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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 classical chromatic semitone. Its [[monzo]] is {{monzo| -3 -1 2 }}, and flipping that yields {{multival| 2 1 -3 }} for the [[wedgie]]. This tells us the [[generator]] is a classical third (major and minor mean the same thing), and that two such thirds give a fifth. In fact, (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.  


Possible tunings for dicot are [[7edo]], [[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 pretending that is 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]].
== 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]].


== 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]] ([[POTE]]): ~2 = 1\1, ~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] (full comma to untempered)
* 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 }}
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}


[[Badness]]: 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, with wedgie {{multival| 2 1 3 -3 -1 4 }} adds 36/35, sharp with wedgie {{multival| 2 1 6 -3 4 11 }} adds 28/27, and dichotic with wedgie {{multival| 2 1 -4 -3 -12 -12 }} 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.
 
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.


Decimal with wedgie {{multival| 4 2 2 -6 -8 -1 }} adds 49/48, sidi with wedgie {{multival| 4 2 9 -3 6 15 }} adds 245/243, and jamesbond with wedgie {{multival| 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.
Temperaments discussed elsewhere are:
* ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]]
* ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]


=== 2.3.5.11 subgroup ===
The rest are considered in each sections below.


Subgroup: 2.3.5.11
==== 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.  


[[Comma list]]: 25/24, 45/44
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.


[[Gencom]]: [2 5/4; 25/24 45/44]
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11


[[Gencom|Gencom mapping]]: [{{val|1 1 2 0 2}}, {{val|0 2 1 0 5}}]
Comma list: 25/24, 45/44


[[Mapping|Sval mapping]]: [{{val|1 1 2 2}}, {{val|0 2 1 5}}]
Subgroup val mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}


[[Tp tuning|POL2 generator]]: ~5/4 = 346.734
Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}


{{Optimal ET sequence|legend=1| 3e, 4e, 7, 24c, 31c, 38cc, 45cce }}
Optimal tunings:
* WE: ~2 = 1206.750{{c}}, ~5/4 = 348.684{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 348.954{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 5.621 cents
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}


Related temperaments: [[Dicot family #Septimal dicot|dicot]], [[Dicot family #Sharp|sharp]], [[Dicot family #Dichotic|dichotic]]
Badness (Sintel): 0.370


==== 2.3.5.11.13 ====
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


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


[[Gencom]]: [2 5/4; 25/24 40/39 45/44]
Subgroup val mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}


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


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


[[Tp tuning|POL2 generator]]: ~5/4 = 350.526
{{Optimal ET sequence|legend=0| 3e, 7, 17 }}


{{Optimal ET sequence|legend=1| 3e, 7, 17, 24c }}
Badness (Sintel): 0.536


[[Tp tuning #T2 tuning|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.
 
Mujannabic was known as ''septimal dicot'' in earlier materials such as [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].  


== Septimal dicot ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 72: Line 92:
{{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:
 
* [[WE]]: ~2 = 1205.532{{c}}, ~6/5 = 337.931{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 336.381
: [[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, 18bc, 25bccd }}
{{Optimal ET sequence|legend=1| 3d, 4, 7 }}


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


=== 11-limit ===
=== 11-limit ===
Line 87: Line 109:
Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}
Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 342.125
Optimal tunings:
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}


{{Optimal ET sequence|legend=1| 3de, 4e, 7 }}
{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}


Badness: 0.019854
Badness (Sintel): 0.656


=== Eudicot ===
=== Eudicot ===
Line 100: Line 124:
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 336.051
Optimal tunings:
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}


{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}


Badness: 0.027114
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 113: Line 139:
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 338.846
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 }}


{{Optimal ET sequence|legend=1| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
Badness (Sintel): 0.985


Badness: 0.023828
== Flattie ==
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.  


== Flat ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 126: Line 156:
{{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:
 
* [[WE]]: ~2 = 1220.466{{c}}, ~6/5 = 337.577{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 331.916
: [[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 141: Line 173:
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 337.532
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=1| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


Badness: 0.024988
Badness (Sintel): 0.826


=== 13-limit ===
=== 13-limit ===
Line 154: Line 188:
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 341.023
Optimal tunings:
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}
 
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


{{Optimal ET sequence|legend=1| 3, 4, 7d }}
Badness (Sintel): 0.968


Badness: 0.023420
== 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.  


== Sharp ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 167: Line 205:
{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}
{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 357.938
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1202.488{{c}}, ~5/4 = 358.680{{c}}
{{Multival|legend=1| 2 1 6 -3 4 11 }}
: [[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 ET sequence|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}
{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}


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


=== 11-limit ===
=== 11-limit ===
Line 182: Line 222:
Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}
Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 356.106
Optimal tunings:
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}


{{Optimal ET sequence|legend=1| 3de, 7d, 10, 17d, 27cde }}
{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}


Badness: 0.022366
Badness (Sintel): 0.739


== Dichotic ==
== Dichotic ==
In dichotic, 7/4 is found at a stack of two perfect fourths.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 195: Line 239:
{{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:
 
* [[WE]]: ~2 = 1200.802{{c}}, ~5/4 = 356.502{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 356.264
: [[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 }}


{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}


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


=== 11-limit ===
=== 11-limit ===
Line 210: Line 256:
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 223: Line 271:
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 236: Line 286:
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness: 0.031719
Badness (Sintel): 1.05


==== 13-limit ====
==== 13-limit ====
Line 249: Line 301:
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness: 0.022741
Badness (Sintel): 0.940


=== Dichosis ===
=== Dichosis ===
Line 262: Line 316:
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness: 0.041361
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 275: Line 331:
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.646
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=1| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness: 0.027938
Badness (Sintel): 1.15


== Decimal ==
== Decimal ==
{{Main| Decimal }}
{{Main| Decimal }}
{{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 292: Line 355:
: mapping generators: ~7/5, ~7/4
: mapping generators: ~7/5, ~7/4


{{Multival|legend=1| 4 2 2 -6 -8 -1 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~7/5 = 603.286{{c}}, ~7/4 = 953.637{{c}} (~7/6 = 252.935{{c}})
[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~7/4 = 948.443 (~7/6 = 251.557)
: [[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 }}


{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}


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


=== 11-limit ===
=== 11-limit ===
Line 307: Line 372:
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 946.507 (~7/6 = 253.493)
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 ET sequence|legend=1| 10, 14c, 24c, 38ccd, 52cccde }}
{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}


Badness: 0.026712
Badness (Sintel): 0.883
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 45/44, 49/48, 91/90
 
Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}
 
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 ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}
 
Badness (Sintel): 0.881


=== Decimated ===
=== Decimated ===
Line 320: Line 402:
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 944.934 (~7/6 = 255.066)
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=1| 4, 10e, 14c }}
{{Optimal ET sequence|legend=0| 4, 10e, 14c }}


Badness: 0.031456
Badness (Sintel): 1.04


=== Decibel ===
=== Decibel ===
Line 333: Line 417:
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 956.507 (~8/7 = 243.493)
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 ET sequence|legend=1| 4, 6, 10 }}
{{Optimal ET sequence|legend=0| 4, 6, 10 }}


Badness: 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, ~14/9
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}


: mapping generators: ~2, ~9/7
[[Badness]] (Sintel): 1.43


{{Multival|legend=1| 4 2 9 -12 3 15 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 427.208
Comma list: 25/24, 45/44, 99/98


{{Optimal ET sequence|legend=1| 3d, 14c, 45cc, 59bcccd }}
Mapping: {{mapping| 1 -1 1 -3 -3 | 0 4 2 9 10 }}


[[Badness]]: 0.056586
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]].
 
[[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 ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 25/24, 45/44, 99/98
Comma list: 25/24, 33/32, 245/242


Mapping: {{mapping| 1 3 3 6 7 | 0 -4 -2 -9 -10 }}
Mapping: {{mapping| 1 3 3 1 2 | 0 -4 -2 5 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 427.273
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=1| 3de, 14c, 45cce, 59bcccdee }}
{{Optimal ET sequence|legend=0| 3, 11c, 14c }}


Badness: 0.032957
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"