Xenharmonic Wiki

Dicot family: Difference between revisions

← Older edit
VisualWikitext
→Septimal dicot: I have no clue what problem you had with what I wrote, but here's my attempt of clarification. -unnecessary formatting complications
Decanonicalize septimal dicot. - 2.3.5.11-subgroup eudicot (no need for explicit documentation if it's canonical)
 
(21 intermediate revisions by 6 users not shown)
Line 1: Line 1:
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)}}.  
{{Technical data page}}
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]].


== Dicot ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 9: Line 14:


{{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:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 354.664
* [[WE]]: ~2 = 1206.283{{c}}, ~5/4 = 350.420{{c}}
: [[error map]]: {{val| 0.000 +7.374 -31.649 }}
: [[error map]]: {{val| +6.283 +5.167 -23.328 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 348.594
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
: error map: {{val| 0.000 -4.766 -37.719 }}
: error map: {{val| 0.000 +0.216 -35.228 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 24: Line 28:
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}


[[Badness]] (Smith): 0.013028
[[Badness]] (Sintel): 0.306


=== Overview to extensions ===
=== 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:  
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]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]
The rest are considered in each sections below.


The rest are considered 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 ===
The 2.3.5.11-subgroup extension is related to [[#Septimal dicot|septimal dicot]], [[#Sharpie|sharpie]], and [[#Dichotic|dichotic]].
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


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


Sval mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}
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 }}
Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}
: gencom: [2 5/4; 25/24 45/44]


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 352.287
* WE: ~2 = 1206.750{{c}}, ~5/4 = 348.684{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 346.734
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 348.954{{c}}


{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c, 38cc, 45cce }}
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}


RMS error: 5.621 cents
Badness (Sintel): 0.370


==== 2.3.5.11.13 subgroup ====
==== 2.3.5.11.13 subgroup ====
Line 62: Line 69:
Comma list: 25/24, 40/39, 45/44
Comma list: 25/24, 40/39, 45/44


Sval mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}
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 }}
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}}, ~5/4 = 351.237{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 350.978{{c}}


Optimal tunings:
{{Optimal ET sequence|legend=0| 3e, 7, 17 }}
* CTE: ~2 = 1200.000, ~5/4 = 352.420
* POTE: ~2 = 1200.000, ~6/5 = 350.526


{{Optimal ET sequence|legend=0| 3e, 7, 17, 24c }}
Badness (Sintel): 0.536


RMS error: 5.916 cents
== 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.  


== Septimal dicot ==
Mujannabic was known as ''septimal dicot'' in earlier materials such as [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].  
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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 84: Line 91:


{{Mapping|legend=1| 1 1 2 2 | 0 2 1 3 }}
{{Mapping|legend=1| 1 1 2 2 | 0 2 1 3 }}
{{Multival|legend=1| 2 1 3 -3 -1 4 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 342.257
* [[WE]]: ~2 = 1205.532{{c}}, ~6/5 = 337.931{{c}}
: [[error map]]: {{val| 0.000 -17.441 -44.056 +57.946 }}
: [[error map]]: {{val| +5.532 -20.561 -37.319 +56.032 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 336.381
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}
: error map: {{val| 0.000 -29.193 -49.933 +40.316 }}
: error map: {{val| 0.000 -24.834 -47.753 +46.856 }}


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


[[Badness]] (Smith): 0.019935
[[Badness]] (Sintel): 0.504


=== 11-limit ===
=== 11-limit ===
Line 105: Line 110:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 345.596
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 342.125
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}


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


Badness (Smith): 0.019854
Badness (Sintel): 0.656


=== Eudicot ===
=== Eudicot ===
Line 120: Line 125:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 340.417
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 336.051
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}


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


Badness (Smith): 0.027114
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 135: Line 140:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 340.835
* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 338.846
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}


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


Badness (Smith): 0.023828
Badness (Sintel): 0.985


== Flattie ==
== 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
[[Subgroup]]: 2.3.5.7
Line 150: Line 155:


{{Mapping|legend=1| 1 1 2 3 | 0 2 1 -1 }}
{{Mapping|legend=1| 1 1 2 3 | 0 2 1 -1 }}
{{Multival|legend=1| 2 1 -1 -3 -7 -5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 346.438
* [[WE]]: ~2 = 1220.466{{c}}, ~6/5 = 337.577{{c}}
: [[error map]]: {{val| 0.000 -9.080 -39.876 -115.264 }}
: [[error map]]: {{val| +20.466 -6.335 -7.804 -45.004 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 331.916
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 335.391{{c}}
: error map: {{val| 0.000 -38.123 -54.398 -100.742 }}
: error map: {{val| 0.000 -31.173 -50.922 -104.217 }}


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


[[Badness]] (Smith): 0.025381
[[Badness]] (Sintel): 0.642


=== 11-limit ===
=== 11-limit ===
Line 171: Line 174:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 343.139
* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 337.532
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}


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


Badness (Smith): 0.024988
Badness (Sintel): 0.826


=== 13-limit ===
=== 13-limit ===
Line 186: Line 189:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 343.655
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 341.023
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}


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


Badness (Smith): 0.023420
Badness (Sintel): 0.968


== Sharpie ==
== 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
[[Subgroup]]: 2.3.5.7
Line 203: Line 206:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 359.564
* [[WE]]: ~2 = 1202.488{{c}}, ~5/4 = 358.680{{c}}
: [[error map]]: {{val| 0.000 +17.173 -26.750 -11.442 }}
: [[error map]]: {{val| +2.488 +17.893 -22.658 -14.258 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 357.938
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}
: error map: {{val| 0.000 +13.921 -28.376 -21.198 }}
: error map: {{val| 0.000 +15.035 -27.818 -17.854 }}
 
{{Multival|legend=1| 2 1 6 -3 4 11 }}


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


[[Badness]] (Smith): 0.028942
[[Badness]] (Sintel): 0.732


=== 11-limit ===
=== 11-limit ===
Line 222: Line 223:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 357.261
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 356.106
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}


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


Badness (Smith): 0.022366
Badness (Sintel): 0.739


== Dichotic ==
== Dichotic ==
Line 237: Line 238:


{{Mapping|legend=1| 1 1 2 4 | 0 2 1 -4 }}
{{Mapping|legend=1| 1 1 2 4 | 0 2 1 -4 }}
{{Multival|legend=1| 2 1 -4 -3 -12 -12 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 356.333
* [[WE]]: ~2 = 1200.802{{c}}, ~5/4 = 356.502{{c}}
: [[error map]]: {{val| 0.000 +10.710 -29.981 +5.844 }}
: [[error map]]: {{val| +0.802 +11.851 -28.208 +8.374 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 356.264
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}
: error map: {{val| 0.000 +10.573 -30.050 +6.119 }}
: error map: {{val| 0.000 +10.595 -30.039 +6.074 }}


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


[[Badness]] (Smith): 0.037565
[[Badness]] (Sintel): 0.951


=== 11-limit ===
=== 11-limit ===
Line 258: Line 257:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 354.183
* WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 354.262
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.236{{c}}


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


Badness (Smith): 0.030680
Badness (Sintel): 1.01


==== 13-limit ====
==== 13-limit ====
Line 273: Line 272:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 354.247
* WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 354.365
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.340{{c}}


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


Badness (Smith): 0.021674
Badness (Sintel): 0.896


=== Dichotomic ===
=== Dichotomic ===
Line 288: Line 287:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 353.751
* WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 354.073
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.024{{c}}


{{Optimal ET sequence|legend=0| 3, 7, 10e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness (Smith): 0.031719
Badness (Sintel): 1.05


==== 13-limit ====
==== 13-limit ====
Line 303: Line 302:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 353.850
* WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 354.313
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.254{{c}}


{{Optimal ET sequence|legend=0| 3, 7, 10e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness (Smith): 0.022741
Badness (Sintel): 0.940


=== Dichosis ===
=== Dichosis ===
Line 318: Line 317:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 361.081
* WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 360.659
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.745{{c}}


{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness (Smith): 0.041361
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 333: Line 332:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 361.061
* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 360.646
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}


{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness (Smith): 0.027938
Badness (Sintel): 1.15


== Decimal ==
== Decimal ==
{{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
Line 351: Line 354:


: mapping generators: ~7/5, ~7/4
: mapping generators: ~7/5, ~7/4
{{Multival|legend=1| 4 2 2 -6 -8 -1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~7/4 = 955.608 (~8/7 = 244.392)
* [[WE]]: ~7/5 = 603.286{{c}}, ~7/4 = 953.637{{c}} (~7/6 = 252.935{{c}})
: [[error map]]: {{val| 0.000 +9.260 -30.706 -13.218 }}
: [[error map]]: {{val| +6.571 +5.318 -22.821 -2.047 }}
* [[POTE]]: ~7/5 = 600.000, ~7/4 = 948.443 (~7/6 = 251.557)
* [[CWE]]: ~7/5 = 600.000{{c}}, ~7/4 = 950.957{{c}} (~7/6 = 249.043{{c}})
: error map: {{val| 0.000 -5.069 -37.871 -20.383 }}
: error map: {{val| 0.000 -0.041 -35.357 -17.869 }}


{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}


[[Badness]] (Smith): 0.028334
[[Badness]] (Sintel): 0.717


=== 11-limit ===
=== 11-limit ===
Line 372: Line 373:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~7/4 = 952.812 (~8/7 = 247.188)
* WE: ~7/5 = 603.558{{c}}, ~7/4 = 952.121{{c}} (~7/6 = 254.996{{c}})
* POTE: ~7/5 = 600.000, ~7/4 = 946.507 (~7/6 = 253.493)
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 948.610{{c}} (~7/6 = 251.390{{c}})


{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}
{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}


Badness (Smith): 0.026712
Badness (Sintel): 0.883


==== 13-limit ====
==== 13-limit ====
Line 387: Line 388:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~7/4 = 954.469 (~8/7 = 245.531)
* WE: ~7/5 = 603.612{{c}}, ~7/4 = 953.663{{c}} (~7/6 = 253.562{{c}})
* POTE: ~7/5 = 600.000, ~7/4 = 947.955 (~7/6 = 252.045)
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 950.116{{c}} (~7/6 = 249.884{{c}})


{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}
{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}


Badness (Smith): 0.021326
Badness (Sintel): 0.881


=== Decimated ===
=== Decimated ===
Line 402: Line 403:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~7/4 = 950.940 (~7/6 = 249.060)
* WE: ~7/5 = 604.535{{c}}, ~7/4 = 952.076{{c}} (~7/6 = 256.994{{c}})
* POTE: ~7/5 = 600.000, ~7/4 = 944.934 (~7/6 = 255.066)
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 946.108{{c}} (~7/6 = 253.892{{c}})


{{Optimal ET sequence|legend=0| 4, 10e, 14c }}
{{Optimal ET sequence|legend=0| 4, 10e, 14c }}


Badness (Smith): 0.031456
Badness (Sintel): 1.04


=== Decibel ===
=== Decibel ===
Line 417: Line 418:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~7/4 = 955.608 (~8/7 = 244.392)
* WE: ~7/5 = 599.404{{c}}, ~7/4 = 955.557{{c}} (~8/7 = 243.251{{c}})
* POTE: ~7/5 = 600.000, ~7/4 = 956.507 (~8/7 = 243.493)
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 956.169{{c}} (~8/7 = 243.831{{c}})


{{Optimal ET sequence|legend=0| 4, 6, 10 }}
{{Optimal ET sequence|legend=0| 4, 6, 10 }}


Badness (Smith): 0.032385
Badness (Sintel): 1.07


== 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


[[Comma list]]: 25/24, 245/243
[[Comma list]]: 25/24, 245/243


{{Mapping|legend=1| 1 3 3 6 | 0 -4 -2 -9 }}
{{Mapping|legend=1| 1 -1 1 -3 | 0 4 2 9 }}
 
: mapping generators: ~2, ~9/7


{{Multival|legend=1| 4 2 9 -12 3 15 }}
: mapping generators: ~2, ~14/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~9/7 = 424.452
* [[WE]]: ~2 = 1207.178{{c}}, ~14/9 = 777.414{{c}}
: [[error map]]: {{val| 0.000 +0.238 -35.217 +11.108 }}
: [[error map]]: {{val| +7.178 +0.523 -24.308 +6.367 }}
* [[POTE]]: ~2 = 1200.000, ~9/7 = 427.208
* [[CWE]]: ~2 = 1200.000{{c}}, ~14/9 = 773.872{{c}}
: error map: {{val| 0.000 -10.789 -40.731 -13.702 }}
: error map: {{val| 0.000 -6.464 -38.569 -3.973 }}


{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}
{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}


[[Badness]] (Smith): 0.056586
[[Badness]] (Sintel): 1.43


=== 11-limit ===
=== 11-limit ===
Line 450: Line 451:
Comma list: 25/24, 45/44, 99/98
Comma list: 25/24, 45/44, 99/98


Mapping: {{mapping| 1 3 3 6 7 | 0 -4 -2 -9 -10 }}
Mapping: {{mapping| 1 -1 1 -3 -3 | 0 4 2 9 10 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 424.587
* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 427.273
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{c}}


{{Optimal ET sequence|legend=0| 3de, …, 11cdee, 14c }}
{{Optimal ET sequence|legend=0| 3de, …, 11cdee, 14c }}


Badness (Smith): 0.032957
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:Dicot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Dicot_family"