Dicot family: Difference between revisions
New temperament: sida (3 & 11c) Tags: Mobile edit Mobile web edit |
Decanonicalize septimal dicot. - 2.3.5.11-subgroup eudicot (no need for explicit documentation if it's canonical) |
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{{Mapping|legend=1| 1 1 2 | 0 2 1 }} | {{Mapping|legend=1| 1 1 2 | 0 2 1 }} | ||
: mapping generators: ~2, ~5/4 | : mapping generators: ~2, ~5/4 | ||
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=== Overview to extensions === | === Overview to extensions === | ||
==== 7-limit extensions ==== | ==== 7-limit extensions ==== | ||
The second comma of the | The second comma of the comma list defines which [[7-limit]] family member we are looking at. Mujannabic adds [[36/35]], flattie adds [[21/20]], sharpie adds [[28/27]], and dichotic adds [[64/63]], all retaining the same period and generator. | ||
The dicot comma, 25/24, factors into the 7-limit as ([[49/48]])⋅([[50/49]]). Since [[49/48]] is the difference between [[8/7]] and [[7/6]], and [[50/49]] is the difference between [[7/5]] and [[10/7]], it makes sense to extend dicot to temper them all out, leading to decimal, a weak extension where the octave and twelfth are split in halves. Other weak extensions include sidi, which adds [[245/243]], and jamesbond, which adds [[16/15]]. Here sidi uses 14/9 as a generator, with two of them making up the combined [[5/2]][[~]][[12/5]] neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator. | The dicot comma, 25/24, factors into the 7-limit as ([[49/48]])⋅([[50/49]]). Since [[49/48]] is the difference between [[8/7]] and [[7/6]], and [[50/49]] is the difference between [[7/5]] and [[10/7]], it makes sense to extend dicot to temper them all out, leading to decimal, a weak extension where the octave and twelfth are split in halves. Other weak extensions include sidi, which adds [[245/243]], and jamesbond, which adds [[16/15]]. Here sidi uses 14/9 as a generator, with two of them making up the combined [[5/2]][[~]][[12/5]] neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator. | ||
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In the 11-limit, we have the identity 25/24 = ([[45/44]])⋅([[55/54]]), so it makes sense to temper out all of them. This leads to the very natural subgroup temperament where [[11/9]]~[[27/22]] is mapped to the neutral third. As such, this is also the path that most of the septimal extensions take to get their 11-limit versions. | In the 11-limit, we have the identity 25/24 = ([[45/44]])⋅([[55/54]]), so it makes sense to temper out all of them. This leads to the very natural subgroup temperament where [[11/9]]~[[27/22]] is mapped to the neutral third. As such, this is also the path that most of the septimal extensions take to get their 11-limit versions. | ||
An alternative identity 25/24 = ([[33/32]])⋅([[100/99]]) | An alternative identity is 25/24 = ([[33/32]])⋅([[100/99]]), and tempering out these commas leads to the 2.3.5.11-subgroup restriction of some of the temperaments below. | ||
=== 2.3.5.11 | === 2.3.5.11 subgroup === | ||
Subgroup: 2.3.5.11 | Subgroup: 2.3.5.11 | ||
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Badness (Sintel): 0.536 | Badness (Sintel): 0.536 | ||
== | == Mujannabic == | ||
Mujannabic extends dicot such that [[7/6]] and [[9/7]] are also conflated with 5/4~6/5. Although 5/4–6/5 covers a giant block of pitches already, 7/6 and 9/7 are often considered as thirds too. On that account one could argue for the utility of this extension despite the relatively poor accuracy. | |||
Mujannabic was known as ''septimal dicot'' in earlier materials such as [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder]. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Flattie == | == Flattie == | ||
This temperament used to be known as ''flat''. Unlike | This temperament used to be known as ''flat''. Unlike mujannabic where 7/6 is added to the neutral third, here [[8/7]] is added instead. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Sidi == | == Sidi == | ||
Sidi tempers out 245/243, and splits 5/2~12/5 in two. Its ploidacot is beta-tetracot. This relates it to [[squares]], to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however. | Sidi tempers out [[245/243]], and splits [[5/2]][[~]][[12/5]] in two. Its [[ploidacot]] is beta-tetracot. This relates it to [[squares]], to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Sida == | == Sida == | ||
Named by [[Xenllium]] in 2026, sida is described as the {{nowrap| 3 & 14c }} temperment, and tempers out [[1323/1280]] and [[4000/3969]]. Its [[ploidacot]] is beta-tetracot, the same as [[#Sidi|sidi]]. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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[[Comma list]]: 25/24, 1323/1280 | [[Comma list]]: 25/24, 1323/1280 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -1 1 6 | 0 4 2 -5 }} | ||
: mapping generators: ~2, ~32/21 | |||
: mapping generators: ~2, ~21 | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1209.021{{c}}, ~21 | * [[WE]]: ~2 = 1209.021{{c}}, ~32/21 = 778.298{{c}} | ||
: [[error map]]: {{val| +9.021 +2.216 -20.696 -6.188 }} | : [[error map]]: {{val| +9.021 +2.216 -20.696 -6.188 }} | ||
* [[CWE]]: ~2 = 1200.000{{c}}, ~21 | * [[CWE]]: ~2 = 1200.000{{c}}, ~32/21 = 772.785{{c}} | ||
: error map: {{val| 0.000 -10.816 -40.744 -32.749 }} | : error map: {{val| 0.000 -10.816 -40.744 -32.749 }} | ||
{{Optimal ET sequence|legend=1| 3 | {{Optimal ET sequence|legend=1| 3, 11c, 14c, 45ccdd }} | ||
[[Badness]] (Sintel): 2.12 | [[Badness]] (Sintel): 2.12 | ||
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Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1209.621{{c}}, ~ | * WE: ~2 = 1209.621{{c}}, ~11/7 = 772.376{{c}} | ||
* CWE: ~2 = 1200.000{{c}}, ~ | * CWE: ~2 = 1200.000{{c}}, ~11/7 = 772.247{{c}} | ||
{{Optimal ET sequence|legend=0| 3 | {{Optimal ET sequence|legend=0| 3, 11c, 14c }} | ||
Badness (Sintel): 1.54 | Badness (Sintel): 1.54 | ||
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[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Dicot family| ]] <!-- main article --> | [[Category:Dicot family| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||