{{!}} Temperaments not supported by 13edt, labelled by comma
{{!)}}
</div><div style{{=}}"display: inline-grid;">
{{(!}} class{{=}}"wikitable"
{{!-}}
{{!}} [[File:357ptslines12c.png|320px]]
{{!-}}
{{!}} Both sets, labeled by comma
{{!)}}
</div>}}
 
= 3.5.11-subgroup temperaments =
== 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]].
 
[[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
 
[[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
 
[[Comma list]]: 1377/1375, 265625/264627
 
{{Mapping|legend=3| 1 0 -11 -15 | 0 1 9 12 }}
 
[[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 ==
{{Main| Deneb }}
Deneb tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]].
 
[[Subgroup]]: 3.5.11
 
[[Comma list]]: 6655/6561
 
{{Mapping|legend=3| 1 2 2 | 0 -3 1 }}
: 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 ET sequence]]: [[5edt|b5]], [[6edt|b6]], [[11edt|b11]], [[17edt|b17]], [[28edt|b28]], [[67edt|b67]], [[95edt|b95]], [[123edt|b123]], [[218edt|b218e]], [[341edt|b341cee]]
 
[[Badness]] (Sintel): 0.255
 
=== Fomalhaut ===
Fomalhaut is an extension of deneb to higher limits that splits the interval of [[11/3]] in three.
 
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-2's temperaments after stars.
 
Subgroup: 3.5.11.13
 
Comma list: 6655/6561, 274625/264627


For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan#Overview to extensions]].
{{Mapping|legend=2| 1 -4 4 9 | 0 9 -3 -11 }}
: mapping generators: ~3, ~65/33


[[Subgroup]]: 3.5.7
Optimal tunings:  
* WE: ~3 = 1905.7441{{c}}, ~65/33 = 1156.2566{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~65/33 = 1153.9402{{c}}


[[Comma list]]: 3125/3087
Optimal ET sequence [[5edt|b5]], [[23edt|b23f]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89f]]


{{Mapping|legend=2| 1 1 1 | 0 3 5 }}
Badness (Sintel): 2.69


: sval mapping generators: ~3, ~25/21
==== 3.5.11.13.17 subgroup ====
Subgroup: 3.5.11.13.17


[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~25/21 = 293.740
Comma list: 1105/1089, 4225/4131, 6655/6561


[[Optimal ET sequence]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]],  [[123edt|b123]], [[136edt|b136]]
{{Mapping|legend=2| 1 -4 4 9 5 | 0 9 -3 -11 -4 }}


=== Mizar ===
Optimal tunings:
This temperament uses a weak extension to 3.5.7.17 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.
* WE: ~3 = 1905.8547{{c}}, ~33/17 = 1156.2973{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.9162{{c}}


[[Subgroup]]: 3.5.7.17
Optimal ET sequence: [[5edt|b5]], [[23edt|b23f]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fg]]


[[Comma list]]: 3125/3087, 2025/2023
Badness (Sintel): 1.15


{{Mapping|legend=2|1 -2 -4 2|0 6 10 1}}
==== 3.5.11.13.17.19 subgroup ====
Subgroup: 3.5.11.13.17.19


: sval mapping generators: ~3, ~17/9
Comma list: 247/243, 325/323, 1105/1089, 4675/4617


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~17/9 = 1097.800
{{Mapping|legend=2| 1 -4 4 9 5 -4 | 0 9 -3 -11 -4 11 }}


[[Support]]ing [[ET]]s: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd
Optimal tunings:  
* WE: ~3 = 1905.9433{{c}}, ~33/17 = 1156.3787{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.9775{{c}}


Badness (Sintel): 0.383
Optimal ET sequence: [[5edt|b5]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fgh]]


==== Full no-twos 17-limit ====
Badness (Sintel): 0.882
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
==== 3.5.11.13.17.19.23 subgroup ====
Subgroup: 3.5.11.13.17.19.23


[[Comma list]]: 275/273, 459/455, 1625/1617, 2025/2023
Comma list: 209/207, 247/243, 255/253, 325/323, 4675/4617


{{Mapping|legend=2|1 -2 -4 12 11 2|0 6 10 -17 -15 1}}
{{Mapping|legend=2| 1 -4 4 9 5 -4 -2 | 0 9 -3 -11 -4 11 8 }}


: sval mapping generators: ~3, ~17/9
Optimal tunings:  
* WE: ~3 = 1905.4597{{c}}, ~33/17 = 1155.9938{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.8954{{c}}


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~17/9 = 1098.298
Optimal ET sequence: [[5edt|b5]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fgh]]


[[Support]]ing [[ET]]s: 26, 71, 45, 19, 97f, 116d
Badness (Sintel): 0.838


Badness (Sintel): 0.841
== Electra ==
[[File:Electra chords.wav|thumb|Just 11:13:15 and 143:165:195 triads on A]]


= 3.5.11 subgroup temperaments =
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>}}.


== Polaris ==
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]].
{{main|Polaris}}


Polaris tempers out the comma 177147/171875, and thus equates 7 5/3's with 15/11, or equivalently 7 9/5's with 11/9.  
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
[[Subgroup]]: 3.5.11


[[Comma list]]: 177147/171875
[[Comma list]]: 2373046875/2357947691


[[Gencom]]: [3/1 5/3; 177147/171875]
{{Mapping|legend=3| 1 -5 -5 | 0 9 10 }}
: mapping generators: ~3, ~11/5


[[Sval]] [[mapping]]: [{{val|1 2 1}}, {{val|0 1 -6}}]
[[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 }}


[[POTE generator]]: ~5/3 = 892.6
[[Optimal ET sequence]]: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[1239edt|b1239e]], [[1310edt|b1310e]], …, [[1665edt|b1665e]], [[1736edt|b1736e]]


[[EDT]]s: 17, 15, 32, 49, 13[+11], 47, 19, 11[+11], 81, 66, 79[+11], 62[+11], 28[+11], 21[-11]
[[Badness]] (Sintel): 0.656


== Deneb ==
=== 3.5.11.13 subgroup ===
{{main|Deneb}}
Subgroup: 3.5.11.13


[[Subgroup]]: 3.5.11
Comma list: 1875/1859, 14641/14625


[[Comma list]]: 6655/6561
{{Mapping|legend=2| 1 -5 -5 -7 | 0 9 10 13 }}


[[Gencom]]: [3/1 11/9; 6655/6561]
Optimal tunings:  
* WE: ~3 = 1901.8899{{c}}, ~11/5 = 1365.9871{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1366.0281{{c}}


[[Sval]] [[mapping]]: [{{val|1 2 2}}, {{val|0 -3 1}}]
Optimal ET sequence: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[181edt|b181]]


[[POTE generator]]: ~[[11/9]] = 340.242
Badness (Sintel): 0.405


[[EDT]]s: 28, 11, 17, 6, 39, 5, 67, 45, 50, 16, 23, 73, 61, 62
; 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 [[User:CompactStar|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


[[Comma list]]: {{monzo|0 -35 9 0 10}}
[[Comma list]]: {{monzo| -35 9 10 }}
 
[[Gencom]]: [3/1 81/55; {{monzo|0 -35 9 0 10}}]


[[Sval]] [[mapping]]: [{{val|1 5 -1}}, {{val|0 -10 9}}]
{{Mapping|legend=3| 1 -5 8 | 0 10 -9 }}
: mapping generators: ~3, ~55/27


[[CTE tuning|CTE generator]]: ~81/55 = 672.410
[[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 }}


[[EDT]]s: 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 [[No-twos subgroup temperaments#Mintra|Mintra]] and [[No-twos subgroup temperaments#3.5.7.11 subgroup|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 335: Line 876:
[[Comma list]]: 1331/1323
[[Comma list]]: 1331/1323


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


Sval mapping generators: ~3, ~21/11
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.4401{{c}}, ~21/11 = 1123.2803{{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 }}


[[Optimal tuning]]s:
[[Optimal ET sequence]]: [[5edt|b5]], [[17edt|b17]], [[22edt|b22]], [[61edt|b61]], [[83edt|b83]], [[105edt|b105]], [[188edt|b188]], [[293edt|b293e]]
* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[11/7]] = 778.961
* [[CWE]]: ~3 = 1\1ed3, ~[[11/7]] = 778.803


[[Support]]ing [[ET]]s: {{EDs|b22, b5, b17, b39, b12, b61, b27, b7, b83, b49, b56, b32, b29, b100|equave=t}}
[[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 pental extension is [[No-twos subgroup temperaments#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


[[Comma list]]: 1331/1323, 218491/216513
Comma list: 1331/1323, 218491/216513


[[Sval]] [[mapping]]: [{{val| 1 0 1 10}}, {{val| 0 3 2 -13}}]
{{Mapping|legend=2| 1 0 1 10 | 0 3 2 -13 }}


Sval mapping generators: ~3, ~21/11
Optimal tunings:
* WE: ~3 = 1903.5668{{c}}, ~11/7 = 1122.7513{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/7 = 1121.7718{{c}}


[[Optimal tuning]]s:
Optimal ET sequence: [[17edt|b17]], [[22edt|b22]], [[39edt|b39]], [[217edt|b217ef]], [[256edt|b256ef]], [[295edt|b295def]], [[334edt|b334deef]], [[373edt|b373deef]]
* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[11/7]] = 780.155
* [[CWE]]: ~3 = 1\1ed3, ~[[11/7]] = 780.183


[[Support]]ing [[ET]]s: {{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 to prime 13 works in the flatter half of the Mintaka tuning range, where the most important pental extension is [[No-twos subgroup temperaments#3.5.7.11 subgroup|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


[[Comma list]]: 1331/1323, 351/343
Comma list: 351/343, 1331/1323


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


Sval mapping generators: ~3, ~21/11
Optimal tunings:
* WE: ~3 = 1899.2045{{c}}, ~21/11 = 1125.8915{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1127.1732{{c}}


[[Optimal tuning]]s:
Optimal ET sequence: [[22edt|b22]], [[27edt|b27]], [[86edt|b86d]], [[113edt|b113d]]
* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[11/7]] = 774.432
* [[CWE]]: ~3 = 1\1ed3, ~[[11/7]] = 774.782


[[Support]]ing [[ET]]s: {{EDs|b5, b27, b22, b32, b17f, b37f, b12ff, b49, b59, b42df, b76, b39ff, b86d, b71f|equave=t}}
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 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]].
[[Subgroup]]: 3.7.11
[[Subgroup]]: 3.7.11


[[Comma list]]: 387420489/386683451
[[Comma list]]: 387420489/386683451


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


Sval mapping generators: ~3, ~81/77
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.8347{{c}}, ~81/77 = 86.9519{{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 tuning]]s:
[[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]]
* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[81/77]] = 86.957
* [[CWE]]: ~3 = 1\1ed3, ~[[81/77]] = 86.957


[[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 ===
[[Subgroup]]: 3.7.11.19
Subgroup: 3.7.11.19
 
Comma list: 3971/3969, 41553/41503
 
{{Mapping|legend=2| 1 2 2 3 | 0 -5 4 -7 }}
 
Optimal tunings:
* WE: ~3 = 1901.8715{{c}}, ~81/77 = 86.9248{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~81/77 = 86.9325{{c}}
 
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 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.
 
Subgroup: 3.7.11.13
 
Comma list: 107811/107653, 387420489/386683451
 
{{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}}


[[Comma list]]: 3971/3969, 41553/41503
Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]]


[[Sval]] [[mapping]]: [{{val| 1 2 2 3}}, {{val| 0 -5 4 -7}}]
Badness (Sintel): 1.26


Sval mapping generators: ~3, ~81/77
==== 3.7.11.13.17 subgroup ====
Subgroup: 3.7.11.13.17


[[Optimal tuning]]s:
Comma list: 14161/14157, 107811/107653, 1108809/1108723
* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[81/77]] = 86.929
 
* [[CWE]]: ~3 = 1\1ed3, ~[[81/77]] = 86.932
{{Mapping|legend=2| 1 2 2 2 2 | 0 -15 12 22 38 }}
 
Optimal tunings:
* WE: ~3 = 1901.7879{{c}}, ~119/117 = 28.9764{{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]]
 
Badness (Sintel): 0.274
 
==== 3.7.11.13.17.19 subgroup ====
Subgroup: 3.7.11.13.17.19
 
Comma list: 3213/3211, 3971/3969, 14161/14157, 41553/41503
 
{{Mapping|legend=2| 1 2 2 2 2 3 | 0 -15 12 22 38 -21 }}
: mapping generators: ~3, ~119/117
 
Optimal tunings:
* WE: ~3 = 1901.8410{{c}}, ~119/117 = 28.9716{{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]]


[[Support]]ing [[ET]]s: {{EDs|b22, b175, b197, b153, b131, b219, b372, b109, b328, b241, b87, b21, b65, b43|equave=t}}
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 418: 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
 
[[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 }}


[[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 ET sequence]]: [[15edt|b15]], [[92edt|b92]], [[107edt|b107]], [[122edt|b122]], [[137edt|b137]], [[152edt|b152]], [[319edt|b319c]], [[471edt|b471c]]


[[CTE tuning|CTE generator]]: ~[[5/3]] = 887.76
[[Badness]] (Sintel): 0.140


== Keladic ==
== Keladic ==
Line 429: 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 444: 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
 
{{Mapping|legend=3| 1 1 1 1 | 0 6 14 17 }}
: mapping generators: ~5, ~343/325


Sval mapping generators: ~3, ~17/9
[[Optimal tuning]]s:
* [[WE]]: ~5 = 2785.855{{c}}, ~343/325 = 97.393{{c}}
: [[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 tuning]]s:
[[Optimal ET sequence]]: TBA
* [[CTE]]: ~3 = 1\1ed3, ~[[17/9]] = 1109.689
* [[CWE]]: ~3 = 1\1ed3, ~[[17/9]] = 1110.376


[[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 462: 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 477: 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:Subgroup temperaments]]
[[Category:Catalogs of subgroup temperaments]]
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Nonoctave]]
[[Category:Tritave]]
[[Category:Tritave-equivalent temperaments]]
Retrieved from "https://en.xen.wiki/w/No-twos_subgroup_temperaments"