Xenharmonic Wiki

Dicot family: Difference between revisions

← Older edit
VisualWikitext
Jerdle (talk | contribs)
autopatrolled
212 edits
Added the 13-limit extension
+ short intro to each temp
 
(24 intermediate revisions by 6 users not shown)
Line 1: Line 1:
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.  


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 ==
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 12: Line 15:
: 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}}, ~6/5 = 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.  
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.


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]]
 
The rest are considered below.  


=== 2.3.5.11 subgroup ===
=== 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
Subgroup: 2.3.5.11


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


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


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


[[Mapping|Sval mapping]]: [{{val|1 1 2 2}}, {{val|0 2 1 5}}]
Optimal tunings:  
* WE: ~2 = 1206.750{{c}}, ~6/5 = 348.684{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 348.954{{c}}


[[Tp tuning|POL2 generator]]: ~5/4 = 346.734
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}


{{Optimal ET sequence|legend=1| 3e, 4e, 7, 24c, 31c, 38cc, 45cce }}
Badness (Sintel): 0.370


[[Tp tuning #T2 tuning|RMS error]]: 5.621 cents
==== 2.3.5.11.13 subgroup ====
 
Related temperaments: [[Dicot family #Septimal dicot|dicot]], [[Dicot family #Sharp|sharp]], [[Dicot family #Dichotic|dichotic]]
 
==== 2.3.5.11.13 ====
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
== 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.  


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


Line 72: Line 85:
{{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}}
: [[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 tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 336.381
{{Optimal ET sequence|legend=1| 3d, 4, 7 }}


{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
[[Badness]] (Sintel): 0.504
 
[[Badness]]: 0.019935


=== 11-limit ===
=== 11-limit ===
Line 87: Line 102:
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 117:
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 132:
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=1| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
{{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 126: Line 149:
{{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 166:
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 181:
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=1| 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 167: Line 198:
{{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}}
: [[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 }}


{{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, 37cd, 47bccd, 57bccdd }}
[[Badness]] (Sintel): 0.732
 
[[Badness]]: 0.028942


=== 11-limit ===
=== 11-limit ===
Line 182: Line 215:
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 232:
{{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 249:
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 264:
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 279:
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 294:
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 309:
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 324:
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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 292: Line 346:
: 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}})
: [[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 tuning]] ([[POTE]]): ~7/5 = 1\2, ~7/4 = 948.443 (~7/6 = 251.557)
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}


{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
[[Badness]] (Sintel): 0.717
 
[[Badness]]: 0.028334


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


Line 308: Line 363:
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 ====
==== 13-limit ====
Line 321: Line 378:
Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}
Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 947.955 (~7/6 = 252.045)
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=1| 10, 14cf, 24cf}}
{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}


Badness: 0.021326
Badness (Sintel): 0.881


=== Decimated ===
=== Decimated ===
Line 334: Line 393:
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 347: Line 408:
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, ~9/7


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}


{{Optimal ET sequence|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 375: 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 tuning (POTE): ~2 = 1\1, ~9/7 = 427.273
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=1| 3de, 14c, 45cce, 59bcccdee }}
{{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]]
Retrieved from "https://en.xen.wiki/w/Dicot_family"