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No-twos subgroup temperaments: Difference between revisions

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''(WIP, [[catalog of 3.5.7 subgroup rank two temperaments]] will eventually be redirected 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]].


== Overview by mapping of 5 ==
= Overview by mapping of 5 =
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 are with 3/1 period and ~5/3 generator. There is one-to-one correspondence between 3.5 subgroup and mapped intervals.
* 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.
* BPS is with 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 is with 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 is with 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 is with a ~7/5 generator, five of which give the ~27/5 (9/5 and an tritave).
* Canopus has a ~7/5 generator, five of which give the ~27/5 (9/5 up a tritave).
* Alnilam is with a ~81/55 generator, ten of which give the ~243/5 (9/5 and three tritave).
* Alnilam has a ~81/55 generator, ten of which give the ~243/5 (9/5 up three tritaves).
* Izar is with a ~16807/10125 generator, twelve of which give the ~2187/5 (9/5 and five tritave).
* Izar has a ~16807/10125 generator, twelve of which give the ~2187/5 (9/5 up five tritaves).
* Nekkar has a ~16807/10935 generator, sixteen of which give the ~6561/5 (9/5 up six 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).
* 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.


= 3.5.7 subgroup temperaments =
== Arcturus ==
== Arcturus ==
''see [[Arcturus]], [[Arcturus clan]]''
{{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 [[Gamelismic clan #Superkleismic|superkleismic]].


Subgroup: 3.5.7
Subgroup: 3.5.7
Line 25: Line 33:
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]]


== Aldebaran ==
[[Badness]] (Sintel): 0.535
''see [[Aldebaran]]''
 
=== Polturus ===
This extension of Arcturus adds [[Polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
 
[[Subgroup]]: 3.5.7.11
 
[[Comma list]]: 15625/15309, 177147/171875


[[Subgroup]]: 3.5.13
[[Gencom]]: [3/1 5/3; 15625/15309 177147/171875]


[[Comma list]]: 3159/3125
[[Mapping]]: [{{val|1 1 -1 5}}, {{val|0 1 6 -6}}]


[[Mapping]]: [{{val|1 0 5}}, {{val|0 1 -2}}]
[[POTE generator]]: ~[[5/3]] = 884.268


[[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]
[[EDT]]s: 15, 13e, 28e, 43dee


[[CTE tuning|CTE generator]]: ~[[5/3]] = 887.76
Badness (Sintel): 2.507


== BPS ==
== BPS ==
{{main|BPS}}
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


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


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


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]]
==Canopus==
 
[[Badness]] (Sintel): 0.066
 
=== Alhena ===
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]]: 196/195, 325/324, 1001/1000
 
[[Sval]] [[mapping]]: [{{val| 1 1 2 -1 2}}, {{val| 0 2 -1 11 -4}}]
 
Sval mapping generators: ~3, ~7/3
 
[[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]], [[194edt|b194d]], [[99edt|b99]], [[181edt|b181d]], [[108edt|b108é]]
(Note that é is used as the wart for 11/2.)
 
Badness (Sintel): 0.187
 
=== Mintra ===
''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]] and [[No-twos subgroup temperaments#Deneb|Deneb]].''
 
This temperament splits 27/7 (the BPS generator up a tritave) into three by means of [[11/7]] or, equivalently, [[7/1]] in three by means of [[21/11]], and is the intersection of BPS, Deneb, 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
 
[[Comma list]]: 245/243, 1331/1323
 
{{Mapping|legend=2|1 5 0 1|0 -6 3 2}}
 
Sval mapping generators: ~3, ~21/11
 
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~11/7 = 780.752
 
[[Support]]ing [[ET]]s: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90
 
Badness (Sintel): 0.302
 
==== Tridecimal Mintra ====
This temperament uses the canonical extension for prime 13 described at [[No-twos subgroup temperaments#Tridecimal Mintaka|Tridecimal Mintaka]].
 
[[Subgroup]]: 3.5.7.11.13
 
[[Comma list]]: 245/243, 275/273, 1575/1573
 
{{Mapping|legend=2|1 5 0 1 10|0 -6 3 2 -13}}
 
Sval mapping generators: ~3, ~21/11
 
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~11/7 = 780.428
 
[[Support]]ing [[ET]]s: 39, 17, 22, 56, 5f, 61, 95, 100, 134, 73f, 139cf, 83cf, 173e, 178cef
 
Badness (Sintel): 0.373
 
=== 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".
 
[[Subgroup]]: 3.5.7.17
 
[[Comma list]]: 245/243, 2025/2023
 
{{Mapping|legend=2|1 1 2 2|0 4 -2 5}}
 
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~17/15 = 220.142
 
[[Support]]ing [[ET]]s: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44
 
Badness (Sintel): 0.177
 
== Canopus ==
{{main|Canopus}}
 
For extensions to this temperament that include the prime 2, see [[Mirkwai clan]]. No-twos extensions will be documented below.
 
[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7


Line 61: Line 161:
{{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:
* [[WE]]: ~3 = 1901.783¢, ~7/5 = 583.905¢
* [[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]]
 
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}}


[[Optimal tuning]] (CTE): ~3 = 1\1edt, ~7/5 = 584.017
Sval mapping generators: ~[[13/9]], ~[[65/63]]


[[Support]]ing [[ET]]s: {{EDs|b13, b10, b75, b62, b23, b49, b36, b16, b7cd, 29, b42c, b19cd, b33cd, b55c|equave=t}}
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 ==
Line 76: Line 216:
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]]
 
[[Badness]] (Sintel): 0.017
 
== 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]].
 
[[Subgroup]]: 3.5.7
 
[[Comma list]]: 35303692060125/33232930569601
 
{{Mapping|legend=2|1 8 3|0 -16 -3}}
 
Sval mapping generators: ~3, ~16807/10935
 
[[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]]
 
[[Badness]] (Sintel): 17.120
 
=== 3.5.7.11 subgroup ===
''See also [[No-twos subgroup temperaments#Mintaka|Mintaka]].''
 
This continues the canonical 11-limit extension of squares.
 
[[Subgroup]]: 3.5.7.11
 
[[Comma list]]: 1331/1323, 120285/117649
 
{{Mapping|legend=2|1 8 3 3|0 -16 -3 -2}}


[[Support]]ing [[ET]]s: {{EDs|b13, b11cd, b193, b15cd, b180, b24c, b167, b37c, b154, 141, b50c, b28cd, b128, b63c|equave=t}}
Sval mapping generators: ~3, ~11/7
== No-twos-or-threes subgroup temperaments ==
 
=== Antipyth ===
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~11/7 = 776.781
''see [[Antipyth]]''
 
[[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 ===
This uses the [[No-twos subgroup temperaments#Minalzidar|Minalzidar]] mapping of 13.
 
[[Subgroup]]: 3.5.7.11.13
 
[[Comma list]]: 169/165, 351/343, 11011/10935
 
{{Mapping|legend=2|1 8 3 3 6|0 -16 -3 -2 -9}}
 
Sval mapping generators: ~3, ~11/7
 
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~11/7 = 776.678
 
[[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 ==
{{main|Sirius}}
 
This tempers out the [[Don Page comma]] between [[5/3]] and [[7/5]], allowing an accurate representation of the 3:5:7 chord, similar to the 5:7:9 in Procyon.
 
For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].
 
[[Subgroup]]: 3.5.7
 
[[Comma list]]: 3125/3087
 
{{Mapping|legend=2| 1 1 1 | 0 3 5 }}
 
: sval mapping generators: ~3, ~25/21
 
[[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
 
Badness (Sintel): 0.286
 
=== Mizar ===
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.
 
[[Subgroup]]: 3.5.7.17
 
[[Comma list]]: 3125/3087, 2025/2023
 
{{Mapping|legend=2|1 -2 -4 2|0 6 10 1}}
 
: sval mapping generators: ~3, ~17/9
 
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~17/9 = 1097.800
 
[[Support]]ing [[ET]]s: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd
 
Badness (Sintel): 0.383
 
==== Full no-twos 17-limit ====
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
 
[[Comma list]]: 275/273, 459/455, 1625/1617, 2025/2023
 
{{Mapping|legend=2|1 -2 -4 12 11 2|0 6 10 -17 -15 1}}
 
: sval mapping generators: ~3, ~17/9
 
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~17/9 = 1098.298
 
[[Support]]ing [[ET]]s: 26, 71, 45, 19, 97f, 116d
 
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;">
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{{!}} [[File:357ptslines12n.png|320px]]
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{{!}} Both sets, labelled by name
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{{center|<div style{{=}}"display: inline-grid; margin-right: 25px;">
{{(!}} class{{=}}"wikitable"
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{{!}} [[File:357ptslines1c.png|320px]]
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{{!}} Temperaments supported by 13edt, labelled by comma
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{{(!}} class{{=}}"wikitable"
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{{!}} [[File:357ptslines2c.png|320px]]
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{{!}} Temperaments not supported by 13edt, labelled by comma
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{{!}} [[File:357ptslines12c.png|320px]]
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{{!}} Both sets, labeled by comma
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= 3.5.11 subgroup temperaments =
 
== Polaris ==
{{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.
 
[[Subgroup]]: 3.5.11
 
[[Comma list]]: 177147/171875
 
[[Gencom]]: [3/1 5/3; 177147/171875]
 
[[Sval]] [[mapping]]: [{{val|1 2 1}}, {{val|0 1 -6}}]
 
[[POTE generator]]: ~5/3 = 892.6
 
[[EDT]]s: 17, 15, 32, 49, 13[+11], 47, 19, 11[+11], 81, 66, 79[+11], 62[+11], 28[+11], 21[-11]
 
== Deneb ==
{{main|Deneb}}
 
[[Subgroup]]: 3.5.11
 
[[Comma list]]: 6655/6561
 
[[Gencom]]: [3/1 11/9; 6655/6561]
 
[[Sval]] [[mapping]]: [{{val|1 2 2}}, {{val|0 -3 1}}]
 
[[POTE generator]]: ~[[11/9]] = 340.242
 
[[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 ==
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]].
 
[[Subgroup]]: 3.5.11
 
[[Comma list]]: {{monzo|0 -35 9 0 10}}
 
[[Gencom]]: [3/1 81/55; {{monzo|0 -35 9 0 10}}]
 
[[Sval]] [[mapping]]: [{{val|1 5 -1}}, {{val|0 -10 9}}]
 
[[CTE tuning|CTE generator]]: ~81/55 = 672.410
 
[[EDT]]s: 99, 17, 82, 116, 181, 65, 14[-5], 280, 48, 215, 31, 133, 314, 263
 
= 3.7.11 subgroup temperaments =
 
== 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
 
[[Comma list]]: 1331/1323
 
[[Sval]] [[mapping]]: [{{val| 1 0 1 }}, {{val| 0 3 2 }}]
 
Sval mapping generators: ~3, ~21/11
 
[[Optimal tuning]]s:
* 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}}
 
=== 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]].
 
[[Subgroup]]: 3.7.11.13
 
[[Comma list]]: 1331/1323, 218491/216513
 
[[Sval]] [[mapping]]: [{{val| 1 0 1 10}}, {{val| 0 3 2 -13}}]
 
Sval mapping generators: ~3, ~21/11
 
[[Optimal tuning]]s:
* 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}}
 
=== 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]].
 
[[Subgroup]]: 3.7.11.13
 
[[Comma list]]: 1331/1323, 351/343
 
[[Sval]] [[mapping]]: [{{val| 1 0 1 -3}}, {{val| 0 3 2 9}}]
 
Sval mapping generators: ~3, ~21/11
 
[[Optimal tuning]]s:
* 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}}
 
== 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
 
[[Comma list]]: 387420489/386683451
 
[[Sval]] [[mapping]]: [{{val| 1 2 2}}, {{val| 0 -5 4 }}]
 
Sval mapping generators: ~3, ~81/77
 
[[Optimal tuning]]s:
* 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}}
 
=== 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
 
[[Comma list]]: 3971/3969, 41553/41503
 
[[Sval]] [[mapping]]: [{{val| 1 2 2 3}}, {{val| 0 -5 4 -7}}]
 
Sval mapping generators: ~3, ~81/77
 
[[Optimal tuning]]s:
* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[81/77]] = 86.929
* [[CWE]]: ~3 = 1\1ed3, ~[[81/77]] = 86.932
 
[[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 =
 
== Aldebaran ==
{{main|Aldebaran}}
 
[[Subgroup]]: 3.5.13
 
[[Comma list]]: 3159/3125
 
[[Sval]] [[mapping]]: [{{val|1 0 5}}, {{val|0 1 -2}}]
 
[[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]
 
[[CTE tuning|CTE generator]]: ~[[5/3]] = 887.76
 
== Keladic ==
[[Subgroup]]: 3.7.13
 
[[Comma list]]: 351/343
 
[[Sval]] [[mapping]]: [{{val| 1 1 0 }}, {{val| 0 1 3 }}]
 
Sval mapping generators: ~3, ~7/3
 
[[Optimal tuning]]s:
* 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}}
 
== Sadalmelik ==
[[Subgroup]]: 3.13.17
 
[[Comma list]]: 85293/83521
 
[[Sval]] [[mapping]]: [{{val| 1 0 2 }}, {{val| 0 4 1 }}]
 
Sval mapping generators: ~3, ~17/9
 
[[Optimal tuning]]s:
* [[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}}
 
= No-twos-or-threes subgroup temperaments =
== Antipyth ==
{{main|Antipyth}}


[[Subgroup]]: 5.7.11
[[Subgroup]]: 5.7.11
Line 95: Line 879:
[[Support]]ing [[ET]]s: {{EDs|c14, c5, c19, c33, c47, c9e, c61, c75, c23e, c24e, c52e, c80e, c89e, c37e|equave=5}}
[[Support]]ing [[ET]]s: {{EDs|c14, c5, c19, c33, c47, c9e, c61, c75, c23e, c24e, c52e, c80e, c89e, c37e|equave=5}}


=== Juggernaut ===
== Juggernaut ==
{{main|Juggernaut}}
 
[[Subgroup]]: 5.7.11
[[Subgroup]]: 5.7.11


Line 108: Line 894:
[[Support]]ing [[ET]]s: {{EDs|c14, c10, c6, c18, c24, c22, c32, c16, c38, c8d, c34, c26d, c46, c52e|equave=5}}
[[Support]]ing [[ET]]s: {{EDs|c14, c10, c6, c18, c24, c22, c32, c16, c38, c8d, c34, c26d, c46, c52e|equave=5}}


==== No-twos-or-threes 13-limit ====
=== Tridecimal juggernaut ===
[[Subgroup]]: 5.7.11.13
[[Subgroup]]: 5.7.11.13


Line 121: Line 907:
[[Support]]ing [[ET]]s: {{EDs|c10, c14, c6, c24, c34, c16f, c44, c18f, c38, c26f, c54, c64|equave=5}}
[[Support]]ing [[ET]]s: {{EDs|c10, c14, c6, c24, c34, c16f, c44, c18f, c38, c26f, c54, c64|equave=5}}


=Graphs=
= Graphs =
 
See: [[Catalog of 3.5.7 subgroup rank two temperaments#Graphs]]
{| class="wikitable"
|-
| | [[File:357plot_cplx_damage.png|alt=357plot_cplx_damage.png|357plot_cplx_damage.png]]
|-
| | Complexity vs. damage plot.  z&lt;1 corresponds to the "Middle Path" inclusion criterion.
|}
 
=Projective tuning space diagrams=


{| class="wikitable"
== Projective tuning space diagrams ==
|-
See: [[Catalog of 3.5.7 subgroup rank two temperaments#Projective tuning space diagrams]]
| | [[File:357ptslines1n.png|320px]]
|-
| | Temperaments with smaller commas, labeled by name
|}
 
{| class="wikitable"
|-
| | [[File:357ptslines1c.png|320px]]
|-
| | Temperaments with smaller commas, labeled by comma
|}
 
{| class="wikitable"
|-
| | [[File:357ptslines2n.png|320px]]
|-
| | Temperaments with larger commas, labeled by name
|}
 
{| class="wikitable"
|-
| | [[File:357ptslines2c.png|320px]]
|-
| | Temperaments with larger commas, labeled by comma
|}
 
{| class="wikitable"
|-
| | [[File:357ptslines12n.png|320px]]
|-
| | Both sets, labeled by name
|}
 
{| class="wikitable"
|-
| | [[File:357ptslines12c.png|320px]]
|-
| | Both sets, labeled by comma
|}
 
<span style="display: block; height: 1px; left: 0px; overflow: hidden; position: absolute; top: 2151px; width: 1px;">
 
{| class="wikitable"
|-
| | 11.675
|}
</span>


[[Category:Subgroup temperaments]]
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Nonoctave]]
[[Category:Non-octave temperaments]]
[[Category:Tritave]]
Retrieved from "https://en.xen.wiki/w/No-twos_subgroup_temperaments"