5L 7s: Difference between revisions

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Spt3125 (talk | contribs)
autopatrolled, Sysops
523 edits
m table edits
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| style="text-align:center;" | 5\12
| style="text-align:center;" | 5\12
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 500
| style="text-align:center;" | 500
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 37\89
| style="text-align:center;" | 37\89
| style="text-align:center;" | 498.876
| style="text-align:center;" | 498.876
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 32\77
| style="text-align:center;" | 32\77
| style="text-align:center;" |  
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| style="text-align:center;" | 498.702
| style="text-align:center;" | 498.702
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 27\65
| style="text-align:center;" | 27\65
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 498.462
| style="text-align:center;" | 498.462
| style="text-align:center;" | Photia
| style="text-align:center;" | Photia
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 49\118
| style="text-align:center;" | 49\118
| style="text-align:center;" |  
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| style="text-align:center;" | 498.305
| style="text-align:center;" | 498.305
| style="text-align:center;" | Helmholtz/Pontiac/Nestoria
| style="text-align:center;" | Helmholtz/Pontiac/Nestoria
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 71\171
| style="text-align:center;" | 71\171
| style="text-align:center;" | 498.246
| style="text-align:center;" | 498.246
| style="text-align:center;" | Helmholtz/Pontiac/Nestoria
| style="text-align:center;" | Helmholtz/Pontiac/Nestoria
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 22\53
| style="text-align:center;" | 22\53
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 498.113
| style="text-align:center;" | 498.113
| style="text-align:center;" | Helenus
| style="text-align:center;" | Helenus
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 39\94
| style="text-align:center;" | 39\94
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 497.872
| style="text-align:center;" | 497.872
| style="text-align:center;" | Garibaldi
| style="text-align:center;" | Garibaldi
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 17\41
| style="text-align:center;" | 17\41
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 497.591
| style="text-align:center;" | 497.591
| style="text-align:center;" | Cassandra
| style="text-align:center;" | Cassandra
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 46\111
| style="text-align:center;" | 46\111
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 497.297
| style="text-align:center;" | 497.297
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 29\70
| style="text-align:center;" | 29\70
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 497.143
| style="text-align:center;" | 497.143
| style="text-align:center;" | Undecental
| style="text-align:center;" | Undecental
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 41\99
| style="text-align:center;" | 41\99
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 496.97
| style="text-align:center;" | 496.97
| style="text-align:center;" | Undecental
| style="text-align:center;" | Undecental
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 12\29
| style="text-align:center;" | 12\29
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 496.552
| style="text-align:center;" | 496.552
| style="text-align:center;" | Optimum rank range (L/s=3/2)
| style="text-align:center;" | Optimum rank range (L/s=3/2)
Line 156: Line 156:
Edson
Edson
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 43\104
| style="text-align:center;" | 43\104
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 496.154
| style="text-align:center;" | 496.154
| style="text-align:center;" |  
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Line 176: Line 176:
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| style="text-align:center;" | 496.157
| style="text-align:center;" | 496.157
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| style="text-align:center;" | L/s = pi/2
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 31\75
| style="text-align:center;" | 31\75
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 496
| style="text-align:center;" | 496
| style="text-align:center;" |  
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Line 198: Line 198:
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| style="text-align:center;" | 495.904
| style="text-align:center;" | 495.904
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| style="text-align:center;" | L/s = phi
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 50\121
| style="text-align:center;" | 50\121
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 495.868
| style="text-align:center;" | 495.868
| style="text-align:center;" | Leapday/[[Peppermint-24|Peppermint]]/Pepperoni
| style="text-align:center;" | Leapday/[[Peppermint-24|Peppermint]]/Pepperoni
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 19\46
| style="text-align:center;" | 19\46
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 495.652
| style="text-align:center;" | 495.652
| style="text-align:center;" | Leapday
| style="text-align:center;" | Leapday
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 45\109
| style="text-align:center;" | 45\109
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 495.413
| style="text-align:center;" | 495.413
| style="text-align:center;" |  
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| style="text-align:center;" | 495.325
| style="text-align:center;" | 495.325
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| style="text-align:center;" | L/s = sqrt(3)
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 26\63
| style="text-align:center;" | 26\63
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 495.238
| style="text-align:center;" | 495.238
| style="text-align:center;" |  
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|-
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 33\80
| style="text-align:center;" | 33\80
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 495
| style="text-align:center;" | 495
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 7\17
| style="text-align:center;" | 7\17
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 494.118
| style="text-align:center;" | 494.118
| style="text-align:center;" | Boundary of propriety (generators larger than this are proper)
| style="text-align:center;" | Boundary of propriety (generators larger than this are proper)
Line 279: Line 279:
Supraphon
Supraphon
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 30\73
| style="text-align:center;" | 30\73
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 493.151
| style="text-align:center;" | 493.151
| style="text-align:center;" |  
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|-
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 23\56
| style="text-align:center;" | 23\56
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 492.857
| style="text-align:center;" | 492.857
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 39\95
| style="text-align:center;" | 39\95
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 492.632
| style="text-align:center;" | 492.632
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 16\39
| style="text-align:center;" | 16\39
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 492.308
| style="text-align:center;" | 492.308
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 41\100
| style="text-align:center;" | 41\100
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 492
| style="text-align:center;" | 492
| style="text-align:center;" |  
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| style="text-align:center;" | 491.946
| style="text-align:center;" | 491.946
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| style="text-align:center;" | L/s = phi+1
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 25\61
| style="text-align:center;" | 25\61
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 491.803
| style="text-align:center;" | 491.803
| style="text-align:center;" |  
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|-
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 491.655
| style="text-align:center;" | 491.655
| style="text-align:center;" |  
| style="text-align:center;" | L/s = e
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| style="text-align:center;" | <span style="display: block; text-align: center;">34\83</span>
| style="text-align:center;" | 34\83
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| style="text-align:center;" | <span style="display: block; text-align: center;">491.566</span>
| style="text-align:center;" | 491.566
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 9\22
| style="text-align:center;" | 9\22
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 490.909
| style="text-align:center;" | 490.909
| style="text-align:center;" | Suprapyth/Supra
| style="text-align:center;" | Suprapyth/Supra
Line 398: Line 398:
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| style="text-align:center;" | 490.569
| style="text-align:center;" | 490.569
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| style="text-align:center;" | L/s = pi
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 29\71
| style="text-align:center;" | 29\71
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 490.141
| style="text-align:center;" | 490.141
| style="text-align:center;" |  
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|-
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 20\49
| style="text-align:center;" | 20\49
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 489.796
| style="text-align:center;" | 489.796
| style="text-align:center;" | Superpyth
| style="text-align:center;" | Superpyth
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 31\76
| style="text-align:center;" | 31\76
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 489.474
| style="text-align:center;" | 489.474
| style="text-align:center;" |  
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|-
|-
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 11\27
| style="text-align:center;" | 11\27
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 488.889
| style="text-align:center;" | 488.889
| style="text-align:center;" | Archy
| style="text-align:center;" | Archy
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|-
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 24\59
| style="text-align:center;" | 24\59
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 488.136
| style="text-align:center;" | 488.136
| style="text-align:center;" |  
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|-
|-
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 13\32
| style="text-align:center;" | 13\32
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 487.500
| style="text-align:center;" | 487.500
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" | 15\37
| style="text-align:center;" | 15\37
| style="text-align:center;" |  
| |
| style="text-align:center;" |  
| |
| style="text-align:center;" | 486.486
| style="text-align:center;" | 486.486
| style="text-align:center;" |  
| |
|-
|-
| style="text-align:center;" |  
| |
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
| |
| style="text-align:center;" |  
| |
| style="text-align:center;" |  
| |
| style="text-align:center;" | 17\42
| style="text-align:center;" | 17\42
| style="text-align:center;" |  
| |
| style="text-align:center;" | 485.714
| style="text-align:center;" | 485.714
| style="text-align:center;" |  
| |
|-
|-
| style="text-align:center;" |  
| |
| style="text-align:center;" |  
| |
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
| |
| style="text-align:center;" |  
| |
| style="text-align:center;" | 19\47
| style="text-align:center;" | 19\47
| style="text-align:center;" | 485.106
| style="text-align:center;" | 485.106
| style="text-align:center;" |  
| |
|-
|-
| style="text-align:center;" | 2\5
| style="text-align:center;" | 2\5
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
| |
| style="text-align:center;" | 480.000
| style="text-align:center;" | 480.000
| style="text-align:center;" |  
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Revision as of 17:46, 30 December 2018

This is the MOS pattern of the Pythagorean/schismatic/Helmholtz/Garibaldi chromatic scale, and also the superpyth chromatic scale. In contrast to the meantone chromatic scale, in which "diatonic" semitones are larger than "chromatic" semitones, here the reverse is true: diatonic semitones are smaller than chromatic semitones, so the diatonic scale subset is actually improper.

The two distinct harmonic entropy minima with this MOS pattern are, on the one hand, scales very close to Pythagorean such that 64/63 is not tempered out, such as the schismatic temperaments known as "Helmholtz" and "Garibaldi", and on the other hand, the much simpler and less accurate scale known as "superpyth" in which 64/63 is tempered out.

The Pythagorean/schismatic version is proper, but the superpyth version is improper (it doesn't become proper until you add 5 more notes to form the superpyth "enharmonic" scale, superpyth[17]).

Generator in cents Comments
5\12 500
37\89 498.876
32\77 498.702
27\65 498.462 Photia
49\118 498.305 Helmholtz/Pontiac/Nestoria
71\171 498.246 Helmholtz/Pontiac/Nestoria
22\53 498.113 Helenus
39\94 497.872 Garibaldi
17\41 497.591 Cassandra
46\111 497.297
29\70 497.143 Undecental
41\99 496.97 Undecental
12\29 496.552 Optimum rank range (L/s=3/2)

Edson

43\104 496.154
496.157 L/s = pi/2
31\75 496
495.904 L/s = phi
50\121 495.868 Leapday/Peppermint/Pepperoni
19\46 495.652 Leapday
45\109 495.413
495.325 L/s = sqrt(3)
26\63 495.238
33\80 495
7\17 494.118 Boundary of propriety (generators larger than this are proper)

Supraphon

30\73 493.151
23\56 492.857
39\95 492.632
16\39 492.308
41\100 492
491.946 L/s = phi+1
25\61 491.803
491.655 L/s = e
34\83 491.566
9\22 490.909 Suprapyth/Supra
490.569 L/s = pi
29\71 490.141
20\49 489.796 Superpyth
31\76 489.474
11\27 488.889 Archy
24\59 488.136
13\32 487.500
15\37 486.486
17\42 485.714
19\47 485.106
2\5 480.000
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