Dicot family: Difference between revisions
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{{Technical data page}} | {{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]]. | 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 | [[Subgroup]]: 2.3.5 | ||
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{{Mapping|legend=1| 1 1 2 | 0 2 1 }} | {{Mapping|legend=1| 1 1 2 | 0 2 1 }} | ||
: mapping generators: ~2, ~5/4 | : mapping generators: ~2, ~5/4 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1206.283{{c}}, ~ | * [[WE]]: ~2 = 1206.283{{c}}, ~5/4 = 350.420{{c}} | ||
: [[error map]]: {{val| +6.283 +5.167 -23.328 }} | : [[error map]]: {{val| +6.283 +5.167 -23.328 }} | ||
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}} | * [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}} | ||
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=== Overview to extensions === | === Overview to extensions === | ||
The second comma of the | ==== 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 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: | Temperaments discussed elsewhere are: | ||
* ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]] | * ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]] | ||
* ''[[Jamesbond]]'' → [[Whitewood family #Jamesbond|Whitewood family]] | |||
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 === | === 2.3.5.11 subgroup === | ||
Subgroup: 2.3.5.11 | Subgroup: 2.3.5.11 | ||
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Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }} | Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1206.750{{c}}, ~ | * WE: ~2 = 1206.750{{c}}, ~5/4 = 348.684{{c}} | ||
* CWE: ~2 = 1200.000{{c}}, ~ | * CWE: ~2 = 1200.000{{c}}, ~5/4 = 348.954{{c}} | ||
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }} | {{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }} | ||
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Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }} | Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1202.433{{c}}, ~ | * WE: ~2 = 1202.433{{c}}, ~5/4 = 351.237{{c}} | ||
* CWE: ~2 = 1200.000{{c}}, ~ | * CWE: ~2 = 1200.000{{c}}, ~5/4 = 350.978{{c}} | ||
{{Optimal ET sequence|legend=0| 3e, 7, 17 }} | {{Optimal ET sequence|legend=0| 3e, 7, 17 }} | ||
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Badness (Sintel): 0.536 | Badness (Sintel): 0.536 | ||
== | == Mujannabic == | ||
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 | [[Subgroup]]: 2.3.5.7 | ||
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== Flattie == | == Flattie == | ||
This temperament used to be known as | 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 | [[Subgroup]]: 2.3.5.7 | ||
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== Sharpie == | == Sharpie == | ||
This temperament used to be known as | 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 | [[Subgroup]]: 2.3.5.7 | ||
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{{Main| Decimal }} | {{Main| Decimal }} | ||
{{See also| Jubilismic clan }} | {{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 | [[Subgroup]]: 2.3.5.7 | ||
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== Sidi == | == 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 | [[Subgroup]]: 2.3.5.7 | ||
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Badness (Sintel): 1.09 | 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:Temperament families]] | ||
[[Category:Dicot family| ]] <!-- main article --> | [[Category:Dicot family| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||