Subgroup temperaments: Difference between revisions
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{{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 == | ||
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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 | |||
Badness (Dirichlet): 0.142 | |||
== 2.9.11 subgroup == | == 2.9.11 subgroup == | ||
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==== Tridecimal guanyintet ==== | ==== 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}. [[40edo]] remains an excellent tuning. | 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 | [[Subgroup]]: 2.5.7/3.11/3.13 | ||
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[[Subgroup]]: 2.3.19/7 | [[Subgroup]]: 2.3.19/7 | ||
[[Comma list]]: [[57/56]] ({{ | [[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 }} | ||
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[[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 == | ||