Dicot family: Difference between revisions
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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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Temperaments discussed elsewhere are: | Temperaments discussed elsewhere are: | ||
* ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]] | * ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]] | ||
* ''[[Jamesbond]]'' → [[ | * ''[[Jamesbond]]'' → [[Whitewood family #Jamesbond|Whitewood family]] | ||
The rest are considered in each sections below. | The rest are considered in each sections below. | ||
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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 is 25/24 = ([[33/32]])⋅([[100/99]]), and tempering out these commas leads to the 2.3.5.11 | 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 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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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, 11c, 14c }} | {{Optimal ET sequence|legend=0| 3, 11c, 14c }} | ||
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Badness (Sintel): 1.54 | Badness (Sintel): 1.54 | ||
[[Category:Dicot family| ]] <!-- main article --> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||