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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]]


[[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]] 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 ==
Line 394: 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]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]], [[123edt|b123]], [[136edt|b136]]
[[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): 0.213
[[Badness]] (Sintel): 0.0166


=== Mizar ===
== Nekkar ==
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]].
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.


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


Subgroup: 3.5.7.11
[[Comma list]]: {{monzo| -24 -3 16 }}


Comma list: 3125/3087, 295245/290521
{{Mapping|legend=3| 1 -8 0 | 0 16 3 }}
: mapping generators: ~3, ~32805/16807


{{Mapping|legend=2| 1 -2 -4 12 | 0 6 10 -17 }}
[[Optimal tuning]]s:
: mapping generators: ~3, ~729/385
* [[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 }}


Optimal tunings:  
[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[311edt|b311dd]]
* WE: ~3 = 1901.0672{{c}}, ~729/385 = 1097.6627{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.1722{{c}}


Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]], [[168edt|b168]]
[[Badness]] (Sintel): 17.1
 
Badness (Sintel): 1.82
 
==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13
 
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.


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


Comma list: 1331/1323, 3125/3087
Comma list: 1331/1323, 120285/117649


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


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1902.6279{{c}}, ~11/5 = 1366.3374{{c}}
* WE: ~3 = 1900.6084{{c}}, ~21/11 = 1124.4193{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8996{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.1738{{c}}


Optimal ET sequence: [[7edt|b7]], [[25edt|b25d]], [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[149edt|b149]]
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[262edt|b262d]]


Badness (Sintel): 0.538
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, 847/845, 1331/1323
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 }}


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 510: Line 510:
[[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 817: Line 817:


=== 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 829: Line 842:


{{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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