Dicot family: Difference between revisions

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: mapping generators: ~2, ~5/4

[[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 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}

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=== Overview to 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.

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.

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==== 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.

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 ===

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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 }}

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}}

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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 }}

Optimal tunings:

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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

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Comma list: 15/14, 22/21, 25/24

~~Mapping:~~ {{~~mapping~~| 1 1 2 2 2 | 0 2 1 3 5 }}

Optimal tunings:

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Comma list: 15/14, 25/24, 33/32

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

Optimal tunings:

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

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

Optimal tunings:

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== 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

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

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

Optimal tunings:

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

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

Optimal tunings:

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Comma list: 25/24, 28/27, 35/33

~~Mapping:~~ {{~~mapping~~| 1 1 2 1 2 | 0 2 1 6 5 }}

Optimal tunings:

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Comma list: 25/24, 45/44, 64/63

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

Optimal tunings:

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Comma list: 25/24, 40/39, 45/44, 64/63

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

Optimal tunings:

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Comma list: 22/21, 25/24, 33/32

~~Mapping:~~ {{~~mapping~~| 1 1 2 4 4 | 0 2 1 -4 -2 }}

Optimal tunings:

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Comma list: 22/21, 25/24, 33/32, 40/39

~~Mapping:~~ {{~~mapping~~| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}

Optimal tunings:

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Comma list: 25/24, 35/33, 64/63

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

Optimal tunings:

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Comma list: 25/24, 35/33, 40/39, 64/63

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

Optimal tunings:

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Comma list: 25/24, 45/44, 49/48

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

Optimal tunings:

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Comma list: 25/24, 45/44, 49/48, 91/90

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

Optimal tunings:

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Comma list: 25/24, 33/32, 49/48

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

Optimal tunings:

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Comma list: 25/24, 35/33, 49/48

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

Optimal tunings:

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== 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

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Comma list: 25/24, 45/44, 99/98

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

Optimal tunings:

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Badness (Sintel): 1.09

[[~~Category~~:~~Temperament families~~]]

== 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 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 }}

[[Badness]] (Sintel): 2.12

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 245/242

Optimal tunings:

* WE: ~2 = 1209.621{{c}}, ~11/7 = 772.376{{c}}

* CWE: ~2 = 1200.000{{c}}, ~11/7 = 772.247{{c}}

Badness (Sintel): 1.54

[[Category:Dicot family| ]]

[[Category:Dicot| ]]

[[Category:~~Rank~~ 2]]

[[Category:Temperament families]]

[[Category:Catalogs of rank-2 temperaments]]

@@ Line 14: / Line 14: @@
 {{Mapping|legend=1| 1 1 2 | 0 2 1 }}
 : mapping generators: ~2, ~5/4
 [[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 }}
 * [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
@@ Line 33: / Line 32: @@
 === Overview to 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.
 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.
@@ Line 45: / Line 45: @@
 ==== 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.
+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 ===
@@ Line 53: / Line 55: @@
 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:
-* 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 }}
@@ Line 70: / Line 72: @@
 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:
@@ Line 80: / Line 82: @@
 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
@@ Line 106: / Line 108: @@
 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:
@@ Line 121: / Line 123: @@
 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:
@@ Line 136: / Line 138: @@
 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:
@@ Line 147: / Line 149: @@
 == 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
@@ Line 170: / Line 172: @@
 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:
@@ Line 185: / Line 187: @@
 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:
@@ Line 219: / Line 221: @@
 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:
@@ Line 253: / Line 255: @@
 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:
@@ Line 268: / Line 270: @@
 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:
@@ Line 283: / Line 285: @@
 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:
@@ Line 298: / Line 300: @@
 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:
@@ Line 313: / Line 315: @@
 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:
@@ Line 328: / Line 330: @@
 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:
@@ Line 369: / Line 371: @@
 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:
@@ Line 384: / Line 386: @@
 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:
@@ Line 399: / Line 401: @@
 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:
@@ Line 414: / Line 416: @@
 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:
@@ Line 425: / Line 427: @@
 == 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
@@ Line 450: / Line 452: @@
 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:
@@ Line 460: / Line 462: @@
 Badness (Sintel): 1.09
-[[Category:Temperament families]]
+== 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|legend=0| 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:Dicot family| ]] <!-- main article -->
 [[Category:Dicot| ]] <!-- key article -->
-[[Category:Rank 2]]
+[[Category:Temperament families]]
+[[Category:Catalogs of rank-2 temperaments]]

Dicot family: Difference between revisions

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