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.

== 2.15.55 subgroup ==

=== Spog ===

This temperament produces [[Slendro_clan#Superpelog|superpelog]]-like [[5L 4s|semiquartal]] scales while being more accurate ([[Subgroup temperaments#2.15.189.55.325.725.279|see]] rational approximations to their intervals).

[[Subgroup]]: 2.15.55

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

[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.655

{{Optimal ET sequence|legend=1|5, 9, 23, 32, 151, 183, 215, 247, 956, 1203, 1450, 3147, 4597 }}

==== 2.15.55.325 ====

[[Comma list]]: [[4225/4224]], [[6656/6655]]

[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.647

[[Support]]ing [[ET]]s: 5, 9, 13[-15], 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311

Related temperament: [[Lehmerismic_temperaments#Lux|lux]]

[[Subgroup]]: 2.15.189.55.325

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

[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.677

[[Support]]ing [[ET]]s: 5, 9, 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311

[[Subgroup]]: 2.15.189.55.325.725

[[Comma list]]: [[1625/1624]], [[2080/2079]], [[3025/3024]], [[4096/4095]]

{{Mapping|legend=2| 1 0 6 5 6 -3 | 0 5 2 1 3 16 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.649

[[Support]]ing [[ET]]s: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119,  142[+725], 151, 183, 206[+725], 311

==== 2.15.189.55.325.725.279 ====

Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.

Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.

{{Mapping|legend=2| 1 0 6 5 6 -3 5 | 0 5 2 1 3 16 4 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.638

[[Support]]ing [[ET]]s: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 151, 183, 206[+725], 311

== 4.3.5 subgroup ==

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

: 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}}

=== Tetrameantone ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

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

[[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}}

 

 

{{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

 

 

==== 4.6.5.7 subgroup (tetrominant) ====

 

[[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

 

 

{{Todo|inline=1|cleanup}}

 

<pre>  

4 6 5

[ ⟨ 1 0 1 ]

⟨ 0 16 2 ] ⟩

⟨2399.3973, 193.8643]

⟨25.21211, 47.81337]

⟨2399.397, 3101.829, 2787.126]

⟨-0.603, -0.126, 0.812]

Complexity 1.369085

Adjusted Error 0.692892 cents

TE Error 0.268047 cents/octave

Unison Vector

[8, 1, -8⟩ (393216:390625)

 

q99, q62, q37, q161, q136, q198, q25, q124, q74, q235

==== 4.6.5.7 ====

4 6 5 7

[ ⟨ 1 0 1 1 ]

⟨ 0 16 2 5 ] ⟩

⟨2399.4195, 193.8654]

⟨25.23883, 47.79592]

⟨2399.420, 3101.846, 2787.150, 3368.747]

⟨-0.580, -0.109, 0.837, -0.079]

Complexity 1.192044

Adjusted Error 0.653313 cents

TE Error 0.232715 cents/octave

[-2, -1, -2, 4⟩ (2401:2400)

[3, 0, -5, 2⟩ (3136:3125)

[5, 1, -3, -2⟩ (6144:6125)

[8, 1, -8, 0⟩ (393216:390625)

q99, q62, q37, q161, q136, q198, q25, q124, q74, q235

==== 4.6.5.7.11 ====

4 6 5 7 11

[ ⟨ 1 0 1 1 1 ]

⟨ 0 16 2 5 9 ] ⟩

⟨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

==== 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]

[0, 1, -1, 0, 1, -1⟩ (66:65)

[-1, -1, -1, 0, 2, 0⟩ (121:120)

[-2, 1, 1, 1, 0, -1⟩ (105:104)

[-3, -1, 1, 1, 1, 0⟩ (385:384)

[1, 3, -1, 0, 0, -2⟩ (864:845)

[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Reduced Mapping

4 6 5 7 11 13 17

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

TE Error 1.977443 cents/octave

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

</pre>

[ ⟨ 1 0 1 1 1 0 1 1 ]

⟨ 0 16 2 5 9 23 13 14 ] ⟩

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

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)

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

==== 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]

[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)

[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)

</pre>

== 4.9.25 subgroup ==

[[Subgroup]]: 4.9.25

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

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

[[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]

[[Subgroup]]: 4.9.49

{{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 ===

[[Subgroup]]: 8.9.7

[[Comma list]]: 64/63

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

: [[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 =

=== Magicaltet ===

{{See also| Chromatic pairs #Magicaltet }}

Magicaltet is related to [[keemic]], [[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|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}

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

[[Optimal tuning]]s:

* [[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* }}

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

=== Starlingtet ===

{{See also | Chromatic pairs #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|legend=2| 1 0 -1 | 0 1 3 }}

: mapping generators: ~2, ~5/3

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

: [[gencom]]: [2 6/5; 126/125]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846

{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents

==== Greeley ====

{{See also| Chromatic pairs #Greeley }}

Greeley is related to [[opossum]] as well as to [[nusecond]].  

[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})

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

{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}

: [[gencom]]: [2 11/10; 121/120 126/125]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776

{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}

: <nowiki/>* wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents

==== Skateboard ====

{{See also| Chromatic pairs #Skateboard }}

Skateboard is related to [[thrasher]].  

[[Subgroup]]: 2.5/3.7/3.11.13/9

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})

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

{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}

: [[gencom]]: [2 6/5; 56/55 91/90 100/99]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158

{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}

[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents

=== Gariberttet ===

Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].

==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====

Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].

[[Subgroup]]: 2.5/3.7/3.13/11

[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})

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

 

{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}

: [[gencom]]: [2 13/11; 275/273 847/845]

* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679

{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}

: <nowiki/>* wart for 13/11

[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents

==== Indium ====

{{See also | Chromatic pairs #Indium }}

Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})

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

{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}

: [[gencom]]: [2 12/11; 3025/3024 3125/3087]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010

{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*^† }}

[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents

==== Ammon ====

{{See also| Chromatic pairs #Ammon }}

Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.

[[Subgroup]]: 2.5/3.7/3.11/3.13/3

[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})

{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}

: [[gencom]]: [2 13/10; 121/120 169/168 275/273]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242

{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents

=== Sentry ===

{{See also | Chromatic pairs #Sentry }}

Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].  

[[Subgroup]]: 2.5/3.9/7

[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})

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

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

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

[[Optimal tuning]]s:  

* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166^†, 197*^†, 215^†, 264*^† }}

[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents

Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name.

[[Subgroup]]: 2.5/3.13/9

 

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

 

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

: [[gencom]]: [2 6/5; 325/324]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.861

{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents

== 2.….7/3… subgroups ==

=== Guanyintet ===

{{See also | Chromatic pairs #Guanyintet }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.  

[[Subgroup]]: 2.5.7/3.11/3

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

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

: mapping generators: ~2, ~12/7

{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}

: [[gencom]]: [2 7/6; 176/175 540/539]