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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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=== 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]].''


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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¢
 
[[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): 0.017


== 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¢
 
[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]]


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


[[Optimal ET sequence]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]],  [[123edt|b123]], [[136edt|b136]]
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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=== Fomalhaut ===
=== 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 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.
Fomalhaut follows the convention of naming no-twos temperaments after stars.
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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 ===
==== 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]].
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]].


Line 501: Line 748:


[[Support]]ing [[ET]]s: {{EDs|b131, b22, b153, b284, b415, b109, b437, b175, b546, b87c, b699, b240, b590, b721|equave=t}}
[[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 =
Line 597: Line 913:
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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