Xenharmonic Wiki

Subgroup temperaments: Difference between revisions

← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,033 edits
decanonicalized "edson" for hemifamity
Overthink (talk | contribs)
autopatrolled
2,671 edits
→Sburb: cent value
 
(10 intermediate revisions by 5 users not shown)
Line 137: Line 137:


{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
== 2.3.25 subgroup ==
=== Shrub ===
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.
Subgroup: 2.3.25
Edo join: 17 & 12
Comma list: [[2048/2025]]
{{Mapping|legend=2| 1 1 7| 0 1 -4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''
Edo join: 17 & 12
Comma list: 2048/2025, 576/575, 82/81
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
===== Sburb =====
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.
Subgroup: 2.3.7.23.25.41.59
Edo join: 17 & 12
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387


== 2.9.5.11 subgroup ==
== 2.9.5.11 subgroup ==
Line 274: Line 313:


Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
== 2.9.7.13.17 subgroup ==
=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].
[[Subgroup]]: 2.9.7.13.17
[[Comma list]]: 729/728, 442/441, 833/832
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836


== 2.9.11 subgroup ==
== 2.9.11 subgroup ==
Line 1,030: Line 1,082:
{{See also | Chromatic pairs #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.  
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.  


[[Subgroup]]: 2.5.7/3.11/3
[[Subgroup]]: 2.5.7/3.11/3
Line 1,036: Line 1,088:
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
[[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|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~12/7
: mapping generators: ~2, ~7/6


{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
Line 1,043: Line 1,095:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093


{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
Line 1,050: Line 1,102:


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])<sup>3</sup> = [[13/10]] and ([[15/14]])<sup>3</sup> = [[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=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7
[[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
Badness (Sintel): 0.329


==== Laz ====
==== Laz ====
Line 1,611: Line 1,682:
[[Subgroup]]: 2.3.19/7
[[Subgroup]]: 2.3.19/7


[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})


{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
Line 1,625: Line 1,696:


[[Badness]] (Sintel): 0.082
[[Badness]] (Sintel): 0.082
=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].
[[Subgroup]]: 2.17/7.19/7
[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})
{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}
{{Optimal ET sequence|legend=1| 7, 18, 25 }}
[[Badness]] (Sintel): 0.005
==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.
Subgroup: 2.17/7.19/7.23/7
Comma list: [[323/322]], [[392/391]]
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}
Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}
{{Optimal ET sequence|legend=0| 7, 18, 25 }}
Badness (Sintel): 0.029
==== 2.25/7.17/7.19/7.23/7 subgroup ====
Subgroup: 2.25/7.17/7.19/7.23/7
Comma list: [[323/322]], [[392/391]], [[476/475]]
Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}
Optimal tunings:
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}
{{Optimal ET sequence|legend=0| 7, 18, 25 }}
Badness (Sintel): 0.053


== 3/2.5/2.… subgroups ==
== 3/2.5/2.… subgroups ==
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"