Dicot family: Difference between revisions

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)² {{=}} (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]].

[[Subgroup]]: 2.3.5

[[Comma list]]: 25/24

: mapping generators: ~2, ~5/4

* [[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}}

[[Tuning ranges]]:

* [[5-odd-limit]] [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)

{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}

[[Badness]] (Sintel): 0.306

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.

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

The rest are considered in each sections below.

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.

Subgroup: 2.3.5.11

Comma list: 25/24, 45/44

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}}, ~5/4 = 348.684{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5/4 = 348.954{{c}}

{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}

Badness (Sintel): 0.370

Subgroup: 2.3.5.11.13

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

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 }}

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

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

{{Optimal ET sequence|legend=0| 3e, 7, 17 }}

Badness (Sintel): 0.536

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

[[Comma list]]: 15/14, 25/24

[[Optimal tuning]]s:  

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

{{Optimal ET sequence|legend=1| 3d, 4, 7 }}

[[Badness]] (Sintel): 0.504

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}

Badness (Sintel): 0.656

Subgroup: 2.3.5.7.11

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

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

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

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

{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}

Badness (Sintel): 0.896

Subgroup: 2.3.5.7.11.13

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:  

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

{{Optimal ET sequence|legend=0| 3d, 4, 7 }}

Badness (Sintel): 0.985

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

[[Comma list]]: 21/20, 25/24

{{Mapping|legend=1| 1 1 2 3 | 0 2 1 -1 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1220.466{{c}}, ~6/5 = 337.577{{c}}

: [[error map]]: {{val| +20.466 -6.335 -7.804 -45.004 }}

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

{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}

[[Badness]] (Sintel): 0.642

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

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

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

{{Optimal ET sequence|legend=0| 3, 4, 7d }}

Badness (Sintel): 0.826

Subgroup: 2.3.5.7.11.13

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:  

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

{{Optimal ET sequence|legend=0| 3, 4, 7d }}

Badness (Sintel): 0.968

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

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1202.488{{c}}, ~5/4 = 358.680{{c}}

: [[error map]]: {{val| +2.488 +17.893 -22.658 -14.258 }}

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

{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}

[[Badness]] (Sintel): 0.732

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}

Badness (Sintel): 0.739

== Dichotic ==

In dichotic, 7/4 is found at a stack of two perfect fourths.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 25/24, 64/63

[[Optimal tuning]]s:  

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

{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}

[[Badness]] (Sintel): 0.951

Subgroup: 2.3.5.7.11

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

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

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

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

{{Optimal ET sequence|legend=0| 7, 10, 17 }}

Badness (Sintel): 1.01

Subgroup: 2.3.5.7.11.13

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:  

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

{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}

Badness (Sintel): 0.896

Subgroup: 2.3.5.7.11

 

 

 

 

Badness (Sintel): 1.05

 

 

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

 

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

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1209.021{{c}}, ~32/21 = 778.298{{c}}

: [[error map]]: {{val| +9.021 +2.216 -20.696 -6.188 }}

: error map: {{val| 0.000 -10.816 -40.744 -32.749 }}

 

 

[[Badness]] (Sintel): 2.12

 

 

 

 

 

 

 

[[Category:Temperament families]]

[[Category:Dicot family| ]] <!-- main article -->

[[Category:Rank 2]]