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Schismatic family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (3/)
Switch to Sintel's badness, WE & CWE tunings (4/)
Line 2,886: Line 2,886:
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.491{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}
: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}
: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.491{{c}} -->


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
[[Badness]] (Sintel): 0.479


==== 2.3.5.17.19 subgroup ====
==== 2.3.5.17.19 subgroup ====
Line 2,901: Line 2,906:
Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.470{{c}}
Optimal tunings:
* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.470{{c}} -->


{{Optimal ET sequence|legend=0| 12, 41, 53, 65 }}
{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}


RMS error: 0.5374 cents
Badness (Sintel): 0.332


=== Nestoria (2.3.5.19) ===
=== Nestoria (2.3.5.19) ===
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''


The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] (, ''[[513/512|S15/S20]]'')}. Strangely, despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is actually sharper than the fifth in optimal schismic. This is likely due to its optimization considering intervals like 19/10 and 19/15.  
The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] , ([[513/512|S15/S20]])}. Strangely, despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is actually sharper than the fifth in optimal schismic. This is likely due to its optimization considering intervals like 19/10 and 19/15.  


[[Subgroup]]: 2.3.5.19
[[Subgroup]]: 2.3.5.19
Line 2,921: Line 2,929:
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.746{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}
: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}
: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.746{{c}} -->


{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
[[Badness]] (Sintel): 0.126


=== Taylor (2.3.5.13) ===
=== Taylor (2.3.5.13) ===
Line 2,939: Line 2,952:
: mapping generators: ~2, ~26/15
: mapping generators: ~2, ~26/15


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~26/15 = 950.855{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}
: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}
: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~26/15 = 950.855{{c}} -->


{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
[[Badness]] (Sintel): 0.334


==== Dakota (2.3.5.13.19) ====
==== Dakota (2.3.5.13.19) ====
Line 2,952: Line 2,970:
Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 950.8199{{c}}
Optimal tunings:
* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8199{{c}} -->


{{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}
{{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}


Badness (Smith): 0.00575
Badness (Sintel): 0.262


===== 2.3.5.13.19.37 subgroup =====
===== 2.3.5.13.19.37 subgroup =====
Line 2,965: Line 2,986:
Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 950.8187{{c}}
Optimal tunings:
* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8187{{c}} -->


{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}
{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}


Badness (Smith): 0.00357
Badness (Sintel): 0.223


=== Quintilischis (2.3.5.17) ===
=== Quintilischis (2.3.5.17) ===
Line 2,983: Line 3,007:
: mapping generators: ~2, ~18/17
: mapping generators: ~2, ~18/17


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~18/17 = 99.649{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}
: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}
: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~18/17 = 99.649{{c}} -->


{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
[[Badness]] (Sintel): 1.34


==== 2.3.5.17.19 subgroup ====
==== 2.3.5.17.19 subgroup ====
Line 2,998: Line 3,027:
Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.652{{c}}
Optimal tunings:
* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.652{{c}} -->


{{Optimal ET sequence|legend=0| 12, 253, 265, 277, 289 }}
{{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }}


RMS error: 0.1636 cents
Badness (Sintel): 1.17


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Schismatic family| ]] <!-- main article -->
[[Category:Schismatic family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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