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Decanonicalize septimal dicot. - 2.3.5.11-subgroup eudicot (no need for explicit documentation if it's canonical)
 
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The [[5-limit]] parent [[comma]] for the dicot family is [[25/24]], the classic chromatic semitone. Its [[monzo]] is {{monzo| -3 -1 2 }}, and flipping that yields {{wedgie| 2 1 -3}} for the [[wedgie]]. This tells us the generator is a third (major and minor mean the same thing), and that two thirds gives a fifth. In fact, (5/4)^2 = 3/2 * 25/24. Possible tunings for dicot are [[7edo]], [[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 pretending that's 5-limit, and like any temperament which seems to involve pretending, dicot is at the edge of what can sensibly be called a temperament at all. In other words, it is an [[exotemperament]].
{{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 ==
== Dicot ==
Subgroup: 2.3.5
{{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]].
 
[[Subgroup]]: 2.3.5


[[Comma list]]: 25/24
[[Comma list]]: 25/24


[[Mapping]]: [{{val|1 1 2}}, {{val|0 2 1}}]
{{Mapping|legend=1| 1 1 2 | 0 2 1 }}
: mapping generators: ~2, ~5/4


[[POTE tuning|POTE generator]]: ~5/4 = 348.594
[[Optimal tuning]]s:
* [[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 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
* [[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]
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314] (full comma to untempered)
* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]
 
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}


{{Val list|legend=1| 3, 4, 7, 17, 24c, 31c }}
[[Badness]] (Sintel): 0.306


[[Badness]]: 0.013028
=== Overview to extensions ===
==== 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.  


=== Seven limit children ===
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.
The second comma of the [[Normal_lists|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot, with wedgie {{wedgie|2 1 3 -3 -1 4}} adds 36/35, sharp with wedgie {{wedgie|2 1 6 -3 4 11}} adds 28/27, and dichotic with wedgie {{wedgie|2 1 -4 -3 -12 -12}} ads 64/63, all retaining the same period and generator. Decimal with wedgie {{wedgie|4 2 2 -6 -8 -1}} adds 49/48, sidi with wedgie {{wedgie|4 2 9 -3 6 15}} adds 245/243, and jamesbond with wedgie {{wedgie|0 0 7 0 11 16}} 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.


== Septimal dicot ==
Temperaments discussed elsewhere are:
Subgroup: 2.3.5.7
* ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]]
* ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]


[[Comma list]]: 15/14, 25/24
The rest are considered in each sections below.


[[Mapping]]: [{{val|1 1 2 2}}, {{val|0 2 1 3}}]
==== 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.


[[Wedgie]]: {{wedgie|2 1 3 -3 -1 4}}
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.


[[POTE generator]]: ~5/4 = 336.381
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11


{{Val list|legend=1| 3d, 4, 7, 18bc, 25bccd }}
Comma list: 25/24, 45/44


[[Badness]]: 0.019935
Subgroup val mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}


=== 11-limit ===
Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}
Subgroup: 2.3.5.7.11


Comma list: 15/14, 22/21, 25/24
Optimal tunings:  
* WE: ~2 = 1206.750{{c}}, ~5/4 = 348.684{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 348.954{{c}}


Mapping: [{{val|1 1 2 2 2}}, {{val|0 2 1 3 5}}]
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}


POTE generator: ~5/4 = 342.125
Badness (Sintel): 0.370


Optimal GPV sequence: {{Val list| 3de, 4e, 7 }}
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13


Badness: 0.019854
Comma list: 25/24, 40/39, 45/44


=== Eudicot ===
Subgroup val mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}
Subgroup: 2.3.5.7.11


Comma list: 15/14, 25/24, 33/32
Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}


Mapping: [{{val|1 1 2 2 4}}, {{val|0 2 1 3 -2}}]
Optimal tunings:  
* WE: ~2 = 1202.433{{c}}, ~5/4 = 351.237{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 350.978{{c}}


POTE generator: ~5/4 = 336.051
{{Optimal ET sequence|legend=0| 3e, 7, 17 }}


Optimal GPV sequence: {{Val list| 3d, 4, 7, 18bc, 25bccd }}
Badness (Sintel): 0.536


Badness: 0.027114
== 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.  


==== 13-limit ====
Mujannabic was known as ''septimal dicot'' in earlier materials such as [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].  
Subgroup: 2.3.5.7.11.13


Comma list: 15/14, 25/24, 33/32, 40/39
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|1 1 2 2 4 4}}, {{val|0 2 1 3 -2 -1}}]
[[Comma list]]: 15/14, 25/24


POTE generator: ~5/4 = 338.846
{{Mapping|legend=1| 1 1 2 2 | 0 2 1 3 }}


Optimal GPV sequence: {{Val list| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
[[Optimal tuning]]s:  
* [[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 }}


Badness: 0.023828
{{Optimal ET sequence|legend=1| 3d, 4, 7 }}


== Flat ==
[[Badness]] (Sintel): 0.504
Subgroup: 2.3.5.7


[[Comma list]]: 21/20, 25/24
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val|1 1 2 3}}, {{val|0 2 1 -1}}]
Comma list: 15/14, 22/21, 25/24


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


[[POTE generator]]: ~5/4 = 331.916
Optimal tunings:  
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}


{{Val list|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}


[[Badness]]: 0.025381
Badness (Sintel): 0.656


=== 11-limit ===
=== Eudicot ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 21/20, 25/24, 33/32
Comma list: 15/14, 25/24, 33/32


Mapping: [{{val|1 1 2 3 4}}, {{val|0 2 1 -1 -2}}]
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}


POTE generator: ~5/4 = 337.532
Optimal tunings:  
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}


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


Badness: 0.024988
Badness (Sintel): 0.896


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 14/13, 21/20, 25/24, 33/32
Comma list: 15/14, 25/24, 33/32, 40/39


Mapping: [{{val|1 1 2 3 4 4}}, {{val|0 2 1 -1 -2 -1}}]
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}


POTE generator: ~5/4 = 341.023
Optimal tunings:  
* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}


Optimal GPV sequence: {{Val list| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3d, 4, 7 }}


Badness: 0.023420
Badness (Sintel): 0.985


== Sharp ==
== Flattie ==
Subgroup: 2.3.5.7
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.  


[[Comma list]]: 25/24, 28/27
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 1 2 1}}, {{val|0 2 1 6}}]
[[Comma list]]: 21/20, 25/24


[[POTE generator]]: ~5/4 = 357.938
{{Mapping|legend=1| 1 1 2 3 | 0 2 1 -1 }}


{{Multival|legend=1|2 1 6 -3 4 11}}
[[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 }}


{{Val list|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}
{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}


[[Badness]]: 0.028942
[[Badness]] (Sintel): 0.642


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 25/24, 28/27, 35/33
Comma list: 21/20, 25/24, 33/32


Mapping: [{{val|1 1 2 1 2}}, {{val|0 2 1 6 5}}]
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}


POTE generator: ~5/4 = 356.106
Optimal tunings:  
* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}


Optimal GPV sequence: {{Val list| 3de, 7d, 10, 17d, 27cde }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


Badness: 0.022366
Badness (Sintel): 0.826


== Decimal ==
=== 13-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 25/24, 49/48
Comma list: 14/13, 21/20, 25/24, 33/32


[[Mapping]]: [{{val|2 0 3 4}}, {{val|0 2 1 1}}]
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}


{{Multival|legend=1|4 2 2 -6 -8 -1}}
Optimal tunings:
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}


[[POTE generator]]: ~7/6 = 251.557
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


{{Val list|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
Badness (Sintel): 0.968


[[Badness]]: 0.028334
== 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.  


=== 11-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 25/24, 45/44, 49/48
[[Comma list]]: 25/24, 28/27


Mapping: [{{val|2 0 3 4 -1}}, {{val|0 2 1 1 5}}]
{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}


POTE generator: ~7/6 = 253.493
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.488{{c}}, ~5/4 = 358.680{{c}}
: [[error map]]: {{val| +2.488 +17.893 -22.658 -14.258 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}
: error map: {{val| 0.000 +15.035 -27.818 -17.854 }}


Optimal GPV sequence: {{Val list| 10, 14c, 24c, 38ccd, 52cccde }}
{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}


Badness: 0.026712
[[Badness]] (Sintel): 0.732


=== Decimated ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 25/24, 33/32, 49/48
Comma list: 25/24, 28/27, 35/33


Mapping: [{{val|2 0 3 4 10}}, {{val|0 2 1 1 -2}}]
Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}


POTE generator: ~7/6 = 255.066
Optimal tunings:  
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}


Optimal GPV sequence: {{Val list| 4, 10e, 14c }}
{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}


Badness: 0.031456
Badness (Sintel): 0.739


=== Decibel ===
== Dichotic ==
Subgroup: 2.3.5.7.11
In dichotic, 7/4 is found at a stack of two perfect fourths.  


Comma list: 25/24, 35/33, 49/48
[[Subgroup]]: 2.3.5.7
 
Mapping: [{{val|2 0 3 4 7}}, {{val|0 2 1 1 0}}]
 
POTE generator: ~8/7 = 243.493
 
Optimal GPV sequence: {{Val list| 4, 6, 10 }}
 
Badness: 0.032385
 
== Dichotic ==
Subgroup: 2.3.5.7


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


[[Mapping]]: [{{val|1 1 2 4}}, {{val|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:
* [[WE]]: ~2 = 1200.802{{c}}, ~5/4 = 356.502{{c}}
: [[error map]]: {{val| +0.802 +11.851 -28.208 +8.374 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}
: error map: {{val| 0.000 +10.595 -30.039 +6.074 }}


[[POTE generator]]: ~5/4 = 356.264
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}


{{Val list|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
[[Badness]] (Sintel): 0.951
 
[[Badness]]: 0.037565


=== 11-limit ===
=== 11-limit ===
Line 219: Line 254:
Comma list: 25/24, 45/44, 64/63
Comma list: 25/24, 45/44, 64/63


Mapping: [{{val|1 1 2 4 2}}, {{val|0 2 1 -4 5}}]
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}


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


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


Badness: 0.030680
Badness (Sintel): 1.01


==== 13-limit ====
==== 13-limit ====
Line 232: Line 269:
Comma list: 25/24, 40/39, 45/44, 64/63
Comma list: 25/24, 40/39, 45/44, 64/63


Mapping: [{{val|1 1 2 4 2 4}}, {{val|0 2 1 -4 5 -1}}]
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}


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


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


Badness: 0.021674
Badness (Sintel): 0.896


=== Dichotomic ===
=== Dichotomic ===
Line 245: Line 284:
Comma list: 22/21, 25/24, 33/32
Comma list: 22/21, 25/24, 33/32


Mapping: [{{val|1 1 2 4 4}}, {{val|0 2 1 -4 -2}}]
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}


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


Optimal GPV sequence: {{Val list| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness: 0.031719
Badness (Sintel): 1.05


==== 13-limit ====
==== 13-limit ====
Line 258: Line 299:
Comma list: 22/21, 25/24, 33/32, 40/39
Comma list: 22/21, 25/24, 33/32, 40/39


Mapping: [{{val|1 1 2 4 4 4}}, {{val|0 2 1 -4 -2 -1}}]
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}


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


Optimal GPV sequence: {{Val list| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness: 0.022741
Badness (Sintel): 0.940


=== Dichosis ===
=== Dichosis ===
Line 271: Line 314:
Comma list: 25/24, 35/33, 64/63
Comma list: 25/24, 35/33, 64/63


Mapping: [{{val|1 1 2 4 5}}, {{val|0 2 1 -4 -5}}]
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}


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


Optimal GPV sequence: {{Val list| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness: 0.041361
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 284: Line 329:
Comma list: 25/24, 35/33, 40/39, 64/63
Comma list: 25/24, 35/33, 40/39, 64/63


Mapping: [{{val|1 1 2 4 5 4}}, {{val|0 2 1 -4 -5 -1}}]
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}
 
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
 
Badness (Sintel): 1.15


POTE generator: ~5/4 = 360.646
== Decimal ==
{{Main| Decimal }}
{{See also| Jubilismic clan }}


Optimal GPV sequence: {{Val list| 3, 7e, 10 }}
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.


Badness: 0.027938
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]].  


== Jamesbond ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 25/24, 81/80
[[Comma list]]: 25/24, 49/48


[[Mapping]]: [{{val|7 11 16 0}}, {{val|0 0 0 1}}]
{{Mapping|legend=1| 2 0 3 4 | 0 2 1 1 }}


{{Multival|legend=1|0 0 7 0 11 16}}
: mapping generators: ~7/5, ~7/4


[[POTE generator]]: ~8/7 = 258.139
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 603.286{{c}}, ~7/4 = 953.637{{c}} (~7/6 = 252.935{{c}})
: [[error map]]: {{val| +6.571 +5.318 -22.821 -2.047 }}
* [[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 }}


{{Val list|legend=1| 7, 14c }}
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}


[[Badness]]: 0.041714
[[Badness]] (Sintel): 0.717


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 25/24, 33/32, 45/44
Comma list: 25/24, 45/44, 49/48


Mapping: [{{val|7 11 16 0 24}}, {{val|0 0 0 1 0}}]
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}


POTE generator: ~8/7 = 258.910
Optimal tunings:
* 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}})


Optimal GPV sequence: {{Val list| 7, 14c }}
{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}


Badness: 0.023524
Badness (Sintel): 0.883


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 25/24, 27/26, 33/32, 40/39
Comma list: 25/24, 45/44, 49/48, 91/90


Mapping: [{{val|7 11 16 0 24 26}}, {{val|0 0 0 1 0 0}}]
Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}


POTE generator: ~8/7 = 250.764
Optimal tunings:
* 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}})


Optimal GPV sequence: {{Val list| 7, 14c }}
{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}


Badness: 0.023003
Badness (Sintel): 0.881


==== Septimal ====
=== Decimated ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 33/32, 49/48
 
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}
 
Optimal tunings:
* 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}})
 
{{Optimal ET sequence|legend=0| 4, 10e, 14c }}
 
Badness (Sintel): 1.04
 
=== Decibel ===
Subgroup: 2.3.5.7.11


Comma list: 25/24, 33/32, 45/44, 65/63
Comma list: 25/24, 35/33, 49/48


Mapping: [{{val|7 11 16 0 24 6}}, {{val|0 0 0 1 0 1}}]
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}


POTE generator: ~8/7 = 247.445
Optimal tunings:  
* 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}})


Optimal GPV sequence: {{Val list| 7, 14cf }}
{{Optimal ET sequence|legend=0| 4, 6, 10 }}


Badness: 0.022569
Badness (Sintel): 1.07


== Sidi ==
== Sidi ==
Subgroup: 2.3.5.7
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


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


[[Mapping]]: [{{val|1 3 3 6}}, {{val|0 -4 -2 -9}}]
{{Mapping|legend=1| 1 -1 1 -3 | 0 4 2 9 }}


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


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


{{Val list|legend=1| 3d, 14c, 45cc, 59bcccd }}
{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}


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


=== 11-limit ===
=== 11-limit ===
Line 366: Line 451:
Comma list: 25/24, 45/44, 99/98
Comma list: 25/24, 45/44, 99/98


Mapping: [{{val|1 3 3 6 7}}, {{val|0 -4 -2 -9 -10}}]
Mapping: {{mapping| 1 -1 1 -3 -3 | 0 4 2 9 10 }}
 
Optimal tunings:
* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{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]].


POTE generator: ~9/7 = 427.273
[[Subgroup]]: 2.3.5.7


Optimal GPV sequence: {{Val list| 3de, 14c, 45cce, 59bcccdee }}
[[Comma list]]: 25/24, 1323/1280


Badness: 0.032957
{{Mapping|legend=1| 1 -1 1 6 | 0 4 2 -5 }}
: mapping generators: ~2, ~32/21


== Quad ==
[[Optimal tuning]]s:
Subgroup: 2.3.5.7
* [[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 }}


[[Comma list]]: 9/8, 25/24
{{Optimal ET sequence|legend=1| 3, 11c, 14c, 45ccdd }}
 
[[Badness]] (Sintel): 2.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val|4 6 9 0}}, {{val|0 0 0 1}}]
Comma list: 25/24, 33/32, 245/242


{{Multival|legend=1|0 0 4 0 6 9}}
Mapping: {{mapping| 1 3 3 1 2 | 0 -4 -2 5 4 }}


[[POTE generator]]: ~8/7 = 324.482
Optimal tunings:  
* WE: ~2 = 1209.621{{c}}, ~11/7 = 772.376{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 772.247{{c}}


{{Val list|legend=1| 4 }}
{{Optimal ET sequence|legend=0| 3, 11c, 14c }}


[[Badness]]: 0.045911
Badness (Sintel): 1.54


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Dicot_family"