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No-twos subgroup temperaments: Difference between revisions

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Complete data for suhail and erigone
 
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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 ~15/7 generator, five of which give the ~45/1 (5/1 up two tritaves).
* Canopus has a ~15/7 generator, five of which give the ~45/1 (5/1 up two tritaves).
* Miaplacidus has a ~7/5 generator, eight of which give the ~15/1 (5/1 up one tritave).
* Alnilam has a ~55/27 generator, ten of which give the ~1215/1 (5/1 up five tritaves).
* Alnilam has a ~55/27 generator, ten of which give the ~1215/1 (5/1 up five tritaves).
* Izar has a ~30375/16807 generator, twelve of which give the ~1215/1 (5/1 up five tritaves).
* Izar has a ~30375/16807 generator, twelve of which give the ~1215/1 (5/1 up five tritaves).
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=== Polturus ===
=== Polturus ===
This extension of arcturus adds [[polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.
This extension of arcturus adds [[#Polaris|polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.


Subgroup: 3.5.7.11
Subgroup: 3.5.7.11
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Badness (Sintel): 2.51
Badness (Sintel): 2.51
== Antares ==
Antares tempers out [[21875/19683]], the amount by which, assuming [[tritave equivalence]], five [[5/3]]'s exceed [[9/7]], or equivalently the amount by which five [[9/5]]'s fall short of [[7/3]]. It is basically to [[#Arcturus|arcturus]] what [[mavila family #Mavila|mavila]] is to [[meantone family #Meantone|meantone]], and is even more inaccurate than mavila. It falls into [[exotemperament]] range with the "comma" it tempers out (~182.8{{cent}}) being greater than a [[Bohlen–Pierce]] step. In terms of diatonic interval regions, the ~[[9/5]] is extremely sharp and becomes a [[neutral seventh]], and the ~[[7/5]] becomes a [[supermajor third]].
Antares was named by {{u|CompactStar}} after the star {{w|Antares}}, following a series of non-octave temperaments that were named after stars.
[[Subgroup]]: 3.5.7
[[Comma list]]: 21875/19683
{{Mapping|legend=3| 1 0 9 | 0 1 -5 }}
: mapping generators: ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1913.7665{{c}}, ~5/3 = 858.1255{{c}}
: [[error map]]: {{val| +11.811 -14.422 -4.388 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 851.8677{{c}}
: error map: {{val| 0.000 -32.391 -20.844 }}
[[Optimal ET sequence]]: [[9edt|b9]], [[20edt|b20]], [[29edt|b29]]
[[Badness]] (Sintel): 2.95


== BPS ==
== BPS ==
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Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


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


{{Mapping|legend=2| 1 5 0 1 10 | 0 -6 3 2 -13 }}
{{Mapping|legend=2| 1 5 0 1 10 | 0 -6 3 2 -13 }}
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{{Main| Dubhe }}
{{Main| Dubhe }}


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


Subgroup: 3.5.7.17
Subgroup: 3.5.7.17


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


{{Mapping|legend=2| 1 1 2 2 | 0 4 -2 5 }}
{{Mapping|legend=2| 1 1 2 2 | 0 4 -2 5 }}
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Badness (Sintel): 0.177
Badness (Sintel): 0.177


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


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


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


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


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
* [[WE]]: ~3 = 1902.4455{{c}}, ~25/21 = 293.7393{{c}}
: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
: [[error map]]: {{val| +0.490 -2.650 +2.316 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~25/21 = 293.7594{{c}}
: error map: {{val| 0.000 -0.381 +1.093 }}
: error map: {{val| 0.000 -3.080 +1.926 }}
 
[[Optimal ET sequence]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]],  [[123edt|b123]], [[136edt|b136]]


[[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): 0.213


[[Badness]] (Sintel): 0.0996
=== 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]].


=== Suhail ===
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.
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
Subgroup: 3.5.7.11


Comma list: 9375/9317, 12005/11979
Comma list: 3125/3087, 295245/290521


{{Mapping|legend=2| 3 4 5 6 | 0 5 4 7 }}
{{Mapping|legend=2| 1 -2 -4 12 | 0 6 10 -17 }}
: mapping generators: ~539/375, ~77/75
: mapping generators: ~3, ~729/385


Optimal tunings:  
Optimal tunings:  
* WE: ~539/375 = 634.9453{{c}}, ~77/75 = 49.8753{{c}}
* WE: ~3 = 1901.0672{{c}}, ~729/385 = 1097.6627{{c}}
* CWE: ~539/375 = 633.9850{{c}}, ~77/75 = 49.8596{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.1722{{c}}


Optimal ET sequence: [[36edt|b36]], [[39edt|b39]], [[75edt|b75]], [[114edt|b114]], [[153edt|b153]], [[267edt|b267]]
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]], [[168edt|b168]]


Badness (Sintel): 0.421
Badness (Sintel): 1.82


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~13/9 = 634.1444{{c}}, ~65/63 = 49.6946{{c}}
* WE: ~3 = 1902.6279{{c}}, ~11/5 = 1366.3374{{c}}
* CWE: ~13/9 = 633.9850{{c}}, ~65/63 = 49.7330{{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]]


Optimal ET sequence: [[36edt|b36]], [[39edt|b39]], [[114edt|b114]], [[153edt|b153]], [[498edt|b498cf]], [[651edt|b651cf]]
Badness (Sintel): 0.538


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


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


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


{{Mapping|legend=3| 1 -5 -2 | 0 12 7 }}
Optimal tunings:
: mapping generators: ~3, ~30375/16807
* WE: ~3 = 1902.4456{{c}}, ~11/5 = 1366.1830{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8649{{c}}


[[Optimal tuning]]s:
Optimal ET sequence: [[7edt|b7]], [[25edt|b25df]], [[32edt|b32]], [[39edt|b39]], [[110edt|b110]], [[149edt|b149]]
* [[WE]]: ~3 = 1901.9584{{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 ET sequence]]: [[13edt|b13]], [[141edt|b141]], [[154edt|b154]], …, [[258edt|b258]], [[271edt|b271]], [[800edt|b800]], [[1071edt|b1071]], [[1342edt|b1342]], [[1613edt|b1613]], [[4568edt|b4568]], [[6181edt|b6181]]
Badness (Sintel): 0.286


[[Badness]] (Sintel): 0.0166
== Canopus ==
{{Main| Canopus }}


== Nekkar ==
For extensions to this temperament that include the prime 2, see [[Canopic clan]]. No-2's extensions will be documented below.
This temperament is the no-2's [[restriction]] of [[squares]], and as such is named after a star that belonged to the obsolete constellation of Quadrans Muralis, whose name has to do with squares. However, seeing the sheer complexity and size of the commas, nekkar is much more naturally thought of as 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]]: {{monzo| -24 -3 16 }}
[[Comma list]]: 16875/16807


{{Mapping|legend=3| 1 -8 0 | 0 16 3 }}
{{Mapping|legend=3| 1 -2 -1 | 0 5 4 }}
: mapping generators: ~3, ~32805/16807
: mapping generators: ~3, ~15/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1900.1550{{c}}, ~32805/16807 = 1124.1916{{c}}
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
: [[error map]]: {{val| -1.800 -0.488 +3.749 }}
: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~32805/16807 = 1125.1876{{c}}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}
: error map: {{val| 0.000 +1.047 +6.737 }}
: 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]]


[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[311edt|b311dd]]
[[Badness]] (Sintel): 0.0996


[[Badness]] (Sintel): 17.1
=== Suhail ===
Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].


=== 3.5.7.11 subgroup ===
Subgroup: 3.5.7.11
Subgroup: 3.5.7.11


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1900.6084{{c}}, ~21/11 = 1124.4193{{c}}
* WE: ~539/375 = 634.9453{{c}}, ~77/75 = 49.8753{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.1738{{c}}
* CWE: ~539/375 = 633.9850{{c}}, ~77/75 = 49.8596{{c}}


Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[262edt|b262d]]
Optimal ET sequence: [[36edt|b36]], [[39edt|b39]], [[75edt|b75]], [[114edt|b114]], [[153edt|b153]], [[267edt|b267]]


Badness (Sintel): 1.37
Badness (Sintel): 0.421


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


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


{{Mapping|legend=2| 1 -8 0 1 -3 | 0 16 3 2 9 }}
{{Mapping|legend=2| 3 4 5 6 7 | 0 5 4 7 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1902.3248{{c}}, ~21/11 = 1125.4837{{c}}
* WE: ~13/9 = 634.1444{{c}}, ~65/63 = 49.6946{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.2772{{c}}
* CWE: ~13/9 = 633.9850{{c}}, ~65/63 = 49.7330{{c}}


Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71f]]
Optimal ET sequence: [[36edt|b36]], [[39edt|b39]], [[114edt|b114]], [[153edt|b153]], [[498edt|b498cf]], [[651edt|b651cf]]


Badness (Sintel): 1.72
Badness (Sintel): 0.330


== Procyon ==
== Procyon ==
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===== Hemigone =====
===== Hemigone =====
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11.13.17). }}
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.
 
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
Subgroup: 3.5.7.11.13.17
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Badness (Sintel): 0.542
Badness (Sintel): 0.542


== Sirius ==
== Miaplacidus ==
{{Main| Sirius }}
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 }}


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]].
[[Optimal ET sequence]]: [[13edt|b13]], [[120edt|b120]], [[133edt|b133]], …, [[211edt|b211]], [[224edt|b224]]


For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].
[[Badness]] (Sintel): 1.24


== Izar ==
[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7


[[Comma list]]: 3125/3087
[[Comma list]]: 13841287201/13839609375


{{Mapping|legend=3| 1 1 1 | 0 3 5 }}
{{Mapping|legend=3| 1 -5 -2 | 0 12 7 }}
: mapping generators: ~3, ~25/21
: mapping generators: ~3, ~30375/16807


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1902.4455{{c}}, ~25/21 = 293.7393{{c}}
* [[WE]]: ~3 = 1901.9584{{c}}, ~30375/16807 = 1024.6759{{c}}
: [[error map]]: {{val| +0.490 -2.650 +2.316 }}
: [[error map]]: {{val| +0.003 +0.005 -0.012 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~25/21 = 293.7594{{c}}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~30375/16807 = 1024.6743{{c}}
: error map: {{val| 0.000 -3.080 +1.926 }}
: error map: {{val| 0.000 +0.002 -0.016 }}
 
[[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]]


[[Optimal ET sequence]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]],  [[123edt|b123]], [[136edt|b136]]
[[Badness]] (Sintel): 0.0166


[[Badness]] (Sintel): 0.213
== Nekkar ==
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.


=== Mizar ===
[[Subgroup]]: 3.5.7
{{Todo|inline=1|complete section|comment= Catalog the intermediate extensions (3.5.7.11, 3.5.7.11.13). }}


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]].
[[Comma list]]: {{monzo| -24 -3 16 }}


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


==== 3.5.7.11.13.17 subgroup ====
[[Optimal tuning]]s:
Subgroup: 3.5.7.11.13.17
* [[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 }}


Comma list: 275/273, 459/455, 1625/1617, 2025/2023
[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[311edt|b311dd]]


{{Mapping|legend=2| 1 -2 -4 12 11 2 | 0 6 10 -17 -15 1 }}
[[Badness]] (Sintel): 17.1
: mapping generators: ~3, ~17/9


Optimal tunings:
=== 3.5.7.11 subgroup ===
* WE: ~3 = 1901.0269{{c}}, ~17/9 = 1097.7583{{c}}
Subgroup: 3.5.7.11
* CWE: ~3 = 1901.9550{{c}}, ~17/9 = 1098.2979{{c}}


Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]]
Comma list: 1331/1323, 120285/117649


Badness (Sintel): 0.841
{{Mapping|legend=2| 1 -8 0 1 | 0 16 3 2 }}


=== Remus ===
Optimal tunings:
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}
* WE: ~3 = 1900.6084{{c}}, ~21/11 = 1124.4193{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.1738{{c}}


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.
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[262edt|b262d]]


This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.
Badness (Sintel): 1.37


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


Comma list: 275/273, 1625/1617, 1575/1573
Comma list: 169/165, 351/343, 11011/10935


{{Mapping|legend=2| 1 -5 -9 -5 -7 | 0 9 15 10 13 }}
{{Mapping|legend=2| 1 -8 0 1 -3 | 0 16 3 2 9 }}
: mapping generators: ~3, ~11/5


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1902.4456{{c}}, ~11/5 = 1366.1830{{c}}
* WE: ~3 = 1902.3248{{c}}, ~21/11 = 1125.4837{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8649{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.2772{{c}}


Optimal ET sequence: [[7edt|b7]], [[25edt|b25df]], [[32edt|b32]], [[39edt|b39]], [[110edt|b110]], [[149edt|b149]]
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71f]]


Badness (Sintel): 0.286
Badness (Sintel): 1.72


== Bohlenic ==
== Bohlenic ==
Line 475: Line 571:
[[Comma list]]: 245/243, 3125/3087
[[Comma list]]: 245/243, 3125/3087


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


Line 491: Line 587:
Comma list: 245/243, 275/273, 847/845
Comma list: 245/243, 275/273, 847/845


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


Optimal tunings:  
Optimal tunings:  
Line 556: Line 652:
= 3.5.11-subgroup temperaments =
= 3.5.11-subgroup temperaments =
== Polaris ==
== Polaris ==
{{Main| Polaris }}
Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]].  It contains [[mos scale]]s of the families {{mos scalesig|link=1|2L 5s<3/1>}}, {{mos scalesig|link=1|2L 7s<3/1>}}, and {{mos scalesig|link=1|2L 9s<3/1>}}. The hendecatonic scale is generally the most usable of these, with the heptatonic and enneatonic scales usually being lopsided. Good tunings for polaris include [[15edt]], [[17edt]] and [[32edt]].
 
Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]].  


[[Subgroup]]: 3.5.11
[[Subgroup]]: 3.5.11
Line 576: Line 670:


[[Badness]] (Sintel): 1.01
[[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 ==
== Deneb ==
{{Main| Deneb }}
{{Main| Deneb }}
Deneb tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]].


[[Subgroup]]: 3.5.11
[[Subgroup]]: 3.5.11
Line 663: Line 801:


Badness (Sintel): 0.838
Badness (Sintel): 0.838
== Electra ==
[[File:Electra chords.wav|thumb|Just 11:13:15 and 143:165:195 triads on A]]
Electra tempers out [[1875/1859]] and [[14641/14625]] in the 3.5.11.13 subgroup, with a generator near [[15/11]] or its tritave complement, [[11/5]]. It possesses [[mos scale]]s of the families {{mos scalesig|link=1|4L 3s<3/1>}}, {{mos scalesig|link=1|7L 4s<3/1>}}, and {{mos scalesig|link=1|7L 11s<3/1>}}.
The 11:13:15 triad is a characteristic feature of many electra scales, and has been suggeested by {{u|CompactStar}} as a fundamental sonority for no-2's music, due to being relatively consonant compared to other no-2's chords. There is also a "minor" version of this triad, 143:165:195. If 11:13:15 is treated as the fundamental sonority of no-2's music, then electra can be viewed as the analogue to [[meantone family #Meantone|meantone]].
The name was given by CompactStar in 2023 after the star {{w|Electra (star)|Electra}}, following a series of non-octave temperaments that were named after stars.
[[Subgroup]]: 3.5.11
[[Comma list]]: 2373046875/2357947691
{{Mapping|legend=3| 1 -5 -5 | 0 9 10 }}
: mapping generators: ~3, ~11/5
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.8666{{c}}, ~11/5 = 1366.1408{{c}}
: [[error map]]: {{val| -0.088 -0.379 +0.757 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~11/5 = 1366.1964{{c}}
: error map: {{val| 0.000 -0.321 +0.8712 }}
[[Optimal ET sequence]]: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[1239edt|b1239e]], [[1310edt|b1310e]], …, [[1665edt|b1665e]], [[1736edt|b1736e]]
[[Badness]] (Sintel): 0.656
=== 3.5.11.13 subgroup ===
Subgroup: 3.5.11.13
Comma list: 1875/1859, 14641/14625
{{Mapping|legend=2| 1 -5 -5 -7 | 0 9 10 13 }}
Optimal tunings:
* WE: ~3 = 1901.8899{{c}}, ~11/5 = 1365.9871{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1366.0281{{c}}
Optimal ET sequence: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[181edt|b181]]
Badness (Sintel): 0.405
; Scales
* [[Electra7]]


== Alnilam ==
== Alnilam ==
Line 782: Line 964:


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


It is also possible to set two-thirds of 11/9 to [[8/7]], giving rise to an add-8 extension.  
It is also possible to set two-thirds of 11/9 to [[8/7]], giving rise to an add-8 extension.  
Subgroup: 3.7.11.13
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}}
Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]]
Badness (Sintel): 1.26


==== 3.7.11.13.17 subgroup ====
==== 3.7.11.13.17 subgroup ====
Line 794: Line 989:


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


Optimal tunings:
Optimal tunings:
Line 878: Line 1,072:


= No-twos no-threes subgroup temperaments =
= No-twos no-threes subgroup temperaments =
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, it makes the 5th harmonic into the equave. Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical.
{{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
 
[[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 ET sequence]]: TBA
 
[[Badness]] (Sintel): 0.277


== Antipyth ==
== Antipyth ==
Line 937: Line 1,175:
Badness (Sintel): 0.116
Badness (Sintel): 0.116


= Graphs =
== Catfish ==
See: [[Catalog of 3.5.7 subgroup rank two temperaments #Graphs]]
[[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
 
=== 5.11.13.17 subgroup ===
Subgroup: 5.11.13.17
 
Comma list: 54925/54043, 1795625/1771561
 
{{Mapping|legend=2| 1 -4 -10 -8 | 0 9 19 16 }}
: mapping generators: ~5, ~325/121
 
Optimal tunings:
* WE: ~5 = 2785.0096{{c}}, ~325/121 = 1699.3010{{c}}
* CWE: ~5 = 2786.3137{{c}}, ~325/121 = 1700.0352{{c}}
 
Optimal ET sequence: [[18ed5|c18]], [[41ed5|c41]], [[59ed5|c59]]
 
Badness (Sintel): 0.905


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


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