No-twos subgroup temperaments: Difference between revisions

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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, 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 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 ~7/5 generator, five of which give the ~27/5 (9/5 up a tritave).
* Canopus has a ~15/7 generator, five of which give the ~45/1 (5/1 up two tritaves).
* Alnilam has a ~81/55 generator, ten of which give the ~243/5 (9/5 up three tritaves).
* Alnilam has a ~55/27 generator, ten of which give the ~1215/1 (5/1 up five tritaves).
* Izar has a ~16807/10125 generator, twelve of which give the ~2187/5 (9/5 up five tritaves).
* Izar has a ~30375/16807 generator, twelve of which give the ~1215/1 (5/1 up five tritaves).
* Nekkar has a ~16807/10935 generator, sixteen of which give the ~6561/5 (9/5 up six tritaves).
* Nekkar has a ~32805/16807 generator, sixteen of which give the ~32805/1 (5/1 up eight 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).
* 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.
* 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.
* Juggernaut uses half-pentave (~11/5) as a period, and has a ~7/5 generator.


= 3.5.7-subgroup temperaments =
= 3.5.7-subgroup temperaments =
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This is a strong extension of BPS to 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]].
This is a strong extension of BPS to 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]].


==== 3.5.7.11/2.13/4 subgroup ====
Subgroup: 3.5.7.11/2.13/4
Subgroup: 3.5.7.11/2.13/4


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=== Mintra ===
=== Mintra ===
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|deneb]], and [[#Mintaka|mintaka]] temperaments as well as the most natural temperament satisfied in the 3.5.7.11 subgroup in [[39edt]].
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|deneb]], and [[#Mintaka|mintaka]] temperaments as well as the most natural temperament satisfied in the 3.5.7.11 subgroup in [[39edt]].
The 13-limit extension uses the canonical extension for prime 13 described at [[#Tridecimal mintaka]].


Subgroup: 3.5.7.11
Subgroup: 3.5.7.11
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==== Tridecimal mintra ====
==== Tridecimal mintra ====
This temperament uses the canonical extension for prime 13 described at [[#Tridecimal mintaka]].
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


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Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].
Tempering out the 3.13-subgroup [[threedie]] splits the tritave into three, meeting 11/1 at seven generators after tempering out the [[sopreisma]].


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


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


{{Mapping|legend=2| 1 7 5 | 0 -12 -7 }}
{{Mapping|legend=2| 1 -5 -2 | 0 12 7 }}
: mapping generators: ~3, ~16807/10125
: mapping generators: ~3, ~30375/16807


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1901.958{{c}}, ~16807/10125 = 877.283{{c}}
* [[WE]]: ~3 = 1901.9584{{c}}, ~30375/16807 = 1024.6759{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~16807/10125 = 877.281{{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]]
[[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
[[Badness]] (Sintel): 0.0166


== Nekkar ==
== 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 3.5.7.11 than 3.5.7, whereupon it becomes a strong extension of [[mintaka]].
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]]: 35303692060125/33232930569601
[[Comma list]]: {{monzo| -24 -3 16 }}


{{Mapping|legend=2|1 8 3|0 -16 -3}}
{{Mapping|legend=2| 1 -8 0 | 0 16 3 }}
: mapping generators: ~3, ~16807/10935
: mapping generators: ~3, ~32805/16807


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1900.155{{c}}, ~16807/10935 = 775.963{{c}}
* [[WE]]: ~3 = 1900.1550{{c}}, ~32805/16807 = 1124.1916{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~16807/10935 = 776.767{{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 ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]]
[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[311edt|b311dd]]


[[Badness]] (Sintel): 17.120
[[Badness]] (Sintel): 17.1


=== 3.5.7.11 subgroup ===
=== 3.5.7.11 subgroup ===
{{See also| Mintaka }}
This continues the canonical 11-limit extension of squares.
Subgroup: 3.5.7.11
Subgroup: 3.5.7.11


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


Subgroup-val mapping: {{mapping| 1 8 3 3 | 0 -16 -3 -2 }}
Subgroup-val mapping: {{mapping| 1 -8 0 1 | 0 16 3 2 }}
: mapping generators: ~3, ~11/7


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 776.781{{c}}
Optimal tunings:
* WE: ~3 = 1900.6084{{c}}, ~21/11 = 1124.4193{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.1738{{c}}


Supporting ETs: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[262edt|b262d]]


Badness (Sintel): 1.375
Badness (Sintel): 1.37


=== 3.5.7.11.13 subgroup ===
=== 3.5.7.11.13 subgroup ===
This uses the [[no-twos subgroup temperaments #Minalzidar|minalzidar]] mapping of 13.
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


Comma list: 169/165, 351/343, 11011/10935
Comma list: 169/165, 351/343, 11011/10935


Subgroup-val mapping: {{mapping| 1 8 3 3 6 | 0 -16 -3 -2 -9 }}
Subgroup-val mapping: {{mapping| 1 -8 0 1 -3 | 0 16 3 2 9 }}
: mapping generators: ~3, ~11/7


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 776.678{{c}}
Optimal tunings:
* WE: ~3 = 1902.3248{{c}}, ~21/11 = 1125.4837{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.2772{{c}}


Supporting ETs: 22, 5c, 27, 49, 71f, 17cf
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71f]]


Badness (Sintel): 1.723
Badness (Sintel): 1.72


== Procyon ==
== 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.
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|sirius]].


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7
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{{Mapping|legend=2| 1 2 2 | 0 -7 -3 }}
{{Mapping|legend=2| 1 2 2 | 0 -7 -3 }}
: mapping generators: ~3, ~17/9
: mapping generators: ~3, ~49/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1902.198{{c}}, ~49/45 = 145.412{{c}}
* [[WE]]: ~3 = 1902.1979{{c}}, ~49/45 = 145.4124{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~49/45 = 145.368{{c}}
: [[error map]]: {{val| +0.243 +0.195 -0.667 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~49/45 = 145.3680{{c}}
: error map: {{val| 0.000 +0.020 -1.020 }}


[[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79
[[Optimal ET sequence]]: [[13edt|b13]], [[92edt|b92]], [[105edt|b105]], [[118edt|b118]], [[131edt|b131]], [[144edt|b144]], [[157edt|b157]], [[327edt|b327]], [[484edt|b484]], [[641edt|b641]], [[1125edt|b1125d]]


[[Badness]] (Sintel): 0.200
[[Badness]] (Sintel): 0.200
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{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}


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.
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.


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


Comma list: 847/845, 1575/1573, 4459/4455
Comma list: 847/845, 1575/1573, 4459/4455


Subgroup-val mapping: {{mapping| 1 9 5 9 7 | 0 -21 -9 -19 -13 }}
Subgroup-val mapping: {{mapping| 1 -12 -4 -10 -6 | 0 21 9 19 13 }}
: mapping generators: ~3, ~49/33
: mapping generators: ~3, ~99/49


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1901.9699{{c}}, ~49/33 = 682.4486{{c}}
* WE: ~3 = 1901.9695{{c}}, ~99/49 = 1219.5210{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~49/33 = 682.4427{{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: [[25edt|b25ce]], [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]]


Badness (Sintel): 0.21396
Badness (Sintel): 0.214


==== Hemigone ====
===== Hemigone =====
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11.13.17). }}
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11.13.17). }}


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]]).
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]]).


[[2277/2275]] may be used in the same way to extend erigone for prime 23.
====== 3.5.7.11.13.17.19 subgroup ======
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
Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 4459/4455


Subgroup-val mapping: {{mapping| 1 30 14 28 20 25 2 | 0 -42 -18 -38 -26 -33 1 }}
Subgroup-val mapping: {{mapping| 1 -12 -4 -10 -6 -8 3 | 0 42 18 38 26 33 -1 }}
: mapping generators: ~3, ~19/9
: mapping generators: ~3, ~27/19


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1902.0918{{c}}, ~19/9 = 1292.3032{{c}}
* WE: ~3 = 1902.0918{{c}}, ~27/19 = 609.7886{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~19/9 = 1292.2083{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~27/19 = 609.7467{{c}}


Optimal ET sequence: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]]
Optimal ET sequence: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]]


Badness (Sintel): 0.45479
Badness (Sintel): 0.455
 
==== 3.5.7.11.13.17.19.23 subgroup ====
[[2277/2275]] may be used in the same way to extend the simpler [[#Erigone|erigone]] to the 3.5.7.11.13.23 subgroup.


====== 3.5.7.11.13.17.19.23 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
Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 2277/2275, 4459/4455


Subgroup-val mapping: {{mapping| 1 30 14 28 20 25 2 64 | 0 -42 -18 -38 -26 -33 1 -90 }}
Subgroup-val mapping: {{mapping| 1 -12 -4 -10 -6 -8 3 -26 | 0 42 18 38 26 33 -1 90 }}
: mapping generators: ~3, ~19/9


Optimal tunings:  
Optimal tunings:  
* WE: ~3 = 1902.0149{{c}}, ~19/9 = 1292.2401{{c}}
* WE: ~3 = 1902.0149{{c}}, ~19/9 = 609.7748{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~19/9 = 1292.1988{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~19/9 = 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]]


Badness (Sintel): 0.54174
Badness (Sintel): 0.542


== Sirius ==
== Sirius ==
{{Main| 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.
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]].
For an overview of extensions to this temperament that include prime 2, see [[Gariboh clan #Overview to extensions]].
Line 360: Line 365:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1902.4455{{c}}, ~25/21 = 293.7393{{c}}
* [[WE]]: ~3 = 1902.4455{{c}}, ~25/21 = 293.7393{{c}}
: [[error map]]: {{val| +0.490 -2.650 +2.316 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~25/21 = 293.7594{{c}}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~25/21 = 293.7594{{c}}
: 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]]: [[6edt|b6]], [[7edt|b7]], [[13edt|b13]], [[71edt|b71]], [[84edt|b84]], [[97edt|b97]], [[110edt|b110]],  [[123edt|b123]], [[136edt|b136]]
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[[Badness]] (Sintel): 0.213
[[Badness]] (Sintel): 0.213


=== Remus ===
=== Mizar ===
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}
{{Todo|inline=1|complete section|comment= Catalog the intermediate extensions (3.5.7.11, 3.5.7.11.13). }}


By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with mos scales of 18, 25, 32, or 39 steps.
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 is essentially [[electra]] but with prime 7, or more accurately, electra is the no-sevens [[restriction]] of this temperament.
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: 3.5.7.11.13
==== 3.5.7.11.13.17 subgroup ====
Subgroup: 3.5.7.11.13.17


Comma list: 275/273, 1625/1617, 1575/1573
Comma list: 275/273, 459/455, 1625/1617, 2025/2023


Subgroup-val mapping: {{mapping| 1 4 6 5 6 | 0 -9 -15 -10 -13 }}
Subgroup-val mapping: {{mapping| 1 -2 -4 12 11 2 | 0 6 10 -17 -15 1 }}
: mapping generators: ~3, ~15/11
: mapping generators: ~3, ~17/9


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~15/11 = 536.090{{c}}
Optimal tunings:
* WE: ~3 = 1901.0269{{c}}, ~17/9 = 1097.7583{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~17/9 = 1098.2979{{c}}


Supporting ETs: 39, 7, 32, 71, 110, 46, 149, 188, 181
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]]


Badness (Sintel): 0.286
Badness (Sintel): 0.841


=== Mizar ===
=== Remus ===
{{Todo|inline=1|complete section|comment= Catalog the intermediate extensions (3.5.7.11, 3.5.7.11.13) instead. }}
{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11). }}


This temperament uses a weak extension to the 3.5.7.17 subgroup 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.
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.


Subgroup: 3.5.7.17
This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.


Comma list: 2025/2023, 3125/3087
==== 3.5.7.11.13 subgroup ====
Subgroup: 3.5.7.11.13


Subgroup-val mapping: {{mapping| 1 -2 -4 2 | 0 6 10 1 }}
Comma list: 275/273, 1625/1617, 1575/1573
: mapping generators: ~3, ~17/9


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~17/9 = 1097.800{{c}}
Subgroup-val mapping: {{mapping| 1 -5 -9 -5 -7 | 0 9 15 10 13 }}
: mapping generators: ~3, ~11/5


Supporting ETs: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd
Optimal tunings:  
* WE: ~3 = 1902.4456{{c}}, ~11/5 = 1366.1830{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8649{{c}}


Badness (Sintel): 0.383
Optimal ET sequence: [[7edt|b7]], [[25edt|b25df]], [[32edt|b32]], [[39edt|b39]], [[110edt|b110]], [[149edt|b149]]


==== 3.5.7.11.13.17 subgroup ====
Badness (Sintel): 0.286
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
 
Subgroup-val mapping: {{mapping| 1 -2 -4 12 11 2 | 0 6 10 -17 -15 1 }}
: mapping generators: ~3, ~17/9
 
Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~17/9 = 1098.298
 
Supporting ETs: 26, 71, 45, 19, 97f, 116d
 
Badness (Sintel): 0.841


== Bohlenic ==
== Bohlenic ==
This temperament is identical to [[13edt]] (equal-tempered [[Bohlen–Pierce scale]]), but has an independent generator for 11.
This temperament is identical to [[13edt]] (equal-tempered [[Bohlen–Pierce scale]]), but has an independent generator for 11.


Subgroup: 3.5.7.11
[[Subgroup]]: 3.5.7.11


Comma list: 245/243, 3125/3087
[[Comma list]]: 245/243, 3125/3087


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


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~27/25 = 146.304{{c}}, ~11 = 4151.318{{c}}
* [[WE]]: ~27/25 = 146.4737{{c}}, ~11/9 = 343.9912{{c}}
* CWE: ~27/25 = 146.304{{c}}, ~11 = 4147.705{{c}}
* [[CWE]]: ~27/25 = 146.3042{{c}}, ~11/9 = 343.7950{{c}}


Optimal ET sequence: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]
[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]], [[299edt|b299ccde]], [[338edt|b338ccde]]


Badness (Sintel): 0.499
[[Badness]] (Sintel): 0.499


=== 3.5.7.11.13 subgroup ===
=== 3.5.7.11.13 subgroup ===
Line 446: Line 445:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 146.304{{c}}, ~11 = 4149.733{{c}}
* WE: ~27/25 = 146.4722{{c}}, ~11/9 = 341.2865{{c}}
* CWE: ~27/25 = 146.304{{c}}, ~11 = 4146.033{{c}}
* CWE: ~27/25 = 146.3042{{c}}, ~11/9 = 342.1231{{c}}


[[Optimal ET sequence]]: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]
Optimal ET sequence: [[13edt|b13]], [[26edt|b26]], [[39edt|b39]]


[[Badness]] (Sintel): 0.365
Badness (Sintel): 0.365


== Tuning diagrams ==
== Tuning diagrams ==
Line 516: Line 515:
[[Comma list]]: 177147/171875
[[Comma list]]: 177147/171875


{{Mapping|legend=2| 1 2 1 | 0 1 -6 }}
{{Mapping|legend=2| 1 0 11 | 0 1 -6 }}
: [[gencom]]: [3/1 5/3; 177147/171875]
: mapping generators: ~3, ~5
 
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1899.1140{{c}}, ~5/3 = 890.5067{{c}}
: [[error map]]: {{val| -2.841 +3.307 +1.212 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 892.8502{{c}}
: error map: {{val| 0.000 +7.691 +6.156 }}


[[Optimal tuning]] ([[POTE]])): ~5/3 = 892.6{{c}}
[[Optimal ET sequence]]: [[15edt|b15]], [[17edt|b17]], [[32edt|b32]], [[113edt|b113]], [[145edt|b145ce]], [[177edt|b177ce]], [[209edt|b209ce]]


[[Support]]ing [[EDT]]s: 17, 15, 32, 49, 13[+11], 47, 19, 11[+11], 81, 66, 79[+11], 62[+11], 28[+11], 21[-11]
[[Badness]] (Sintel): 1.01


== Deneb ==
== Deneb ==
Line 531: Line 536:


{{Mapping|legend=2| 1 2 2 | 0 -3 1 }}
{{Mapping|legend=2| 1 2 2 | 0 -3 1 }}
: [[gencom]]: [3/1 11/9; 6655/6561]
: mapping generators: ~3, ~11/9
 
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1903.7592{{c}}, ~11/9 = 340.5646{{c}}
: [[error map]]: {{val| +1.804 -0.489 -3.235 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~11/9 = 340.0519{{c}}
: error map: {{val| 0.000 -2.559 -7.356 }}


[[Optimal tuning]] ([[POTE]])): ~11/9 = 340.242{{c}}
[[Optimal ET sequence]]: [[5edt|b5]], [[6edt|b6]], [[11edt|b11]], [[17edt|b17]], [[28edt|b28]], [[67edt|b67]], [[95edt|b95]], [[123edt|b123]], [[218edt|b218e]], [[341edt|b341cee]]


[[Support]]ing [[EDT]]s: 28, 11, 17, 6, 39, 5, 67, 45, 50, 16, 23, 73, 61, 62
[[Badness]] (Sintel): 0.255


=== Fomalhaut ===
=== Fomalhaut ===
Line 542: Line 553:
Fomalhaut was considered in the 23-limit from the start, as an attempt to approximate the no-2's, no-7's 23-limit as accurately as possible using 25 to 35 notes per equave, defined as the b28 & b33 temperament in this limit.  
Fomalhaut was considered in the 23-limit from the start, as an attempt to approximate the no-2's, no-7's 23-limit as accurately as possible using 25 to 35 notes per equave, defined as the b28 & b33 temperament in this limit.  


Fomalhaut follows the convention of naming no-twos temperaments after stars.
Fomalhaut follows the convention of naming no-2's temperaments after stars.


Subgroup: 3.5.11.13
Subgroup: 3.5.11.13
Line 548: Line 559:
Comma list: 6655/6561, 274625/264627
Comma list: 6655/6561, 274625/264627


Subgroup-val mapping: {{mapping| 1 5 1 -2 | 0 -9 3 11 }}
Subgroup-val mapping: {{mapping| 1 -4 4 9 | 0 9 -3 -11 }}
: gencom: [3/1 99/65; 6655/6561 274625/264627]
: mapping generators: ~3, ~65/33
 
Optimal tunings:
* WE: ~3 = 1905.7441{{c}}, ~65/33 = 1156.2566{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~65/33 = 1153.9402{{c}}


Optimal tuning (POTE): ~99/65 = 748.0156{{c}}
Optimal ET sequence [[5edt|b5]], [[23edt|b23f]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89f]]


Supporting EDTs: {{EDs| b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ff |equave=t}}
Badness (Sintel): 2.69


==== 3.5.11.13.17 subgroup ====
==== 3.5.11.13.17 subgroup ====
Line 560: Line 575:
Comma list: 1105/1089, 4225/4131, 6655/6561
Comma list: 1105/1089, 4225/4131, 6655/6561


Subgroup-val mapping: {{mapping| 1 5 1 -2 1 | 0 -9 3 11 4 }}
Subgroup-val mapping: {{mapping| 1 -4 4 9 5 | 0 9 -3 -11 -4 }}
: gencom: [3/1 99/65; 1105/1089 4225/4131 6655/6561]


Optimal tuning (POTE): ~17/11 = 748.0236{{c}}
Optimal tunings:  
* WE: ~3 = 1905.8547{{c}}, ~33/17 = 1156.2973{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.9162{{c}}


Supporting EDTs: {{EDs| b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ffg |equave=t}}
Optimal ET sequence: [[5edt|b5]], [[23edt|b23f]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fg]]
 
Badness (Sintel): 1.15


==== 3.5.11.13.17.19 subgroup ====
==== 3.5.11.13.17.19 subgroup ====
Line 572: Line 590:
Comma list: 247/243, 325/323, 1105/1089, 4675/4617
Comma list: 247/243, 325/323, 1105/1089, 4675/4617


Subgroup-val mapping: {{mapping| 1 5 1 -2 1 7 | 0 -9 3 11 4 -11 }}
Subgroup-val mapping: {{mapping| 1 -4 4 9 5 -4 | 0 9 -3 -11 -4 11 }}
: gencom: [3/1 99/65; 247/243 325/323 1105/1089 4675/4617]
 
Optimal tunings:  
* WE: ~3 = 1905.9433{{c}}, ~33/17 = 1156.3787{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.9775{{c}}


Optimal tuning (POTE): ~17/11 = 747.9960{{c}}
Optimal ET sequence: [[5edt|b5]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fgh]]


Supporting EDTs: {{EDs| b28, b33, b5, b61, b56f, b23f, b38ch, b66ch, b89fgh, b10cfh |equave=t}}
Badness (Sintel): 0.882


==== 3.5.11.13.17.19.23 subgroup ====
==== 3.5.11.13.17.19.23 subgroup ====
Line 584: Line 605:
Comma list: 209/207, 247/243, 255/253, 325/323, 4675/4617
Comma list: 209/207, 247/243, 255/253, 325/323, 4675/4617


Subgroup-val mapping: {{mapping| 1 5 1 -2 1 7 6 | 0 -9 3 11 4 -11 -8 }}
Subgroup-val mapping: {{mapping| 1 -4 4 9 5 -4 -2 | 0 9 -3 -11 -4 11 8 }}
: gencom: [3/1 99/65; 209/207 247/243 255/253 325/323 4675/4617]
 
Optimal tunings:  
* WE: ~3 = 1905.4597{{c}}, ~33/17 = 1155.9938{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~33/17 = 1153.8954{{c}}


Optimal tuning (POTE): ~17/11 = 748.0874{{c}}
Optimal ET sequence: [[5edt|b5]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fgh]]


Supporting EDTs: {{EDs| b28, b5, b33, b23f, b61, b56f, b38ch, b10cfhi, b66ch, b51ffg |equave=t}}
Badness (Sintel): 0.838


== Alnilam ==
== 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 {{u|CompactStar}} to continue with the theme of naming no-twos temperaments after proper star names, but also to indirectly reference [[mavila]].
Nearly a [[microtemperament]], alnilam takes a generator of [[~]][[55/27]] and equates nine of them tritave reduced with [[27/11]]. The name was given by {{u|CompactStar}} to continue with the theme of naming no-2's temperaments after proper star names, but also to indirectly reference [[mavila]] as its tritave-complement generator is ~[[81/55]], a flat fifth.


[[Subgroup]]: 3.5.11
[[Subgroup]]: 3.5.11
Line 598: Line 622:
[[Comma list]]: {{monzo| -35 9 10 }}
[[Comma list]]: {{monzo| -35 9 10 }}


{{Mapping|legend=2| 1 5 -1 | 0 -10 9 }}
{{Mapping|legend=2| 1 -5 8 | 0 10 -9 }}
: [[gencom]]: [3/1 81/55; {{monzo| 0 -35 9 0 10 }}]
: mapping generators: ~3, ~55/27


[[Optimal tuning]] ([[CTE]]): ~81/55 = 672.410{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.3576{{c}}, ~55/27 = 1229.7879{{c}}
: [[error map]]: {{val| +0.403 -0.222 -0.548 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~55/27 = 1229.5321{{c}}
: error map: {{val| 0.000 -0.768 -1.467 }}


[[Support]]ing EDTs: {{EDs| 99, 17, 82, 116, 181, 65, 14[-5], 280, 48, 215, 31, 133, 314, 263
[[Optimal ET sequence]]: [[17edt|b17]], [[65edt|b65]], [[82edt|b82]], [[99edt|b99]], [[577edt|b577]], [[676edt|b676]], [[775edt|b775e]], [[874edt|b874e]], [[973edt|b973e]], [[2045edt|b2045ceee]]


= 3.7.11 subgroup temperaments =
[[Badness]] (Sintel): 3.84
 
= 3.7.11-subgroup temperaments =
== Mintaka ==
== Mintaka ==
{{Main| Mintaka }}
{{Main| Mintaka }}


Extensions to prime 5 are covered at [[#Mintra]] and [[#Nekkar]].
Mintaka tempers out [[1331/1323]] in the 3.7.11 subgroup. It is the common [[restriction]] of [[#Mintra|mintra]] and [[#Nekkar|nekkar]].


[[Subgroup]]: 3.7.11
[[Subgroup]]: 3.7.11
Line 619: Line 649:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[POTE]]: ~3 = 1901.955{{c}}, ~11/7 = 778.961{{c}}
* [[WE]]: ~3 = 1902.4401{{c}}, ~21/11 = 1123.2803{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~11/7 = 778.803{{c}}
: [[error map]]: {{val| +0.485 +1.015 -2.317 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~21/11 = 1123.1517{{c}}
: error map: {{val| 0.000 +0.629 -3.060 }}


[[Support]]ing [[ET]]s: {{EDs| b22, b5, b17, b39, b12, b61, b27, b7, b83, b49, b56, b32, b29, b100 |equave=t}}
[[Optimal ET sequence]]: [[5edt|b5]], [[17edt|b17]], [[22edt|b22]], [[61edt|b61]], [[83edt|b83]], [[105edt|b105]], [[188edt|b188]], [[293edt|b293e]]
 
[[Badness]] (Sintel): 0.0528


=== Tridecimal mintaka ===
=== Tridecimal mintaka ===
This extension to prime 13 works in the sharper half of the mintaka tuning range, where the most important add-5 extension is [[#Mintra|mintra]].
This extension for prime 13 works in the sharper half of the mintaka tuning range. It is the no-5 restriction of [[#Mintra|tridecimal mintra]].


Subgroup: 3.7.11.13
Subgroup: 3.7.11.13
Line 632: Line 666:


Subgroup-val mapping: {{mapping| 1 0 1 10 | 0 3 2 -13 }}
Subgroup-val mapping: {{mapping| 1 0 1 10 | 0 3 2 -13 }}
: mapping generators: ~3, ~21/11


Optimal tunings:
Optimal tunings:
* POTE: ~3 = 1901.955{{c}}, ~11/7 = 780.155{{c}}
* WE: ~3 = 1903.5668{{c}}, ~11/7 = 1122.7513{{c}}
* CWE: ~3 = 1901.955{{c}}, ~11/7 = 780.183{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~11/7 = 1121.7718{{c}}


Supporting ETs: {{EDs| b39, b22, b17, b5f, b61, b56, b100, b139f, b95, b178ef, b83f, b134, b73f, b217ef |equave=t}}
Optimal ET sequence: [[17edt|b17]], [[22edt|b22]], [[39edt|b39]], [[217edt|b217ef]], [[256edt|b256ef]], [[295edt|b295def]], [[334edt|b334deef]], [[373edt|b373deef]]
 
Badness (Sintel): 0.604


=== Minalzidar ===
=== Minalzidar ===
This extension for prime 13 works in the flatter half of the mintaka tuning range, where the most important add-5 extension is [[#Nekkar|Nekkar]].
This extension for prime 13 works in the flatter half of the mintaka tuning range. It is the no-5 restriction of [[#Nekkar|tridecimal nekkar]].


Subgroup: 3.7.11.13
Subgroup: 3.7.11.13
Line 651: Line 686:


Optimal tunings:
Optimal tunings:
* POTE: ~3 = 1901.955{{c}}, ~11/7 = 774.432{{c}}
* WE: ~3 = 1899.2045{{c}}, ~21/11 = 1125.8915{{c}}
* CWE: ~3 = 1901.955{{c}}, ~11/7 = 774.782{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1127.1732{{c}}
 
Optimal ET sequence: [[22edt|b22]], [[27edt|b27]], [[86edt|b86d]], [[113edt|b113d]]


Supporting ETs: {{EDs| b5, b27, b22, b32, b17f, b37f, b12ff, b49, b59, b42df, b76, b39ff, b86d, b71f |equave=t}}
Badness (Sintel): 0.472


== Mebsuta ==
== 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]].
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]].
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]].


It is also possible to set 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-subgroup [[restriction]] of [[mohaha]].
It is also possible to set 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-subgroup [[restriction]] of [[mohaha]].
Line 669: Line 708:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[POTE]]: ~3 = 1901.955{{c}}, ~81/77 = 86.957{{c}}
* [[WE]]: ~3 = 1901.8347{{c}}, ~81/77 = 86.9519{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~81/77 = 86.957{{c}}
: [[error map]]: {{val| -0.120 +0.084 +0.159 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~81/77 = 86.8601{{c}}
: error map: {{val| 0.000 +0.284 +0.433 }}
 
[[Optimal ET sequence]]: [[21edt|b21]], [[22edt|b22]], [[87edt|b87]], [[109edt|b109]], [[131edt|b131]], [[153edt|b153]], [[175edt|b175]], [[503edt|b503]], [[678edt|b678]], [[853edt|b853]], [[1028edt|b1028]], [[2909edt|b2909e]], [[3937edt|b3937de]], [[4965edt|b4965dee]]


[[Support]]ing [[ET]]s: {{EDs| b22, b175, b153, b197, b131, b328, b109, b21, b219, b87, b43, b372, b65, b23 |equave=t}}
[[Badness]] (Sintel): 0.128


=== 3.7.11.19 subgroup ===
=== 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
Subgroup: 3.7.11.19


Line 682: Line 723:


Subgroup-val mapping: {{mapping| 1 2 2 3 | 0 -5 4 -7 }}
Subgroup-val mapping: {{mapping| 1 2 2 3 | 0 -5 4 -7 }}
: mapping generators: ~3, ~81/77


Optimal tunings:
Optimal tunings:
* POTE: ~3 = 1901.955{{c}}, ~[[81/77]] = 86.929{{c}}
* WE: ~3 = 1901.8715{{c}}, ~81/77 = 86.9248{{c}}
* CWE: ~3 = 1901.955{{c}}, ~[[81/77]] = 86.932{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~81/77 = 86.9325{{c}}
 
Optimal ET sequence: [[21edt|b21]], [[22edt|b22]], [[109edt|b109]], [[131edt|b131]], [[153edt|b153]], [[175edt|b175]], [[372edt|b372]], [[547edt|b547]], [[1269edt|b1269]], [[1816edt|b1816]]


Supporting ETs: {{EDs| b22, b175, b197, b153, b131, b219, b372, b109, b328, b241, b87, b21, b65, b43 |equave=t}}
Badness (Sintel): 0.128


=== Adhara ===
=== Adhara ===
Line 697: Line 739:
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]]: 3.7.11.13.17
==== 3.7.11.13.17 subgroup ====
Subgroup: 3.7.11.13.17


[[Comma list]]: 14161/14157, 107811/107653, 1108809/1108723
Comma list: 14161/14157, 107811/107653, 1108809/1108723


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


[[Optimal tuning]]s:
Optimal tunings:
* [[POTE]]: ~3 = 1901.955{{c}}, ~119/117 = 28.979{{c}}
* WE: ~3 = 1901.7879{{c}}, ~119/117 = 28.9764{{c}}
* [[CTE]]: ~3 = 1901.955{{c}}, ~119/117 = 28.970{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~119/117 = 28.9772{{c}}
 
Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247f]]


[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247f]], [[3216edt|b3216defff]]
Badness (Sintel): 0.274


==== 3.7.11.13.17.19 subgroup ====
==== 3.7.11.13.17.19 subgroup ====
This includes the natural extension of mebsuta to prime 19.
Subgroup: 3.7.11.13.17.19
Subgroup: 3.7.11.13.17.19


Line 721: Line 764:


Optimal tunings:
Optimal tunings:
* POTE: ~3 = 1901.955{{c}}, ~119/117 = 28.973{{c}}
* WE: ~3 = 1901.8410{{c}}, ~119/117 = 28.9716{{c}}
* CTE: ~3 = 1901.955{{c}}, ~119/117 = 28.970{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~119/117 = 28.9729{{c}}


Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[525edt|b525]], [[722edt|b722]], [[919edt|b919]], [[2035edt|b2035df]]
Optimal ET sequence: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[525edt|b525]], [[722edt|b722]], [[919edt|b919]], [[2035edt|b2035df]]
Badness (Sintel): 0.225


= Other tritave-based subgroups =
= Other tritave-based subgroups =
Line 735: Line 780:


{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1900.8391{{c}}, ~5/3 = 887.8617{{c}}
: [[error map]]: {{val| -1.116 +2.387 -1.219 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 888.1251{{c}}
: error map: {{val| 0.000 +3.766 +0.098 }}


[[Optimal tuning]] ([[CTE]]): ~5/3 = 887.76{{c}}
[[Optimal ET sequence]]: [[15edt|b15]], [[92edt|b92]], [[107edt|b107]], [[122edt|b122]], [[137edt|b137]], [[152edt|b152]], [[319edt|b319c]], [[471edt|b471c]]


[[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]
[[Badness]] (Sintel): 0.140


== Keladic ==
== Keladic ==
Line 745: Line 797:
[[Comma list]]: 351/343
[[Comma list]]: 351/343


{{Mapping|legend=2| 1 1 0 | 0 1 3 }}
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~3, ~7/3
: mapping generators: ~3, ~7


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[POTE]]: ~3 = 1901.955{{c}}, ~7/3 = 1480.661{{c}}
* [[WE]]: ~3 = 1899.1302{{c}}, ~7/3 = 1478.4616{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~7/3 = 1479.487{{c}}
: [[error map]]: {{val| -2.825 +8.766 -5.143 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~7/3 = 1479.4872{{c}}
: error map: {{val| 0.000 +12.616 -2.066 }}
 
[[Optimal ET sequence]]: [[4edt|b4]], [[5edt|b5]], [[9edt|b9]], [[86edt|b86d]], [[95edt|b95d]], [[104edt|b104d]], [[113edt|b113d]], [[122edt|b122d]], [[131edt|b131d]]


[[Support]]ing [[ET]]s: {{EDs| b9, b5, b14, b13, b23, b22, b32, b6f, b31, b19f, b17f, b41, b7ff, b40 |equave=t}}
[[Badness]] (Sintel): 0.125


== Sadalmelik ==
== Sadalmelik ==
Line 763: Line 819:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~3 = 1901.955{{c}}, ~17/9 = 1109.689{{c}}
* [[WE]]: ~3 = 1900.2977{{c}}, ~17/9 = 1109.8480{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~17/9 = 1110.376{{c}}
: [[error map]]: {{val| -1.657 -1.136 +5.488 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~17/9 = 1110.3763{{c}}
: error map: {{val| 0.000 +0.978 +9.331 }}


[[Support]]ing [[ET]]s: {{EDs| b12, b5, b7, b17, b29, b19, b41, b53, b31, b65, b22f, b9ff, b77, b43 |equave=t}}
[[Optimal ET sequence]]: [[5edt|b5]], [[7edt|b7]], [[12edt|b12]], [[65edt|b65]], [[77edt|b77]], [[89edt|b89]], [[101edt|b101g]], [[113edt|b113g]], [[238edt|b238gg]]


= No-twos-or-threes subgroup temperaments =
[[Badness]] (Sintel): 0.322
 
= No-2's no-3's subgroup temperaments =
== Antipyth ==
== Antipyth ==
{{Main| Antipyth }}
{{Main| Antipyth }}
Line 776: Line 836:
[[Comma list]]: 859375/823543
[[Comma list]]: 859375/823543


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


[[Optimal tuning]]s ([[CTE]]): ~5 = 2786.314{{c}}, ~7/5 = 592.728{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~5 = 2782.0966{{c}}, ~7/5 = 592.8517{{c}}
: [[error map]]: {{val| -4.217 +6.122 -1.356 }}
* [[CWE]]: ~5 = 2786.3137{{c}}, ~7/5 = 593.3647{{c}}
: error map: {{val| 0.000 +10.852 +2.235 }}


[[Support]]ing [[ET]]s: {{EDs| c14, c5, c19, c33, c47, c9e, c61, c75, c23e, c24e, c52e, c80e, c89e, c37e |equave=5}}
[[Optimal ET sequence]]: [[5ed5|c5]], [[9ed5|c9e]], [[14ed5|c14]], [[33ed5|c33]], [[47ed5|c47]], [[61ed5|c61]]
 
[[Badness]] (Sintel): 1.04


== Juggernaut ==
== Juggernaut ==
Line 790: Line 856:
[[Comma list]]: 125/121
[[Comma list]]: 125/121


{{Mapping|legend=2| 2 4 3 | 0 1 0 }}
{{Mapping|legend=2| 2 0 3 | 0 1 0 }}
: mapping generators: ~11/5, ~7/25
: mapping generators: ~11/5, ~7


[[Optimal tuning]] ([[CTE]]): ~11/5 = 1393.157{{c}}, ~[[7/5]] = 582.512{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~11/5 = 1388.4013{{c}}, ~7/5 = 591.9463{{c}}
: [[error map]]: {{val| -9.511 -0.077 +13.886 }}
* [[CWE]]: ~11/5 = 1393.1569{{c}}, ~7/5 = 590.1275{{c}}
: error map: {{val| 0.000 +7.615 +28.153 }}


[[Support]]ing [[ET]]s: {{EDs| c14, c10, c6, c18, c24, c22, c32, c16, c38, c8d, c34, c26d, c46, c52e |equave=5}}
[[Optimal ET sequence]]: [[2ed5|c2]], [[4ed5|c4]], [[10ed5|c10]], [[14ed5|c14]], [[66ed5|c66e]], [[80ed5|c80e]], [[94ed5|c94e]], [[108ed5|c108ee]], [[122ed5|c122ee]]
 
[[Badness]] (Sintel): 0.116


=== Tridecimal juggernaut ===
=== Tridecimal juggernaut ===
Subgroup: 5.7.11.13
Subgroup: 5.7.11.13


Comma list: 125/121, 637/625
Comma list: 125/121, 637/605


{{Mapping|legend=2| 2 4 3 0 | 0 1 0 -2 }}
Subgroup-val mapping: {{mapping| 2 0 3 8 | 0 1 0 -2 }}
 
Optimal tunings:
* WE: ~11/5 = 1392.6466{{c}}, ~7/5 = 570.6655{{c}}
* CWE: ~11/5 = 1393.1569{{c}}, ~7/5 = 570.9139{{c}}


[[Optimal tuning]] ([[CTE]]): ~11/5 = 1393.157{{c}}, ~[[7/5]] = 582.512{{c}}
Optimal ET sequence: [[4ed5|c4]], [[6ed5|c6]], [[10ed5|c10]], [[24ed5|c24]], [[34d5|c34]], [[44ed5|c44]]


[[Support]]ing [[ET]]s: {{EDs| c10, c14, c6, c24, c34, c16f, c44, c18f, c38, c26f, c54, c64 |equave=5}}
Badness (Sintel): 0.116


= Graphs =
= Graphs =

Latest revision as of 14:42, 13 September 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
Todo: WIP

Further entries in the catalog of 3.5.7 subgroup rank two temperaments will eventually be documented here.

This is a collection of subgroup temperaments 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

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.
  • 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.
  • 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).
  • 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).
  • Nekkar has a ~32805/16807 generator, sixteen of which give the ~32805/1 (5/1 up eight 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

Main article: Arcturus

Strong extensions of this temperament that include the octave are opossum and crepuscular. Weak extensions are catalan, bunya, superkleismic, and bohpier. Non-octave extensions are documented below.

Subgroup: 3.5.7

Comma list: 15625/15309

Subgroup-val mapping: [⟨1 0 -7], ⟨0 1 6]]

mapping generators: ~3, ~5

Optimal tunings:

  • WE: ~3 = 1903.8634 ¢, ~5/3 = 878.9230 ¢
error map: ⟨+1.908 -3.527 +0.849]
  • CWE: ~3 = 1901.9550 ¢, ~5/3 = 878.2907 ¢
error map: ⟨0.000 -6.068 -1.037]

Optimal ET sequence: b2, b9d, b11, b13, b340cc

Badness (Sintel): 0.535

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-val mapping: [⟨1 1 -1 5], ⟨0 1 6 -6]]

Optimal tunings:

  • WE: ~3 = 1890.5039 ¢, ~5/3 = 879.7029 ¢
  • CWE: ~3 = 1901.9550 ¢, ~5/3 = 884.8724 ¢

Optimal ET sequence: b13e, b15, b28e, b43dee

Badness (Sintel): 2.51

BPS

Main article: BPS

For extensions to this temperament that include the octave, see Sensamagic clan. Non-octave extensions are documented below.

Subgroup: 3.5.7

Comma list: 245/243

Subgroup-val mapping: [⟨1 1 2], ⟨0 2 -1]]

mapping generators: ~3, ~9/7

Optimal tunings:

  • WE: ~3 = 1903.7398 ¢, ~9/7 = 440.9014 ¢
error map: ⟨+1.785 -0.771 -2.248]
  • CWE: ~3 = 1901.9550 ¢, ~9/7 = 440.6646 ¢
error map: ⟨0.000 -3.030 -5.580]

Optimal ET sequence: b4, b9, b13, b56, b69, b82, b95, b367cdd, b462cdd

Badness (Sintel): 0.0659

Alhena

Todo: complete section

Catalog the intermediate extension (3.5.7.11/2).

This is a strong extension of BPS to 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.

3.5.7.11/2.13/4 subgroup

Subgroup: 3.5.7.11/2.13/4

Comma list: 196/195, 325/324, 1001/1000

Subgroup-val mapping: [⟨1 1 2 -1 2], ⟨0 2 -1 11 -4]]

mapping generators: ~3, ~9/7

Optimal tunings:

  • Subgroup WE: ~3 = 1903.5584 ¢, ~9/7 = 441.3534 ¢
  • Subgroup CWE: ~3 = 1901.9550 ¢, ~9/7 = 441.0249 ¢

Optimal ET sequence: b13, b43, b56, b69, b289d, b358ddé, b427cddé *

* é is used as the wart for 11/2.

Badness (Sintel): 0.187

Mintra

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.

The 13-limit extension uses the canonical extension for prime 13 described at #Tridecimal mintaka.

Subgroup: 3.5.7.11

Comma list: 245/243, 1331/1323

Subgroup-val mapping: [⟨1 -1 3 3], ⟨0 6 -3 -2]]

mapping generators: ~3, ~11/7

Optimal tunings:

  • WE: ~3 = 1904.1393 ¢, ~11/7 = 781.6168 ¢
  • CWE: ~3 = 1901.9550 ¢, ~11/7 = 780.7525 ¢

Optimal ET sequence: b17, b39, b95, b134, b229de

Badness (Sintel): 0.302

Tridecimal mintra

Subgroup: 3.5.7.11.13

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

Subgroup-val mapping: [⟨1 5 0 1 10], ⟨0 -6 3 2 -13]]

Optimal tunings:

  • WE: ~3 = 1903.9326 ¢, ~11/7 = 781.1856 ¢
  • CWE: ~3 = 1901.9550 ¢, ~11/7 = 780.4280 ¢

Optimal ET sequence: b17, b22, b39, b229cde

Badness (Sintel): 0.373

Dubhe

Main article: 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

Subgroup-val mapping: [⟨1 1 2 2], ⟨0 4 -2 5]]

mapping generators: ~3, ~17/15

Optimal tunings:

  • WE: ~3 = 1903.4591 ¢, ~17/15 = 220.2094 ¢
  • CWE: ~3 = 1901.9550 ¢, ~17/15 = 220.1420 ¢

Optimal ET sequence: b8, b9, b17, b26, b69, b95, b121

Badness (Sintel): 0.177

Canopus

Main article: Canopus

For extensions to this temperament that include the prime 2, see Canopic clan. No-2's extensions will be documented below.

Subgroup: 3.5.7

Comma list: 16875/16807

Subgroup-val mapping: [⟨1 -2 -1], ⟨0 5 4]]

mapping generators: ~3, ~15/7

Optimal tunings:

  • WE: ~3 = 1901.7826 ¢, ~15/7 = 1317.8771 ¢
error map: ⟨+1.785 -0.771 -2.248]
  • CWE: ~3 = 1901.9550 ¢, ~15/7 = 1317.9686 ¢
error map: ⟨0.000 -0.381 +1.093]

Optimal ET sequence: b13, b62, b75, b88, b101, b114, b355, b469, b583, b697

Badness (Sintel): 0.0996

Suhail

Todo: complete section

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.

3.5.7.11.13 subgroup

Subgroup: 3.5.7.11.13

Comma list: 1575/1573, 1625/1617, 4459/4455

Subgroup-val mapping: [⟨3 4 5 6 7], ⟨0 5 4 7 0]]

mapping generators: ~13/9, ~65/63

Optimal tunings:

  • WE: ~13/9 = 634.1444 ¢, ~65/63 = 49.6946 ¢
  • CWE: ~13/9 = 633.9850 ¢, ~65/63 = 49.7330 ¢

Optimal ET sequence: b39, b114, b153, b498cf, b651cf

Badness (Sintel): 0.330

Izar

Subgroup: 3.5.7

Comma list: 13841287201/13839609375

Subgroup-val mapping: [⟨1 -5 -2], ⟨0 12 7]]

mapping generators: ~3, ~30375/16807

Optimal tunings:

  • WE: ~3 = 1901.9584 ¢, ~30375/16807 = 1024.6759 ¢
error map: ⟨+0.003 +0.005 -0.012]
  • CWE: ~3 = 1901.9550 ¢, ~30375/16807 = 1024.6743 ¢
error map: ⟨0.000 +0.002 -0.016]

Optimal ET sequence: b13, b141, b154, …, b258, b271, b800, b1071, b1342, b1613, b4568, b6181

Badness (Sintel): 0.0166

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. However, the 13-limit extension restricts to minalzidar rather than tridecimal mintaka.

Subgroup: 3.5.7

Comma list: [-24 -3 16⟩

Subgroup-val mapping: [⟨1 -8 0], ⟨0 16 3]]

mapping generators: ~3, ~32805/16807

Optimal tunings:

  • WE: ~3 = 1900.1550 ¢, ~32805/16807 = 1124.1916 ¢
error map: ⟨-1.800 -0.488 +3.749]
  • CWE: ~3 = 1901.9550 ¢, ~32805/16807 = 1125.1876 ¢
error map: ⟨0.000 +1.047 +6.737]

Optimal ET sequence: b22, b49, b71, b120, b191d, b311dd

Badness (Sintel): 17.1

3.5.7.11 subgroup

Subgroup: 3.5.7.11

Comma list: 1331/1323, 120285/117649

Subgroup-val mapping: [⟨1 -8 0 1], ⟨0 16 3 2]]

Optimal tunings:

  • WE: ~3 = 1900.6084 ¢, ~21/11 = 1124.4193 ¢
  • CWE: ~3 = 1901.9550 ¢, ~21/11 = 1125.1738 ¢

Optimal ET sequence: b22, b49, b71, b120, b191d, b262d

Badness (Sintel): 1.37

3.5.7.11.13 subgroup

Subgroup: 3.5.7.11.13

Comma list: 169/165, 351/343, 11011/10935

Subgroup-val mapping: [⟨1 -8 0 1 -3], ⟨0 16 3 2 9]]

Optimal tunings:

  • WE: ~3 = 1902.3248 ¢, ~21/11 = 1125.4837 ¢
  • CWE: ~3 = 1901.9550 ¢, ~21/11 = 1125.2772 ¢

Optimal ET sequence: b22, b49, b71f

Badness (Sintel): 1.72

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

Subgroup-val mapping: [⟨1 2 2], ⟨0 -7 -3]]

mapping generators: ~3, ~49/45

Optimal tunings:

  • WE: ~3 = 1902.1979 ¢, ~49/45 = 145.4124 ¢
error map: ⟨+0.243 +0.195 -0.667]
  • CWE: ~3 = 1901.9550 ¢, ~49/45 = 145.3680 ¢
error map: ⟨0.000 +0.020 -1.020]

Optimal ET sequence: b13, b92, b105, b118, b131, b144, b157, b327, b484, b641, b1125d

Badness (Sintel): 0.200

Erigone

Todo: complete section

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.

3.5.7.11.13 subgroup

Subgroup: 3.5.7.11.13

Comma list: 847/845, 1575/1573, 4459/4455

Subgroup-val mapping: [⟨1 -12 -4 -10 -6], ⟨0 21 9 19 13]]

mapping generators: ~3, ~99/49

Optimal tunings:

  • WE: ~3 = 1901.9695 ¢, ~99/49 = 1219.5210 ¢
  • CWE: ~3 = 1901.9550 ¢, ~99/49 = 1219.5123 ¢

Optimal ET sequence: b25ce, b39, b92, b131, b170, b301, b471

Badness (Sintel): 0.214

Hemigone
Todo: complete section

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. Then, 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.

3.5.7.11.13.17.19 subgroup

Subgroup: 3.5.7.11.13.17.19

Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 4459/4455

Subgroup-val mapping: [⟨1 -12 -4 -10 -6 -8 3], ⟨0 42 18 38 26 33 -1]]

mapping generators: ~3, ~27/19

Optimal tunings:

  • WE: ~3 = 1902.0918 ¢, ~27/19 = 609.7886 ¢
  • CWE: ~3 = 1901.9550 ¢, ~27/19 = 609.7467 ¢

Optimal ET sequence: b25ce, b53, b78, b131, b209, b340

Badness (Sintel): 0.455

3.5.7.11.13.17.19.23 subgroup

Subgroup: 3.5.7.11.13.17.19.23

Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 2277/2275, 4459/4455

Subgroup-val mapping: [⟨1 -12 -4 -10 -6 -8 3 -26], ⟨0 42 18 38 26 33 -1 90]]

Optimal tunings:

  • WE: ~3 = 1902.0149 ¢, ~19/9 = 609.7748 ¢
  • CWE: ~3 = 1901.9550 ¢, ~19/9 = 609.7562 ¢

Optimal ET sequence: b53i, b78i, b131, b340, b471

Badness (Sintel): 0.542

Sirius

Main article: 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

Subgroup-val mapping: [⟨1 1 1], ⟨0 3 5]]

mapping generators: ~3, ~25/21

Optimal tunings:

  • WE: ~3 = 1902.4455 ¢, ~25/21 = 293.7393 ¢
error map: ⟨+0.490 -2.650 +2.316]
  • CWE: ~3 = 1901.9550 ¢, ~25/21 = 293.7594 ¢
error map: ⟨0.000 -3.080 +1.926]

Optimal ET sequence: b6, b7, b13, b71, b84, b97, b110, b123, b136

Badness (Sintel): 0.213

Mizar

Todo: complete section

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.

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.

3.5.7.11.13.17 subgroup

Subgroup: 3.5.7.11.13.17

Comma list: 275/273, 459/455, 1625/1617, 2025/2023

Subgroup-val mapping: [⟨1 -2 -4 12 11 2], ⟨0 6 10 -17 -15 1]]

mapping generators: ~3, ~17/9

Optimal tunings:

  • WE: ~3 = 1901.0269 ¢, ~17/9 = 1097.7583 ¢
  • CWE: ~3 = 1901.9550 ¢, ~17/9 = 1098.2979 ¢

Optimal ET sequence: b19, b26, b45, b71

Badness (Sintel): 0.841

Remus

Todo: complete section

Catalog the intermediate extension (3.5.7.11).

By splitting the generator of sirius into three, remus efficiently represents the no-2's 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-7's restriction of this temperament.

3.5.7.11.13 subgroup

Subgroup: 3.5.7.11.13

Comma list: 275/273, 1625/1617, 1575/1573

Subgroup-val mapping: [⟨1 -5 -9 -5 -7], ⟨0 9 15 10 13]]

mapping generators: ~3, ~11/5

Optimal tunings:

  • WE: ~3 = 1902.4456 ¢, ~11/5 = 1366.1830 ¢
  • CWE: ~3 = 1901.9550 ¢, ~11/5 = 1365.8649 ¢

Optimal ET sequence: b7, b25df, b32, b39, b110, b149

Badness (Sintel): 0.286

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 }}

mapping generators: ~27/25, ~11

Optimal tunings:

  • WE: ~27/25 = 146.4737 ¢, ~11/9 = 343.9912 ¢
  • CWE: ~27/25 = 146.3042 ¢, ~11/9 = 343.7950 ¢

Optimal ET sequence: b13, b26, b39, b299ccde, b338ccde

Badness (Sintel): 0.499

3.5.7.11.13 subgroup

Subgroup: 3.5.7.11.13

Comma list: 245/243, 275/273, 847/845

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

Optimal tunings:

  • WE: ~27/25 = 146.4722 ¢, ~11/9 = 341.2865 ¢
  • CWE: ~27/25 = 146.3042 ¢, ~11/9 = 342.1231 ¢

Optimal ET sequence: b13, b26, b39

Badness (Sintel): 0.365

Tuning diagrams

357plot_cplx_damage.png
Complexity vs. damage plot. z < 1 corresponds to the "Middle Path" inclusion criterion.
Temperaments supported by 13edt, labelled by name
Temperaments not supported by 13edt, labelled by name
Both sets, labelled by name
Temperaments supported by 13edt, labelled by comma
Temperaments not supported by 13edt, labelled by comma
Both sets, labeled by comma

3.5.11-subgroup temperaments

Polaris

Main article: 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.

Subgroup: 3.5.11

Comma list: 177147/171875

Subgroup-val mapping: [⟨1 0 11], ⟨0 1 -6]]

mapping generators: ~3, ~5

Optimal tunings:

  • WE: ~3 = 1899.1140 ¢, ~5/3 = 890.5067 ¢
error map: ⟨-2.841 +3.307 +1.212]
  • CWE: ~3 = 1901.9550 ¢, ~5/3 = 892.8502 ¢
error map: ⟨0.000 +7.691 +6.156]

Optimal ET sequence: b15, b17, b32, b113, b145ce, b177ce, b209ce

Badness (Sintel): 1.01

Deneb

Main article: Deneb

Subgroup: 3.5.11

Comma list: 6655/6561

Subgroup-val mapping: [⟨1 2 2], ⟨0 -3 1]]

mapping generators: ~3, ~11/9

Optimal tunings:

  • WE: ~3 = 1903.7592 ¢, ~11/9 = 340.5646 ¢
error map: ⟨+1.804 -0.489 -3.235]
  • CWE: ~3 = 1901.9550 ¢, ~11/9 = 340.0519 ¢
error map: ⟨0.000 -2.559 -7.356]

Optimal ET sequence: b5, b6, b11, b17, b28, b67, b95, b123, b218e, b341cee

Badness (Sintel): 0.255

Fomalhaut

Fomalhaut is an extension of deneb to higher limits that splits the interval of 11/3 in three.

Fomalhaut was considered in the 23-limit from the start, as an attempt to approximate the no-2's, no-7's 23-limit as accurately as possible using 25 to 35 notes per equave, defined as the b28 & b33 temperament in this limit.

Fomalhaut follows the convention of naming no-2's temperaments after stars.

Subgroup: 3.5.11.13

Comma list: 6655/6561, 274625/264627

Subgroup-val mapping: [⟨1 -4 4 9], ⟨0 9 -3 -11]]

mapping generators: ~3, ~65/33

Optimal tunings:

  • WE: ~3 = 1905.7441 ¢, ~65/33 = 1156.2566 ¢
  • CWE: ~3 = 1901.9550 ¢, ~65/33 = 1153.9402 ¢

Optimal ET sequence b5, b23f, b28, b61, b89f

Badness (Sintel): 2.69

3.5.11.13.17 subgroup

Subgroup: 3.5.11.13.17

Comma list: 1105/1089, 4225/4131, 6655/6561

Subgroup-val mapping: [⟨1 -4 4 9 5], ⟨0 9 -3 -11 -4]]

Optimal tunings:

  • WE: ~3 = 1905.8547 ¢, ~33/17 = 1156.2973 ¢
  • CWE: ~3 = 1901.9550 ¢, ~33/17 = 1153.9162 ¢

Optimal ET sequence: b5, b23f, b28, b61, b89fg

Badness (Sintel): 1.15

3.5.11.13.17.19 subgroup

Subgroup: 3.5.11.13.17.19

Comma list: 247/243, 325/323, 1105/1089, 4675/4617

Subgroup-val mapping: [⟨1 -4 4 9 5 -4], ⟨0 9 -3 -11 -4 11]]

Optimal tunings:

  • WE: ~3 = 1905.9433 ¢, ~33/17 = 1156.3787 ¢
  • CWE: ~3 = 1901.9550 ¢, ~33/17 = 1153.9775 ¢

Optimal ET sequence: b5, b28, b61, b89fgh

Badness (Sintel): 0.882

3.5.11.13.17.19.23 subgroup

Subgroup: 3.5.11.13.17.19.23

Comma list: 209/207, 247/243, 255/253, 325/323, 4675/4617

Subgroup-val mapping: [⟨1 -4 4 9 5 -4 -2], ⟨0 9 -3 -11 -4 11 8]]

Optimal tunings:

  • WE: ~3 = 1905.4597 ¢, ~33/17 = 1155.9938 ¢
  • CWE: ~3 = 1901.9550 ¢, ~33/17 = 1153.8954 ¢

Optimal ET sequence: b5, b28, b61, b89fgh

Badness (Sintel): 0.838

Alnilam

Nearly a microtemperament, alnilam takes a generator of ~55/27 and equates nine of them tritave reduced with 27/11. The name was given by CompactStar to continue with the theme of naming no-2's temperaments after proper star names, but also to indirectly reference mavila as its tritave-complement generator is ~81/55, a flat fifth.

Subgroup: 3.5.11

Comma list: [-35 9 10⟩

Subgroup-val mapping: [⟨1 -5 8], ⟨0 10 -9]]

mapping generators: ~3, ~55/27

Optimal tunings:

  • WE: ~3 = 1902.3576 ¢, ~55/27 = 1229.7879 ¢
error map: ⟨+0.403 -0.222 -0.548]
  • CWE: ~3 = 1901.9550 ¢, ~55/27 = 1229.5321 ¢
error map: ⟨0.000 -0.768 -1.467]

Optimal ET sequence: b17, b65, b82, b99, b577, b676, b775e, b874e, b973e, b2045ceee

Badness (Sintel): 3.84

3.7.11-subgroup temperaments

Mintaka

Main article: Mintaka

Mintaka tempers out 1331/1323 in the 3.7.11 subgroup. It is the common restriction of mintra and nekkar.

Subgroup: 3.7.11

Comma list: 1331/1323

Subgroup-val mapping: [⟨1 0 1], ⟨0 3 2]]

mapping generators: ~3, ~21/11

Optimal tunings:

  • WE: ~3 = 1902.4401 ¢, ~21/11 = 1123.2803 ¢
error map: ⟨+0.485 +1.015 -2.317]
  • CWE: ~3 = 1901.9550 ¢, ~21/11 = 1123.1517 ¢
error map: ⟨0.000 +0.629 -3.060]

Optimal ET sequence: b5, b17, b22, b61, b83, b105, b188, b293e

Badness (Sintel): 0.0528

Tridecimal mintaka

This extension for prime 13 works in the sharper half of the mintaka tuning range. It is the no-5 restriction of tridecimal mintra.

Subgroup: 3.7.11.13

Comma list: 1331/1323, 218491/216513

Subgroup-val mapping: [⟨1 0 1 10], ⟨0 3 2 -13]]

Optimal tunings:

  • WE: ~3 = 1903.5668 ¢, ~11/7 = 1122.7513 ¢
  • CWE: ~3 = 1901.9550 ¢, ~11/7 = 1121.7718 ¢

Optimal ET sequence: b17, b22, b39, b217ef, b256ef, b295def, b334deef, b373deef

Badness (Sintel): 0.604

Minalzidar

This extension for prime 13 works in the flatter half of the mintaka tuning range. It is the no-5 restriction of tridecimal nekkar.

Subgroup: 3.7.11.13

Comma list: 351/343, 1331/1323

Subgroup-val mapping: [⟨1 0 1 -3], ⟨0 3 2 9]]

mapping generators: ~3, ~21/11

Optimal tunings:

  • WE: ~3 = 1899.2045 ¢, ~21/11 = 1125.8915 ¢
  • CWE: ~3 = 1901.9550 ¢, ~21/11 = 1127.1732 ¢

Optimal ET sequence: b22, b27, b86d, b113d

Badness (Sintel): 0.472

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.

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.

It is also possible to set 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-subgroup restriction of mohaha.

Subgroup: 3.7.11

Comma list: 387420489/386683451

Subgroup-val mapping: [⟨1 2 2], ⟨0 -5 4]]

mapping generators: ~3, ~81/77

Optimal tunings:

  • WE: ~3 = 1901.8347 ¢, ~81/77 = 86.9519 ¢
error map: ⟨-0.120 +0.084 +0.159]
  • CWE: ~3 = 1901.9550 ¢, ~81/77 = 86.8601 ¢
error map: ⟨0.000 +0.284 +0.433]

Optimal ET sequence: b21, b22, b87, b109, b131, b153, b175, b503, b678, b853, b1028, b2909e, b3937de, b4965dee

Badness (Sintel): 0.128

3.7.11.19 subgroup

Subgroup: 3.7.11.19

Comma list: 3971/3969, 41553/41503

Subgroup-val mapping: [⟨1 2 2 3], ⟨0 -5 4 -7]]

Optimal tunings:

  • WE: ~3 = 1901.8715 ¢, ~81/77 = 86.9248 ¢
  • CWE: ~3 = 1901.9550 ¢, ~81/77 = 86.9325 ¢

Optimal ET sequence: b21, b22, b109, b131, b153, b175, b372, b547, b1269, b1816

Badness (Sintel): 0.128

Adhara

Todo: complete section

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).

It is also possible to set two-thirds of 11/9 to 8/7, giving rise to an add-8 extension.

3.7.11.13.17 subgroup

Subgroup: 3.7.11.13.17

Comma list: 14161/14157, 107811/107653, 1108809/1108723

Subgroup-val mapping: [⟨1 2 2 2 2], ⟨0 -15 12 22 38]]

mapping generators: ~3, ~119/117

Optimal tunings:

  • WE: ~3 = 1901.7879 ¢, ~119/117 = 28.9764 ¢
  • CWE: ~3 = 1901.9550 ¢, ~119/117 = 28.9772 ¢

Optimal ET sequence: b65, b66, b131, b197, b328, b525, b722, b1247f

Badness (Sintel): 0.274

3.7.11.13.17.19 subgroup

Subgroup: 3.7.11.13.17.19

Comma list: 3213/3211, 3971/3969, 14161/14157, 41553/41503

Subgroup-val mapping: [⟨1 2 2 2 2 3], ⟨0 -15 12 22 38 -21]]

mapping generators: ~3, ~119/117

Optimal tunings:

  • WE: ~3 = 1901.8410 ¢, ~119/117 = 28.9716 ¢
  • CWE: ~3 = 1901.9550 ¢, ~119/117 = 28.9729 ¢

Optimal ET sequence: b65, b66, b131, b197, b525, b722, b919, b2035df

Badness (Sintel): 0.225

Other tritave-based subgroups

Aldebaran

Main article: Aldebaran

Subgroup: 3.5.13

Comma list: 3159/3125

Subgroup-val mapping: [⟨1 0 5], ⟨0 1 -2]]

mapping generators: ~3, ~5

Optimal tunings:

  • WE: ~3 = 1900.8391 ¢, ~5/3 = 887.8617 ¢
error map: ⟨-1.116 +2.387 -1.219]
  • CWE: ~3 = 1901.9550 ¢, ~5/3 = 888.1251 ¢
error map: ⟨0.000 +3.766 +0.098]

Optimal ET sequence: b15, b92, b107, b122, b137, b152, b319c, b471c

Badness (Sintel): 0.140

Keladic

Subgroup: 3.7.13

Comma list: 351/343

Subgroup-val mapping: [⟨1 0 -3], ⟨0 1 3]]

mapping generators: ~3, ~7

Optimal tunings:

  • WE: ~3 = 1899.1302 ¢, ~7/3 = 1478.4616 ¢
error map: ⟨-2.825 +8.766 -5.143]
  • CWE: ~3 = 1901.9550 ¢, ~7/3 = 1479.4872 ¢
error map: ⟨0.000 +12.616 -2.066]

Optimal ET sequence: b4, b5, b9, b86d, b95d, b104d, b113d, b122d, b131d

Badness (Sintel): 0.125

Sadalmelik

Subgroup: 3.13.17

Comma list: 85293/83521

Subgroup-val mapping: [⟨1 0 2], ⟨0 4 1]]

mapping generators: ~3, ~17/9

Optimal tunings:

  • WE: ~3 = 1900.2977 ¢, ~17/9 = 1109.8480 ¢
error map: ⟨-1.657 -1.136 +5.488]
  • CWE: ~3 = 1901.9550 ¢, ~17/9 = 1110.3763 ¢
error map: ⟨0.000 +0.978 +9.331]

Optimal ET sequence: b5, b7, b12, b65, b77, b89, b101g, b113g, b238gg

Badness (Sintel): 0.322

No-2's no-3's subgroup temperaments

Antipyth

Main article: Antipyth

Subgroup: 5.7.11

Comma list: 859375/823543

Subgroup-val mapping: [⟨1 0 -7], ⟨0 1 7]]

mapping generators: ~5, ~7

Optimal tunings:

  • WE: ~5 = 2782.0966 ¢, ~7/5 = 592.8517 ¢
error map: ⟨-4.217 +6.122 -1.356]
  • CWE: ~5 = 2786.3137 ¢, ~7/5 = 593.3647 ¢
error map: ⟨0.000 +10.852 +2.235]

Optimal ET sequence: c5, c9e, c14, c33, c47, c61

Badness (Sintel): 1.04

Juggernaut

Main article: Juggernaut

Subgroup: 5.7.11

Comma list: 125/121

Subgroup-val mapping: [⟨2 0 3], ⟨0 1 0]]

mapping generators: ~11/5, ~7

Optimal tunings:

  • WE: ~11/5 = 1388.4013 ¢, ~7/5 = 591.9463 ¢
error map: ⟨-9.511 -0.077 +13.886]
  • CWE: ~11/5 = 1393.1569 ¢, ~7/5 = 590.1275 ¢
error map: ⟨0.000 +7.615 +28.153]

Optimal ET sequence: c2, c4, c10, c14, c66e, c80e, c94e, c108ee, c122ee

Badness (Sintel): 0.116

Tridecimal juggernaut

Subgroup: 5.7.11.13

Comma list: 125/121, 637/605

Subgroup-val mapping: [⟨2 0 3 8], ⟨0 1 0 -2]]

Optimal tunings:

  • WE: ~11/5 = 1392.6466 ¢, ~7/5 = 570.6655 ¢
  • CWE: ~11/5 = 1393.1569 ¢, ~7/5 = 570.9139 ¢

Optimal ET sequence: c4, c6, c10, c24, c34, c44

Badness (Sintel): 0.116

Graphs

See: Catalog of 3.5.7 subgroup rank two temperaments #Graphs

Projective tuning space diagrams

See: Catalog of 3.5.7 subgroup rank two temperaments #Projective tuning space diagrams

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