Dicot family: Difference between revisions

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The [[5-limit]] parent [[comma]] for the '''dicot family''' is [[25/24]], the classical chromatic semitone. The [[generator]] is 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.  


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 pretending that this is 5-limit, and like any temperament which seems to involve "pretending", dicot is close to the edge of what can sensibly be called a temperament at all. In other words, it is an [[exotemperament]].
== 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 13: Line 16:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 354.664
* [[WE]]: ~2 = 1206.283{{c}}, ~6/5 = 350.420{{c}}
: [[error map]]: {{val| 0.000 +7.374 -31.649 }}
: [[error map]]: {{val| +6.283 +5.167 -23.328 }}
* [[POTE]]: ~2 = 1\1, ~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 27:
{{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 adds 36/35, sharp adds 28/27, and dichotic adds 64/63, all retaining the same period and generator.  
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]]


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 rest are considered 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]], [[#Sharp|sharp]], and [[#Dichotic|dichotic]].  
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
Subgroup: 2.3.5.11
Line 38: Line 47:
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 = 1\1, ~5/4 = 352.287
* WE: ~2 = 1206.750{{c}}, ~6/5 = 348.684{{c}}
* POTE: ~2 = 1\1, ~6/5 = 346.734
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 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 57: Line 64:
Comma list: 25/24, 40/39, 45/44
Comma list: 25/24, 40/39, 45/44


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


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 352.420
* WE: ~2 = 1202.433{{c}}, ~5/4 = 351.237{{c}}
* POTE: ~2 = 1\1, ~6/5 = 350.526
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 350.978{{c}}


{{Optimal ET sequence|legend=0| 3e, 7, 17, 24c }}
{{Optimal ET sequence|legend=0| 3e, 7, 17 }}


RMS error: 5.916 cents
Badness (Sintel): 0.536


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


Line 77: Line 84:


{{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 = 1\1, ~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 = 1\1, ~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]]: 0.019935
[[Badness]] (Sintel): 0.504


=== 11-limit ===
=== 11-limit ===
Line 98: Line 103:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 345.596
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* POTE: ~2 = 1\1, ~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: 0.019854
Badness (Sintel): 0.656


=== Eudicot ===
=== Eudicot ===
Line 113: Line 118:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 340.417
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* POTE: ~2 = 1\1, ~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: 0.027114
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 128: Line 133:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 340.835
* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
* POTE: ~2 = 1\1, ~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: 0.023828
Badness (Sintel): 0.985
 
== 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.


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


Line 141: Line 148:


{{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 = 1\1, ~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 = 1\1, ~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]]: 0.025381
[[Badness]] (Sintel): 0.642


=== 11-limit ===
=== 11-limit ===
Line 162: Line 167:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 343.139
* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
* POTE: ~2 = 1\1, ~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: 0.024988
Badness (Sintel): 0.826


=== 13-limit ===
=== 13-limit ===
Line 177: Line 182:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 343.655
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* POTE: ~2 = 1\1, ~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: 0.023420
Badness (Sintel): 0.968
 
== 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 192: Line 199:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~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 = 1\1, ~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]]: 0.028942
[[Badness]] (Sintel): 0.732


=== 11-limit ===
=== 11-limit ===
Line 211: Line 216:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 357.261
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* POTE: ~2 = 1\1, ~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: 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 224: Line 231:


{{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 = 1\1, ~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 = 1\1, ~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]]: 0.037565
[[Badness]] (Sintel): 0.951


=== 11-limit ===
=== 11-limit ===
Line 245: Line 250:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 354.183
* WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}
* POTE: ~2 = 1\1, ~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: 0.030680
Badness (Sintel): 1.01


==== 13-limit ====
==== 13-limit ====
Line 260: Line 265:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 354.247
* WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}
* POTE: ~2 = 1\1, ~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: 0.021674
Badness (Sintel): 0.896


=== Dichotomic ===
=== Dichotomic ===
Line 275: Line 280:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 353.751
* WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}
* POTE: ~2 = 1\1, ~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: 0.031719
Badness (Sintel): 1.05


==== 13-limit ====
==== 13-limit ====
Line 290: Line 295:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 353.850
* WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}
* POTE: ~2 = 1\1, ~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: 0.022741
Badness (Sintel): 0.940


=== Dichosis ===
=== Dichosis ===
Line 305: Line 310:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 361.081
* WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}
* POTE: ~2 = 1\1, ~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: 0.041361
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 320: Line 325:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 361.061
* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
* POTE: ~2 = 1\1, ~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: 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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 337: Line 345:


: 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 = 1\2, ~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 = 1\2, ~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]]: 0.028334
[[Badness]] (Sintel): 0.717


=== 11-limit ===
=== 11-limit ===
Line 358: Line 364:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 0.026712
Badness (Sintel): 0.883


==== 13-limit ====
==== 13-limit ====
Line 373: Line 379:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 0.021326
Badness (Sintel): 0.881


=== Decimated ===
=== Decimated ===
Line 388: Line 394:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 0.031456
Badness (Sintel): 1.04


=== Decibel ===
=== Decibel ===
Line 403: Line 409:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 1\2, ~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 = 1\2, ~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: 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 = 1\1, ~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 = 1\1, ~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]]: 0.056586
[[Badness]] (Sintel): 1.43


=== 11-limit ===
=== 11-limit ===
Line 436: Line 442:
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 = 1\1, ~9/7 = 424.587
* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
* POTE: ~2 = 1\1, ~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: 0.032957
Badness (Sintel): 1.09


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot| ]] <!-- key article -->
[[Category:Dicot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 14:10, 17 August 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The dicot family of temperaments tempers out 25/24, the classical chromatic semitone.

Dicot

The head of this family, dicot, is generated by a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, (5/4)2 = (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 ⟨24 38 55] (24c), and 31edo using the 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: [⟨1 1 2], ⟨0 2 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1206.283 ¢, ~6/5 = 350.420 ¢
error map: ⟨+6.283 +5.167 -23.328]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 351.086 ¢
error map: ⟨0.000 +0.216 -35.228]

Tuning ranges:

Optimal ET sequence: 3, 4, 7, 17, 24c, 31c

Badness (Sintel): 0.306

Overview to extensions

The second comma of the 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 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:

The rest are considered below.

2.3.5.11 subgroup

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: [⟨1 1 2 2], ⟨0 2 1 5]]

Gencom mapping: [⟨1 1 2 0 2], ⟨0 2 1 0 5]]

Optimal tunings:

  • WE: ~2 = 1206.750 ¢, ~6/5 = 348.684 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 348.954 ¢

Optimal ET sequence: 3e, 4e, 7, 24c, 31c

Badness (Sintel): 0.370

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

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

Subgroup val mapping: [⟨1 1 2 2 4], ⟨0 2 1 5 -1]]

Gencom mapping: [⟨1 1 2 0 2 4], ⟨0 2 1 0 5 -1]]

Optimal tunings:

  • WE: ~2 = 1202.433 ¢, ~5/4 = 351.237 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 350.978 ¢

Optimal ET sequence: 3e, 7, 17

Badness (Sintel): 0.536

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: [⟨1 1 2 2], ⟨0 2 1 3]]

Optimal tunings:

  • WE: ~2 = 1205.532 ¢, ~6/5 = 337.931 ¢
error map: ⟨+5.532 -20.561 -37.319 +56.032]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 338.561 ¢
error map: ⟨0.000 -24.834 -47.753 +46.856]

Optimal ET sequence: 3d, 4, 7

Badness (Sintel): 0.504

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 25/24

Mapping: [⟨1 1 2 2 2], ⟨0 2 1 3 5]]

Optimal tunings:

  • WE: ~2 = 1203.346 ¢, ~6/5 = 343.078 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 343.260 ¢

Optimal ET sequence: 3de, 4e, 7

Badness (Sintel): 0.656

Eudicot

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 2 2 4], ⟨0 2 1 3 -2]]

Optimal tunings:

  • WE: ~2 = 1205.828 ¢, ~6/5 = 337.683 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 336.909 ¢

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness (Sintel): 0.896

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 2 4 4], ⟨0 2 1 3 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1202.660 ¢, ~6/5 = 339.597 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 339.104 ¢

Optimal ET sequence: 3d, 4, 7

Badness (Sintel): 0.985

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.

Subgroup: 2.3.5.7

Comma list: 21/20, 25/24

Mapping: [⟨1 1 2 3], ⟨0 2 1 -1]]

Optimal tunings:

  • WE: ~2 = 1220.466 ¢, ~6/5 = 337.577 ¢
error map: ⟨+20.466 -6.335 -7.804 -45.004]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 335.391 ¢
error map: ⟨0.000 -31.173 -50.922 -104.217]

Optimal ET sequence: 3, 4, 7d, 11cd, 18bcddd

Badness (Sintel): 0.642

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 2 3 4], ⟨0 2 1 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1216.069 ¢, ~6/5 = 342.052 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 338.467 ¢

Optimal ET sequence: 3, 4, 7d

Badness (Sintel): 0.826

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 3 4 4], ⟨0 2 1 -1 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1211.546 ¢, ~6/5 = 344.304 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 341.373 ¢

Optimal ET sequence: 3, 4, 7d

Badness (Sintel): 0.968

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.

Subgroup: 2.3.5.7

Comma list: 25/24, 28/27

Mapping: [⟨1 1 2 1], ⟨0 2 1 6]]

Optimal tunings:

  • WE: ~2 = 1202.488 ¢, ~5/4 = 358.680 ¢
error map: ⟨+2.488 +17.893 -22.658 -14.258]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 358.495 ¢
error map: ⟨0.000 +15.035 -27.818 -17.854]

Optimal ET sequence: 3d, 7d, 10

Badness (Sintel): 0.732

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 28/27, 35/33

Mapping: [⟨1 1 2 1 2], ⟨0 2 1 6 5]]

Optimal tunings:

  • WE: ~2 = 1201.518 ¢, ~5/4 = 356.557 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 356.457 ¢

Optimal ET sequence: 3de, 7d, 10, 17d

Badness (Sintel): 0.739

Dichotic

In dichotic, 7/4 is found at a stack of two perfect fourths.

Subgroup: 2.3.5.7

Comma list: 25/24, 64/63

Mapping: [⟨1 1 2 4], ⟨0 2 1 -4]]

Optimal tunings:

  • WE: ~2 = 1200.802 ¢, ~5/4 = 356.502 ¢
error map: ⟨+0.802 +11.851 -28.208 +8.374]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 356.275 ¢
error map: ⟨0.000 +10.595 -30.039 +6.074]

Optimal ET sequence: 3, 7, 10, 17, 27c

Badness (Sintel): 0.951

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 64/63

Mapping: [⟨1 1 2 4 2], ⟨0 2 1 -4 5]]

Optimal tunings:

  • WE: ~2 = 1199.504 ¢, ~5/4 = 354.115 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.236 ¢

Optimal ET sequence: 7, 10, 17

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 40/39, 45/44, 64/63

Mapping: [⟨1 1 2 4 2 4], ⟨0 2 1 -4 5 -1]]

Optimal tunings:

  • WE: ~2 = 1199.289 ¢, ~5/4 = 354.156 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.340 ¢

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness (Sintel): 0.896

Dichotomic

Subgroup: 2.3.5.7.11

Comma list: 22/21, 25/24, 33/32

Mapping: [⟨1 1 2 4 4], ⟨0 2 1 -4 -2]]

Optimal tunings:

  • WE: ~2 = 1203.949 ¢, ~5/4 = 355.239 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.024 ¢

Optimal ET sequence: 3, 7, 10e

Badness (Sintel): 1.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 22/21, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 4 4 4], ⟨0 2 1 -4 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1202.979 ¢, ~5/4 = 355.193 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.254 ¢

Optimal ET sequence: 3, 7, 10e

Badness (Sintel): 0.940

Dichosis

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 64/63

Mapping: [⟨1 1 2 4 5], ⟨0 2 1 -4 -5]]

Optimal tunings:

  • WE: ~2 = 1197.526 ¢, ~5/4 = 359.915 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 360.745 ¢

Optimal ET sequence: 3, 7e, 10

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 35/33, 40/39, 64/63

Mapping: [⟨1 1 2 4 5 4], ⟨0 2 1 -4 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1197.922 ¢, ~5/4 = 360.021 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 360.722 ¢

Optimal ET sequence: 3, 7e, 10

Badness (Sintel): 1.15

Decimal

Main article: 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.

Subgroup: 2.3.5.7

Comma list: 25/24, 49/48

Mapping: [⟨2 0 3 4], ⟨0 2 1 1]]

mapping generators: ~7/5, ~7/4

Optimal tunings:

  • WE: ~7/5 = 603.286 ¢, ~7/4 = 953.637 ¢ (~7/6 = 252.935 ¢)
error map: ⟨+6.571 +5.318 -22.821 -2.047]
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 950.957 ¢ (~7/6 = 249.043 ¢)
error map: ⟨0.000 -0.041 -35.357 -17.869]

Optimal ET sequence: 4, 10, 14c, 24c, 38ccd

Badness (Sintel): 0.717

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 0 3 4 -1], ⟨0 2 1 1 5]]

Optimal tunings:

  • WE: ~7/5 = 603.558 ¢, ~7/4 = 952.121 ¢ (~7/6 = 254.996 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 948.610 ¢ (~7/6 = 251.390 ¢)

Optimal ET sequence: 4e, 10, 14c, 24c

Badness (Sintel): 0.883

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 45/44, 49/48, 91/90

Mapping: [⟨2 0 3 4 -1 1], ⟨0 2 1 1 5 4]]

Optimal tunings:

  • WE: ~7/5 = 603.612 ¢, ~7/4 = 953.663 ¢ (~7/6 = 253.562 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 950.116 ¢ (~7/6 = 249.884 ¢)

Optimal ET sequence: 4ef, 10, 14cf, 24cf

Badness (Sintel): 0.881

Decimated

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 0 3 4 10], ⟨0 2 1 1 -2]]

Optimal tunings:

  • WE: ~7/5 = 604.535 ¢, ~7/4 = 952.076 ¢ (~7/6 = 256.994 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 946.108 ¢ (~7/6 = 253.892 ¢)

Optimal ET sequence: 4, 10e, 14c

Badness (Sintel): 1.04

Decibel

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 49/48

Mapping: [⟨2 0 3 4 7], ⟨0 2 1 1 0]]

Optimal tunings:

  • WE: ~7/5 = 599.404 ¢, ~7/4 = 955.557 ¢ (~8/7 = 243.251 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 956.169 ¢ (~8/7 = 243.831 ¢)

Optimal ET sequence: 4, 6, 10

Badness (Sintel): 1.07

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

Comma list: 25/24, 245/243

Mapping: [⟨1 -1 1 -3], ⟨0 4 2 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1207.178 ¢, ~14/9 = 777.414 ¢
error map: ⟨+7.178 +0.523 -24.308 +6.367]
  • CWE: ~2 = 1200.000 ¢, ~14/9 = 773.872 ¢
error map: ⟨0.000 -6.464 -38.569 -3.973]

Optimal ET sequence: 3d, …, 11cd, 14c

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 99/98

Mapping: [⟨1 -1 1 -3 -3], ⟨0 4 2 9 10]]

Optimal tunings:

  • WE: ~2 = 1207.200 ¢, ~11/7 = 777.363 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/7 = 773.777 ¢

Optimal ET sequence: 3de, …, 11cdee, 14c

Badness (Sintel): 1.09

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