Diaschismic family: Difference between revisions
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{{Main| Diaschismic }} | {{Main| Diaschismic }} | ||
The [[period]] of diaschismic is half an [[ | The [[period]] of diaschismic is half an [[octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities. | ||
This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list. | This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list. | ||
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{{Mapping|legend=1| 2 0 11 | 0 1 -2 }} | {{Mapping|legend=1| 2 0 11 | 0 1 -2 }} | ||
: mapping generators: ~45/32, ~3 | : mapping generators: ~45/32, ~3 | ||
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==== Subgroup extensions ==== | ==== Subgroup extensions ==== | ||
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', | Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions_2|#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}. | ||
== Septimal diaschismic == | == Septimal diaschismic == | ||
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[[Badness]] (Sintel): 0.507 | [[Badness]] (Sintel): 0.507 | ||
=== 2.3.5.7.17 subgroup === | |||
Subgroup: 2.3.5.7.17 | |||
Comma list: 50/49, 64/63, 85/84 | |||
Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }} | |||
Optimal tunings: | |||
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}} | |||
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}} | |||
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }} | |||
Badness (Sintel): 0.438 | |||
=== 11-limit === | === 11-limit === | ||
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Badness (Sintel): 0.673 | Badness (Sintel): 0.673 | ||
==== 2.3.5.7.11.17 subgroup ==== | |||
Subgroup: 2.3.5.7.11.17 | |||
Comma list: 50/49, 64/63, 85/84, 99/98 | |||
Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }} | |||
Optimal tunings: | |||
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}} | |||
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}} | |||
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }} | |||
Badness (Sintel): 0.645 | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Badness (Sintel): 0.937 | Badness (Sintel): 0.937 | ||
==== 2.3.5.7.11.17 subgroup ==== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63 | |||
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }} | |||
Optimal tunings: | |||
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}} | |||
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}} | |||
{{Optimal ET sequence|legend=0| 10, 12e, 22 }} | |||
Badness (Sintel): 0.766 | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }} | Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }} | ||
: mapping generators: ~2, ~55/32 | : mapping generators: ~2, ~55/32 | ||
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{{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }} | {{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }} | ||
: mapping generators: ~25/21, ~3 | : mapping generators: ~25/21, ~3 | ||
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{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }} | {{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }} | ||
: mapping generators: ~45/32, ~35/24 | : mapping generators: ~45/32, ~35/24 | ||
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{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }} | {{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }} | ||
: mapping generators: ~45/32, ~10/7 | : mapping generators: ~45/32, ~10/7 | ||
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== Sruti == | == Sruti == | ||
{{Redirect|Sruti|the tuning concept in Indian music|Shruti}} | |||
Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning. | Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning. | ||
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{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }} | {{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }} | ||
: mapping generators: ~45/32, ~140/81 | : mapping generators: ~45/32, ~140/81 | ||
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{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }} | {{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }} | ||
: mapping generators: ~45/32, ~7/4 | : mapping generators: ~45/32, ~7/4 | ||
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{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }} | {{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }} | ||
: mapping generators: ~45/32, ~9/7 | : mapping generators: ~45/32, ~9/7 | ||
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{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }} | {{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }} | ||
: mapping generators: ~45/32, ~8/7 | : mapping generators: ~45/32, ~8/7 | ||
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{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }} | {{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }} | ||
: mapping generators: ~45/32, ~21/16 | : mapping generators: ~45/32, ~21/16 | ||
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Badness (Sintel): 1.13 | Badness (Sintel): 1.13 | ||
== Subgroup extensions == | |||
=== Srutal archagall (2.3.5.17) === | |||
{{See also | Fiventeen }} | |||
This extension of 5-limit diaschismic adds prime 17 and which with respect to [[MVP archagall]] is able to express the harmonics [[75/1|75]] and [[85/1|85]] in their appropriate prime subgroup. It achieves this by equating [[85/64]] with [[4/3]] by tempering out their difference of [[256/255]] (S16). Therefore it also tempers out [[289/288]] (S17) and thus equates [[17/15]] with [[9/8]] due to tempering out [[136/135]] (S16⋅S17). It could be described as the 10 & 12 temperament with strong emphasis on 12edo being the better tuning on the 2.3.5.17 subgroup, implying ideal tunings of 34edo, 46edo or 80edo. | |||
Subgroup: 2.3.5.17 | |||
Comma list: 136/135, 256/255 | |||
Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }} | |||
: mapping generators: ~17/12, ~3 | |||
Optimal tunings: | |||
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}} | |||
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}} | |||
{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }} | |||
Badness (Sintel): 0.212 | |||
[[Category:Diaschismic family| ]] <!-- main article --> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||