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

[[Comma list]]: [[100663296/100656875]]

{{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 }}

[[Subgroup]]: 2.15.55.325

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

[[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

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

{{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

==== 2.15.189.55.325.725 ====

[[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

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

[[Subgroup]]: 2.15.189.55.325.725.279

[[Comma list]]: [[1625/1624]], [[2016/2015]], [[2080/2079]], [[3025/3024]], [[4096/4095]]

{{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 ==

{{Main| Tetrahanson }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

{{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}}

[[Subgroup]]: 4.3.5

 

[[Comma list]]: 256/243

 

{{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}}

[[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) ====

[[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

 

 

 

=== 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.

 

 

{{Todo|inline=1|cleanup}}

 

<pre>  

Reduced Mapping

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 ====

Reduced Mapping

4 6 5 7 11 13

⟨ 0 16 2 5 9 23 ] ⟩

TE Error 1.810487 cents/octave

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

</pre>

[ ⟨ 1 0 1 1 1 0 1 ]

⟨ 0 16 2 5 9 23 13 ] ⟩

TE Generator Tunings (cents)

⟨2400.4701, 193.4599]

⟨43.39350, 35.55764]

TE Tuning Map (cents)

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

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

==== 4.6.5.7.11.13.17.19 ====

⟨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]

Adjusted Error 8.712222 cents

TE Error 2.050935 cents/octave

[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)

[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)

</pre>

==== 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]

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

</pre>

[[Subgroup]]: 4.9.25

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

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]

[[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]]): ~9/4 = 1419.190

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]]): ~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

== 2.5/3.… subgroups ==

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 generators: ~2, ~5/3

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

* [[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

{{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 }}

 

{{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

 

{{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]

* [[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*^† }}

: <nowiki/>* wart for 5/3

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

=== Marveltwintri ===

{{See also| Chromatic pairs #Marveltwintri }}

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

[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})

{{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