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Ragismic microtemperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (8/)
Switch to Sintel's badness, WE & CWE tunings (9/)
Line 1,023: Line 1,023:
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}


{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
: mapping generators: ~2278125/1605632, ~448/405
: mapping generators: ~2278125/1605632, ~7168/5625


[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 600.000{{c}}, ~448/405 = 176.805{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2278125/1605632 = 600.0888{{c}}, ~7168/5625 = 423.2574{{c}}
: [[error map]]: {{val| +0.178 -0.141 -0.343 +0.165 }}
* [[CWE]]: ~2278125/1605632 = 600.0000{{c}}, ~7168/5625 = 423.1986{{c}}
: error map: {{val| 0.000 -0.374 -0.725 -0.111 }}


{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}


[[Badness]] (Sintel): 4.454
[[Badness]] (Sintel): 4.45


=== 11-limit ===
=== 11-limit ===
Line 1,037: Line 1,041:
Comma list: 3025/3024, 4375/4374, 1265625/1261568
Comma list: 3025/3024, 4375/4374, 1265625/1261568


Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~448/405 = 176.806{{c}}
Optimal tunings:
* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}


{{Optimal ET sequence|legend=0| 190, 224, 414, 638, 1052c }}
{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}


Badness (Sintel): 1.357
Badness (Sintel): 1.36


=== 13-limit ===
=== 13-limit ===
Line 1,050: Line 1,056:
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
Comma list: 625/624, 729/728, 1575/1573, 2200/2197


Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~195/176 = 176.804{{c}}
Optimal tunings:
* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}


{{Optimal ET sequence|legend=0| 190, 224, 414, 638, 1690bcc, 2328bccde }}
{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}


Badness (Sintel): 0.936
Badness (Sintel): 0.936


== Moulin ==
== Moulin ==
Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the {{nowrap| 494 & 1619 }} temperament. Since 11/8 is within 23 generators, the 25-tone mos (4L 21s) of this temperament contains the 8:11:13 triad.
Moulin can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~[[13/11]], and 73 of them octave reduced reach the [[3/2|perfect fifth]]. Since [[11/8]] is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,065: Line 1,073:
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}


{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
: Mapping generators: ~2, ~6422528/3796875
: mapping generators: ~2, ~3796875/3211264


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6422528/3796875 = 910.9323{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}
: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}


{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}


[[Badness]] (Sintel): 5.931
[[Badness]] (Sintel): 5.93


=== 11-limit ===
=== 11-limit ===
Line 1,079: Line 1,091:
Comma list: 4375/4374, 759375/758912, 100663296/100656875
Comma list: 4375/4374, 759375/758912, 100663296/100656875


Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~1024/605 = 910.9323{{c}}
Optimal tunings:
* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}


{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}


Badness (Sintel): 2.240
Badness (Sintel): 2.24


=== 13-limit ===
=== 13-limit ===
Line 1,092: Line 1,106:
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078


Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~22/13 = 910.9323{{c}}
Optimal tunings:
* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}


{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}


Badness (Sintel): 1.118
Badness (Sintel): 1.12


== Palladium ==
== Palladium ==
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments #Palladium]]''.
: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.


The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
Line 1,110: Line 1,126:


{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
: Mapping generators: ~83349/81920, ~3
: mapping generators: ~83349/81920, ~3


[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6074{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~83349/81920 = 26.0910{{c}}, ~3/2 = 701.7155{{c}}
: [[error map]]: {{val| +0.185 -0.055 -0.061 +0.349 }}
* [[CWE]]: ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6491{{c}}
: error map: {{val| 0.000 -0.306 -0.407 -0.910 }}


{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}
{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}


[[Badness]] (Sintel): 7.807
[[Badness]] (Sintel): 7.81


=== 11-limit ===
=== 11-limit ===
Line 1,125: Line 1,145:
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}


Optimal tuning (POTE): ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.5951{{c}}
Optimal tunings:
* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}


{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de }}
{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}


Badness (Sintel): 2.439
Badness (Sintel): 2.44


=== 13-limit ===
=== 13-limit ===
Line 1,138: Line 1,160:
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}


Optimal tuning (POTE): ~65/64 = 26.0870{{c}}, ~3/2 = 701.6419{{c}}
Optimal tunings:
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}}


{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334de }}
{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}


Badness (Sintel): 1.684
Badness (Sintel): 1.68


=== 17-limit ===
=== 17-limit ===
Line 1,151: Line 1,175:
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}


Optimal tuning (POTE): ~65/64 = 26.0870{{c}}, ~3/2 = 701.6425{{c}}
Optimal tunings:
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}


{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334deg }}
{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}


Badness (Sintel): 1.143
Badness (Sintel): 1.14


== Oviminor ==
== Oviminor ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''  
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''  


Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.  
Oviminor is named after the facts that it takes 184 minor thirds of [[6/5]] to reach the interval class of [[4/3]], the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,166: Line 1,192:
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}


{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
{{Mapping|legend=1| 1 -134 -134 -401 | 0 184 185 548 }}
: Mapping generators: ~2, ~6/5
: mapping generators: ~2, ~5/3


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 315.7501{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}
: [[error map]]: {{val| +0.019 +0.010 -0.085 +0.032 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}
: error map: {{val| 0.000 -0.011 -0.120 +0.008 }}


{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}


[[Badness]] (Sintel): 14.739
[[Badness]] (Sintel): 14.7


== Octoid ==
== Octoid ==
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