No-twos subgroup temperaments: Difference between revisions

{{Todo|WIP|inline=1|text=Further entries in the [[catalog of 3.5.7 subgroup rank two temperaments]] will eventually be documented here.}}

= Overview by mapping of 5 =

* Arcturus, antares, aldebaran, delta centauri and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals.

* BPS has a ~9/7 generator, two of which give the ~5/3.

* 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).

* Miaplacidus has a ~7/5 generator, eight of which give the ~15/1 (5/1 up one tritave).  

* Alnilam has a ~55/27 generator, ten of which give the ~1215/1 (5/1 up five tritaves).

* 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).

= 3.5.7-subgroup temperaments =

== Arcturus ==

[[Strong extension]]s of this temperament that include the octave are [[trienstonic clan #Opossum|opossum]] and [[jubilismic clan #Crepuscular|crepuscular]]. [[Weak extension]]s are [[kleismic family #Catalan|catalan]], [[tetracot family #Bunya|bunya]], [[gamelismic clan #Superkleismic|superkleismic]], and [[sensamagic clan #Bohpier|bohpier]]. Non-octave extensions are documented below.

[[Subgroup]]: 3.5.7

 

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1903.8634{{c}}, ~5/3 = 878.9230{{c}}

: [[error map]]: {{val| +1.908 -3.527 +0.849 }}

: error map: {{val| 0.000 -6.068 -1.037 }}

 

 

 

 

Subgroup: 3.5.7.11

 

 

 

* CWE: ~3 = 1901.9550{{c}}, ~5/3 = 884.8724{{c}}

 

 

 

 

 

 

 

{{Mapping|legend=3| 1 0 9 | 0 1 -5 }}

 

: [[error map]]: {{val| +11.811 -14.422 -4.388 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 851.8677{{c}}

[[Optimal ET sequence]]: [[9edt|b9]], [[20edt|b20]], [[29edt|b29]]

[[Badness]] (Sintel): 2.95

{{Main| BPS }}

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

 

 

 

 

 

 

 

 

 

Optimal ET sequence: [[13edt|b13]], [[43edt|b43]], [[56edt|b56]], [[69edt|b69]], [[151edt|b151]], [[220edt|b220d]], [[289edt|b289dé]] *

 

 

 

 

 

 

Optimal ET sequence: [[13edt|b13]], [[43edt|b43]], [[56edt|b56]], [[69edt|b69]], [[289edt|b289d]], [[358edt|b358ddé]], [[427edt|b427cddé]] *

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: mapping generators: ~3, ~81/35

 

[[Optimal tuning]]s:  

* [[WE]]: ~3 = 1891.9431{{c}}, ~81/35 = 1397.0975{{c}}

: [[error map]]: {{val| -10.012 +7.881 +7.654 }}

 

 

[[Badness]] (Sintel): 1.34

 

 

 

 

 

 

{{Mapping|legend=3| 1 1 1 | 0 3 5 }}

 

: [[error map]]: {{val| +0.490 -2.650 +2.316 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~25/21 = 293.7594{{c}}

: error map: {{val| 0.000 -3.080 +1.926 }}

 

 

 

 

The mapping for prime 17 is similar to what [[dubhe]] does: tempering out [[2025/2023]] to split the 7-limit generator in half; in this case, 25/7 is split into two intervals of [[17/9]], which turns out to occupy the position of a [[macrodiatonic]] fifth, specifically a macro-[[flattone]] fifth.

 

 

 

{{Mapping|legend=2| 1 -2 -4 12 | 0 6 10 -17 }}

 

 

 

 

 

 

 

* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.2837{{c}}

 

Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

Badness (Sintel): 0.286

{{Main| Canopus }}

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

{{Mapping|legend=3| 1 -2 -1 | 0 5 4 }}

 

: [[error map]]: {{val| +1.785 -0.771 -2.248 }}

 

[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]

 

 

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

 

 

{{Mapping|legend=2| 3 4 5 6 | 0 5 4 7 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

{{Mapping|legend=3| 1 2 2 | 0 -7 -3 }}

 

 

 

 

 

Subgroup: 3.5.7.11

 

 

{{Mapping|legend=2| 1 -12 -4 -10 | 0 21 9 19 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: mapping generators: ~3, ~7/5

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1901.8789{{c}}, ~7/5 = 585.8428{{c}}

: [[error map]]: {{val| -0.076 -1.450 +1.880 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~7/5 = 585.8590{{c}}

: error map: {{val| 0.000 -1.396 +1.950 }}

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

 

== Izar ==

{{Mapping|legend=3| 1 -5 -2 | 0 12 7 }}

 

 

 

 

 

 

 

 

: [[error map]]: {{val| -1.800 -0.488 +3.749 }}

 

 

 

=== 3.5.7.11 subgroup ===

Subgroup: 3.5.7.11

 

 

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

 

 

 

 

 

 

{{Mapping|legend=2| 1 -8 0 1 -3 | 0 16 3 2 9 }}

 

* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.2772{{c}}

 

 

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=3| 13 19 23 0 | 0 0 0 1 }}

: mapping generators: ~27/25, ~11

* [[WE]]: ~27/25 = 146.4737{{c}}, ~11/9 = 343.9912{{c}}

* [[CWE]]: ~27/25 = 146.3042{{c}}, ~11/9 = 343.7950{{c}}

 

 

 

 

 

 

 

 

 

 

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

 

[[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}}

 

 

 

; Scales

 

 

Delta centauri is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The slightly less accurate extension to the 3.5.11.17 subgroup equates twelve generators with [[17/9]].

 

 

 

 

 

 

 

 

 

[[Optimal tuning]]s:  

* [[WE]]: ~3 = 1901.9930{{c}}, ~5/3 = 884.1054{{c}}

: [[error map]]: {{val| +0.038 -0.215 +1.645 -1.669 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 884.0911{{c}}

: error map: {{val| 0.000 -0.268 +1.592 -1.727 }}

[[Optimal ET sequence]]: [[13edt|b13g]], [[15edt|b15]], [[28edt|b28]], [[43edt|b43]], [[71edt|b71]], [[185edt|b185]], [[256edt|b256]], [[327edt|b327]]

[[Badness]] (Sintel): 0.146

== Deneb ==

{{Main| Deneb }}

Deneb tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]].

 

[[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}}

 

 

 

 

 

 

 

 

 

 

 

 

 

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

 

 

 

Optimal ET sequence: [[5edt|b5]], [[23edt|b23f]], [[28edt|b28]], [[61edt|b61]], [[89edt|b89fg]]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Optimal ET sequence: [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[181edt|b181]]

Badness (Sintel): 0.405

* [[Electra7]]

== 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 {{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.

[[Comma list]]: {{monzo| -35 9 10 }}

 

{{Mapping|legend=3| 1 -5 8 | 0 10 -9 }}

 

* [[WE]]: ~3 = 1902.3576{{c}}, ~55/27 = 1229.7879{{c}}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

Comma list: 351/343, 1331/1323

: mapping generators: ~3, ~21/11

* WE: ~3 = 1899.2045{{c}}, ~21/11 = 1125.8915{{c}}

* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1127.1732{{c}}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

* WE: ~3 = 1901.5487{{c}}, ~121/119 = 29.0313{{c}}

* CWE: ~3 = 1901.9550{{c}}, ~121/119 = 29.0392{{c}}

 

 

 

 

 

 

 

 

 

 

 

 

* WE: ~3 = 1901.8410{{c}}, ~119/117 = 28.9716{{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 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 ET sequence]]: [[15edt|b15]], [[92edt|b92]], [[107edt|b107]], [[122edt|b122]], [[137edt|b137]], [[152edt|b152]], [[319edt|b319c]], [[471edt|b471c]]

 

 

 

: mapping generators: ~3, ~7

[[Optimal tuning]]s:

* [[WE]]: ~3 = 1899.1302{{c}}, ~7/3 = 1478.4616{{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]]

[[Badness]] (Sintel): 0.125

[[Subgroup]]: 3.13.17

[[Comma list]]: 85293/83521

: mapping generators: ~3, ~17/9

* [[WE]]: ~3 = 1900.2977{{c}}, ~17/9 = 1109.8480{{c}}

: [[error map]]: {{val| -1.657 -1.136 +5.488 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~17/9 = 1110.3763{{c}}

 

[[Optimal ET sequence]]: [[5edt|b5]], [[7edt|b7]], [[12edt|b12]], [[65edt|b65]], [[77edt|b77]], [[89edt|b89]], [[101edt|b101g]], [[113edt|b113g]], [[238edt|b238gg]]

[[Badness]] (Sintel): 0.322

= No-twos no-threes subgroup temperaments =

 

 

 

 

 

 

 

* [[WE]]: ~5 = 2785.665{{c}}, ~55/49 = 194.939{{c}}

 

 

 

=== Pentapassion ===

 

 

 

 

* [[WE]]: ~5 = 2785.855{{c}}, ~343/325 = 97.393{{c}}

* [[CWE]]: ~5 = 2786.314{{c}}, ~343/325 = 97.386{{c}}

 

[[Optimal ET sequence]]: TBA

 

[[Badness]] (Sintel): 0.277

 

{{Mapping|legend=3| 1 0 -7 | 0 1 7 }}

 

: error map: {{val| 0.000 +10.852 +2.235 }}

[[Optimal ET sequence]]: [[5ed5|c5]], [[9ed5|c9e]], [[14ed5|c14]], [[33ed5|c33]], [[47ed5|c47]], [[61ed5|c61]]

[[Badness]] (Sintel): 1.04

{{Todo|inline=1|rename|comment=CompactStar: possibly rename juggernaut (Cole suggested the name "Macrodec", not sure about it yet though)}}

{{Main| Juggernaut }}

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

 

 

 

 

Comma list: 125/121, 637/605

{{Mapping|legend=2| 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 ET sequence: [[4ed5|c4]], [[6ed5|c6]], [[10ed5|c10]], [[24ed5|c24]], [[34d5|c34]], [[44ed5|c44]]

 

Badness (Sintel): 0.116

[[File:Salmon7.wav|thumb|Catfish[7] scale and 11:13:17 chord on A in 20ed11/5 tuning]]

Named by {{u|CompactStar}} in 2023, catfish tempers out 54925/54043 and 1795625/1771561 in the 5.11.13.17 subgroup.

CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{mos scalesig|2L 3s<11/5>}}, {{mos scalesig|2L 5s<11/5>}}, {{mos scalesig|2L 7s<11/5>}}, and {{mos scalesig|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord.

[[Subgroup]]: 5.11.13

[[Comma list]]: {{monzo| 14 -19 9 }}

{{Mapping|legend=3| 1 -4 -10 | 0 9 19 }}

* [[WE]]: ~5 = 2785.1693{{c}}, ~325/121 = 1699.4922{{c}}

: [[error map]]: {{val| -1.144 +3.435 -1.869 }}

* [[CWE]]: ~5 = 2786.3137{{c}}, ~325/121 = 1700.1381{{c}}

: error map: {{val| 0.000 +4.670 -1.042 }}

Badness (Sintel): 10.3

=== 5.11.13.17 subgroup ===

Subgroup: 5.11.13.17

Comma list: 54925/54043, 1795625/1771561

{{Mapping|legend=2| 1 -4 -10 -8 | 0 9 19 16 }}

: mapping generators: ~5, ~325/121