Dicot family: Difference between revisions

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

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|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}}

: [[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 }}

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

* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314] (full comma to untempered)

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

[[Badness]] (Sintel): 0.306

=== 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]], 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.

Temperaments discussed elsewhere are:  

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

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

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

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 }}

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 }}

 

 

== Septimal 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

 

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

 

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

 

* [[WE]]: ~2 = 1205.532{{c}}, ~6/5 = 337.931{{c}}

: [[error map]]: {{val| +5.532 -20.561 -37.319 +56.032 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}

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

 

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

 

 

=== 11-limit ===

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 }}

 

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

* 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:  

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

* 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 septimal dicot 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 }}

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

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

 

 

[[Badness]] (Sintel): 0.642

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Dicot| ]] <!-- key article -->