Dicot family: Difference between revisions

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== Dicot ==

{{~~Redirect~~|~~Dicot|the ploidacot signature|Ploidacot/~~Dicot}}

The head of this family, dicot, is [[generator|generated]] by a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, {{nowrap|(5/4)<sup>2</sup> {{=}} (3/2)(25/24)}}. Its [[ploidacot]] is the same as its name, dicot.

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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 ===

~~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 the most sense to temper out both of them, leading to decimal. In the 11-limit, we have the identity 25/24=~~([[45/44]])([[55/54]]), but also 50/49~~=([[99/98]])([[100/99]]), so it makes sense to temper out all of them. That interestingly leads to a rank-1 temperament, the 14c val of [[14edo]]. However, since dicot is an exotemperament anyway, the other extensions should also be considered.

==== 7-limit extensions ====

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 second comma of the ~~[[normal lists|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.

~~Decimal adds~~ [[49/48]]~~, sidi adds~~ [[~~245~~/~~243~~]], and ~~jamesbond adds~~ [[16/15]]~~. Here decimal divides~~ the [[~~period~~]] to a [[~~sqrt(2)|semi-octave~~]], and 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:

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* ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]

The rest are considered below.

The rest are considered in each sections below.

==== 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 ===

~~The 2.3.5.11-subgroup extension maps [[11/9]]~[[27/22]] to the neutral third. As such, it is related to most of the septimal extensions.~~

Subgroup: 2.3.5.11

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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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Badness (Sintel): 0.536

== ~~Septimal dicot~~ ==

== Mujannabic ==

~~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 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

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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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Decimal tempers out 49/48 and [[50/49]], and has a semi-octave period for 7/5~10/7 and a hemitwelfth generator for 7/4~12/7. Its ploidacot is diploid dicot. [[10edo]] makes for a good tuning, from which it derives its name. [[14edo]] in the 14c val and [[24edo]] in the 24c val are also among the possibilities.

Decimal can be extended to the 11-limit by the usual path of tempering out 45/44 and 55/54. There is an alternative due to the identity 50/49 = ([[99/98]])⋅([[100/99]]), in which case it also tempers out 33/32. The two mappings meet at the 14c val of [[14edo]].

[[Subgroup]]: 2.3.5.7

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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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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 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

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}}

Badness (Sintel): 1.54

[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Dicot family| ]]

~~[[Category:Dicot| ]] ~~

[[Category:Rank 2]]

@@ Line 3: / Line 3: @@
 == Dicot ==
-{{Redirect|Dicot|the ploidacot signature|Ploidacot/Dicot}}
+{{Main| Dicot }}
 The head of this family, dicot, is [[generator|generated]] by a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, {{nowrap|(5/4)<sup>2</sup> {{=}} (3/2)(25/24)}}. Its [[ploidacot]] is the same as its name, dicot.
@@ 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 32: / Line 31: @@
 === Overview to extensions ===
-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 the most sense to temper out both of them, leading to decimal. In the 11-limit, we have the identity 25/24=([[45/44]])([[55/54]]), but also 50/49=([[99/98]])([[100/99]]), so it makes sense to temper out all of them. That interestingly leads to a rank-1 temperament, the 14c val of [[14edo]]. However, since dicot is an exotemperament anyway, the other extensions should also be considered.
+==== 7-limit extensions ====
+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 second comma of the [[normal lists|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.
-Decimal adds [[49/48]], sidi adds [[245/243]], and jamesbond adds [[16/15]]. Here decimal divides the [[period]] to a [[sqrt(2)|semi-octave]], and 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:
@@ Line 42: / Line 40: @@
 * ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]
-The rest are considered below.
+The rest are considered in each sections below.
+==== 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 ===
-The 2.3.5.11-subgroup extension maps [[11/9]]~[[27/22]] to the neutral third. As such, it is related to most of the septimal extensions.
 Subgroup: 2.3.5.11
@@ Line 56: / Line 57: @@
 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 80: / Line 81: @@
 Badness (Sintel): 0.536
-== Septimal dicot ==
+== Mujannabic ==
-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 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
@@ Line 145: / Line 148: @@
 == 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 341: / Line 344: @@
 Decimal tempers out 49/48 and [[50/49]], and has a semi-octave period for 7/5~10/7 and a hemitwelfth generator for 7/4~12/7. Its ploidacot is diploid dicot. [[10edo]] makes for a good tuning, from which it derives its name. [[14edo]] in the 14c val and [[24edo]] in the 24c val are also among the possibilities.
+Decimal can be extended to the 11-limit by the usual path of tempering out 45/44 and 55/54. There is an alternative due to the identity 50/49 = ([[99/98]])⋅([[100/99]]), in which case it also tempers out 33/32. The two mappings meet at the 14c val of [[14edo]].
 [[Subgroup]]: 2.3.5.7
@@ Line 421: / Line 426: @@
 == 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 455: / Line 460: @@
 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:Pages with mostly numerical content]]
 [[Category:Dicot family| ]] <!-- main article -->
-[[Category:Dicot| ]] <!-- key article -->
 [[Category:Rank 2]]