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Hemimean clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (1/)
Switch to Sintel's badness, WE & CWE tunings (2/); restore the name *semihemiwürschmidt*, as there's no internal conflict
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{{See also| Würschmidt family }}
{{See also| Würschmidt family }}


Hemiwürschmidt (sometimes spelled ''hemiwuerschmidt'') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.
Hemiwürschmidt (sometimes spelled ''hemiwuerschmidt'') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher-limit temperament, mapping 11 to 40 generators and 13 to -39.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Badness (Sintel): 1.15
Badness (Sintel): 1.15


=== Semihemiwür ===
=== Semihemiwürschmidt ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3136/3125, 9801/9800
Comma list: 2401/2400, 3136/3125, 9801/9800


{{Mapping|legend=0| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
{{Mapping|legend=0| 2 -2 4 4 15 | 0 16 2 5 -25 }}
: mapping generators: ~99/70, ~28/25


: mapping generators: ~99/70, ~495/392
Optimal tunings:  
 
* WE: ~99/70 = 599.8485{{c}}, ~28/25 = 193.8531{{c}}
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021
* CWE: ~99/70 = 600.0000{{c}}, ~28/25 = 193.9002{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
{{Optimal ET sequence|legend=0| 62e, 68, 130, 198, 328 }}


Badness: 0.044848
Badness (Sintel): 1.48


==== 13-limit ====
==== 13-limit ====
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Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125


{{Mapping|legend=0| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 | 0 16 2 5 -25 26 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
Optimal tunings:
* WE: ~99/70 = 599.8514{{c}}, ~28/25 = 193.8555{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/25 = 193.9006{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
{{Optimal ET sequence|legend=0| 62e, 68, 130, 198, 328 }}


Badness: 0.023388
Badness (Sintel): 0.966


===== Semihemiwürat =====
===== Semihemiwürat =====
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Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625


{{Mapping|legend=0| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 3 | 0 16 2 5 -25 26 16 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112
Optimal tunings:
* WE: ~17/12 = 599.9069{{c}}, ~28/25 = 193.8811{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9092{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198 }}


Badness: 0.028987
Badness (Sintel): 1.48


====== 19-limit ======
====== 19-limit ======
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Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625


{{Mapping|legend=0| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 3 3 | 0 16 2 5 -25 26 16 17 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145
Optimal tunings:
* WE: ~17/12 = 599.9292{{c}}, ~19/17 = 193.8916{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~19/17 = 193.9128{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198 }}


Badness: 0.021707
Badness (Sintel): 1.32


===== Semihemiwüram =====
===== Semihemiwüram =====
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Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224


{{Mapping|legend=0| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 14 | 0 16 2 5 -25 26 -18 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112
Optimal tunings:  
* WE: ~17/12 = 599.7759{{c}}, ~28/25 = 193.8534{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9238{{c}}


{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
{{Optimal ET sequence|legend=1| 68, 130g, 198g }}


Badness: 0.029718
Badness (Sintel): 1.51


====== 19-limit ======
====== 19-limit ======
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Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224


{{Mapping|legend=0| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 14 14 | 0 16 2 5 -25 26 -18 -17 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428
Optimal tunings:
* WE: ~17/12 = 599.7147{{c}}, ~28/25 = 193.8506{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9417{{c}}


{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}
{{Optimal ET sequence|legend=1| 68, 130gh, 198gh }}


Badness: 0.029545
Badness (Sintel): 1.80


== Hemithirds ==
== Hemithirds ==
Retrieved from "https://en.xen.wiki/w/Hemimean_clan"