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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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= 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]].
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 [[Gamelismic clan #Superkleismic|superkleismic]].


Subgroup: 3.5.7
Subgroup: 3.5.7
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Sval mapping generators: ~3, ~5
Sval mapping generators: ~3, ~5


[[POTE generator]]: ~5/3 = 878.042
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1903.863¢, ~5/3 = 878.923¢
* [[CWE]]: ~3 = 1901.955¢, ~5/3 = 878.291¢


[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]
[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]
[[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]]'s mapping for [[11/9]], mapping it to 5 generators down.


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[[EDT]]s: 15, 13e, 28e, 43dee
[[EDT]]s: 15, 13e, 28e, 43dee
Badness (Sintel): 2.507


== 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 will be documented below.


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7
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Sval mapping generators: ~3, ~9/7
Sval mapping generators: ~3, ~9/7


[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~9/7 = 440.4881
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1903.740¢, ~9/7 = 440.901¢
* [[CWE]]: ~3 = 1901.955¢, ~9/7 = 440.665¢


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


Badness (Sintel): 0.066
[[Badness]] (Sintel): 0.066


=== Alhena ===
=== Alhena ===
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[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~9/7 = 441.025
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~9/7 = 441.025


Supporting ETs: [[13edt|b13]], [[69edt|b69]], [[56edt|b56]], [[82edt|b82]], [[43edt|b43]], [[125edt|b125]], [[30edt|b30]], [[151edt|b151]], [[95edt|b95]], [[17edt|b17é]][+11/2], [[194edt|b194d]], [[99edt|b99]], [[181edt|b181d]], [[108edt|b108é]]
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]]
(Note that é is used as the wart for 11/2.)
(Note that é is used as the wart for 11/2.)


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=== Mintra ===
=== Mintra ===
''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]] and [[No-twos subgroup temperaments#Deneb|Deneb]].''
''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]], 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]].
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]].


[[Subgroup]]: 3.5.7.11
[[Subgroup]]: 3.5.7.11
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==== Tridecimal Mintra ====
==== Tridecimal Mintra ====
This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments#Tridecimal Mintaka|Tridecimal Mintaka]].
This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments#Tridecimal Mintaka|Tridecimal Mintaka]].


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[[Comma list]]: 245/243, 2025/2023
[[Comma list]]: 245/243, 2025/2023


{{Mapping|legend=2|1 1 2 2|0 4 -2 3}}
{{Mapping|legend=2|1 1 2 2|0 4 -2 5}}


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~17/15 = 220.142
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~17/15 = 220.142
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{{Mapping|legend=2|1 3 3|0 -5 -4}}
{{Mapping|legend=2|1 3 3|0 -5 -4}}


Sval mapping generators: ~3, ~7/5
Sval mapping generators: ~[[3/1|3]], ~[[7/5]]


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: ~3 = 1\1edt, ~7/5 = 584.017
* [[WE]]: ~3 = 1901.783¢, ~7/5 = 583.905¢
* [[POTE|PETE]]: ~3 = 1\1edt, ~7/5 = 583.9584
* [[CWE]]: ~3 = 1901.955¢, ~7/5 = 583.986¢


[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
[[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): 0.100
=== Suhail ===
Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].
[[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}}
Sval mapping generators: ~[[13/9]], ~[[65/63]]
Generator tunings:
: {| class="wikitable right-1"
|-
!
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| 634.144, 49.695
| 634.1448, 49.6946
|-
| [[Constrained_tuning|Constrained]]
| 1\b3 = 633.985, 49.733
| 1\b3 = 633.985, 49.839
|-
| [[POTE|Destretched]]
| 1\b3 = 633.985, 49.6825
| 1\b3 = 633.985, 49.6821
|}
[[Optimal ET sequence]]: [[39edt|b39]], [[114edt|b114]], [[153edt|b153]], [[498edt|b498cf]], [[651edt|b651cf]] <!-- b804cff is not a GPV -->
Badness (Sintel): 0.330


== Izar ==
== Izar ==
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Sval mapping generators: ~3, ~16807/10125
Sval mapping generators: ~3, ~16807/10125


[[Optimal tuning]] (CTE): ~3 = 1\1edt, ~16807/10125 = 877.280
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.958¢, ~16807/10125 = 877.283¢
* [[CWE]]: ~3 = 1901.955¢, ~16807/10125 = 877.281¢


[[Support]]ing [[ET]]s: {{EDs|b13, b11cd, b193, b15cd, b180, b24c, b167, b37c, b154, 141, b50c, b28cd, b128, b63c|equave=t}}
[[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): 0.017


== Nekkar ==
== Nekkar ==
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Sval mapping generators: ~3, ~16807/10935
Sval mapping generators: ~3, ~16807/10935


[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~16807/10935 = 776.767
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1900.155¢, ~16807/10935 = 775.963¢
* [[CWE]]: ~3 = 1901.955¢, ~16807/10935 = 776.767¢


[[Support]]ing [[ET]]s: 22, 49, 5c, 71, 27, 17c, 120, 93, 76c, 32cc, 169d, 115, 191d, 164d
[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]]
 
[[Badness]] (Sintel): 17.120


=== 3.5.7.11 subgroup ===
=== 3.5.7.11 subgroup ===
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[[Support]]ing [[ET]]s: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd
[[Support]]ing [[ET]]s: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd
Badness (Sintel): 1.375


=== 3.5.7.11.13 subgroup ===
=== 3.5.7.11.13 subgroup ===
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[[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf
[[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf
Badness (Sintel): 1.723
== 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.
[[Subgroup]]: 3.5.7
[[Comma list]]: 823543/820125
{{Mapping|legend=2|1 2 2|0 -7 -3}}
: sval mapping generators: ~3, ~17/9
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.198¢, ~49/45 = 145.412¢
* [[CWE]]: ~3 = 1901.955¢, ~49/45 = 145.368¢
[[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79
[[Badness]] (Sintel): 0.200
=== Erigone ===
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.
[[Subgroup]]: [[3.5.7.11.13_subgroup|3.5.7.11.13]]
[[Comma list]]: 847/845, 1575/1573, 4459/4455
{{Mapping|legend=2|1 9 5 9 7|0 -21 -9 -19 -13}}
: [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[49/33]]
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.9699, ~49/33 = 682.4486 <!-- IDK why the second rounded value disagrees with Sintel's calculator, I got it from FloraC's -->
* [[CWE]]: ~3 = 1\1edt, ~49/33 = 682.4427
[[Optimal ET sequence]]: [[25edt|b25ce]], [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]]
[[Badness]] (Sintel): 0.21396
==== Hemigone ====
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]]).
[[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
{{Mapping|legend=2|1 30 14 28 20 25 2|0 -42 -18 -38 -26 -33 1}}
: [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[19/9]]
[[Optimal tuning]] ([[WE]]): ~3 = 1902.0918, ~19/9 = 1292.3032
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~19/9 = 1292.2083
[[Optimal ET sequence]]: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]]
[[Badness]] (Sintel): 0.45479
==== <small>(no-2s) </small>23-limit ====
[[2277/2275]] may be used in the same way to extend the simpler [[#Erigone|erigone]] to the 3.5.7.11.13.23 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
{{Mapping|legend=2|1 30 14 28 20 25 2 64|0 -42 -18 -38 -26 -33 1 -90}}
: [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[19/9]]
[[Optimal tuning]] ([[WE]]): ~3 = 1902.0149, ~19/9 = 1292.2401
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~19/9 = 1292.1988
[[Optimal ET sequence]]: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]
[[Badness]] (Sintel): 0.54174


== Sirius ==
== Sirius ==
{{main|Sirius}}
{{main|Sirius}}


For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan#Overview to extensions]].
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]].


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7
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: sval mapping generators: ~3, ~25/21
: sval mapping generators: ~3, ~25/21


[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~25/21 = 293.740
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.445¢, ~25/21 = 293.739¢
* [[CWE]]: ~3 = 1901.955¢, ~25/21 = 293.759¢


[[Optimal ET sequence]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]], [[123edt|b123]], [[136edt|b136]]
[[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
 
=== Remus ===
{{main|Electra}}
 
By splitting the generator of Sirius into three, remus efficiently represents the no-2s 13-limit with MOS scales of 18, 25, 32, or 39 steps.
 
This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-sevens restriction of this temperament.
 
[[Subgroup]]: 3.5.7.11.13
 
[[Comma list]]: 275/273, 1625/1617, 1575/1573
 
{{Mapping|legend=2|1 4 6 5 6|0 -9 -15 -10 -13}}
 
: sval mapping generators: ~3, ~15/11
 
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~15/11 = 536.090
 
[[Support]]ing [[ET]]s: 39, 7, 32, 71, 110, 46, 149, 188, 181
 
Badness (Sintel): 0.286


=== Mizar ===
=== Mizar ===
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Badness (Sintel): 0.841
Badness (Sintel): 0.841
== Bohlenic ==
This temperament is identical to [[13edt]] (equal-tempered [[Bohlen–Pierce scale]]), but has an independent generator for 11.
[[Subgroup]]: 3.5.7.11
[[Comma list]]: 245/243, 3125/3087
{{Mapping|legend=2| 13 19 23 0 | 0 0 0 1 }}
: sval mapping generators: ~27/25, ~11
[[Optimal tuning]]s:
* [[CTE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4151.318¢
* [[CWE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4147.705¢
[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]
[[Badness]] (Sintel): 0.499
=== Full no-twos 13-limit ===
[[Subgroup]]: 3.5.7.11.13
[[Comma list]]: 245/243, 275/273, 847/845
{{Mapping|legend=2| 13 19 23 0 2 | 0 0 0 1 1 }}
[[Optimal tuning]]s:
* [[CTE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4149.733¢
* [[CWE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4146.033¢
[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]
[[Badness]] (Sintel): 0.365
== 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.
|}
{{center|<div style{{=}}"display: inline-grid; margin-right: 25px;">
{{(!}} 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 =
= 3.5.11 subgroup temperaments =
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[[EDT]]s: 28, 11, 17, 6, 39, 5, 67, 45, 50, 16, 23, 73, 61, 62
[[EDT]]s: 28, 11, 17, 6, 39, 5, 67, 45, 50, 16, 23, 73, 61, 62
=== Fomalhaut ===
Fomalhaut is an extension of Deneb to higher limits that splits the interval of [[11/3]] in three.
The 23-limit version of Fomalhaut was created first, as an attempt to approximate the no-2s, no-7s 23-limit as accurately as possible using 25 to 35 notes per equave, defined as the b28 & b33 temperament in this limit. Then the lower limit versions were created by simply extrapolating the temperament downwards.
Fomalhaut follows the convention of naming no-twos temperaments after stars.
[[Subgroup]]: 3.5.11.13
[[Comma list]]: 6655/6561, 274625/264627
[[Gencom]]: [3/1 99/65; 6655/6561 274625/264627]
[[Sval]] [[mapping]]: [{{val|1 5 1 -2}}, {{val|0 -9 3 11}}]
[[POTE generator]]: ~[[99/65]] = 748.0156
[[EDT]]s: {{EDs|b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ff|equave=t}}
* Complexity: 1.561892
* Adjusted Error: 6.495941 cents
* TE Error: 1.755451 cents/octave
==== 3.5.11.13.17 ====
[[Subgroup]]: 3.5.11.13.17
[[Comma list]]: 1105/1089, 4225/4131, 6655/6561
[[Gencom]]: [3/1 99/65; 1105/1089 4225/4131 6655/6561]
[[Sval]] [[mapping]]: [{{val|1 5 1 -2 1}}, {{val|0 -9 3 11 4}}]
[[POTE generator]]: ~[[17/11]] = 748.0236
[[EDT]]s: {{EDs|b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ffg|equave=t}}
* Complexity: 1.418914
* Adjusted Error: 6.431616 cents
* TE Error: 1.573498 cents/octave
==== 3.5.11.13.17.19 ====
[[Subgroup]]: 3.5.11.13.17.19
[[Comma list]]: 247/243, 325/323, 1105/1089, 4675/4617
[[Gencom]]: [3/1 99/65; 247/243 325/323 1105/1089 4675/4617]
[[Sval]] [[mapping]]: [{{val|1 5 1 -2 1 7}}, {{val|0 -9 3 11 4 -11}}]
[[POTE generator]]: ~[[17/11]] = 747.9960
[[EDT]]s: {{EDs|b28, b33, b5, b61, b56f, b23f, b38ch, b66ch, b89fgh, b10cfh|equave=t}}
* Complexity: 1.449992
* Adjusted Error: 6.125446 cents
* TE Error: 1.441985 cents/octave
==== 3.5.11.13.17.19.23 ====
[[Subgroup]]: 3.5.11.13.17.19.23
[[Comma list]]: 209/207, 247/243, 255/253, 325/323, 4675/4617
[[Gencom]]: [3/1 99/65; 209/207 247/243 255/253 325/323 4675/4617]
[[Sval]] [[mapping]]: [{{val|1 5 1 -2 1 7 6}}, {{val|0 -9 3 11 4 -11 -8}}]
[[POTE generator]]: ~[[17/11]] = 748.0874
[[EDT]]s: {{EDs|b28, b5, b33, b23f, b61, b56f, b38ch, b10cfhi, b66ch, b51ffg|equave=t}}
* Complexity: 1.382541
* Adjusted Error: 7.087107 cents
* TE Error: 1.566709 cents/octave


== Alnilam ==
== Alnilam ==
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== 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]].


[[Subgroup]]: 3.7.11
[[Subgroup]]: 3.7.11
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== 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]].
[[Subgroup]]: 3.7.11
[[Subgroup]]: 3.7.11


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=== 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


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[[Support]]ing [[ET]]s: {{EDs|b22, b175, b197, b153, b131, b219, b372, b109, b328, b241, b87, b21, b65, b43|equave=t}}
[[Support]]ing [[ET]]s: {{EDs|b22, b175, b197, b153, b131, b219, b372, b109, b328, b241, b87, b21, b65, b43|equave=t}}
==== 3.5.7.11.19 subgroup ====
Tempering out [[12005/11979]], the unisquary comma, sets 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 restriction of [[mohaha]].
[[Subgroup]]: 3.5.7.11.19
[[Comma list]]: 3971/3969, 12005/11979, 41553/41503
[[Sval]] [[mapping]]: [{{val| 1 0 2 2 3}}, {{val| 0 32 -5 4 -7}}]
Sval mapping generators: ~3, ~81/77
[[Optimal tuning]]s:
* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[81/77]] = 87.065
* [[CWE]]: ~3 = 1\1ed3, ~[[81/77]] = 87.066
[[Support]]ing [[ET]]s: {{EDs|b131, b22, b153, b284, b415, b109, b437, b175, b546, b87c, b699, b240, b590, b721|equave=t}}
=== 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]]~[[41edo]], [[131edt]], and [[197edt]]).
[[Subgroup]]: 3.7.11.13.17
[[Comma list]]: 14161/14157, 107811/107653, 1108809/1108723
[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2}}, {{val| 0 -15 12 22 38}}]
Sval mapping generators: ~3, ~119/117
[[Optimal tuning]]s:
* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[119/117]] = 28.979
* [[CTE]]: ~3 = 1\1ed3, ~[[119/117]] = 28.970
[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247f]], [[3216edt|b3216defff]]
==== 3.7.11.13.17.19 subgroup ====
This includes the natural extension of Mebsuta to prime 19.
[[Subgroup]]: 3.7.11.13.17.19
[[Comma list]]: 3213/3211, 3971/3969, 14161/14157, 41553/41503
[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 3}}, {{val| 0 -15 12 22 38 -21}}]
Sval mapping generators: ~3, ~119/117
[[Optimal tuning]]s:
* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[119/117]] = 28.973
* [[CTE]]: ~3 = 1\1ed3, ~[[119/117]] = 28.970
[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[525edt|b525]], [[722edt|b722]], [[919edt|b919]], [[2035edt|b2035df]]
==== 3.7.8.11.13.17.19 subgroup ====
This sets two-thirds of 11/9 to [[8/7]].
[[Subgroup]]: 3.7.8.11.13.17.19
[[Comma list]]: 513/512, 729/728, 833/832, 969/968, 3971/3969
[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2 3}}, {{val| 0 -15 -7 12 22 38 -21}}]
Sval mapping generators: ~3, ~64/63
[[Optimal tuning]]s:
* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[64/63]] = 28.978
* [[CTE]]: ~3 = 1\1ed3, ~[[64/63]] = 28.975
[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247âf]], [[1969edt|b1969ââf]]
(â is the wart for 8.)
==== 3.5.7.8.11.13.17.19.23 subgroup ====
At the cost of lower accuracy, [[Procyon]] can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit.
[[Subgroup]]: 3.5.7.8.11.13.17.19.23
[[Comma list]]: 361/360, 441/440, 513/512, 729/728, 833/832, 969/968, 1127/1125
[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2 2 3 4}}, {{val| 0 -35 -15 -7 12 22 38 -21 -75}}]
Sval mapping generators: ~3, ~64/63
[[Optimal tuning]]s:
* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[64/63]] = 29.032
* [[CTE]]: ~3 = 1\1ed3, ~[[64/63]] = 29.041
[[Optimal ET sequence]]: [[65edt|b65i]], [[66edt|b66i]], [[131edt|b131]]


= Other tritave-based subgroups =
= Other tritave-based subgroups =
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See: [[Catalog of 3.5.7 subgroup rank two temperaments#Projective tuning space diagrams]]
See: [[Catalog of 3.5.7 subgroup rank two temperaments#Projective tuning space diagrams]]


[[Category:Subgroup temperaments]]
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Nonoctave]]
[[Category:Non-octave temperaments]]
[[Category:Tritave]]
[[Category:Tritave-equivalent temperaments]]
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