Subgroup temperaments: Difference between revisions

 

<nowiki/>* Wart for 7/5

 

<nowiki/>* Wart for 49/5

 

<nowiki/>* Wart for 13/5

 

<nowiki/>* Wart for 11/5

 

<nowiki/>* Wart for 13/5

 

<nowiki/>* Wart for 19/7

 

<nowiki/>* Wart for 15

 

<nowiki/>* Wart for 6

 

<nowiki/>* Wart for 6

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]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.  

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

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

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

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

* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455

* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093

: <nowiki/>* wart for 7/3

Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])³ = [[13/10]] and ([[15/14]])³ = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.

[[Subgroup]]: 2.5.7/3.11/3.13

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})

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

[[Optimal tuning]]s:  

* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152

* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>

: <nowiki/>* wart for 7/3

Laz is related to [[avalokita]] as well as to [[winston]].  

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}

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

{{Mapping|legend=5| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}

[[Optimal tuning]]s:  

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700

{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}

: <nowiki/>* wart for 7/3

: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents

Kryptonite is related to [[krypton]].  

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})

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

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

: [[gencom]]: [2 13/12; 56/55 78/77 91/90]

[[Optimal tuning]]s:

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428

 

[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents

Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.  

: [[gencom]]: [2 5; 225/224]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638

: <nowiki/>* wart for 9/7

Sulis is related to [[minerva]] and [[würschmidt]].  

[[Subgroup]]: 2.5.9/7.11/9

[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})

{{Mapping|legend=3| 1 0 5 -9 | 0 1 -2 4 }}]

[[Optimal tuning]]s:  

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558

{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}

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

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

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

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

: <nowiki/>* wart for 5/3

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.  

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

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

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

==== Greeley ====

{{See also| Chromatic pairs #Greeley }}

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

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

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

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

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

[[Optimal tuning]]s:  

* [[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* }}

[[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=3| 1 0 -1 2 2 | 0 1 3 2 -2 }}

{{Mapping|legend=5| 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]

[[Optimal tuning]]s:  

* [[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 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) ====

[[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=3| 1 0 0 0 | 0 3 5 1 }}

{{Mapping|legend=5| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}

[[Optimal tuning]]s:  

* [[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

{{See also | Chromatic pairs #Indium }}

[[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=3| 1 0 0 2 | 0 6 10 -1 }}

{{Mapping|legend=5| 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]

[[Optimal tuning]]s:  

* [[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*^† }}

: <nowiki/>* wart for 7/3

: ^† wart for 11/3

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

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=3| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=5| 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]

[[Optimal tuning]]s:  

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

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

: [[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*^† }}

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. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].

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

{{Mapping|legend=5| 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 = 882.886

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861

Kiribati is related to [[nakika]] as well as to [[octacot]].  

: [[gencom]]: [2 21/20; 100/99 245/242]

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892

[[Comma list]]: [[24576/24565]] ({{monzo| 13 1 -3 }})

Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.  

[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695

* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174

Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].  

[[Comma list]]: [[847/845]], [[1001/1000]]

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

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

: [[gencom]]: [2 13/10; 847/845 1001/1000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}

[[Subgroup]]: 2.7/5.11/5.13/5.17/5

[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]

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

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

: [[gencom]]: [2 13/10; 170/169 221/220 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95^t, 124^t }}

: ^t wart for 17/5

[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents