Dicot family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Switch to Sintel's badness, WE & CWE tunings, per community consensus
+ short intro to each temp
 
(2 intermediate revisions by the same user not shown)
Line 1: Line 1:
{{Technical data page}}
{{Technical data page}}
The '''dicot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[25/24]], the classical chromatic semitone. The head of this family, dicot, is [[generator|generated]] by a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, {{nowrap|(5/4)<sup>2</sup> {{=}} (3/2)(25/24)}}.  
The '''dicot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[25/24]], the classical chromatic semitone.  
 
== Dicot ==
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 18: Line 20:
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
: error map: {{val| 0.000 +0.216 -35.228 }}
: error map: {{val| 0.000 +0.216 -35.228 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 354.664{{c}}
: [[error map]]: {{val| 0.000 +7.374 -31.649 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 348.594{{c}}
: error map: {{val| 0.000 -4.766 -37.719 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 32: Line 30:


=== 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.  
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 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.
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:  
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 below.  
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]], [[#Sharpie|sharpie]], 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 51: Line 50:


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:  
* WE: ~2 = 1206.750{{c}}, ~6/5 = 348.684{{c}}
* WE: ~2 = 1206.750{{c}}, ~6/5 = 348.684{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 348.954{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 348.954{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 352.287{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 346.734{{c}} -->


{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}
Line 72: Line 67:


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


{{Optimal ET sequence|legend=0| 3e, 7, 17 }}
{{Optimal ET sequence|legend=0| 3e, 7, 17 }}
Line 86: Line 77:


== 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 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.  
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 99: Line 90:
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}
: error map: {{val| 0.000 -24.834 -47.753 +46.856 }}
: error map: {{val| 0.000 -24.834 -47.753 +46.856 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 342.257{{c}}
: [[error map]]: {{val| 0.000 -17.441 -44.056 +57.946 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 336.381{{c}}
: error map: {{val| 0.000 -29.193 -49.933 +40.316 }} -->


{{Optimal ET sequence|legend=1| 3d, 4, 7 }}
{{Optimal ET sequence|legend=1| 3d, 4, 7 }}
Line 118: Line 105:
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 345.596{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 342.125{{c}} -->


{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}
{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}
Line 135: Line 120:
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 340.417{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 336.051{{c}} -->


{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}
Line 152: Line 135:
* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 340.835{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 338.846{{c}} -->


{{Optimal ET sequence|legend=0| 3d, 4, 7 }}
{{Optimal ET sequence|legend=0| 3d, 4, 7 }}
Line 160: Line 141:


== 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 septimal dicot 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 173: Line 154:
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 335.391{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 335.391{{c}}
: error map: {{val| 0.000 -31.173 -50.922 -104.217 }}
: error map: {{val| 0.000 -31.173 -50.922 -104.217 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 346.438{{c}}
: [[error map]]: {{val| 0.000 -9.080 -39.876 -115.264 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 331.916{{c}}
: error map: {{val| 0.000 -38.123 -54.398 -100.742 }} -->


{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
Line 192: Line 169:
* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 343.139{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 337.532{{c}} -->


{{Optimal ET sequence|legend=0| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}
Line 209: Line 184:
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 343.655{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 341.023{{c}} -->


{{Optimal ET sequence|legend=0| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}
Line 217: Line 190:


== 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 230: Line 203:
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}
: error map: {{val| 0.000 +15.035 -27.818 -17.854 }}
: error map: {{val| 0.000 +15.035 -27.818 -17.854 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 359.564{{c}}
: [[error map]]: {{val| 0.000 +17.173 -26.750 -11.442 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 357.938{{c}}
: error map: {{val| 0.000 +13.921 -28.376 -21.198 }} -->


{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}
{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}
Line 249: Line 218:
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 357.261{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 356.106{{c}} -->


{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}
{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}
Line 270: Line 237:
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}
: error map: {{val| 0.000 +10.595 -30.039 +6.074 }}
: error map: {{val| 0.000 +10.595 -30.039 +6.074 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.333{{c}}
: [[error map]]: {{val| 0.000 +10.710 -29.981 +5.844 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.264{{c}}
: error map: {{val| 0.000 +10.573 -30.050 +6.119 }} -->


{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}
Line 289: Line 252:
* WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}
* WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.236{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.236{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 354.183{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.262{{c}} -->


{{Optimal ET sequence|legend=0| 7, 10, 17 }}
{{Optimal ET sequence|legend=0| 7, 10, 17 }}
Line 306: Line 267:
* WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}
* WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.340{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.340{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 354.247{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.365{{c}} -->


{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}
Line 323: Line 282:
* WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}
* WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.024{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.024{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 353.751{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.073{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7, 10e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}
Line 340: Line 297:
* WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}
* WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.254{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.254{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 353.850{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.313{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7, 10e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}
Line 357: Line 312:
* WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}
* WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.745{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.745{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 361.081{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 360.659{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
Line 374: Line 327:
* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 361.061{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 360.646{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
Line 384: Line 335:
{{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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 398: Line 351:
* [[CWE]]: ~7/5 = 600.000{{c}}, ~7/4 = 950.957{{c}} (~7/6 = 249.043{{c}})
* [[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 }}
: error map: {{val| 0.000 -0.041 -35.357 -17.869 }}
<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~7/4 = 955.608{{c}} (~8/7 = 244.392{{c}})
: [[error map]]: {{val| 0.000 +9.260 -30.706 -13.218 }}
* [[POTE]]: ~7/5 = 600.000{{c}}, ~7/4 = 948.443{{c}} (~7/6 = 251.557{{c}})
: error map: {{val| 0.000 -5.069 -37.871 -20.383 }} -->


{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}
Line 417: Line 366:
* WE: ~7/5 = 603.558{{c}}, ~7/4 = 952.121{{c}} (~7/6 = 254.996{{c}})
* 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}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 948.610{{c}} (~7/6 = 251.390{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 952.812{{c}} (~8/7 = 247.188{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 946.507{{c}} (~7/6 = 253.493{{c}}) -->


{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}
{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}
Line 434: Line 381:
* WE: ~7/5 = 603.612{{c}}, ~7/4 = 953.663{{c}} (~7/6 = 253.562{{c}})
* 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}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 950.116{{c}} (~7/6 = 249.884{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 954.469{{c}} (~8/7 = 245.531{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 947.955{{c}} (~7/6 = 252.045{{c}}) -->


{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}
{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}
Line 451: Line 396:
* WE: ~7/5 = 604.535{{c}}, ~7/4 = 952.076{{c}} (~7/6 = 256.994{{c}})
* 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}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 946.108{{c}} (~7/6 = 253.892{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 950.940{{c}} (~7/6 = 249.060{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 944.934{{c}} (~7/6 = 255.066{{c}}) -->


{{Optimal ET sequence|legend=0| 4, 10e, 14c }}
{{Optimal ET sequence|legend=0| 4, 10e, 14c }}
Line 468: Line 411:
* WE: ~7/5 = 599.404{{c}}, ~7/4 = 955.557{{c}} (~8/7 = 243.251{{c}})
* 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}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 956.169{{c}} (~8/7 = 243.831{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 955.608{{c}} (~8/7 = 244.392{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 956.507{{c}} (~8/7 = 243.493{{c}}) -->


{{Optimal ET sequence|legend=0| 4, 6, 10 }}
{{Optimal ET sequence|legend=0| 4, 6, 10 }}
Line 476: Line 417:


== 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


Line 489: Line 432:
* [[CWE]]: ~2 = 1200.000{{c}}, ~14/9 = 773.872{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~14/9 = 773.872{{c}}
: error map: {{val| 0.000 -6.464 -38.569 -3.973 }}
: error map: {{val| 0.000 -6.464 -38.569 -3.973 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~14/9 = 775.548{{c}}
: [[error map]]: {{val| 0.000 +0.238 -35.217 +11.108 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 772.792{{c}}
: error map: {{val| 0.000 -10.789 -40.731 -13.702 }} -->


{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}
{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}
Line 508: Line 447:
* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/7 = 775.413{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/7 = 772.727{{c}} -->


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

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

Retrieved from "https://en.xen.wiki/index.php?title=Dicot_family&oldid=207341"