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Dicot family: Difference between revisions

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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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1206.283{{c}}, ~6/5 = 350.420{{c}}
* [[WE]]: ~2 = 1206.283{{c}}, ~5/4 = 350.420{{c}}
: [[error map]]: {{val| +6.283 +5.167 -23.328 }}
: [[error map]]: {{val| +6.283 +5.167 -23.328 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
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=== Overview to extensions ===
=== Overview to extensions ===
==== 7-limit extensions ====
==== 7-limit extensions ====
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot adds [[36/35]], flattie adds [[21/20]], sharpie adds [[28/27]], and dichotic adds [[64/63]], all retaining the same period and generator.  
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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==== Subgroup extensions ====
==== Subgroup extensions ====
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-subgroup restriction of some of the temperaments below.


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1206.750{{c}}, ~6/5 = 348.684{{c}}
* WE: ~2 = 1206.750{{c}}, ~5/4 = 348.684{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 348.954{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 348.954{{c}}


{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}
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Badness (Sintel): 0.536
Badness (Sintel): 0.536


== Septimal dicot ==
== Mujannabic ==
{{Main| Dicot }}
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.


Septimal dicot is the extension where [[7/6]] and [[9/7]] are also conflated into 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 canonicity 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 septimal dicot where 7/6 is added to the neutral third, here [[8/7]] is added instead.  
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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Badness (Sintel): 1.09
Badness (Sintel): 1.09
== 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
[[Comma list]]: 25/24, 1323/1280
{{Mapping|legend=1| 1 -1 1 6 | 0 4 2 -5 }}
: mapping generators: ~2, ~32/21
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1209.021{{c}}, ~32/21 = 778.298{{c}}
: [[error map]]: {{val| +9.021 +2.216 -20.696 -6.188 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~32/21 = 772.785{{c}}
: error map: {{val| 0.000 -10.816 -40.744 -32.749 }}
{{Optimal ET sequence|legend=1| 3, 11c, 14c, 45ccdd }}
[[Badness]] (Sintel): 2.12
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 25/24, 33/32, 245/242
Mapping: {{mapping| 1 3 3 1 2 | 0 -4 -2 5 4 }}
Optimal tunings:
* WE: ~2 = 1209.621{{c}}, ~11/7 = 772.376{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 772.247{{c}}
{{Optimal ET sequence|legend=0| 3, 11c, 14c }}
Badness (Sintel): 1.54


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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