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→Dicot: - links to geryon ({8/7, 25/21}, deleted), replace with ternary and quad
 
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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


Line 33: Line 32:
=== 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.


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]]
* ''[[Ternary]]'' → [[Very low accuracy temperaments #Ternary|Very low accuracy temperaments]]
* ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]
* ''[[Quad]]'' → [[Very low accuracy temperaments #Quad|Very low accuracy temperaments]]
* ''[[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 version of eudicot.
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.


=== Dicot (2.3.5.11 subgroup) ===
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


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Subgroup val mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}
Subgroup val mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}


Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}
{{Mapping|legend=4| 1 1 2 0 2 | 0 2 1 0 5 }}


Optimal tunings:  
Optimal tunings:  
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Subgroup val mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}
Subgroup val mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}


Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}
{{Mapping|legend=4| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}


Optimal tunings:  
Optimal tunings:  
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Badness (Sintel): 0.536
Badness (Sintel): 0.536


=== Eudicot (2.3.5.11 subgroup) ===
== Mujannabic ==
Subgroup: 2.3.5.11
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.
 
Comma list: 25/24, 33/32
 
Subgroup val mapping: {{mapping| 1 1 2 4 | 0 2 1 -2 }}
 
Gencom mapping: {{mapping| 1 1 2 0 4 | 0 2 1 0 -2 }}
 
Optimal tunings:
* WE: ~2 = 1209.224{{c}}, ~5/4 = 347.612{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 346.056{{c}}
 
{{Optimal ET sequence|legend=0| 3, 4, 7, 45cceee }}
 
Badness (Sintel): 0.493
 
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13
 
Comma list: 25/24, 33/32, 40/39
 
Subgroup val mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -2 -1 }}
 
Gencom mapping: {{mapping| 1 1 2 0 4 4 | 0 2 1 0 -2 -1 }}


Optimal tunings:
Mujannabic was known as ''septimal dicot'' in earlier materials such as [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].  
* WE: ~2 = 1205.934{{c}}, ~5/4 = 349.530{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 348.213{{c}}
 
{{Optimal ET sequence|legend=0| 3, 4, 7 }}
 
Badness (Sintel): 0.575
 
== Septimal dicot ==
{{Main| Dicot }}
 
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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Comma list: 15/14, 22/21, 25/24
Comma list: 15/14, 22/21, 25/24


Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}
{{Mapping|legend=0| 1 1 2 2 2 | 0 2 1 3 5 }}


Optimal tunings:  
Optimal tunings:  
Line 157: Line 123:
Comma list: 15/14, 25/24, 33/32
Comma list: 15/14, 25/24, 33/32


Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}
{{Mapping|legend=0| 1 1 2 2 4 | 0 2 1 3 -2 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 15/14, 25/24, 33/32, 40/39
Comma list: 15/14, 25/24, 33/32, 40/39


Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}
{{Mapping|legend=0| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
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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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Comma list: 21/20, 25/24, 33/32
Comma list: 21/20, 25/24, 33/32


Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}
{{Mapping|legend=0| 1 1 2 3 4 | 0 2 1 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 14/13, 21/20, 25/24, 33/32
Comma list: 14/13, 21/20, 25/24, 33/32


Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}
{{Mapping|legend=0| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 255: Line 221:
Comma list: 25/24, 28/27, 35/33
Comma list: 25/24, 28/27, 35/33


Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}
{{Mapping|legend=0| 1 1 2 1 2 | 0 2 1 6 5 }}


Optimal tunings:  
Optimal tunings:  
Line 289: Line 255:
Comma list: 25/24, 45/44, 64/63
Comma list: 25/24, 45/44, 64/63


Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}
{{Mapping|legend=0| 1 1 2 4 2 | 0 2 1 -4 5 }}


Optimal tunings:  
Optimal tunings:  
Line 304: Line 270:
Comma list: 25/24, 40/39, 45/44, 64/63
Comma list: 25/24, 40/39, 45/44, 64/63


Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}
{{Mapping|legend=0| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 319: Line 285:
Comma list: 22/21, 25/24, 33/32
Comma list: 22/21, 25/24, 33/32


Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}
{{Mapping|legend=0| 1 1 2 4 4 | 0 2 1 -4 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 334: Line 300:
Comma list: 22/21, 25/24, 33/32, 40/39
Comma list: 22/21, 25/24, 33/32, 40/39


Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}
{{Mapping|legend=0| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 349: Line 315:
Comma list: 25/24, 35/33, 64/63
Comma list: 25/24, 35/33, 64/63


Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}
{{Mapping|legend=0| 1 1 2 4 5 | 0 2 1 -4 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 364: Line 330:
Comma list: 25/24, 35/33, 40/39, 64/63
Comma list: 25/24, 35/33, 40/39, 64/63


Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}
{{Mapping|legend=0| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 405: Line 371:
Comma list: 25/24, 45/44, 49/48
Comma list: 25/24, 45/44, 49/48


Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}
{{Mapping|legend=0| 2 0 3 4 -1 | 0 2 1 1 5 }}


Optimal tunings:  
Optimal tunings:  
Line 420: Line 386:
Comma list: 25/24, 45/44, 49/48, 91/90
Comma list: 25/24, 45/44, 49/48, 91/90


Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}
{{Mapping|legend=0| 2 0 3 4 -1 1| 0 2 1 1 5 4}}


Optimal tunings:  
Optimal tunings:  
Line 435: Line 401:
Comma list: 25/24, 33/32, 49/48
Comma list: 25/24, 33/32, 49/48


Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}
{{Mapping|legend=0| 2 0 3 4 10 | 0 2 1 1 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 450: Line 416:
Comma list: 25/24, 35/33, 49/48
Comma list: 25/24, 35/33, 49/48


Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}
{{Mapping|legend=0| 2 0 3 4 7 | 0 2 1 1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 486: Line 452:
Comma list: 25/24, 45/44, 99/98
Comma list: 25/24, 45/44, 99/98


Mapping: {{mapping| 1 -1 1 -3 -3 | 0 4 2 9 10 }}
{{Mapping|legend=0| 1 -1 1 -3 -3 | 0 4 2 9 10 }}


Optimal tunings:  
Optimal tunings:  
Line 521: Line 487:
Comma list: 25/24, 33/32, 245/242
Comma list: 25/24, 33/32, 245/242


Mapping: {{mapping| 1 3 3 1 2 | 0 -4 -2 5 4 }}
{{Mapping|legend=0| 1 3 3 1 2 | 0 -4 -2 5 4 }}


Optimal tunings:  
Optimal tunings:  
Line 531: Line 497:
Badness (Sintel): 1.54
Badness (Sintel): 1.54


[[Category:Dicot family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Dicot_family"