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No-twos subgroup temperaments: Difference between revisions

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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 }}
Line 153: Line 153:
{{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 }}
Line 170: Line 170:
Badness (Sintel): 0.177
Badness (Sintel): 0.177


== Canopus ==
== Sirius ==
{{Main| Canopus }}
{{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]]
 
[[Badness]] (Sintel): 0.213
 
=== Mizar ===
Mizar exploits the sirius tuning of the 25/21 generator being close to [[13/11]] (in order to split 7/5 evenly); additionally this tempers out [[459/455]], equating [[17/13]] to [[35/27]].
 
The mapping for prime 17 is similar to what [[dubhe]] does: tempering out [[2025/2023]] to split the 7-limit generator in half; in this case, 25/7 is split into two intervals of [[17/9]], which turns out to occupy the position of a [[macrodiatonic]] fifth, specifically a macro-[[flattone]] fifth.
 
Subgroup: 3.5.7.11
 
Comma list: 3125/3087, 295245/290521


[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
{{Mapping|legend=2| 1 -2 -4 12 | 0 6 10 -17 }}
: mapping generators: ~3, ~729/385


[[Badness]] (Sintel): 0.0996
Optimal tunings:  
* WE: ~3 = 1901.0672{{c}}, ~729/385 = 1097.6627{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.1722{{c}}


=== Suhail ===
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]], [[168edt|b168]]
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}


Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].
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]] 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: ~13/9, ~65/63
: 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: [[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 ==
Line 302: Line 355:


=== Erigone ===
=== Erigone ===
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}
Erigone splits the 9/1-complement generator of procyon into three, allowing for an accurate representation of 11/9 at 19 generators and 13/9 at 13 generators.
 
Subgroup: 3.5.7.11
 
Comma list: 12005/11979, 456533/455625
 
{{Mapping|legend=2| 1 -12 -4 -10 | 0 21 9 19 }}
: mapping generators: ~3, ~99/49


Erigone splits the 9/1-complement generator of procyon into three, allowing for an accurate representation of 11/9 at 19 generators and 13/9 at 13 generators.
Optimal tunings:
* WE: ~3 = 1902.1806{{c}}, ~99/49 = 1219.6482{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~99/49 = 1219.5121{{c}}
 
Optimal ET sequence: [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[471edt|b471]], [[641edt|b641]], [[811edt|b811]], [[1452edt|b1452d]]
 
Badness (Sintel): 0.312


==== 3.5.7.11.13 subgroup ====
==== 3.5.7.11.13 subgroup ====
Line 312: Line 378:


{{Mapping|legend=2| 1 -12 -4 -10 -6 | 0 21 9 19 13 }}
{{Mapping|legend=2| 1 -12 -4 -10 -6 | 0 21 9 19 13 }}
: mapping generators: ~3, ~99/49


Optimal tunings:  
Optimal tunings:  
Line 318: Line 383:
* CWE: ~3 = 1901.9550{{c}}, ~99/49 = 1219.5123{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~99/49 = 1219.5123{{c}}


Optimal ET sequence: [[25edt|b25ce]], [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]]
Optimal ET sequence: [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]]


Badness (Sintel): 0.214
Badness (Sintel): 0.214


===== 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.
 
Subgroup: 3.5.7.11.13.17
 
Comma list: 847/845, 1575/1573, 3185/3179, 4459/4455
 
{{Mapping|legend=2| 1 -12 -4 -10 -6 -8 | 0 42 18 38 26 33 }}
: mapping generators: ~3, ~121/85
 
Optimal tunings:
* WE: ~3 = 1901.8706{{c}}, ~121/85 = 609.7295{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~121/85 = 609.7549{{c}}


By tempering out [[3971/3969]], erigone's generator ([[~]][[99/49]]) is split into two [[27/19]]'s. Then, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]).
Optimal ET sequence: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]


[[2277/2275]] may be used in the same way to extend erigone for prime 23.
Badness (Sintel): 0.453


====== 3.5.7.11.13.17.19 subgroup ======
====== 3.5.7.11.13.17.19 subgroup ======
Line 335: Line 411:


{{Mapping|legend=2| 1 -12 -4 -10 -6 -8 3 | 0 42 18 38 26 33 -1 }}
{{Mapping|legend=2| 1 -12 -4 -10 -6 -8 3 | 0 42 18 38 26 33 -1 }}
: mapping generators: ~3, ~27/19


Optimal tunings:  
Optimal tunings:  
Line 353: Line 428:


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1902.0149{{c}}, ~19/9 = 609.7748{{c}}
* WE: ~3 = 1902.0149{{c}}, ~27/19 = 609.7748{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~19/9 = 609.7562{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~27/19 = 609.7562{{c}}


Optimal ET sequence: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]
Optimal ET sequence: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]]
Line 360: Line 435:
Badness (Sintel): 0.542
Badness (Sintel): 0.542


== Sirius ==
== Izar ==
{{Main| Sirius }}
 
This tempers out the [[Don Page comma]] between [[5/3]] and [[7/5]], allowing an accurate representation of the 3:5:7 chord, similar to the 5:7:9 in [[#Procyon|procyon]].
 
For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].
 
[[Subgroup]]: 3.5.7
[[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 ==
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[[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


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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:  
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== Deneb ==
== Deneb ==
{{Main| Deneb }}
{{Main| Deneb }}
Deneb tempers out the comma [[6655/6561]] and equates 3 [[11/9]]s to [[9/5]].


[[Subgroup]]: 3.5.11
[[Subgroup]]: 3.5.11
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[[Badness]] (Sintel): 3.84
[[Badness]] (Sintel): 3.84
== Delta centauri ==
{{Main| Delta centauri }}
Delta centauri is a [[microtemperament]] that is generated by ~[[5/3]], and 9 generators stacked represents [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The slightly less accurate extension to the 3.5.11.17 subgroup equates 12 generators to [[17/9]].
[[Subgroup]]: 3.5.11
[[Comma list]]: [[1953125/1948617]]
{{Mapping|legend=3| 1 1 -2 | 0 1 9 }}
: mapping generators: ~3, ~5/3
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.102{{c}}, ~5/3 = 883.954{{c}}
: [[error map]]: {{val| +0.147 -0.258 +0.064 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 883.901{{c}}
: error map: {{val| 0.000 -0.458 -0.121}}
[[Optimal ET sequence]]: TBA
[[Badness]] (Sintel): 0.114
=== 3.5.11.17 subgroup ===
[[Subgroup]]: 3.5.11.17
[[Comma list]]: [[1377/1375]], [[265625/264627]]
{{Mapping|legend=3| 1 1 -2 -3 | 0 1 9 12 }}
: mapping generators: ~3, ~5/3
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1901.933{{c}}, ~5/3 = 884.105{{c}}
: [[error map]]: {{val| +0.038 -0.215 +1.645 -1.669 }}
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 884.091{{c}}
: error map: {{val| 0.000 -0.268 +1.592 -1.727}}
[[Optimal ET sequence]]: TBA
[[Badness]] (Sintel): 0.146


= 3.7.11-subgroup temperaments =
= 3.7.11-subgroup temperaments =
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=== 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 ====
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{{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:
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