Dicot family: Difference between revisions

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The '''dicot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[25/24]], the classical chromatic semitone. 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)}}.

The '''dicot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[25/24]], the classical chromatic semitone. Dicot was likely the first named of the temperaments ending in -cot, as it is the only one to correspond with a proper botanical term (referring to plants with two embryonic leaves) and it is the most inaccurate.

== 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.

Possible tunings for dicot are [[7edo]], [[10edo]], [[17edo]], [[24edo]] using the val {{val| 24 38 55 }} (24c), and [[31edo]] using the val {{val| 31 49 71 }} (31c). In a sense, what dicot is all about is using neutral thirds and sixths and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", dicot is close to the edge of what can be sensibly called a temperament at all. In other words, it is an [[exotemperament]].

~~== Dicot ==~~

[[Subgroup]]: 2.3.5

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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}}

: error map: {{val| 0.000 +0.216 -35.228 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 354.664{{c}}~~

~~: [[error map]]: {{val| 0.000 +7.374 -31.649 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 348.594{{c}}

~~: error map: {{val| 0.000 -4.766 -37.719 }} -->~~

[[Tuning ranges]]:

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=== Overview to extensions ===

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, sharpie adds 28/27, and dichotic adds 64/63, all retaining the same period and generator.

==== 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.

~~Decimal adds~~ 49/48, sidi adds 245/243, and jamesbond adds 81/80. Here ~~decimal divides the period to 1/2 octave, and~~ sidi uses 9/7 as a generator, with two of them making up the combined 5/~~3 and 8~~/5 neutral ~~sixth~~. 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:

* ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]]

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

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.

~~The rest are considered~~ below.

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 is related to [[#Septimal dicot|septimal dicot]], [[#Sharpie|sharpie]], and [[#Dichotic|dichotic]].~~

Subgroup: 2.3.5.11

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Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}

~~: gencom: [2 5/4; 25/24 45/44]~~

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}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 352.287{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~6/5 = 346.734{{c}} -->

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

~~: gencom: [2 5/4; 25/24 40/39 45/44]~~

Optimal tunings:

* WE: ~2 = 1202.433{{c}}, ~6/5 = 351.237{{c}}

* WE: ~2 = 1202.433{{c}}, ~5/4 = 351.237{{c}}

* CWE~~: ~2 = 1200.000{{c}}, ~6/5 = 350.978{{c}}~~

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 350.978{{c}}

~~<!-- * CTE~~: ~2 = 1200.000{{c}}, ~5/4 ~~= 352.420{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~6/5 = 350.~~526~~{{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 is a giant block 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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* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}

: error map: {{val| 0.000 -24.834 -47.753 +46.856 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 342.257{{c}}~~

~~: [[error map]]: {{val| 0.000 -17.441 -44.056 +57.946 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 336.381{{c}}

~~: error map: {{val| 0.000 -29.193 -49.933 +40.316 }} -->~~

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* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}

* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 345.596{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~6/5 = 342.125{{c}} -->

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* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}

* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 340.417{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~6/5 = 336.051{{c}} -->

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* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}

* CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 340.835{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~6/5 = 338.846{{c}} -->

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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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* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 335.391{{c}}

: error map: {{val| 0.000 -31.173 -50.922 -104.217 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 346.438{{c}}~~

~~: [[error map]]: {{val| 0.000 -9.080 -39.876 -115.264 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 331.916{{c}}

~~: error map: {{val| 0.000 -38.123 -54.398 -100.742 }} -->~~

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* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}

* CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 343.139{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~6/5 = 337.532{{c}} -->

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* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}

* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 343.655{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~6/5 = 341.023{{c}} -->

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== Sharpie ==

This temperament used to be known as '''sharp'''. This is where you find 7/6 at the major second and 7/4 at the major sixth.

This temperament used to be known as ''sharp''. This is where you find 7/6 at the major second and [[7/4]] at the major sixth.

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}

: error map: {{val| 0.000 +15.035 -27.818 -17.854 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 359.564{{c}}~~

~~: [[error map]]: {{val| 0.000 +17.173 -26.750 -11.442 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 357.938{{c}}

~~: error map: {{val| 0.000 +13.921 -28.376 -21.198 }} -->~~

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* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 357.261{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~5/4 = 356.106{{c}} -->

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* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}

: error map: {{val| 0.000 +10.595 -30.039 +6.074 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.333{{c}}~~

~~: [[error map]]: {{val| 0.000 +10.710 -29.981 +5.844 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.264{{c}}

~~: error map: {{val| 0.000 +10.573 -30.050 +6.119 }} -->~~

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* WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.236{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 354.183{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.262{{c}} -->

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* WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.340{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 354.247{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.365{{c}} -->

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* WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.024{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 353.751{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.073{{c}} -->

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* WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.254{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 353.850{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.313{{c}} -->

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* WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.745{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 361.081{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~5/4 = 360.659{{c}} -->

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* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 361.061{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~5/4 = 360.646{{c}} -->

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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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* [[CWE]]: ~7/5 = 600.000{{c}}, ~7/4 = 950.957{{c}} (~7/6 = 249.043{{c}})

: error map: {{val| 0.000 -0.041 -35.357 -17.869 }}

~~<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~7/4 = 955.608{{c}} (~8/7 = 244.392{{c}})~~

~~: [[error map]]: {{val| 0.000 +9.260 -30.706 -13.218 }}~~

* [[POTE]]: ~7/5 = 600.000{{c}}, ~7/4 = 948.443{{c}} (~7/6 = 251.557{{c}})

~~: error map: {{val| 0.000 -5.069 -37.871 -20.383 }} -->~~

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* WE: ~7/5 = 603.558{{c}}, ~7/4 = 952.121{{c}} (~7/6 = 254.996{{c}})

* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 948.610{{c}} (~7/6 = 251.390{{c}})

~~<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 952.812{{c}} (~8/7 = 247.188{{c}})~~

* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 946.507{{c}} (~7/6 = 253.493{{c}}) -->

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* WE: ~7/5 = 603.612{{c}}, ~7/4 = 953.663{{c}} (~7/6 = 253.562{{c}})

* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 950.116{{c}} (~7/6 = 249.884{{c}})

~~<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 954.469{{c}} (~8/7 = 245.531{{c}})~~

* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 947.955{{c}} (~7/6 = 252.045{{c}}) -->

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* WE: ~7/5 = 604.535{{c}}, ~7/4 = 952.076{{c}} (~7/6 = 256.994{{c}})

* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 946.108{{c}} (~7/6 = 253.892{{c}})

~~<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 950.940{{c}} (~7/6 = 249.060{{c}})~~

* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 944.934{{c}} (~7/6 = 255.066{{c}}) -->

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* WE: ~7/5 = 599.404{{c}}, ~7/4 = 955.557{{c}} (~8/7 = 243.251{{c}})

* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 956.169{{c}} (~8/7 = 243.831{{c}})

~~<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 955.608{{c}} (~8/7 = 244.392{{c}})~~

* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 956.507{{c}} (~8/7 = 243.493{{c}}) -->

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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.

[[Subgroup]]: 2.3.5.7

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* [[CWE]]: ~2 = 1200.000{{c}}, ~14/9 = 773.872{{c}}

: error map: {{val| 0.000 -6.464 -38.569 -3.973 }}

~~<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~14/9 = 775.548{{c}}~~

~~: [[error map]]: {{val| 0.000 +0.238 -35.217 +11.108 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 772.792{{c}}

~~: error map: {{val| 0.000 -10.789 -40.731 -13.702 }} -->~~

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* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}

* CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~11/7 = 775.413{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~11/7 = 772.727{{c}} -->

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 1: / Line 1: @@
 {{Technical data page}}
-The '''dicot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[25/24]], the classical chromatic semitone. 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)}}.
+The '''dicot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[25/24]], the classical chromatic semitone. Dicot was likely the first named of the temperaments ending in -cot, as it is the only one to correspond with a proper botanical term (referring to plants with two embryonic leaves) and it is the most inaccurate.
+== 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.
 Possible tunings for dicot are [[7edo]], [[10edo]], [[17edo]], [[24edo]] using the val {{val| 24 38 55 }} (24c), and [[31edo]] using the val {{val| 31 49 71 }} (31c). In a sense, what dicot is all about is using neutral thirds and sixths and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", dicot is close to the edge of what can be sensibly called a temperament at all. In other words, it is an [[exotemperament]].
-== Dicot ==
 [[Subgroup]]: 2.3.5
@@ Line 10: / 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}}
 : error map: {{val| 0.000 +0.216 -35.228 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 354.664{{c}}
-: [[error map]]: {{val| 0.000 +7.374 -31.649 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 348.594{{c}}
-: error map: {{val| 0.000 -4.766 -37.719 }} -->
 [[Tuning ranges]]:
@@ Line 32: / Line 31: @@
 === Overview to extensions ===
-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, sharpie adds 28/27, and dichotic adds 64/63, all retaining the same period and generator.
+==== 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.
-Decimal adds 49/48, sidi adds 245/243, and jamesbond adds 81/80. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth. 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:
 * ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]]
+* ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]
+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.
-The rest are considered below.
+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 is related to [[#Septimal dicot|septimal dicot]], [[#Sharpie|sharpie]], and [[#Dichotic|dichotic]].
 Subgroup: 2.3.5.11
@@ Line 51: / Line 55: @@
 Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}
-: gencom: [2 5/4; 25/24 45/44]
 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}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 352.287{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~6/5 = 346.734{{c}} -->
 {{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}
@@ Line 72: / Line 72: @@
 Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}
-: gencom: [2 5/4; 25/24 40/39 45/44]
 Optimal tunings:
-* WE: ~2 = 1202.433{{c}}, ~6/5 = 351.237{{c}}
+* WE: ~2 = 1202.433{{c}}, ~5/4 = 351.237{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~6/5 = 350.978{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~5/4 = 350.978{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 352.420{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~6/5 = 350.526{{c}} -->
 {{Optimal ET sequence|legend=0| 3e, 7, 17 }}
@@ Line 85: / 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 is a giant block 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 99: / Line 97: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}
 : error map: {{val| 0.000 -24.834 -47.753 +46.856 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 342.257{{c}}
-: [[error map]]: {{val| 0.000 -17.441 -44.056 +57.946 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 336.381{{c}}
-: error map: {{val| 0.000 -29.193 -49.933 +40.316 }} -->
 {{Optimal ET sequence|legend=1| 3d, 4, 7 }}
@@ Line 118: / Line 112: @@
 * WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 345.596{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~6/5 = 342.125{{c}} -->
 {{Optimal ET sequence|legend=0| 3de, 4e, 7 }}
@@ Line 135: / Line 127: @@
 * WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 340.417{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~6/5 = 336.051{{c}} -->
 {{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}
@@ Line 152: / Line 142: @@
 * WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 340.835{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~6/5 = 338.846{{c}} -->
 {{Optimal ET sequence|legend=0| 3d, 4, 7 }}
@@ Line 160: / 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 173: / Line 161: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 335.391{{c}}
 : error map: {{val| 0.000 -31.173 -50.922 -104.217 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 346.438{{c}}
-: [[error map]]: {{val| 0.000 -9.080 -39.876 -115.264 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 331.916{{c}}
-: error map: {{val| 0.000 -38.123 -54.398 -100.742 }} -->
 {{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
@@ Line 192: / Line 176: @@
 * WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 343.139{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~6/5 = 337.532{{c}} -->
 {{Optimal ET sequence|legend=0| 3, 4, 7d }}
@@ Line 209: / Line 191: @@
 * WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 343.655{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~6/5 = 341.023{{c}} -->
 {{Optimal ET sequence|legend=0| 3, 4, 7d }}
@@ Line 217: / Line 197: @@
 == Sharpie ==
-This temperament used to be known as '''sharp'''. This is where you find 7/6 at the major second and 7/4 at the major sixth.
+This temperament used to be known as ''sharp''. This is where you find 7/6 at the major second and [[7/4]] at the major sixth.
 [[Subgroup]]: 2.3.5.7
@@ Line 230: / Line 210: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}
 : error map: {{val| 0.000 +15.035 -27.818 -17.854 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 359.564{{c}}
-: [[error map]]: {{val| 0.000 +17.173 -26.750 -11.442 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 357.938{{c}}
-: error map: {{val| 0.000 +13.921 -28.376 -21.198 }} -->
 {{Optimal ET sequence|legend=1| 3d, 7d, 10 }}
@@ Line 249: / Line 225: @@
 * WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 357.261{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~5/4 = 356.106{{c}} -->
 {{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}
@@ Line 270: / Line 244: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}
 : error map: {{val| 0.000 +10.595 -30.039 +6.074 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.333{{c}}
-: [[error map]]: {{val| 0.000 +10.710 -29.981 +5.844 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.264{{c}}
-: error map: {{val| 0.000 +10.573 -30.050 +6.119 }} -->
 {{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}
@@ Line 289: / Line 259: @@
 * WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.236{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 354.183{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.262{{c}} -->
 {{Optimal ET sequence|legend=0| 7, 10, 17 }}
@@ Line 306: / Line 274: @@
 * WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.340{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 354.247{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.365{{c}} -->
 {{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}
@@ Line 323: / Line 289: @@
 * WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.024{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 353.751{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.073{{c}} -->
 {{Optimal ET sequence|legend=0| 3, 7, 10e }}
@@ Line 340: / Line 304: @@
 * WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.254{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 353.850{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.313{{c}} -->
 {{Optimal ET sequence|legend=0| 3, 7, 10e }}
@@ Line 357: / Line 319: @@
 * WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.745{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 361.081{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~5/4 = 360.659{{c}} -->
 {{Optimal ET sequence|legend=0| 3, 7e, 10 }}
@@ Line 374: / Line 334: @@
 * WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 361.061{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~5/4 = 360.646{{c}} -->
 {{Optimal ET sequence|legend=0| 3, 7e, 10 }}
@@ Line 384: / Line 342: @@
 {{Main| Decimal }}
 {{See also| Jubilismic clan }}
+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 398: / Line 360: @@
 * [[CWE]]: ~7/5 = 600.000{{c}}, ~7/4 = 950.957{{c}} (~7/6 = 249.043{{c}})
 : error map: {{val| 0.000 -0.041 -35.357 -17.869 }}
-<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~7/4 = 955.608{{c}} (~8/7 = 244.392{{c}})
-: [[error map]]: {{val| 0.000 +9.260 -30.706 -13.218 }}
-* [[POTE]]: ~7/5 = 600.000{{c}}, ~7/4 = 948.443{{c}} (~7/6 = 251.557{{c}})
-: error map: {{val| 0.000 -5.069 -37.871 -20.383 }} -->
 {{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}
@@ Line 417: / Line 375: @@
 * WE: ~7/5 = 603.558{{c}}, ~7/4 = 952.121{{c}} (~7/6 = 254.996{{c}})
 * CWE: ~7/5 = 600.000{{c}}, ~7/4 = 948.610{{c}} (~7/6 = 251.390{{c}})
-<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 952.812{{c}} (~8/7 = 247.188{{c}})
-* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 946.507{{c}} (~7/6 = 253.493{{c}}) -->
 {{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}
@@ Line 434: / Line 390: @@
 * WE: ~7/5 = 603.612{{c}}, ~7/4 = 953.663{{c}} (~7/6 = 253.562{{c}})
 * CWE: ~7/5 = 600.000{{c}}, ~7/4 = 950.116{{c}} (~7/6 = 249.884{{c}})
-<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 954.469{{c}} (~8/7 = 245.531{{c}})
-* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 947.955{{c}} (~7/6 = 252.045{{c}}) -->
 {{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}
@@ Line 451: / Line 405: @@
 * WE: ~7/5 = 604.535{{c}}, ~7/4 = 952.076{{c}} (~7/6 = 256.994{{c}})
 * CWE: ~7/5 = 600.000{{c}}, ~7/4 = 946.108{{c}} (~7/6 = 253.892{{c}})
-<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 950.940{{c}} (~7/6 = 249.060{{c}})
-* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 944.934{{c}} (~7/6 = 255.066{{c}}) -->
 {{Optimal ET sequence|legend=0| 4, 10e, 14c }}
@@ Line 468: / Line 420: @@
 * WE: ~7/5 = 599.404{{c}}, ~7/4 = 955.557{{c}} (~8/7 = 243.251{{c}})
 * CWE: ~7/5 = 600.000{{c}}, ~7/4 = 956.169{{c}} (~8/7 = 243.831{{c}})
-<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 955.608{{c}} (~8/7 = 244.392{{c}})
-* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 956.507{{c}} (~8/7 = 243.493{{c}}) -->
 {{Optimal ET sequence|legend=0| 4, 6, 10 }}
@@ Line 476: / 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.
 [[Subgroup]]: 2.3.5.7
@@ Line 489: / Line 441: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~14/9 = 773.872{{c}}
 : error map: {{val| 0.000 -6.464 -38.569 -3.973 }}
-<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~14/9 = 775.548{{c}}
-: [[error map]]: {{val| 0.000 +0.238 -35.217 +11.108 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 772.792{{c}}
-: error map: {{val| 0.000 -10.789 -40.731 -13.702 }} -->
 {{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}
@@ Line 508: / Line 456: @@
 * WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
 * CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/7 = 775.413{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~11/7 = 772.727{{c}} -->
 {{Optimal ET sequence|legend=0| 3de, …, 11cdee, 14c }}
 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]]