Subgroup temperaments: Difference between revisions

Baldy results from taking every other generator of the [[garibaldi temperament]]. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

== 2.9.5.11 subgroup ==

=== Glacial ===

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

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

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==

Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====

Subgroup: 2.9.7.11

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

== 2.9.7.11 subgroup ==

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

[[Badness]]: 0.00263

{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].  

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

=== Machine ===

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

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

: [[gencom]]: [2 8/7; 64/63 99/98]

* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128

* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

[[Badness]]: 0.00233

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

[[Badness]]: 0.00439

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.11 subgroup ==

Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

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

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

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

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

== 2.9.15.7 subgroup ==

Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

==== 2.9.15.7.11.13 ====

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

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

: sval mapping generators: ~2, ~21/16

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

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

==== 2.9.5.21 ====

''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

==== B-team ====

B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

[[Subgroup]]: 4.3.5

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

{{Main| Tetrameantone }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

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

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

{{Mapping|legend=2| 1 0 1 | 0 5 1 }}

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

=== Blacktetra ===

[[Subgroup]]: 4.3.5

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

== 4.6.5 subgroup ==

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

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

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

<nowiki />* Wart for 4

[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

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

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

<nowiki />* Wart for 4

=== Fourwar ===

The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

[ ⟨ 1 0 1 ]

⟨ 0 16 2 ] ⟩

TE Generator Tunings (cents)

⟨2399.3973, 193.8643]

⟨25.21211, 47.81337]

⟨2399.397, 3101.829, 2787.126]

⟨-0.603, -0.126, 0.812]

[8, 1, -8⟩ (393216:390625)

[ ⟨ 1 0 1 1 ]

⟨ 0 16 2 5 ] ⟩

⟨2399.4195, 193.8654]

TE Error 0.232715 cents/octave

[5, 1, -3, -2⟩ (6144:6125)

</pre>

==== 4.6.5.7.11 ====

4 6 5 7 11

[ ⟨ 1 0 1 1 1 ]

⟨ 0 16 2 5 9 ] ⟩

TE Generator Tunings (cents)

⟨2400.1097, 193.9498]

⟨24.18752, 48.52491]

⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]

⟨0.110, 1.241, 1.696, 1.033, -5.660]

Complexity 1.068792

Adjusted Error 2.926965 cents

TE Error 0.846083 cents/octave

[-1, -1, -1, 0, 2⟩ (121:120)

[2, 0, -2, -1, 1⟩ (176:175)

[-3, -1, 1, 1, 1⟩ (385:384)

[-1, 0, 3, -3, 1⟩ (1375:1372)

[-2, -1, -2, 4, 0⟩ (2401:2400)

[1, 0, 1, -4, 2⟩ (2420:2401)

q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124

</pre>

==== 4.6.5.7.11.13 ====

<pre>

Reduced Mapping

4 6 5 7 11 13

[ ⟨ 1 0 1 1 1 0 ]

⟨ 0 16 2 5 9 23 ] ⟩

⟨2401.2305, 193.5378]

⟨42.79107, 35.98524]

⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

Complexity 1.219191

Adjusted Error 6.699599 cents

TE Error 1.810487 cents/octave

Unison Vectors

[0, 1, -1, 0, 1, -1⟩ (66:65)

[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff

==== 4.6.5.7.11.13.17 ====

4 6 5 7 11 13 17

[ ⟨ 1 0 1 1 1 0 1 ]

⟨ 0 16 2 5 9 23 13 ] ⟩

⟨2400.4701, 193.4599]

⟨43.39350, 35.55764]

⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

Complexity 1.129881

Adjusted Error 8.082725 cents

TE Error 1.977443 cents/octave

[0, 1, -1, 0, 1, -1, 0⟩ (66:65)

[1, 1, 1, -1, 0, 0, -1⟩ (120:119)

[1, 2, 0, 0, -1, -1, 0⟩ (144:143)

[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)

[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)

[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg

==== 4.6.5.7.11.13.17.19 ====

4 6 5 7 11 13 17 19

[ ⟨ 1 0 1 1 1 0 1 1 ]

⟨ 0 16 2 5 9 23 13 14 ] ⟩

⟨2399.9219, 193.3952]

⟨44.14256, 35.03670]

⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

Complexity 1.058472

Adjusted Error 8.712222 cents

TE Error 2.050935 cents/octave

[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)

[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)

[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)

[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)

[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)

[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)

[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh

==== 4.6.5.7.11.13.17.19.23 ====

4 6 5 7 11 13 17 19 23

[ ⟨ 1 0 1 1 1 0 1 1 0 ]

⟨ 0 16 2 5 9 23 13 14 28 ] ⟩

⟨2399.3286, 193.5316]

⟨37.31613, 39.63311]

⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920

Adjusted Error 9.502017 cents

TE Error 2.100561 cents/octave

[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)

[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)

[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)

[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)

[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)

[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)

[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)

[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)

[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)

[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii

== 4.9.25 subgroup ==

=== Meansquared ===

[[Subgroup]]: 4.9.25

 

[[Comma list]]: [[6561/6400]]

Mapping generators: ~4, ~9/64

 

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

 

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

 

== 4.9.49 subgroup ==

[[Subgroup]]: 4.9.49

 

[[Comma list]]: 4096/3969

 

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

 

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~[[9/8]] = 219.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

 

== 8.9.7 subgroup ==

Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

 

[[Subgroup]]: 8.9.7

 

[[Comma list]]: 64/63

 

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

 

: sval mapping generators: ~8, ~9

 

: [[gencom]]: [8 9/8; 64/63]

 

[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898

 

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10^-3

= Fractional subgroup temperaments =

== 2.5/3… subgroups ==

=== Magicaltet ===

Magicaltet is related to [[supermagic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

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

: mapping generators: ~2, ~5/3

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

: [[gencom]]: [2 6/5; 100/99 385/384]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}

<nowiki/>* Wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

=== Starlingtet ===

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.  

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

: mapping generators: ~2, ~5/3

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