No-twos subgroup temperaments: Difference between revisions

* 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.

* Canopus has a ~7/5 generator, five of which give the ~27/5 (9/5 up a tritave).

* Alnilam has a ~81/55 generator, ten of which give the ~243/5 (9/5 up three tritaves).

* Izar has a ~16807/10125 generator, twelve of which give the ~2187/5 (9/5 up five tritaves).

* Nekkar has a ~16807/10935 generator, sixteen of which give the ~6561/5 (9/5 up six tritaves).

* Juggernaut uses half-pentave(~11/5) as a period, and has a ~7/5 generator.

For extensions of this temperament that include the prime 2, see [[trienstonic clan #Opossum|opossum]], [[jubilismic clan #Crepuscular|crepuscular]], [[kleismic family #Catalan|catalan]], [[tetracot family #Bunya|bunya]], [[sensamagic clan #Bohpier|bohpier]], and [[gamelismic clan #Superkleismic|superkleismic]].

* [[WE]]: ~3 = 1903.863{{c}}, ~5/3 = 878.923{{c}}

* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 878.291{{c}}

[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]

Subgroup-val mapping: 1 1 -1 5 | 0 1 6 -6 }}

Optimal tuning (POTE): ~5/3 = 884.268{{c}}

Badness (Sintel): 2.507

For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions will be documented below.

This is a strong extension to BPS in the subgroup 3.5.7.11/2.13/4 that equates the "semitone" of [[27/25]]~[[49/45]] to [[13/12]], and then three of these intervals to [[14/11]].

: mapping generators: ~3, ~7/3

Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~9/7 = 441.025{{c}}

Supporting ETs: [[13edt|b13]], [[69edt|b69]], [[56edt|b56]], [[82edt|b82]], [[43edt|b43]], [[125edt|b125]], [[30edt|b30]], [[151edt|b151]], [[95edt|b95]], [[17edt|b17é]], [[194edt|b194d]], [[99edt|b99]], [[181edt|b181d]], [[108edt|b108é]] *

<nowiki/>* é is used as the wart for 11/2.

: ''See also: [[No-twos subgroup temperaments #Mintaka|Mintaka]] and [[No-twos subgroup temperaments #Deneb|Deneb]].''

This temperament splits 27/7 (the BPS generator up a tritave) into three by means of [[11/7]] or, equivalently, [[7/1]] in three by means of [[21/11]], and is the intersection of BPS, Deneb, and Mintaka temperaments as well as the most natural temperament satisfied in the 3.5.7.11 subgroup in [[39edt]].

Subgroup-val mapping: {{mapping| 1 5 0 1 | 0 -6 3 2 }}

mapping generators: ~3, ~21/11

Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 780.752{{c}}

Supporting ETs: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90

Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 780.428{{c}}

Supporting ETs: 39, 17, 22, 56, 5f, 61, 95, 100, 134, 73f, 139cf, 83cf, 173e, 178cef

Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~17/15 = 220.142{{c}}

Supporting ETs: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44

* WE: ~13/9 = 634.144{{c}}, ~65/63 = 49.695{{c}}

* CWE: ~13/9 = 633.985{{c}}, ~65/63 = 49.733{{c}}

{{Mapping|legend=2| 1 7 5 | 0 -12 -7 }}

: mapping generators: ~3, ~16807/10125

* [[WE]]: ~3 = 1901.958{{c}}, ~16807/10125 = 877.283{{c}}

* [[CWE]]: ~3 = 1901.955{{c}}, ~16807/10125 = 877.281{{c}}

[[Badness]] (Sintel): 0.017

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]].

[[Comma list]]: 35303692060125/33232930569601

{{Mapping|legend=2|1 8 3|0 -16 -3}}

: mapping generators: ~3, ~16807/10935

* [[WE]]: ~3 = 1900.155{{c}}, ~16807/10935 = 775.963{{c}}

* [[CWE]]: ~3 = 1901.955{{c}}, ~16807/10935 = 776.767{{c}}

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

[[Badness]] (Sintel): 17.120

Subgroup-val mapping: {{mapping| 1 8 3 3 | 0 -16 -3 -2 }}

Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 776.781{{c}}

Supporting ETs: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd

Badness (Sintel): 1.375

Subgroup-val mapping: {{mapping| 1 8 3 3 6 | 0 -16 -3 -2 -9 }}

Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 776.678{{c}}

Supporting ETs: 22, 5c, 27, 49, 71f, 17cf

Badness (Sintel): 1.723

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.

: mapping generators: ~3, ~17/9

* [[WE]]: ~3 = 1902.198{{c}}, ~49/45 = 145.412{{c}}

* [[CWE]]: ~3 = 1901.955{{c}}, ~49/45 = 145.368{{c}}

[[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79

Erigone splits the (tritave-augmented) generator of [[no-twos subgroup temperaments #Procyon|procyon]] into three, allowing for an accurate representation of 11/9 at -19 generators and 13/9 at -13 generators.

Subgroup-val mapping: {{mapping| 1 9 5 9 7 | 0 -21 -9 -19 -13 }}

: mapping generators: ~3, ~49/33

* WE: ~3 = 1901.9699{{c}}, ~49/33 = 682.4486{{c}}

* CWE: ~3 = 1901.9550{{c}}, ~49/33 = 682.4427{{c}}

Badness (Sintel): 0.21396

==== Hemigone ====

By tempering out [[3971/3969]], erigone's tritave-augmented generator ([[49/11]]) is split into two [[19/9]]s. Then, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]).

Subgroup-val mapping: {{mapping| 1 30 14 28 20 25 2 | 0 -42 -18 -38 -26 -33 1 }}

: mapping generators: ~3, ~19/9

* WE: ~3 = 1902.0918{{c}}, ~19/9 = 1292.3032{{c}}

* CWE: ~3 = 1901.9550{{c}}, ~19/9 = 1292.2083{{c}}

Badness (Sintel): 0.45479

 

Subgroup-val mapping: {{mapping| 1 30 14 28 20 25 2 64 | 0 -42 -18 -38 -26 -33 1 -90 }}

* WE: ~3 = 1902.0149{{c}}, ~19/9 = 1292.2401{{c}}

* CWE: ~3 = 1901.9550{{c}}, ~19/9 = 1292.1988{{c}}

Badness (Sintel): 0.54174

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.

=== Remus ===

{{Todo|inline=1|complete section|comment= 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-sevens [[restriction]] of this temperament.

Subgroup: 3.5.7.11.13

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

Subgroup-val mapping: {{mapping| 1 4 6 5 6 | 0 -9 -15 -10 -13 }}

: mapping generators: ~3, ~15/11

Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~15/11 = 536.090{{c}}

Supporting ETs: 39, 7, 32, 71, 110, 46, 149, 188, 181

Badness (Sintel): 0.286

=== Mizar ===

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

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.

Subgroup: 3.5.7.17

Comma list: 2025/2023, 3125/3087

Subgroup-val mapping: {{mapping| 1 -2 -4 2 | 0 6 10 1 }}

: mapping generators: ~3, ~17/9

Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~17/9 = 1097.800{{c}}

Supporting ETs: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd

Badness (Sintel): 0.383

 

 

 

 

 

 

Badness (Sintel): 0.841

Subgroup: 3.5.7.11

Comma list: 245/243, 3125/3087

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

Optimal tunings:  

* CTE: ~27/25 = 146.304{{c}}, ~11 = 4151.318{{c}}

* CWE: ~27/25 = 146.304{{c}}, ~11 = 4147.705{{c}}

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

Badness (Sintel): 0.499

* CTE: ~27/25 = 146.304{{c}}, ~11 = 4149.733{{c}}

* CWE: ~27/25 = 146.304{{c}}, ~11 = 4146.033{{c}}

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

[[Badness]] (Sintel): 0.365

{{Mapping|legend=2| 1 2 1 | 0 1 -6 }}

: [[gencom]]: [3/1 5/3; 177147/171875]

[[Optimal tuning]] ([[POTE]])): ~5/3 = 892.6{{c}}

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

: [[gencom]]: [3/1 11/9; 6655/6561]

[[Optimal tuning]] ([[POTE]])): ~11/9 = 340.242{{c}}

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

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

Subgroup-val mapping: {{mapping| 1 5 1 -2 | 0 -9 3 11 }}

: gencom: [3/1 99/65; 6655/6561 274625/264627]

Optimal tuning (POTE): ~99/65 = 748.0156{{c}}

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

Subgroup-val mapping: {{mapping| 1 5 1 -2 1 | 0 -9 3 11 4 }}

: gencom: [3/1 99/65; 1105/1089 4225/4131 6655/6561]

Optimal tuning (POTE): ~17/11 = 748.0236{{c}}

Supporting EDTs: {{EDs| b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ffg |equave=t}}

Subgroup-val mapping: {{mapping| 1 5 1 -2 1 7 | 0 -9 3 11 4 -11 }}

: gencom: [3/1 99/65; 247/243 325/323 1105/1089 4675/4617]

Optimal tuning (POTE): ~17/11 = 747.9960{{c}}

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

Subgroup-val mapping: {{mapping| 1 5 1 -2 1 7 6 | 0 -9 3 11 4 -11 -8 }}

: gencom: [3/1 99/65; 209/207 247/243 255/253 325/323 4675/4617]

Optimal tuning (POTE): ~17/11 = 748.0874{{c}}

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

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]].

{{Mapping|legend=2| 1 5 -1 | 0 -10 9 }}

: [[gencom]]: [3/1 81/55; {{monzo| 0 -35 9 0 10 }}]

[[Optimal tuning]] ([[CTE]]): ~81/55 = 672.410{{c}}

[[Support]]ing EDTs: {{EDs| 99, 17, 82, 116, 181, 65, 14[-5], 280, 48, 215, 31, 133, 314, 263

= 3.7.11 subgroup temperaments =

Extensions to prime 5 are covered at [[#Mintra]] and [[#Nekkar]].

* [[POTE]]: ~3 = 1901.955{{c}}, ~11/7 = 778.961{{c}}

* [[CWE]]: ~3 = 1901.955{{c}}, ~11/7 = 778.803{{c}}

[[Support]]ing [[ET]]s: {{EDs| b22, b5, b17, b39, b12, b61, b27, b7, b83, b49, b56, b32, b29, b100 |equave=t}}

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]].

* POTE: ~3 = 1901.955{{c}}, ~11/7 = 780.155{{c}}

* CWE: ~3 = 1901.955{{c}}, ~11/7 = 780.183{{c}}

Supporting ETs: {{EDs| b39, b22, b17, b5f, b61, b56, b100, b139f, b95, b178ef, b83f, b134, b73f, b217ef |equave=t}}

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]].

* POTE: ~3 = 1901.955{{c}}, ~11/7 = 774.432{{c}}

* CWE: ~3 = 1901.955{{c}}, ~11/7 = 774.782{{c}}

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

* [[POTE]]: ~3 = 1901.955{{c}}, ~81/77 = 86.957{{c}}

* [[CWE]]: ~3 = 1901.955{{c}}, ~81/77 = 86.957{{c}}

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

* POTE: ~3 = 1901.955{{c}}, ~[[81/77]] = 86.929{{c}}

* CWE: ~3 = 1901.955{{c}}, ~[[81/77]] = 86.932{{c}}

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

[[Subgroup]]: 3.7.11.13.17

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

{{Mapping|legend=2| 1 2 2 2 2 | 0 -15 12 22 38 }}

[[Optimal tuning]]s:

* [[POTE]]: ~3 = 1901.955{{c}}, ~119/117 = 28.979{{c}}

* [[CTE]]: ~3 = 1901.955{{c}}, ~119/117 = 28.970{{c}}

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

* POTE: ~3 = 1901.955{{c}}, ~119/117 = 28.973{{c}}

* CTE: ~3 = 1901.955{{c}}, ~119/117 = 28.970{{c}}

{{main|Aldebaran}}

[[Sval]] [[mapping]]: [{{val|1 0 5}}, {{val|0 1 -2}}]

[[Support]]ing [[ET]]s: 15, 17, 13, 32, 47, 28, 11[-13], 19[+13], 43, 9[-13], 7[-13], 49[+13], 21[+13], 41[-13]

[[CTE tuning|CTE generator]]: ~[[5/3]] = 887.76

[[Sval]] [[mapping]]: [{{val| 1 1 0 }}, {{val| 0 1 3 }}]

Sval mapping generators: ~3, ~7/3

[[Optimal tuning]]s:

* PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~[[7/3]] = 1480.661

* [[CWE]]: ~3 = 1\1ed3, ~[[7/3]] = 1479.487

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

[[Sval]] [[mapping]]: [{{val| 1 0 2 }}, {{val| 0 4 1 }}]

Sval mapping generators: ~3, ~17/9

[[Optimal tuning]]s:

* [[CTE]]: ~3 = 1\1ed3, ~[[17/9]] = 1109.689

* [[CWE]]: ~3 = 1\1ed3, ~[[17/9]] = 1110.376

[[Support]]ing [[ET]]s: {{EDs|b12, b5, b7, b17, b29, b19, b41, b53, b31, b65, b22f, b9ff, b77, b43|equave=t}}

= No-twos-or-threes subgroup temperaments =

{{main|Antipyth}}

{{Mapping|legend=2| 1 2 7 | 0 1 7 }}

Mapping generators: ~5, ~7/25

[[Optimal tuning]] ([[CTE]]): ~5 = 1\1ed5, ~[[7/5]] = 592.728

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

{{main|Juggernaut}}

{{Mapping|legend=2| 2 4 3 | 0 1 0 }}

Mapping generators: ~11/5, ~7/25

[[Optimal tuning]] ([[CTE]]): ~11/5 = 1\2ed5, ~[[7/5]] = 582.512

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

[[Subgroup]]: 5.7.11.13

[[Comma list]]: 125/121, 637/625

{{Mapping|legend=2| 2 4 3 0 | 0 1 0 -2 }}

Mapping generators: ~11/5, ~7/25

[[Optimal tuning]] ([[CTE]]): ~11/5 = 1\2ed5, ~[[7/5]] = 582.512

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

See: [[Catalog of 3.5.7 subgroup rank two temperaments#Graphs]]

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