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No-fives subgroup temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (5/)
Switch to Sintel's badness, WE & CWE tunings (6/)
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9871{{c}}, ~8/7 = 231.1001{{c}}
* [[WE]]: ~2 = 1199.9871{{c}}, ~8/7 = 231.1001{{c}}
: [[error map]]: {{val| -0.013 +0.000 +0.035 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 231.1024{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 231.1024{{c}}
: error map: {{val| 0.000 +0.014 +0.072 }}


{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1106, 2077, 5260, 7337, 9414d }}
{{Optimal ET sequence|legend=0| 26, 83, 109, 135, 566, 701, 836, 971, 1106, 2077, 5260, 7337, 9414d }}
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=== Hectosaros leap week ===
=== Hectosaros leap week ===
Defined as the {{nowrap| 320 & 1803 }} temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.
This temperament may be described as the {{nowrap| 320 & 1803 }} temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7


[[Comma list]]: {{monzo|-50 -746 439}}
[[Comma list]]: {{monzo| -50 -746 439 }}


[[Mapping]]: [{{val|1 313 532}}, {{val|0 -439 -746}}]
{{Mapping|legend=2| 1 -126 -214 | 0 439 746 }}
: mapping generators: ~2, ~{{monzo| -16 -243 143 }}


[[Optimal tuning]] ([[CTE]]): ~{{monzo|17 343 143}} = 851.248
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0010{{c}}, ~{{monzo| -16 -243 143 }} = 348.7520{{c}}
: [[error map]]: {{val| +0.001 +0.036 -0.067 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -16 -243 143 }} = 348.7517{{c}}
: error map: {{val| 0.000 +0.035 -0.068 }}


{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0164 cents
[[Badness]] (Sintel): 17.7 × 10<sup>3</sup>


==== 2.3.7.13 subgroup ====
==== 2.3.7.13 subgroup ====
Subgroup: 2.3.7.13
Subgroup: 2.3.7.13


Comma list: {{monzo|-42 -2 -5 16}}, {{monzo|10 -46 29 -5}}
Comma list: {{monzo| -42 -2 -5 16 }}, {{monzo| 10 -46 29 -5 }}


Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 | 0 439 746 288 }}


Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
Optimal tunings:  
* WE: ~2 = 1200.0058{{c}}, ~{{monzo| 18 -9 8 -7 }} = 348.7534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~{{monzo| 18 -9 8 -7 }} = 348.8517{{c}}


{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}  
{{Optimal ET sequence|legend=0| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689, 11255d }}  
 
Badness (Sintel): 53.2


==== 2.3.7.13.17 subgroup ====
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17
Subgroup: 2.3.7.13.17


Comma list: 39337984/39328497, {{monzo|0 -14 7 4 -3}}, {{monzo|-18 -24 14 -1 5}}
Comma list: 39337984/39328497, {{monzo| 0 -14 7 4 -3 }}, {{monzo| -18 -24 14 -1 5 }}


Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 | 0 439 746 288 76 }}


Optimal tuning (CTE): ~6144/3757 = 851.248
Optimal tunings:  
* WE: ~2 = 1200.9870{{c}}, ~3757/3072 = 348.7480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3757/3072 = 348.7517{{c}}


{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
{{Optimal ET sequence|legend=0| 320, 1483bd, 1803, 2123 }}
 
Badness (Sintel): 13.4


==== 2.3.7.13.17.19 subgroup ====
==== 2.3.7.13.17.19 subgroup ====
Subgroup: 2.3.7.13.17.19
Subgroup: 2.3.7.13.17.19


Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256
Comma list: 10081799/10077696, 10754912/10744731, 39337984/39328497, 480024727/480020256
 
Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 -171 | 0 439 746 288 76 603 }}


Mapping: [{{val|1 313 532 208 58 432}}, {{val|0 -439 -746 -288 -76 -603}}]
Optimal tunings:  
* WE: ~2 = 1200.9961{{c}}, ~3757/3072 = 348.7506{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3757/3072 = 348.7517{{c}}


Optimal tuning (CTE): ~6144/3757 = 851.248
{{Optimal ET sequence|legend=0| 320, 1483bd, 1803, 2123 }}


{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
Badness (Sintel): 7.46


=== Purpleheart ===
=== Purpleheart ===
{{See also| Whitewood }}
[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7


[[Comma list]]: 2187/2048
[[Comma list]]: 2187/2048


{{Mapping|legend=1| 7 11 0 | 0 0 1 }}
{{Mapping|legend=2| 7 11 0 | 0 0 1 }}
 
: mapping generators: ~9/8, ~7
: mapping generators: ~9/8, ~7


[[Optimal tuning]] ([[CTE]]): ~9/8 = 1\7, ~7/4 = 968.826 (~64/63 = 59.746)
[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 172.1541{{c}}, ~7/4 = 958.5433{{c}} (~64/63 = 74.3805{{c}})
: [[error map]]: {{val| +5.079 -8.260 -0.124 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~7/4 = 959.2372{{c}} (~64/63 = 69.3373{{c}})
: error map: {{val| 0.000 -16.241 -9.589 }}


{{Optimal ET sequence|legend=1| 7, 14, 35, 49bd }}
{{Optimal ET sequence|legend=1| 7, 14, 35, 49bd }}


[[Badness]]: 0.0875
[[Badness]] (Sintel): 3.00
 
=== Ennea ===
Subgroup: 2.3.7.11
 
[[Comma list]]: 41503/41472, 43923/43904
 
[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
 
[[Gencom|Gencom mapping]]: [{{val|1 14/9 0 25/9 31/9}}, {{val|0 2 0 2 1}}]
 
[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
 
[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
 
{{Optimal ET sequence|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents


=== Superslendric ===
=== Superslendric ===
In superslendric, 8 [[8/7]]s are equated to [[3/1]] (related to [[8edt]]).
In superslendric, eight [[8/7]]'s are equated to [[3/1]]. This relates it to [[8edt]].


Subgroup: 2.3.7
[[Subgroup]]: 2.3.7


[[Comma list]]: 17294403/16777216
[[Comma list]]: 17294403/16777216


[[Gencom]]: [2 8/7; 17294403/16777216]
{{Mapping|legend=2| 1 0 3 | 0 8 -1 }}
: mapping generators: ~2, ~8/7


{{mapping|legend=2|1 0 3|0 8 -1}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.1628{{c}}, ~8/7 = 237.7287{{c}}
: [[error map]]: {{val| +1.163 -0.125 -3.066 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 237.5664{{c}}
: error map: {{val| 0.000 -1.424 -6.392 }}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[8/7]] = 237.712
{{Optimal ET sequence|legend=1| 5, …, 66b, 71b, 76, 81, 86, 91, 96d }}


=== Parapyth (rank 3) ===
[[Badness]] (Sintel): 6.15
{{see also| Pentacircle temperaments #Parapyth }}
 
Subgroup: 2.3.7.11
 
[[Comma list]]: 896/891
 
[[Gencom]]: [2 3/2 28/27; 896/891]
 
[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 }}, {{val| 0 1 0 3 -1 }}, {{val| 0 0 0 1 1 }}]
 
[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
 
[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
 
{{Optimal ET sequence|legend=1| 17, 36, 41, 58, 63, 104 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
 
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 352/351, 364/363
 
The gencom below gives [[Margo Schulter]]'s favored basis
 
[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
 
[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 6 }}, {{val| 0 1 0 3 -1 -4 }}, {{val| 0 0 0 1 1 1 }}]
 
[[Sval]] [[mapping]]: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
 
[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
 
{{Optimal ET sequence|legend=1| 17, 41, 46, 58, 87, 104 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents


=== Heartland (rank 3) ===
=== Heartland (rank 3) ===
{{Main| Heartland }}
{{Main| Heartland }}


Heartland, with a generator of ~21/19, is named for its tempering of the heartlandisma, [[3971/3969]]. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).  
Heartland, with a generator of [[~]][[21/19]], is named for its tempering of the heartlandisma, [[3971/3969]]. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).  


Subgroup: 2.3.7.11.19
[[Subgroup]]: 2.3.7.11.19


[[Comma list]]: 243/242, 1083/1078
[[Comma list]]: 243/242, 1083/1078


[[Gencom]]: [2 21/19 7; 243/242 1083/1078]
{{Mapping|legend=2| 1 1 0 2 1 | 0 4 0 10 3 | 0 0 1 0 1 }}
 
: mapping generators: ~2, ~21/19, ~7
[[Mapping|Sval mapping]]: [{{val|1 1 0 2 1}}, {{val|0 4 0 10 3}}, {{val| 0 0 1 0 1 }}]


[[Tp tuning|POL2 generator]]: ~21/19 = 175.2713, ~7 = 3369.3784
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0983{{c}}, ~21/19 = 175.2856{{c}}, ~7/4 = 969.4578{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/19 = 175.2894{{c}}, ~7/4 = 969.5203{{c}}


{{Optimal ET sequence|legend=1| 7, 14, 27e, 34dh, 41, 89, 130 }}
{{Optimal ET sequence|legend=1| 14, 27e, 34dh, 41, 89, 130, 219 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents
[[Badness]] (Sintel): 0.615


== Temperaments with a 2.3.11 gene ==
== Temperaments with a 2.3.11 gene ==
Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"