Ditonmic family: Difference between revisions

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-This tempers out the ditonma, 1220703125/1207959552.
+{{Technical data page}}
+The '''ditonmic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ditonma]] ({{monzo|legend=1| -27 -2 13 }}, [[ratio]]: 1 220 703 125 / 1 207 959 552).
-=Ditonic=
+== Ditonic ==
-Commas: 1220703125/1207959552
+Named by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, ditonic splits [[~]][[5/2]] in two for a generator, which happens to be an interval very close in size to the [[octave complement]] of the ditone (i.e. the Pythagorean major third, [[81/64]]). It can be described as 3 & 53, and is part of the [[schismic–Mercator equivalence continuum]] with equivalence number ''n'' = 13/2. Its [[ploidacot]] is eta-13-cot. Note that the ditone itself is 52 generator steps away.
-POTE generator: ~15625/12288 = 407.574
+[[Subgroup]]: 2.3.5
-Map: [&lt;1 6 3|, &lt;0 -13 -2|]
+[[Comma list]]: 1220703125/1207959552
-EDOs: 47, 50, 53, 474c, 527c
+{{Mapping|legend=1| 1 -7 1 | 0 13 2 }}
+: mapping generators: ~2, ~24576/15625
-Badness: 0.1671
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2971{{c}}, ~24576/15625 = 792.6223{{c}}
+: [[error map]]: {{val| +0.297 +0.055 -0.772 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~24576/15625 = 792.4403{{c}}
+: error map: {{val| 0.000 -0.231 -1.433 }}
-==7-limit==
+{{Optimal ET sequence|legend=1| 3, …, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc }}
-Commas: 126/125, 8751645/8388608
-POTE generator: ~2625/2048 = 407.954
+[[Badness]] (Sintel): 3.92
-Map: [&lt;1 6 3 -4|, &lt;0 -13 -2 20|]
+== Coditone ==
+[[Subgroup]]: 2.3.5.7
-EDOs: 3, 47, 50
+[[Comma list]]: 225/224, 2125764/2100875
-Badness: 0.2421
+{{Mapping|legend=1| 1 -7 1 -17 | 0 13 2 30 }}
-==11-limit==
+[[Optimal tuning]]s:
-Commas: 126/125, 245/242, 2079/2048
+* [[WE]]: ~2 = 1200.4411{{c}}, ~1944/1225 = 792.6016{{c}}
+: [[error map]]: {{val| +0.441 -1.223 -0.669 +1.722 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~1944/1225 = 792.3270{{c}}
+: error map: {{val| 0.000 -1.704 -1.660 +0.985 }}
-POTE generator: ~14/11 = 407.892
+{{Optimal ET sequence|legend=1| 50, 53, 103, 156 }}
-Map: [&lt;1 6 3 -4 -3|, &lt;0 -13 -2 20 19|]
+[[Badness]] (Sintel): 1.63
-EDOs: 3, 47, 50, 53d
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness: 0.1009
+Comma list: 225/224, 385/384, 78408/78125
-==13-limit==
+{{Mapping|legend=0| 1 -7 1 -17 16 | 0 13 2 30 -19 }}
-Commas: 105/104, 126/125, 245/242, 1287/1280
-POTE generator: ~14/11 = 407.887
+Optimal tunings:
+* WE: ~2 = 1200.8016{{c}}, ~198/125 = 792.7878{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2692{{c}}
-Map: [&lt;1 6 3 -4 -3 2|, &lt;0 -13 -2 20 19 5|]
+{{Optimal ET sequence|legend=0| 50, 53, 103, 259be, 362bcee }}
-EDOs: 3, 47, 50, 53d
+Badness (Sintel): 1.47
-Badness: 0.0550
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=Coditone=
+Comma list: 225/224, 351/350, 385/384, 847/845
-Commas: 225/224, 2125764/2100875
-POTE generator: ~1225/972 = 407.690
+{{Mapping|legend=0| 1 -7 1 -17 16 7 | 0 13 2 30 -19 -5 }}
-Map: [&lt;1 6 3 13|, &lt;0 -13 -2 -30|]
+Optimal tunings:
+* WE: ~2 = 1200.7372{{c}}, ~198/125 = 792.7511{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~198/125 = 792.2715{{c}}
-EDOs: 50, 53, 103, 156
+{{Optimal ET sequence|legend=0| 50, 53, 103, 259be, 362bceef }}
-==11-limit==
+Badness (Sintel): 1.01
-Commas: 99/98, 176/175, 6655/6561
-POTE generator: ~242/189 = 407.567
+=== Coditonic ===
+Subgroup: 2.3.5.7.11
-Map: [&lt;1 6 3 13 15|, &lt;0 -13 -2 -30 -34|]
+Comma list: 99/98, 176/175, 6655/6561
-EDOs: 53
+{{Mapping|legend=0| 1 -7 1 -17 -19 | 0 13 2 30 34 }}
-Badness: 0.0639
+Optimal tunings:
+* WE: ~2 = 1200.3578{{c}}, ~189/121 = 792.6693{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~189/121 = 792.4463{{c}}
-==13-limit==
+{{Optimal ET sequence|legend=0| 3de, 50e, 53 }}
-Commas: 99/98, 176/175, 325/324, 847/845
-POTE generator: ~33/26 = 407.541
+Badness (Sintel): 2.11
-Map: [&lt;1 6 3 13 15 20|, &lt;0 -13 -2 -30 -34 -48|]
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-EDOs: 53
+Comma list: 99/98, 176/175, 325/324, 847/845
-Badness: 0.0440
+{{Mapping|legend=0| 1 -7 1 -17 -19 -28 | 0 13 2 30 34 48 }}
+Optimal tunings:
+* WE: ~2 = 1200.2797{{c}}, ~52/33 = 792.6439{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~52/33 = 792.4682{{c}}
+{{Optimal ET sequence|legend=0| 3def, 50eff, 53 }}
+Badness (Sintel): 1.82
+== Diton ==
+This low-accuracy temperament was considered the canonical extension of ditonic, and catalogued as so in [[Graham Breed]]'s Temperament Finder.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 126/125, 8751645/8388608
+{{Mapping|legend=1| 1 -7 1 16 | 0 13 2 -20 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.9455{{c}}, ~2048/1323 = 793.3304{{c}}
+: [[error map]]: {{val| +1.946 -2.278 +2.293 -4.305 }}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~2048/1323 = 792.0527{{c}}
+: error map: {{val| 0.000 -5.270 -2.208 -9.879 }}
+{{Optimal ET sequence|legend=1| 3, 47, 50 }}
+[[Badness]] (Sintel): 6.13
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 126/125, 245/242, 2079/2048
+{{Mapping|legend=0| 1 -7 1 16 16 | 0 13 2 -20 -19 }}
+Optimal tunings:
+* WE: ~2 = 1201.6076{{c}}, ~11/7 = 793.1692{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 792.1016{{c}}
+{{Optimal ET sequence|legend=0| 3, 47, 50 }}
+Badness (Sintel): 3.34
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 105/104, 126/125, 245/242, 1287/1280
+{{Mapping|legend=0| 1 -7 1 16 16 7 | 0 13 2 -20 -19 -5 }}
+Optimal tunings:
+* WE: ~2 = 1201.3885{{c}}, ~11/7 = 793.0294{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 792.1096{{c}}
+{{Optimal ET sequence|legend=0| 3, 47, 50 }}
+Badness (Sintel): 2.27
+== References ==
+[[Category:Ditonmic family| ]] <!-- main article -->
+[[Category:Ditonic]] <!-- key article -->
+[[Category:Temperament families]]
+[[Category:Catalogs of rank-2 temperaments]]