Subgroup temperaments: Difference between revisions
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== 2.3.… subgroups == | == 2.3.… subgroups == | ||
=== Shrub === | === Shrub === | ||
Named by [[User: VectorGraphics|Vector]] in 2026, shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. It has an equal-temperament join of 12 & 17. | Named by [[User: VectorGraphics|Vector]] in 2026, shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. Specifically, the diatonic major third in shrub temperament is [[32/25]]. It has an equal-temperament join of 12 & 17. | ||
It can also admit a [[41/1|41st harmonic]] by tempering out [[82/81]] (same mapping as [[reversed meantone]]). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error. | It can also admit a [[41/1|41st harmonic]] by tempering out [[82/81]] (same mapping as [[reversed meantone]]). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error. | ||
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=== Barbados === | === Barbados === | ||
Barbados | Barbados tempers out [[676/675]] alone in the 2.3.13/5 subgroup. It melodically resembles [[semaphoresmic clan #Semaphore|semaphore]], but targets specifically the interseptimal intervals like [[13/10]] and [[15/13]]. [[#Bridgetown|Bridgetown]], which adds 11/5, is a notable extension of it. | ||
The {1, 3, 13/5} and {1, 3, 9, 13/5} [[minimax tuning]]s have 13/5 and 3 tuned pure, respectively. Edos which may be used for it include but are not limited to [[24edo]], [[29edo]], and [[53edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales. If tuned near-optimally, it is just barely a [[microtemperament]], with errors of under one cent for 3 and 13/5, and given its ultra-low complexity, this makes barbados an extremely [[badness|low-badness]] temperament within its subgroup. | |||
Several related temperaments, all tempering out 676/675, can be found in [[The Archipelago]]. | Several related temperaments, all tempering out 676/675, can be found in [[The Archipelago]]. | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9502{{c}}, ~26/15 = 951.0713{{c}} | * [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9502{{c}}, ~26/15 = 951.0713{{c}} | ||
: [[error map]]: {{val| -0.050 +0.188 -0.950 }} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.0933{{c}} | * [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.0933{{c}} | ||
: error map: {{val| 0.000 +0.232 -0.934 }} | |||
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }} | {{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 0.022 | ||
; Music | ; Music | ||
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=== Demon === | === Demon === | ||
Demon is a 2.9.11 subgroup temperament which equates three [[11/9]]'s with [[16/9]], or equivalently three [[18/11]]'s with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third. It was discovered by {{u|CompactStar}} while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") [[mos]], known for its lack of representation of simple temperaments. The optimal tuning for the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports this temperament. It is technically a restriction of 11-limit [[porcupine]]. | Demon is a 2.9.11 subgroup temperament which equates three [[11/9]]'s with [[16/9]], or equivalently three [[18/11]]'s with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third. It was discovered by {{u|CompactStar}} while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") [[mos]], known for its lack of representation of simple temperaments. The optimal tuning for the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports this temperament. It is technically a weak restriction of 11-limit [[porcupine]]. | ||
[[#Mabon|Mabon]] is a strong extension of this temperament to the 2.9.7.11 subgroup. | [[#Mabon|Mabon]] is a strong extension of this temperament to the 2.9.7.11 subgroup. | ||
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== 5/2-repeating subgroups == | == 5/2-repeating subgroups == | ||
=== Frosthog === | === Frosthog === | ||
Frosthog is a [[5/2]]-repeating non-octave temperament which tempers out the frostma, [[245/242]], in the 5/2.7/2.11/2 subgroup. Tempering out the frostma equates two [[11/7]]'s to 5/2, thus the period is sqrt(5/2) (which represents both 11/7 and [[35/22]]), while the generator is ~[[7/5]]. | |||
It was | It was discovered and named by {{u|CompactStar}} in 2026, after [[hedgehog]] and the frostma, as it is a "macro-hedgehog" temperament producing the same [[mos]] patterns as hedgehog (compare [[macrodiatonic and microdiatonic scales]]). | ||
The 5/2.7/2.11/2 subgroup could be seen as a somewhat more practical version of the 5.7.11 subgroup (the essential | The 5/2.7/2.11/2 subgroup could be seen as a somewhat more practical version of the 5.7.11 subgroup (the essential no-2's no-3's subgroup) on account of both containing the 5:7:11 chord, and [[5/2]] being a much more workable period size than [[5/1]], although there is still the issue of how to make 5/2 sound equivalent. | ||
[[Subgroup]]: 5/2.7/2.11/2 | [[Subgroup]]: 5/2.7/2.11/2 | ||
[[Comma list]]: 245/242 ({{monzo| 1 2 -2}}) | [[Comma list]]: 245/242 ({{monzo| 1 2 -2 }}) | ||
{{Mapping|legend=3| 2 | {{Mapping|legend=3| 2 0 1 | 0 1 1 }} | ||
: mapping generators: ~11/7, ~7/ | : mapping generators: ~11/7, ~7/2 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~11/7 = 792. | * [[Tp tuning|Subgroup]] [[WE]]: ~11/7 = 792.3594{{c}}, ~7/5 = 579.9314{{c}} | ||
* [[Tp tuning|Subgroup]] [[CWE]]: ~11/7 = 793. | * [[Tp tuning|Subgroup]] [[CWE]]: ~11/7 = 793.1569{{c}}, ~7/5 = 578.7807{{c}} | ||
[[Optimal ET sequence]]: | [[Optimal ET sequence]]: [[8ed5/2|*8]], [[14ed5/2|*14]], [[22ed5/2|*22]], [[30ed5/2|*30]], [[52ed5/2|*52]], [[82ed5/2|*82]] | ||
[[Badness]] (Sintel): 0.060 | [[Badness]] (Sintel): 0.060 | ||