186zpi: Difference between revisions

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autopatrolled
2,880 edits
Line 1,384: Line 1,384:
!Error (rel, [[Relative cent| %]])
!Error (rel, [[Relative cent| %]])
|-
|-
|[[17/13]]
| [[17/13]]
|0.030
| 0.030
|0.102
| 0.102
|-
|-
|'''[[5/1]]'''
| '''[[5/1]]'''
|'''0.075'''
| '''0.075'''
|'''0.259'''
| '''0.259'''
|-
|-
|[[25/17]]
| [[25/17]]
|0.100
| 0.100
|0.344
| 0.344
|-
|-
|[[25/13]]
| [[25/13]]
|0.129
| 0.129
|0.446
| 0.446
|-
|-
|[[23/11]]
| [[23/11]]
|0.138
| 0.138
|0.477
| 0.477
|-
|-
|[[25/1]]
| [[25/1]]
|0.150
| 0.150
|0.517
| 0.517
|-
|-
|''[[11/8]]''
| ''[[11/8]]''
|''0.155''
| ''0.155''
|
| ''0.533''
''0.533''
|-
|-
|[[17/5]]
| [[17/5]]
|0.175
| 0.175
|0.602
| 0.602
|-
|-
|[[13/5]]
| [[13/5]]
|0.204
| 0.204
|0.704
| 0.704
|-
|-
|'''[[17/1]]'''
| '''[[17/1]]'''
|'''0.250'''
| '''0.250'''
|'''0.861'''
| '''0.861'''
|-
|-
|'''[[13/1]]'''
| '''[[13/1]]'''
|'''0.279'''
| '''0.279'''
|'''0.963'''
| '''0.963'''
|-
|-
|
| ''[[9/7]]''
''[[9/7]]''
| ''0.289''
|''0.289''
| ''0.996''
|''0.996''
|-
|-
|''[[23/8]]''
| ''[[23/8]]''
|''0.293''
| ''0.293''
|''1.011''
| ''1.011''
|-
|-
|'''[[23/1]]'''
| '''[[23/1]]'''
|'''0.621'''
| '''0.621'''
|'''2.140'''
| '''2.140'''
|-
|-
|[[31/29]]
| [[31/29]]
|0.641
| 0.641
|2.209
| 2.209
|-
|-
|[[30/29]]
| [[30/29]]
|0.642
| 0.642
|2.211
| 2.211
|-
|-
|[[23/5]]
| [[23/5]]
|0.696
| 0.696
|2.399
| 2.399
|-
|-
|[[29/6]]
| [[29/6]]
|0.717
| 0.717
|2.470
| 2.470
|-
|-
|
| ''[[9/8]]''
''[[9/8]]''
| ''0.736''
|''0.736''
| ''2.535''
|''2.535''
|-
|-
|'''[[11/1]]'''
| '''[[11/1]]'''
|'''0.760'''
| '''0.760'''
|'''2.617'''
| '''2.617'''
|-
|-
|[[25/23]]
| [[25/23]]
|0.771
| 0.771
|2.657
| 2.657
|-
|-
|[[11/5]]
| [[11/5]]
|0.835
| 0.835
|2.876
| 2.876
|-
|-
|[[23/17]]
| [[23/17]]
|0.871
| 0.871
|3.001
| 3.001
|-
|-
|[[21/19]]
| [[21/19]]
|0.881
| 0.881
|3.037
| 3.037
|-
|-
|''[[11/9]]''
| ''[[11/9]]''
|''0.891''
| ''0.891''
|''3.069''
| ''3.069''
|-
|-
|[[23/13]]
| [[23/13]]
|0.901
| 0.901
|3.103
| 3.103
|-
|-
|[[25/11]]
| [[25/11]]
|0.910
| 0.910
|3.135
| 3.135
|-
|-
|
| ''[[8/1]]''
''[[8/1]]''
| ''0.914''
|''0.914''
| ''3.151''
|''3.151''
|-
|-
|
| ''[[8/5]]''
''[[8/5]]''
| ''0.990''
|''0.990''
| ''3.409''
|''3.409''
|-
|-
|[[17/11]]
| [[17/11]]
|1.009
| 1.009
|3.478
| 3.478
|-
|-
|
| ''[[8/7]]''
''[[8/7]]''
| ''1.025''
|''1.025''
| ''3.531''
|''3.531''
|-
|-
|''[[23/9]]''
| ''[[23/9]]''
|''1.029''
| ''1.029''
|''3.546''
| ''3.546''
|-
|-
|[[13/11]]
| [[13/11]]
|1.039
| 1.039
|3.580
| 3.580
|-
|-
|''[[25/8]]''
| ''[[25/8]]''
|''1.065''
| ''1.065''
|''3.668''
| ''3.668''
|-
|-
|''[[17/8]]''
| ''[[17/8]]''
|''1.164''
| ''1.164''
|''4.012''
| ''4.012''
|-
|-
|''[[27/19]]''
| ''[[27/19]]''
|''1.171''
| ''1.171''
|
| ''4.033''
''4.033''
|-
|-
|[[11/7]]
| [[11/7]]
|1.180
| 1.180
|4.065
| 4.065
|-
|-
|''[[13/8]]''
| ''[[13/8]]''
|''1.194''
| ''1.194''
|''4.114''
| ''4.114''
|-
|-
|[[31/30]]
| [[31/30]]
|1.283
| 1.283
|4.420
| 4.420
|-
|-
|[[23/7]]
| [[23/7]]
|1.318
| 1.318
|4.542
| 4.542
|-
|-
|[[31/6]]
| [[31/6]]
|1.358
| 1.358
|4.679
| 4.679
|-
|-
|
| ''[[9/1]]''
''[[9/1]]''
| ''1.650''
|''1.650''
| ''5.686''
|''5.686''
|-
|-
|
| ''[[9/5]]''
''[[9/5]]''
| ''1.725''
|''1.725''
| ''5.944''
|''5.944''
|-
|-
|''[[20/19]]''
| ''[[20/19]]''
|''1.726''
| ''1.726''
|''5.947''
| ''5.947''
|-
|-
|''[[25/9]]''
| ''[[25/9]]''
|''1.800''
| ''1.800''
|''6.203''
| ''6.203''
|-
|-
|''[[19/4]]''
| ''[[19/4]]''
|''1.801''
| ''1.801''
|''6.205''
| ''6.205''
|-
|-
|''[[17/9]]''
| ''[[17/9]]''
|''1.900''
| ''1.900''
|''6.547''
| ''6.547''
|-
|-
|''[[24/19]]''
| ''[[24/19]]''
|''1.906''
| ''1.906''
|''6.568''
| ''6.568''
|-
|-
|''[[13/9]]''
| ''[[13/9]]''
|''1.930''
| ''1.930''
|''6.649''
| ''6.649''
|-
|-
|'''[[7/1]]'''
| '''[[7/1]]'''
|'''1.939'''
| '''1.939'''
|'''6.682'''
| '''6.682'''
|-
|-
|
| [[7/5]]
[[7/5]]
| 2.015
|2.015
| 6.941
|6.941
|-
|-
|''[[31/28]]''
| ''[[31/28]]''
|''2.060''
| ''2.060''
|
| ''7.099''
''7.099''
|-
|-
|[[25/7]]
| [[25/7]]
|2.090
| 2.090
|7.199
| 7.199
|-
|-
|[[17/7]]
| [[17/7]]
|2.189
| 2.189
|7.543
| 7.543
|-
|-
|[[13/7]]
| [[13/7]]
|2.219
| 2.219
|7.645
| 7.645
|-
|-
|''[[21/20]]''
| ''[[21/20]]''
|''2.607''
| ''2.607''
|''8.984''
| ''8.984''
|-
|-
|''[[21/4]]''
| ''[[21/4]]''
|''2.683''
| ''2.683''
|''9.242''
| ''9.242''
|-
|-
|''[[29/28]]''
| ''[[29/28]]''
|''2.702''
| ''2.702''
|''9.308''
| ''9.308''
|-
|-
|''[[32/19]]''
| ''[[32/19]]''
|''2.716''
| ''2.716''
|
| ''9.356''
''9.356''
|-
|-
|[[19/3]]
| [[19/3]]
|2.821
| 2.821
|9.719
| 9.719
|-
|-
|[[19/15]]
| [[19/15]]
|2.896
| 2.896
|9.977
| 9.977
|-
|-
|''[[27/20]]''
| ''[[27/20]]''
|''2.897''
| ''2.897''
|''9.980''
| ''9.980''
|-
|-
|''[[27/4]]''
| ''[[27/4]]''
|''2.972''
| ''2.972''
|''10.238''
| ''10.238''
|-
|-
|''[[32/31]]''
| ''[[32/31]]''
|''3.085''
| ''3.085''
|''10.630''
| ''10.630''
|-
|-
|''[[15/14]]''
| ''[[15/14]]''
|''3.343''
| ''3.343''
|''11.519''
| ''11.519''
|-
|-
|''[[14/3]]''
| ''[[14/3]]''
|''3.418''
| ''3.418''
|''11.777''
| ''11.777''
|-
|-
|[[13/6]]
| [[13/6]]
|3.428
| 3.428
|11.811
| 11.811
|-
|-
|[[17/6]]
| [[17/6]]
|3.458
| 3.458
| 11.913
| 11.913
|-
|-
|[[30/13]]
| [[30/13]]
|3.503
| 3.503
| 12.069
| 12.069
|-
|-
|[[30/17]]
| [[30/17]]
|3.533
| 3.533
|12.171
| 12.171
|-
|-
|[[25/6]]
| [[25/6]]
|3.557
| 3.557
|12.256
| 12.256
|-
|-
|''[[32/21]]''
| ''[[32/21]]''
|''3.597''
| ''3.597''
|
| ''12.393''
''12.393''
|-
|-
|[[6/5]]
| [[6/5]]
|3.632
| 3.632
|12.515
| 12.515
|-
|-
|[[6/1]]
| [[6/1]]
|3.708
| 3.708
|12.774
| 12.774
|-
|-
|''[[32/29]]''
| ''[[32/29]]''
|''3.726''
| ''3.726''
|''12.839''
| ''12.839''
|-
|-
|''[[28/19]]''
| ''[[28/19]]''
|''3.741''
| ''3.741''
|''12.887''
| ''12.887''
|-
|-
|[[30/1]]
| [[30/1]]
|3.783
| 3.783
|13.032
| 13.032
|-
|-
|''[[32/27]]''
| ''[[32/27]]''
|''3.886''
| ''3.886''
|''13.389''
| ''13.389''
|-
|-
|''[[31/4]]''
| ''[[31/4]]''
|''4.000''
| ''4.000''
|''13.781''
| ''13.781''
|-
|-
|''[[31/20]]''
| ''[[31/20]]''
|''4.075''
| ''4.075''
|''14.039''
| ''14.039''
|-
|-
|[[29/13]]
| [[29/13]]
|4.145
| 4.145
| 14.280
| 14.280
|-
|-
|[[29/17]]
| [[29/17]]
|4.174
| 4.174
| 14.382
| 14.382
|-
|-
|[[29/25]]
| [[29/25]]
|4.274
| 4.274
|14.726
| 14.726
|-
|-
|[[23/6]]
| [[23/6]]
|4.329
| 4.329
|14.914
| 14.914
|-
|-
|[[12/7]]
| [[12/7]]
|4.333
| 4.333
|14.928
| 14.928
|-
|-
|[[29/5]]
| [[29/5]]
|4.349
| 4.349
|14.985
| 14.985
|-
|-
|''[[16/15]]''
| ''[[16/15]]''
|''4.368''
| ''4.368''
|''15.050''
| ''15.050''
|-
|-
|[[30/23]]
| [[30/23]]
|4.404
| 4.404
|15.172
| 15.172
|-
|-
|'''[[29/1]]'''
| '''[[29/1]]'''
|'''4.424'''
| '''4.424'''
|'''15.243'''
| '''15.243'''
|-
|-
|''[[16/3]]''
| ''[[16/3]]''
|''4.443''
| ''4.443''
|''15.309''
| ''15.309''
|-
|-
|[[11/6]]
| [[11/6]]
|4.467
| 4.467
|15.391
| 15.391
|-
|-
|''[[22/15]]''
| ''[[22/15]]''
|''4.523''
| ''4.523''
|''15.583''
| ''15.583''
|-
|-
|[[30/11]]
| [[30/11]]
|4.542
| 4.542
|15.649
| 15.649
|-
|-
|''[[20/3]]''
| ''[[20/3]]''
|''4.547''
| ''4.547''
|''15.666''
| ''15.666''
|-
|-
|''[[22/3]]''
| ''[[22/3]]''
|''4.598''
| ''4.598''
|''15.842''
| ''15.842''
|-
|-
|
| ''[[4/3]]''
''[[4/3]]''
| ''4.622''
|''4.622''
| ''15.924''
|''15.924''
|-
|-
|''[[29/4]]''
| ''[[29/4]]''
|''4.641''
| ''4.641''
|''15.990''
| ''15.990''
|-
|-
|''[[15/4]]''
| ''[[15/4]]''
|''4.697''
| ''4.697''
|''16.183''
| ''16.183''
|-
|-
|''[[29/20]]''
| ''[[29/20]]''
|''4.716''
| ''4.716''
|''16.248''
| ''16.248''
|-
|-
|[[31/13]]
| [[31/13]]
|4.786
| 4.786
| 16.489
| 16.489
|-
|-
|[[31/17]]
| [[31/17]]
|4.816
| 4.816
|16.591
| 16.591
|-
|-
|''[[28/27]]''
| ''[[28/27]]''
|''4.911''
| ''4.911''
|''16.920''
| ''16.920''
|-
|-
|[[31/25]]
| [[31/25]]
|4.915
| 4.915
|16.935
| 16.935
|-
|-
|[[31/5]]
| [[31/5]]
|4.990
| 4.990
| 17.194
| 17.194
|-
|-
|[[29/23]]
| [[29/23]]
|5.046
| 5.046
|17.383
| 17.383
|-
|-
|'''[[31/1]]'''
| '''[[31/1]]'''
|'''5.066'''
| '''5.066'''
|'''17.452'''
| '''17.452'''
|-
|-
|''[[27/14]]''
| ''[[27/14]]''
|''5.069''
| ''5.069''
|''17.463''
| ''17.463''
|-
|-
|[[29/11]]
| [[29/11]]
|5.184
| 5.184
|17.860
| 17.860
|-
|-
|''[[15/2]]''
| ''[[15/2]]''
|''5.283''
| ''5.283''
|''18.201''
| ''18.201''
|-
|-
|''[[29/8]]''
| ''[[29/8]]''
|''5.339''
| ''5.339''
|''18.394''
| ''18.394''
|-
|-
|
| ''[[3/2]]''
''[[3/2]]''
| ''5.358''
|''5.358''
| ''18.459''
|''18.459''
|-
|-
|''[[10/3]]''
| ''[[10/3]]''
|''5.433''
| ''5.433''
|''18.718''
| ''18.718''
|-
|-
|[[12/11]]
| [[12/11]]
|5.513
| 5.513
|18.993
| 18.993
|-
|-
|''[[32/3]]''
| ''[[32/3]]''
|''5.536''
| ''5.536''
|''19.075''
| ''19.075''
|-
|-
|''[[26/15]]''
| ''[[26/15]]''
|''5.562''
| ''5.562''
|''19.164''
| ''19.164''
|-
|-
|''[[32/15]]''
| ''[[32/15]]''
|''5.612''
| ''5.612''
|''19.334''
| ''19.334''
|-
|-
|''[[26/3]]''
| ''[[26/3]]''
|''5.637''
| ''5.637''
|
| ''19.422''
''19.422''
|-
|-
|[[7/6]]
| [[7/6]]
|5.647
| 5.647
| 19.456
| 19.456
|-
|-
|[[23/12]]
| [[23/12]]
|5.651
| 5.651
| 19.470
| 19.470
|-
|-
|[[31/23]]
| [[31/23]]
|5.687
| 5.687
|19.592
| 19.592
|-
|-
|[[30/7]]
| [[30/7]]
|5.722
| 5.722
| 19.714
| 19.714
|-
|-
|[[31/19]]
| [[31/19]]
|5.801
| 5.801
| 19.986
| 19.986
|-
|-
|[[31/11]]
| [[31/11]]
|5.825
| 5.825
|20.069
| 20.069
|-
|-
|''[[31/8]]''
| ''[[31/8]]''
|''5.980''
| ''5.980''
|''20.603''
| ''20.603''
|-
|-
|''[[29/9]]''
| ''[[29/9]]''
|''6.075''
| ''6.075''
|''20.929''
| ''20.929''
|-
|-
|''[[27/16]]''
| ''[[27/16]]''
|''6.094''
| ''6.094''
|''20.994''
| ''20.994''
|-
|-
|''[[19/14]]''
| ''[[19/14]]''
|''6.239''
| ''6.239''
|''21.496''
| ''21.496''
|-
|-
|''[[27/22]]''
| ''[[27/22]]''
|''6.248''
| ''6.248''
|''21.528''
| ''21.528''
|-
|-
|[[12/1]]
| [[12/1]]
|6.272
| 6.272
|21.610
| 21.610
|-
|-
|[[12/5]]
| [[12/5]]
|6.347
| 6.347
|21.869
| 21.869
|-
|-
|[[29/7]]
| [[29/7]]
|6.364
| 6.364
|21.925
| 21.925
|-
|-
|''[[21/16]]''
| ''[[21/16]]''
|''6.383''
| ''6.383''
|''21.991''
| ''21.991''
|-
|-
|[[25/12]]
| [[25/12]]
|6.422
| 6.422
| 22.127
| 22.127
|-
|-
|[[29/19]]
| [[29/19]]
|6.442
| 6.442
| 22.195
| 22.195
|-
|-
|[[17/12]]
| [[17/12]]
|6.522
| 6.522
| 22.471
| 22.471
|-
|-
|[[19/18]]
| [[19/18]]
|6.528
| 6.528
|22.492
| 22.492
|-
|-
|''[[22/21]]''
| ''[[22/21]]''
|''6.538''
| ''6.538''
|''22.524''
| ''22.524''
|-
|-
|[[13/12]]
| [[13/12]]
|6.552
| 6.552
|22.573
| 22.573
|-
|-
|''[[28/3]]''
| ''[[28/3]]''
|''6.561''
| ''6.561''
|''22.606''
| ''22.606''
|-
|-
|''[[28/15]]''
| ''[[28/15]]''
|''6.637''
| ''6.637''
|''22.865''
| ''22.865''
|-
|-
|[[31/21]]
| [[31/21]]
|6.682
| 6.682
|23.023
| 23.023
|-
|-
|''[[31/9]]''
| ''[[31/9]]''
|''6.716''
| ''6.716''
|''23.138''
| ''23.138''
|-
|-
|''[[28/13]]''
| ''[[28/13]]''
|''6.846''
| ''6.846''
|''23.588''
| ''23.588''
|-
|-
|''[[28/17]]''
| ''[[28/17]]''
|''6.876''
| ''6.876''
|''23.690''
| ''23.690''
|-
|-
|''[[31/27]]''
| ''[[31/27]]''
|''6.972''
| ''6.972''
|''24.019''
| ''24.019''
|-
|-
|''[[28/25]]''
| ''[[28/25]]''
|''6.976''
| ''6.976''
|''24.034''
| ''24.034''
|-
|-
|[[31/7]]
| [[31/7]]
|7.005
| 7.005
|24.134
| 24.134
|-
|-
|''[[27/2]]''
| ''[[27/2]]''
|''7.008''
| ''7.008''
|''24.145''
| ''24.145''
|-
|-
|''[[28/5]]''
| ''[[28/5]]''
|''7.051''
| ''7.051''
|''24.292''
| ''24.292''
|-
|-
|''[[27/10]]''
| ''[[27/10]]''
|''7.083''
| ''7.083''
|''24.404''
| ''24.404''
|-
|-
|[[30/19]]
| [[30/19]]
|7.084
| 7.084
|24.406
| 24.406
|-
|-
|''[[28/1]]''
| ''[[28/1]]''
|''7.126''
| ''7.126''
|''24.551''
| ''24.551''
|-
|-
|[[19/6]]
| [[19/6]]
|7.159
| 7.159
|24.665
| 24.665
|-
|-
|''[[19/16]]''
| ''[[19/16]]''
|''7.264''
| ''7.264''
|''25.027''
| ''25.027''
|-
|-
|''[[27/26]]''
| ''[[27/26]]''
|''7.288''
| ''7.288''
|''25.108''
| ''25.108''
|-
|-
|''[[21/2]]''
| ''[[21/2]]''
|''7.297''
| ''7.297''
|''25.141''
| ''25.141''
|-
|-
|[[29/21]]
| [[29/21]]
|7.324
| 7.324
|25.232
| 25.232
|-
|-
|''[[21/10]]''
| ''[[21/10]]''
|''7.372''
| ''7.372''
|''25.400''
| ''25.400''
|-
|-
|''[[22/19]]''
| ''[[22/19]]''
|''7.419''
| ''7.419''
|''25.561''
| ''25.561''
|-
|-
|''[[26/21]]''
| ''[[26/21]]''
|''7.577''
| ''7.577''
|''26.104''
| ''26.104''
|-
|-
|''[[29/27]]''
| ''[[29/27]]''
|''7.613''
| ''7.613''
|''26.228''
| ''26.228''
|-
|-
|''[[31/24]]''
| ''[[31/24]]''
|''7.707''
| ''7.707''
|''26.554''
| ''26.554''
|-
|-
|''[[28/23]]''
| ''[[28/23]]''
|''7.747''
| ''7.747''
|''26.691''
| ''26.691''
|-
|-
|[[26/7]]
| [[26/7]]
|7.761
| 7.761
|26.739
| 26.739
|-
|-
|''[[32/13]]''
| ''[[32/13]]''
|''7.871''
| ''7.871''
|''27.119''
| ''27.119''
|-
|-
|''[[28/11]]''
| ''[[28/11]]''
|''7.886''
| ''7.886''
|''27.168''
| ''27.168''
|-
|-
|''[[32/17]]''
| ''[[32/17]]''
|''7.901''
| ''7.901''
|''27.221''
| ''27.221''
|-
|-
|[[10/7]]
| [[10/7]]
|7.965
| 7.965
|27.443
| 27.443
|-
|-
|''[[32/25]]''
| ''[[32/25]]''
|''8.001''
| ''8.001''
|
| ''27.565''
''27.565''
|-
|-
|[[7/2]]
| [[7/2]]
|8.040
| 8.040
|27.702
| 27.702
|-
|-
|''[[26/9]]''
| ''[[26/9]]''
|''8.050''
| ''8.050''
|''27.735''
| ''27.735''
|-
|-
|''[[32/5]]''
| ''[[32/5]]''
|''8.076''
| ''8.076''
|''27.824''
| ''27.824''
|-
|-
|''[[32/1]]''
| ''[[32/1]]''
|''8.151''
| ''8.151''
|''28.082''
| ''28.082''
|-
|-
|''[[19/2]]''
| ''[[19/2]]''
|''8.179''
| ''8.179''
|''28.178''
| ''28.178''
|-
|-
|''[[19/10]]''
| ''[[19/10]]''
|''8.254''
| ''8.254''
|''28.437''
| ''28.437''
|-
|-
|''[[10/9]]''
| ''[[10/9]]''
|''8.254''
| ''8.254''
|''28.439''
| ''28.439''
|-
|-
|
| ''[[9/2]]''
''[[9/2]]''
| ''8.329''
|''8.329''
| ''28.698''
|''28.698''
|-
|-
|''[[29/24]]''
| ''[[29/24]]''
|''8.348''
| ''8.348''
|''28.763''
| ''28.763''
|-
|-
|''[[26/19]]''
| ''[[26/19]]''
|''8.458''
| ''8.458''
|''29.141''
| ''29.141''
|-
|-
|[[31/3]]
| [[31/3]]
|8.622
| 8.622
| 29.705
| 29.705
|-
|-
|[[31/15]]
| [[31/15]]
|8.697
| 8.697
|29.964
| 29.964
|-
|-
|''[[32/23]]''
| ''[[32/23]]''
|''8.772''
| ''8.772''
|''30.222''
| ''30.222''
|-
|-
|''[[28/9]]''
| ''[[28/9]]''
|''8.776''
| ''8.776''
|''30.237''
| ''30.237''
|-
|-
|''[[13/4]]''
| ''[[13/4]]''
|''8.786''
| ''8.786''
|''30.270''
| ''30.270''
|-
|-
|[[22/7]]
| [[22/7]]
|8.800
| 8.800
|30.319
| 30.319
|-
|-
|''[[17/4]]''
| ''[[17/4]]''
|''8.815''
| ''8.815''
|''30.372''
| ''30.372''
|-
|-
|''[[20/13]]''
| ''[[20/13]]''
|''8.861''
| ''8.861''
|''30.529''
| ''30.529''
|-
|-
|''[[20/17]]''
| ''[[20/17]]''
|''8.891''
| ''8.891''
|''30.631''
| ''30.631''
|-
|-
|''[[32/11]]''
| ''[[32/11]]''
|''8.910''
| ''8.910''
|''30.699''
| ''30.699''
|-
|-
|''[[25/4]]''
| ''[[25/4]]''
|''8.915''
| ''8.915''
|''30.716''
| ''30.716''
|-
|-
|[[26/11]]
| [[26/11]]
|8.941
| 8.941
|30.803
| 30.803
|-
|-
|''[[16/7]]''
| ''[[16/7]]''
|''8.955''
| ''8.955''
|''30.852''
| ''30.852''
|-
|-
|
| ''[[5/4]]''
''[[5/4]]''
| ''8.990''
|''8.990''
| ''30.974''
|''30.974''
|-
|-
|
| ''[[4/1]]''
''[[4/1]]''
| ''9.065''
|''9.065''
| ''31.233''
|''31.233''
|-
|-
|[[26/23]]
| [[26/23]]
|9.079
| 9.079
|31.281
| 31.281
|-
|-
|''[[22/9]]''
| ''[[22/9]]''
|''9.089''
| ''9.089''
|''31.315''
| ''31.315''
|-
|-
|''[[20/1]]''
| ''[[20/1]]''
|''9.140''
| ''9.140''
|''31.492''
| ''31.492''
|-
|-
|[[11/10]]
| [[11/10]]
|9.145
| 9.145
|31.508
| 31.508
|-
|-
|[[11/2]]
| [[11/2]]
|9.220
| 9.220
|31.766
| 31.766
|-
|-
|''[[16/9]]''
| ''[[16/9]]''
|''9.244''
| ''9.244''
|''31.848''
| ''31.848''
|-
|-
|[[29/3]]
| [[29/3]]
|9.263
| 9.263
| 31.914
| 31.914
|-
|-
|[[23/10]]
| [[23/10]]
|9.284
| 9.284
| 31.985
| 31.985
|-
|-
|[[29/15]]
| [[29/15]]
|9.338
| 9.338
|32.173
| 32.173
|-
|-
|[[23/2]]
| [[23/2]]
|9.359
| 9.359
|32.243
| 32.243
|-
|-
|''[[23/4]]''
| ''[[23/4]]''
|''9.686''
| ''9.686''
|''33.373''
| ''33.373''
|-
|-
|''[[18/7]]''
| ''[[18/7]]''
|''9.691''
| ''9.691''
|''33.387''
| ''33.387''
|-
|-
|[[26/1]]
| [[26/1]]
|9.700
| 9.700
|33.421
| 33.421
|-
|-
|''[[23/20]]''
| ''[[23/20]]''
|''9.762''
| ''9.762''
|''33.632''
| ''33.632''
|-
|-
|[[26/5]]
| [[26/5]]
|9.775
| 9.775
|33.679
| 33.679
|-
|-
|''[[32/9]]''
| ''[[32/9]]''
|''9.801''
| ''9.801''
|''33.768''
| ''33.768''
|-
|-
|''[[11/4]]''
| ''[[11/4]]''
|''9.825''
| ''9.825''
|''33.850''
| ''33.850''
|-
|-
|[[26/25]]
| [[26/25]]
|9.850
| 9.850
|33.938
| 33.938
|-
|-
|''[[20/11]]''
| ''[[20/11]]''
|''9.900''
| ''9.900''
|''34.109''
| ''34.109''
|-
|-
|[[10/1]]
| [[10/1]]
|9.905
| 9.905
| 34.125
| 34.125
|-
|-
|[[26/17]]
| [[26/17]]
|9.950
| 9.950
|34.282
| 34.282
|-
|-
|'''[[2/1]]'''
| '''[[2/1]]'''
|'''9.980'''
| '''9.980'''
|'''34.384'''
| '''34.384'''
|-
|-
|[[5/2]]
| [[5/2]]
|10.055
| 10.055
|34.642
| 34.642
|-
|-
|''[[32/7]]''
| ''[[32/7]]''
|
| ''10.090''
''10.090''
| ''34.764''
|''34.764''
|-
|-
|[[23/22]]
| [[23/22]]
|10.118
| 10.118
|34.861
| 34.861
|-
|-
|[[25/2]]
| [[25/2]]
|10.130
| 10.130
|34.901
| 34.901
|-
|-
|''[[16/11]]''
| ''[[16/11]]''
|
| ''10.135''
''10.135''
| ''34.917''
|''34.917''
|-
|-
|[[17/10]]
| [[17/10]]
|10.155
| 10.155
| 34.986
| 34.986
|-
|-
|[[13/10]]
| [[13/10]]
|10.184
| 10.184
|35.088
| 35.088
|-
|-
|[[17/2]]
| [[17/2]]
|10.230
| 10.230
|35.244
| 35.244
|-
|-
|[[13/2]]
| [[13/2]]
|10.259
| 10.259
|35.346
| 35.346
|-
|-
|''[[14/9]]''
| ''[[14/9]]''
|
| ''10.269''
''10.269''
| ''35.380''
|''35.380''
|-
|-
|''[[23/16]]''
| ''[[23/16]]''
|
| ''10.273''
''10.273''
| ''35.394''
|''35.394''
|-
|-
|[[19/13]]
| [[19/13]]
|10.587
| 10.587
| 36.475
| 36.475
|-
|-
|[[19/17]]
| [[19/17]]
|10.617
| 10.617
| 36.577
| 36.577
|-
|-
|[[29/12]]
| [[29/12]]
|10.697
| 10.697
|36.853
| 36.853
|-
|-
|''[[9/4]]''
| ''[[9/4]]''
|
| ''10.716''
''10.716''
| ''36.919''
|''36.919''
|-
|-
|[[25/19]]
| [[25/19]]
|10.716
| 10.716
|36.921
| 36.921
|-
|-
|[[22/1]]
| [[22/1]]
|10.739
| 10.739
|37.001
| 37.001
|-
|-
|''[[20/9]]''
| ''[[20/9]]''
|
| ''10.791''
''10.791''
| ''37.177''
|''37.177''
|-
|-
|[[19/5]]
| [[19/5]]
|10.791
| 10.791
|37.180
| 37.180
|-
|-
|[[22/5]]
| [[22/5]]
|10.814
| 10.814
|37.259
| 37.259
|-
|-
|'''[[19/1]]'''
| '''[[19/1]]'''
|'''10.866'''
| '''10.866'''
|'''37.438'''
| '''37.438'''
|-
|-
|''[[18/11]]''
| ''[[18/11]]''
|
| ''10.870''
''10.870''
| ''37.452''
|''37.452''
|-
|-
|[[25/22]]
| [[25/22]]
|10.890
| 10.890
|37.518
| 37.518
|-
|-
|''[[16/1]]''
| ''[[16/1]]''
|
| ''10.894''
''10.894''
| ''37.534''
|''37.534''
|-
|-
|''[[16/5]]''
| ''[[16/5]]''
|
| ''10.969''
''10.969''
| ''37.793''
|''37.793''
|-
|-
|[[22/17]]
| [[22/17]]
|10.989
| 10.989
|37.862
| 37.862
|-
|-
|''[[7/4]]''
| ''[[7/4]]''
|
| ''11.005''
''11.005''
| ''37.915''
|''37.915''
|-
|-
|''[[23/18]]''
| ''[[23/18]]''
|
| ''11.009''
''11.009''
| ''37.929''
|''37.929''
|-
|-
|[[22/13]]
| [[22/13]]
|11.019
| 11.019
|37.964
| 37.964
|-
|-
|''[[25/16]]''
| ''[[25/16]]''
|
| ''11.044''
''11.044''
| ''38.052''
|''38.052''
|-
|-
|''[[20/7]]''
| ''[[20/7]]''
|
| ''11.080''
''11.080''
| ''38.174''
|''38.174''
|-
|-
|''[[17/16]]''
| ''[[17/16]]''
|
| ''11.144''
''11.144''
| ''38.395''
|''38.395''
|-
|-
|[[14/11]]
| [[14/11]]
|11.160
| 11.160
|38.448
| 38.448
|-
|-
|''[[16/13]]''
| ''[[16/13]]''
|
| ''11.174''
''11.174''
| ''38.497''
|''38.497''
|-
|-
|[[23/14]]
| [[23/14]]
|11.298
| 11.298
| 38.925
| 38.925
|-
|-
|[[31/12]]
| [[31/12]]
|11.338
| 11.338
| 39.062
| 39.062
|-
|-
|[[21/13]]
| [[21/13]]
|11.468
| 11.468
| 39.512
| 39.512
|-
|-
|[[23/19]]
| [[23/19]]
|11.488
| 11.488
| 39.579
| 39.579
|-
|-
|[[21/17]]
| [[21/17]]
|11.498
| 11.498
| 39.614
| 39.614
|-
|-
|[[25/21]]
| [[25/21]]
|11.598
| 11.598
| 39.958
| 39.958
|-
|-
|[[19/11]]
| [[19/11]]
|11.626
| 11.626
|40.056
| 40.056
|-
|-
|''[[18/1]]''
| ''[[18/1]]''
|
| ''11.630''
''11.630''
| ''40.069''
|''40.069''
|-
|-
|[[21/5]]
| [[21/5]]
|11.673
| 11.673
|40.216
| 40.216
|-
|-
|''[[18/5]]''
| ''[[18/5]]''
|
| ''11.705''
''11.705''
| ''40.328''
|''40.328''
|-
|-
|[[21/1]]
| [[21/1]]
|11.748
| 11.748
|40.475
| 40.475
|-
|-
|''[[27/13]]''
| ''[[27/13]]''
|
| ''11.758''
''11.758''
| ''40.508''
|''40.508''
|-
|-
|''[[25/18]]''
| ''[[25/18]]''
|
| ''11.780''
''11.780''
| ''40.587''
|''40.587''
|-
|-
|''[[19/8]]''
| ''[[19/8]]''
|
| ''11.781''
''11.781''
| ''40.589''
|''40.589''
|-
|-
|''[[27/17]]''
| ''[[27/17]]''
|
| ''11.787''
''11.787''
| ''40.610''
|''40.610''
|-
|-
|''[[18/17]]''
| ''[[18/17]]''
|
| ''11.880''
''11.880''
| ''40.930''
|''40.930''
|-
|-
|''[[19/12]]''
| ''[[19/12]]''
|
| ''11.886''
''11.886''
| ''40.952''
|''40.952''
|-
|-
|''[[27/25]]''
| ''[[27/25]]''
|
| ''11.887''
''11.887''
| ''40.954''
|''40.954''
|-
|-
|''[[18/13]]''
| ''[[18/13]]''
|
| ''11.910''
''11.910''
| ''41.032''
|''41.032''
|-
|-
|[[14/1]]
| [[14/1]]
|11.919
| 11.919
|41.066
| 41.066
|-
|-
|''[[27/5]]''
| ''[[27/5]]''
|
| ''11.962''
''11.962''
| ''41.213''
|''41.213''
|-
|-
|[[14/5]]
| [[14/5]]
|11.994
| 11.994
|41.324
| 41.324
|-
|-
|''[[27/1]]''
| ''[[27/1]]''
|
| ''12.037''
''12.037''
| ''41.471''
|''41.471''
|-
|-
|''[[31/14]]''
| ''[[31/14]]''
|
| ''12.040''
''12.040''
| ''41.482''
|''41.482''
|-
|-
|[[25/14]]
| [[25/14]]
|12.069
| 12.069
| 41.583
| 41.583
|-
|-
|[[17/14]]
| [[17/14]]
|12.169
| 12.169
| 41.926
| 41.926
|-
|-
|[[14/13]]
| [[14/13]]
|12.199
| 12.199
| 42.028
| 42.028
|-
|-
|[[31/18]]
| [[31/18]]
|12.329
| 12.329
| 42.478
| 42.478
|-
|-
|[[23/21]]
| [[23/21]]
|12.369
| 12.369
|42.615
| 42.615
|-
|-
|''[[24/13]]''
| ''[[24/13]]''
|
| ''12.493''
''12.493''
| ''43.044''
|''43.044''
|-
|-
|[[21/11]]
| [[21/11]]
|12.507
| 12.507
|43.092
| 43.092
|-
|-
|''[[19/9]]''
| ''[[19/9]]''
|
| ''12.517''
''12.517''
| ''43.124''
|''43.124''
|-
|-
|''[[24/17]]''
| ''[[24/17]]''
|
| ''12.523''
''12.523''
| ''43.146''
|''43.146''
|-
|-
|''[[25/24]]''
| ''[[25/24]]''
|
| ''12.623''
''12.623''
| ''43.489''
|''43.489''
|-
|-
|''[[27/23]]''
| ''[[27/23]]''
|
| ''12.658''
''12.658''
| ''43.611''
|''43.611''
|-
|-
|''[[21/8]]''
| ''[[21/8]]''
|
| ''12.662''
''12.662''
| ''43.626''
|''43.626''
|-
|-
|''[[29/14]]''
| ''[[29/14]]''
|
| ''12.681''
''12.681''
| ''43.691''
|''43.691''
|-
|-
|''[[24/5]]''
| ''[[24/5]]''
|
| ''12.698''
''12.698''
| ''43.748''
|''43.748''
|-
|-
|''[[24/1]]''
| ''[[24/1]]''
|
| ''12.773''
''12.773''
| ''44.006''
|''44.006''
|-
|-
|''[[27/11]]''
| ''[[27/11]]''
|
| ''12.797''
''12.797''
| ''44.089''
|''44.089''
|-
|-
|[[19/7]]
| [[19/7]]
|12.806
| 12.806
|44.120
| 44.120
|-
|-
|''[[27/8]]''
| ''[[27/8]]''
|
| ''12.951''
''12.951''
| ''44.622''
|''44.622''
|-
|-
|[[29/18]]
| [[29/18]]
|12.970
| 12.970
|44.687
| 44.687
|-
|-
|''[[31/16]]''
| ''[[31/16]]''
|
| ''13.065''
''13.065''
| ''45.014''
|''45.014''
|-
|-
|''[[31/22]]''
| ''[[31/22]]''
|
| ''13.220''
''13.220''
| ''45.547''
|''45.547''
|-
|-
|''[[15/7]]''
| ''[[15/7]]''
|
| ''13.323''
''13.323''
| ''45.902''
|''45.902''
|-
|-
|''[[24/23]]''
| ''[[24/23]]''
|
| ''13.394''
''13.394''
| ''46.147''
|''46.147''
|-
|-
|''[[7/3]]''
| ''[[7/3]]''
|
| ''13.398''
''13.398''
| ''46.161''
|''46.161''
|-
|-
|[[13/3]]
| [[13/3]]
|13.408
| 13.408
|46.194
| 46.194
|-
|-
|[[17/3]]
| [[17/3]]
|13.437
| 13.437
| 46.296
| 46.296
|-
|-
|[[15/13]]
| [[15/13]]
|13.483
| 13.483
| 46.453
| 46.453
|-
|-
|[[17/15]]
| [[17/15]]
|13.513
| 13.513
|46.555
| 46.555
|-
|-
|''[[24/11]]''
| ''[[24/11]]''
|
| ''13.532''
''13.532''
| ''46.624''
|''46.624''
|-
|-
|[[25/3]]
| [[25/3]]
|13.537
| 13.537
|46.640
| 46.640
|-
|-
|[[5/3]]
| [[5/3]]
|13.612
| 13.612
|46.898
| 46.898
|-
|-
|'''[[3/1]]'''
| '''[[3/1]]'''
|'''13.687'''
| '''13.687'''
|'''47.157'''
| '''47.157'''
|-
|-
|''[[29/16]]''
| ''[[29/16]]''
|
| ''13.706''
''13.706''
| ''47.223''
|''47.223''
|-
|-
|[[15/1]]
| [[15/1]]
|13.762
| 13.762
|47.416
| 47.416
|-
|-
|''[[29/22]]''
| ''[[29/22]]''
|
| ''13.861''
''13.861''
| ''47.756''
|''47.756''
|-
|-
|''[[27/7]]''
| ''[[27/7]]''
|
| ''13.976''
''13.976''
| ''48.153''
|''48.153''
|-
|-
|''[[31/2]]''
| ''[[31/2]]''
|
| ''13.980''
''13.980''
| ''48.164''
|''48.164''
|-
|-
|''[[31/10]]''
| ''[[31/10]]''
|
| ''14.055''
''14.055''
| ''48.423''
|''48.423''
|-
|-
|[[29/26]]
| [[29/26]]
|14.125
| 14.125
|48.664
| 48.664
|-
|-
|''[[31/26]]''
| ''[[31/26]]''
|
| ''14.259''
''14.259''
| ''49.127''
|''49.127''
|-
|-
|[[23/3]]
| [[23/3]]
|14.308
| 14.308
|49.297
| 49.297
|-
|-
|[[24/7]]
| [[24/7]]
|14.313
| 14.313
| 49.312
| 49.312
|-
|-
|[[29/10]]
| [[29/10]]
|14.329
| 14.329
|49.368
| 49.368
|-
|-
|''[[15/8]]''
| ''[[15/8]]''
|
| ''14.348''
''14.348''
| ''49.434''
|''49.434''
|-
|-
|[[23/15]]
| [[23/15]]
|14.384
| 14.384
|49.556
| 49.556
|-
|-
|[[29/2]]
| [[29/2]]
|14.404
| 14.404
|49.627
| 49.627
|-
|-
|''[[8/3]]''
| ''[[8/3]]''
|
| ''14.423''
''14.423''
| ''49.692''
|''49.692''
|-
|-
|[[11/3]]
| [[11/3]]
|14.447
| 14.447
|49.774
| 49.774
|-
|-
|''[[15/11]]''
| ''[[15/11]]''
|
| ''14.503''
''14.503''
| ''49.967''
|''49.967''
|}
|}


[[Category:Zeta peak indexes]]
[[Category:Zeta peak indexes]]

Revision as of 00:23, 11 August 2024

186 zeta peak index (abbreviated 186zpi), is the equal-step tuning system obtained from the 186st peak of the Riemann zeta function.

Tuning Strength Closest EDO Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap EDO Octave (cents) Consistent Distinct
186zpi 41.3438354846780 29.0248832971658 1.876590 0.241233 11.567493 41edo 1190.02021518380 2 2

Theory

Record on the Riemann zeta function with primes 2 and 3 removed

186zpi sets a height record on the Riemann zeta function with primes 2 and 3 removed. The previous record is 125zpi and the next one is 565zpi. It is important to highlight that the optimal equal tunings obtained by excluding the prime numbers 2 and 3 from the Riemann zeta function differs very slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.

Unmodified Riemann zeta function Riemann zeta function with primes 2 and 3 removed
Tuning Strength Closest EDO Tuning Strength Closest EDO
ZPI Steps per octave Step size (cents) Height EDO Octave (cents) Steps per octave Step size (cents) Height EDO Octave (cents)
125zpi 30.6006474885974 39.2148564976330 1.468164 31edo 1215.66055142662 30.5974484926723 39.2189564527704 3.769318 31edo 1215.78765003588
186zpi 41.3438354846780 29.0248832971658 1.876590 41edo 1190.02021518380 41.3477989230936 29.0221010852836 4.469823 41edo 1189.90614449663
565zpi 98.6209462564991 12.1678005084130 2.305330 99edo 1204.61225033289 98.6257548378926 12.1672072570942 4.883729 99edo 1204.55351845233

Harmonic series

As a non-octave, non-tritave scale, 186zpi features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.

Approximation of harmonics in 186zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -10.0 +13.7 +9.1 +0.1 +3.7 -1.9 -0.9 -1.7 -9.9 -0.8 -6.3 +0.3 -11.9 +13.8 -10.9
Relative (%) -34.4 +47.2 +31.2 +0.3 +12.8 -6.7 -3.2 -5.7 -34.1 -2.6 -21.6 +1.0 -41.1 +47.4 -37.5
Step 41 66 83 96 107 116 124 131 137 143 148 153 157 162 165
Approximation of harmonics in 186zpi
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Error Absolute (¢) +0.2 -11.6 +10.9 +9.1 +11.7 -10.7 -0.6 +12.8 +0.2 -9.7 +12.0 +7.1 +4.4 +3.8 +5.1 +8.2
Relative (%) +0.9 -40.1 +37.4 +31.5 +40.5 -37.0 -2.1 +44.0 +0.5 -33.4 +41.5 +24.6 +15.2 +13.0 +17.5 +28.1
Step 169 172 176 179 182 184 187 190 192 194 197 199 201 203 205 207

Approximation of EDONOIs

Based on harmonics with less than 1 cent of error, 186zpi can be approximated by 96ed5, 124ed8 (or every 3 steps of 124edo), 143ed11, 153ed13, 169ed17, 187ed23, and 192ed25.

Intervals and notation

There are several ways to approach notation. The simplest method involves using the notations from 41edo. However, this method does not preserve octave compression when rendered by notation software. To address this issue, consider using the ups and downs notation from 124edo at every 3-degree step (i.e., the edonoi 124ed8).

It is important to note that 124edo provides two possible fifths (3/2). The closest one, from the val <124 197] (i.e. the patent val), is the fifth mapped to 73 steps of 124edo with a relative error of +46.465%. The second closest, from the val <124 196] (i.e. the val 124b), is mapped to 72 steps of 124edo with a relative error of -53.535%. This second fifth, which appears in 124ed8, also corresponds to the fifth of 31edo. Therefore, we choose to use the ups and downs notation of the 124b temperament, denoted as <124 196].

JI ratios are comprised of 32-integer limit ratios,
and are stylized as follows to indicate their accuracy:
  • Bold Underlined: relative error < 8.333 %
  • Bold: relative error < 16.667 %
  • Normal: relative error < 25 %
  • Small: relative error < 33.333 %
  • Small Small: relative error < 41.667 %
  • Small Small Small: relative error < 50 %
124b at every 3 steps

Whole tone = 20 steps
Limma = 12 steps
Apotome = 8 steps
Degree Cents Ratios Ups and Downs Notation Step
0 0.000 P1 0
1 29.025 ^^^1 3
2 58.050 32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24 vvA1, ^^d2 6
3 87.075 24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17 vvvm2 9
4 116.100 17/16, 16/15, 31/29, 15/14, 29/27, 14/13 m2 12
5 145.124 27/25, 13/12, 25/23, 12/11, 23/21 ^^^m2 15
6 174.149 11/10, 32/29, 21/19, 31/28, 10/9 vvM2 18
7 203.174 29/26, 19/17, 28/25, 9/8, 26/23, 17/15 ^M2 21
8 232.199 25/22, 8/7, 31/27, 23/20 ^4M2 24
9 261.224 15/13, 22/19, 29/25, 7/6 ^^^d3 27
10 290.249 27/23, 20/17, 13/11, 32/27, 19/16, 25/21, 31/26 vvm3 30
11 319.274 6/5, 29/24, 23/19 ^m3 33
12 348.299 17/14, 28/23, 11/9, 27/22, 16/13 ~3 36
13 377.323 21/17, 26/21, 31/25, 5/4 vM3 39
14 406.348 29/23, 24/19, 19/15, 14/11 ^^M3 42
15 435.373 23/18, 32/25, 9/7, 31/24, 22/17 vvvA3 45
16 464.398 13/10, 30/23, 17/13, 21/16, 25/19, 29/22 v44 48
17 493.423 4/3 v4 51
18 522.448 31/23, 27/20, 23/17, 19/14, 15/11 ^^4 54
19 551.473 26/19, 11/8, 29/21, 18/13 vvvA4 57
20 580.498 25/18, 32/23, 7/5, 31/22 A4 60
21 609.523 24/17, 17/12, 27/19, 10/7 vd5 63
22 638.547 23/16, 13/9, 29/20, 16/11 ^^d5 66
23 667.572 19/13, 22/15, 25/17, 28/19, 31/21 vvv5 69
24 696.597 3/2 P5 72
25 725.622 32/21, 29/19, 26/17, 23/15 ^^^5 75
26 754.647 20/13, 17/11, 31/20, 14/9 vvA5, ^^d6 78
27 783.672 25/16, 11/7, 30/19, 19/12 vvvm6 81
28 812.697 27/17, 8/5, 29/18 m6 84
29 841.722 21/13, 13/8, 31/19, 18/11 ^^^m6 87
30 870.746 23/14, 28/17, 5/3 vvM6 90
31 899.771 32/19, 27/16, 22/13 ^M6 93
32 928.796 17/10, 29/17, 12/7, 31/18 ^4M6 96
33 957.821 19/11, 26/15, 7/4 ^^^d7 99
34 986.846 30/17, 23/13, 16/9 vvm7 102
35 1015.871 25/14, 9/5, 29/16 ^m7 105
36 1044.896 20/11, 31/17, 11/6 ~7 108
37 1073.921 24/13, 13/7, 28/15, 15/8 vM7 111
38 1102.946 32/17, 17/9, 19/10 ^^M7 114
39 1131.970 21/11, 23/12, 25/13, 27/14, 29/15, 31/16 vvvA7 117
40 1160.995 v41 +1oct 120
41 1190.020 2/1 v1 +1oct 123
42 1219.045 ^^1 +1oct 126
43 1248.070 31/15, 29/14 vvvA1 +1oct 129
44 1277.095 27/13, 25/12, 23/11, 21/10 v4m2 +1oct 132
45 1306.120 19/9, 17/8, 32/15, 15/7 vm2 +1oct 135
46 1335.145 28/13, 13/6 ^^m2 +1oct 138
47 1364.170 24/11, 11/5, 31/14 vvvM2 +1oct 141
48 1393.194 20/9, 29/13, 9/4 M2 +1oct 144
49 1422.219 25/11, 16/7 ^^^M2 +1oct 147
50 1451.244 23/10, 30/13 vvA2 +1oct, ^^d3 +1oct 150
51 1480.269 7/3, 26/11 vvvm3 +1oct 153
52 1509.294 19/8, 31/13, 12/5 m3 +1oct 156
53 1538.319 29/12, 17/7, 22/9 ^^^m3 +1oct 159
54 1567.344 27/11, 32/13 vvM3 +1oct 162
55 1596.369 5/2 ^M3 +1oct 165
56 1625.393 28/11, 23/9, 18/7 ^4M3 +1oct 168
57 1654.418 31/12, 13/5 ^^^d4 +1oct 171
58 1683.443 21/8, 29/11 vv4 +1oct 174
59 1712.468 8/3, 27/10 ^4 +1oct 177
60 1741.493 19/7, 30/11, 11/4 ~4 +1oct 180
61 1770.518 25/9, 14/5 vA4 +1oct 183
62 1799.543 31/11, 17/6 ^^A4 +1oct, vvd5 +1oct 186
63 1828.568 20/7, 23/8, 26/9 ^d5 +1oct 189
64 1857.593 29/10, 32/11 ~5 +1oct 192
65 1886.617 v5 +1oct 195
66 1915.642 3/1 ^^5 +1oct 198
67 1944.667 31/10 vvvA5 +1oct 201
68 1973.692 28/9, 25/8, 22/7 v4m6 +1oct 204
69 2002.717 19/6, 16/5 vm6 +1oct 207
70 2031.742 29/9, 13/4 ^^m6 +1oct 210
71 2060.767 23/7 vvvM6 +1oct 213
72 2089.792 10/3 M6 +1oct 216
73 2118.816 27/8, 17/5, 24/7 ^^^M6 +1oct 219
74 2147.841 31/9 vvA6 +1oct, ^^d7 +1oct 222
75 2176.866 7/2 vvvm7 +1oct 225
76 2205.891 32/9, 25/7, 18/5 m7 +1oct 228
77 2234.916 29/8, 11/3 ^^^m7 +1oct 231
78 2263.941 26/7 vvM7 +1oct 234
79 2292.966 15/4 ^M7 +1oct 237
80 2321.991 19/5, 23/6 ^4M7 +1oct 240
81 2351.016 27/7, 31/8 ^^^d1 +2oct 243
82 2380.040 vv1 +2oct 246
83 2409.065 4/1 ^1 +2oct 249
84 2438.090 ^41 +2oct 252
85 2467.115 29/7, 25/6 ^^^d2 +2oct 255
86 2496.140 21/5, 17/4 vvm2 +2oct 258
87 2525.165 30/7, 13/3 ^m2 +2oct 261
88 2554.190 22/5 ~2 +2oct 264
89 2583.215 31/7 vM2 +2oct 267
90 2612.239 9/2 ^^M2 +2oct 270
91 2641.264 32/7, 23/5 vvvA2 +2oct 273
92 2670.289 14/3 v4m3 +2oct 276
93 2699.314 19/4 vm3 +2oct 279
94 2728.339 24/5, 29/6 ^^m3 +2oct 282
95 2757.364 vvvM3 +2oct 285
96 2786.389 5/1 M3 +2oct 288
97 2815.414 ^^^M3 +2oct 291
98 2844.439 31/6, 26/5 vvA3 +2oct, ^^d4 +2oct 294
99 2873.463 21/4 vvv4 +2oct 297
100 2902.488 16/3 P4 +2oct 300
101 2931.513 27/5 ^^^4 +2oct 303
102 2960.538 11/2 vvA4 +2oct 306
103 2989.563 28/5, 17/3 ^A4 +2oct 309
104 3018.588 23/4 d5 +2oct 312
105 3047.613 29/5 ^^^d5 +2oct 315
106 3076.638 vv5 +2oct 318
107 3105.663 6/1 ^5 +2oct 321
108 3134.687 ^45 +2oct 324
109 3163.712 31/5, 25/4 ^^^d6 +2oct 327
110 3192.737 19/3 vvm6 +2oct 330
111 3221.762 32/5 ^m6 +2oct 333
112 3250.787 13/2 ~6 +2oct 336
113 3279.812 20/3 vM6 +2oct 339
114 3308.837 27/4 ^^M6 +2oct 342
115 3337.862 vvvA6 +2oct 345
116 3366.886 7/1 v4m7 +2oct 348
117 3395.911 vm7 +2oct 351
118 3424.936 29/4 ^^m7 +2oct 354
119 3453.961 22/3 vvvM7 +2oct 357
120 3482.986 15/2 M7 +2oct 360
121 3512.011 23/3 ^^^M7 +2oct 363
122 3541.036 31/4 vvA7 +2oct, ^^d1 +3oct 366
123 3570.061 vvv1 +3oct 369
124 3599.086 8/1 P1 +3oct 372
125 3628.110 ^^^1 +3oct 375
126 3657.135 25/3 vvA1 +3oct, ^^d2 +3oct 378
127 3686.160 vvvm2 +3oct 381
128 3715.185 17/2 m2 +3oct 384
129 3744.210 26/3 ^^^m2 +3oct 387
130 3773.235 vvM2 +3oct 390
131 3802.260 9/1 ^M2 +3oct 393
132 3831.285 ^4M2 +3oct 396
133 3860.309 28/3 ^^^d3 +3oct 399
134 3889.334 19/2 vvm3 +3oct 402
135 3918.359 29/3 ^m3 +3oct 405
136 3947.384 ~3 +3oct 408
137 3976.409 10/1 vM3 +3oct 411
138 4005.434 ^^M3 +3oct 414
139 4034.459 31/3 vvvA3 +3oct 417
140 4063.484 21/2 v44 +3oct 420
141 4092.509 32/3 v4 +3oct 423
142 4121.533 ^^4 +3oct 426
143 4150.558 11/1 vvvA4 +3oct 429
144 4179.583 A4 +3oct 432
145 4208.608 vd5 +3oct 435
146 4237.633 23/2 ^^d5 +3oct 438
147 4266.658 vvv5 +3oct 441
148 4295.683 12/1 P5 +3oct 444
149 4324.708 ^^^5 +3oct 447
150 4353.732 vvA5 +3oct, ^^d6 +3oct 450
151 4382.757 25/2 vvvm6 +3oct 453
152 4411.782 m6 +3oct 456
153 4440.807 13/1 ^^^m6 +3oct 459
154 4469.832 vvM6 +3oct 462
155 4498.857 27/2 ^M6 +3oct 465
156 4527.882 ^4M6 +3oct 468
157 4556.907 14/1 ^^^d7 +3oct 471
158 4585.932 vvm7 +3oct 474
159 4614.956 ^m7 +3oct 477
160 4643.981 29/2 ~7 +3oct 480
161 4673.006 vM7 +3oct 483
162 4702.031 15/1 ^^M7 +3oct 486
163 4731.056 31/2 vvvA7 +3oct 489
164 4760.081 v41 +4oct 492
165 4789.106 16/1 v1 +4oct 495
166 4818.131 ^^1 +4oct 498
167 4847.156 vvvA1 +4oct 501
168 4876.180 v4m2 +4oct 504
169 4905.205 17/1 vm2 +4oct 507
170 4934.230 ^^m2 +4oct 510
171 4963.255 vvvM2 +4oct 513
172 4992.280 18/1 M2 +4oct 516
173 5021.305 ^^^M2 +4oct 519
174 5050.330 vvA2 +4oct, ^^d3 +4oct 522
175 5079.355 vvvm3 +4oct 525
176 5108.379 19/1 m3 +4oct 528
177 5137.404 ^^^m3 +4oct 531
178 5166.429 vvM3 +4oct 534
179 5195.454 20/1 ^M3 +4oct 537
180 5224.479 ^4M3 +4oct 540
181 5253.504 ^^^d4 +4oct 543
182 5282.529 21/1 vv4 +4oct 546
183 5311.554 ^4 +4oct 549
184 5340.579 22/1 ~4 +4oct 552
185 5369.603 vA4 +4oct 555
186 5398.628 ^^A4 +4oct, vvd5 +4oct 558
187 5427.653 23/1 ^d5 +4oct 561
188 5456.678 ~5 +4oct 564
189 5485.703 v5 +4oct 567
190 5514.728 24/1 ^^5 +4oct 570
191 5543.753 vvvA5 +4oct 573
192 5572.778 25/1 v4m6 +4oct 576
193 5601.802 vm6 +4oct 579
194 5630.827 26/1 ^^m6 +4oct 582
195 5659.852 vvvM6 +4oct 585
196 5688.877 M6 +4oct 588
197 5717.902 27/1 ^^^M6 +4oct 591
198 5746.927 vvA6 +4oct, ^^d7 +4oct 594
199 5775.952 28/1 vvvm7 +4oct 597
200 5804.977 m7 +4oct 600
201 5834.002 29/1 ^^^m7 +4oct 603
202 5863.026 vvM7 +4oct 606
203 5892.051 30/1 ^M7 +4oct 609
204 5921.076 ^4M7 +4oct 612
205 5950.101 31/1 ^^^d1 +5oct 615
206 5979.126 vv1 +5oct 618
207 6008.151 32/1 ^1 +5oct 621

Approximation to JI

The following table illustrates the representation of the 32-integer limit intervals in 186zpi. Prime harmonics are in bold; inconsistent intervals are in italic.

Intervals by direct approximation (even if inconsistent)
Ratio Error (abs, ¢) Error (rel,  %)
17/13 0.030 0.102
5/1 0.075 0.259
25/17 0.100 0.344
25/13 0.129 0.446
23/11 0.138 0.477
25/1 0.150 0.517
11/8 0.155 0.533
17/5 0.175 0.602
13/5 0.204 0.704
17/1 0.250 0.861
13/1 0.279 0.963
9/7 0.289 0.996
23/8 0.293 1.011
23/1 0.621 2.140
31/29 0.641 2.209
30/29 0.642 2.211
23/5 0.696 2.399
29/6 0.717 2.470
9/8 0.736 2.535
11/1 0.760 2.617
25/23 0.771 2.657
11/5 0.835 2.876
23/17 0.871 3.001
21/19 0.881 3.037
11/9 0.891 3.069
23/13 0.901 3.103
25/11 0.910 3.135
8/1 0.914 3.151
8/5 0.990 3.409
17/11 1.009 3.478
8/7 1.025 3.531
23/9 1.029 3.546
13/11 1.039 3.580
25/8 1.065 3.668
17/8 1.164 4.012
27/19 1.171 4.033
11/7 1.180 4.065
13/8 1.194 4.114
31/30 1.283 4.420
23/7 1.318 4.542
31/6 1.358 4.679
9/1 1.650 5.686
9/5 1.725 5.944
20/19 1.726 5.947
25/9 1.800 6.203
19/4 1.801 6.205
17/9 1.900 6.547
24/19 1.906 6.568
13/9 1.930 6.649
7/1 1.939 6.682
7/5 2.015 6.941
31/28 2.060 7.099
25/7 2.090 7.199
17/7 2.189 7.543
13/7 2.219 7.645
21/20 2.607 8.984
21/4 2.683 9.242
29/28 2.702 9.308
32/19 2.716 9.356
19/3 2.821 9.719
19/15 2.896 9.977
27/20 2.897 9.980
27/4 2.972 10.238
32/31 3.085 10.630
15/14 3.343 11.519
14/3 3.418 11.777
13/6 3.428 11.811
17/6 3.458 11.913
30/13 3.503 12.069
30/17 3.533 12.171
25/6 3.557 12.256
32/21 3.597 12.393
6/5 3.632 12.515
6/1 3.708 12.774
32/29 3.726 12.839
28/19 3.741 12.887
30/1 3.783 13.032
32/27 3.886 13.389
31/4 4.000 13.781
31/20 4.075 14.039
29/13 4.145 14.280
29/17 4.174 14.382
29/25 4.274 14.726
23/6 4.329 14.914
12/7 4.333 14.928
29/5 4.349 14.985
16/15 4.368 15.050
30/23 4.404 15.172
29/1 4.424 15.243
16/3 4.443 15.309
11/6 4.467 15.391
22/15 4.523 15.583
30/11 4.542 15.649
20/3 4.547 15.666
22/3 4.598 15.842
4/3 4.622 15.924
29/4 4.641 15.990
15/4 4.697 16.183
29/20 4.716 16.248
31/13 4.786 16.489
31/17 4.816 16.591
28/27 4.911 16.920
31/25 4.915 16.935
31/5 4.990 17.194
29/23 5.046 17.383
31/1 5.066 17.452
27/14 5.069 17.463
29/11 5.184 17.860
15/2 5.283 18.201
29/8 5.339 18.394
3/2 5.358 18.459
10/3 5.433 18.718
12/11 5.513 18.993
32/3 5.536 19.075
26/15 5.562 19.164
32/15 5.612 19.334
26/3 5.637 19.422
7/6 5.647 19.456
23/12 5.651 19.470
31/23 5.687 19.592
30/7 5.722 19.714
31/19 5.801 19.986
31/11 5.825 20.069
31/8 5.980 20.603
29/9 6.075 20.929
27/16 6.094 20.994
19/14 6.239 21.496
27/22 6.248 21.528
12/1 6.272 21.610
12/5 6.347 21.869
29/7 6.364 21.925
21/16 6.383 21.991
25/12 6.422 22.127
29/19 6.442 22.195
17/12 6.522 22.471
19/18 6.528 22.492
22/21 6.538 22.524
13/12 6.552 22.573
28/3 6.561 22.606
28/15 6.637 22.865
31/21 6.682 23.023
31/9 6.716 23.138
28/13 6.846 23.588
28/17 6.876 23.690
31/27 6.972 24.019
28/25 6.976 24.034
31/7 7.005 24.134
27/2 7.008 24.145
28/5 7.051 24.292
27/10 7.083 24.404
30/19 7.084 24.406
28/1 7.126 24.551
19/6 7.159 24.665
19/16 7.264 25.027
27/26 7.288 25.108
21/2 7.297 25.141
29/21 7.324 25.232
21/10 7.372 25.400
22/19 7.419 25.561
26/21 7.577 26.104
29/27 7.613 26.228
31/24 7.707 26.554
28/23 7.747 26.691
26/7 7.761 26.739
32/13 7.871 27.119
28/11 7.886 27.168
32/17 7.901 27.221
10/7 7.965 27.443
32/25 8.001 27.565
7/2 8.040 27.702
26/9 8.050 27.735
32/5 8.076 27.824
32/1 8.151 28.082
19/2 8.179 28.178
19/10 8.254 28.437
10/9 8.254 28.439
9/2 8.329 28.698
29/24 8.348 28.763
26/19 8.458 29.141
31/3 8.622 29.705
31/15 8.697 29.964
32/23 8.772 30.222
28/9 8.776 30.237
13/4 8.786 30.270
22/7 8.800 30.319
17/4 8.815 30.372
20/13 8.861 30.529
20/17 8.891 30.631
32/11 8.910 30.699
25/4 8.915 30.716
26/11 8.941 30.803
16/7 8.955 30.852
5/4 8.990 30.974
4/1 9.065 31.233
26/23 9.079 31.281
22/9 9.089 31.315
20/1 9.140 31.492
11/10 9.145 31.508
11/2 9.220 31.766
16/9 9.244 31.848
29/3 9.263 31.914
23/10 9.284 31.985
29/15 9.338 32.173
23/2 9.359 32.243
23/4 9.686 33.373
18/7 9.691 33.387
26/1 9.700 33.421
23/20 9.762 33.632
26/5 9.775 33.679
32/9 9.801 33.768
11/4 9.825 33.850
26/25 9.850 33.938
20/11 9.900 34.109
10/1 9.905 34.125
26/17 9.950 34.282
2/1 9.980 34.384
5/2 10.055 34.642
32/7 10.090 34.764
23/22 10.118 34.861
25/2 10.130 34.901
16/11 10.135 34.917
17/10 10.155 34.986
13/10 10.184 35.088
17/2 10.230 35.244
13/2 10.259 35.346
14/9 10.269 35.380
23/16 10.273 35.394
19/13 10.587 36.475
19/17 10.617 36.577
29/12 10.697 36.853
9/4 10.716 36.919
25/19 10.716 36.921
22/1 10.739 37.001
20/9 10.791 37.177
19/5 10.791 37.180
22/5 10.814 37.259
19/1 10.866 37.438
18/11 10.870 37.452
25/22 10.890 37.518
16/1 10.894 37.534
16/5 10.969 37.793
22/17 10.989 37.862
7/4 11.005 37.915
23/18 11.009 37.929
22/13 11.019 37.964
25/16 11.044 38.052
20/7 11.080 38.174
17/16 11.144 38.395
14/11 11.160 38.448
16/13 11.174 38.497
23/14 11.298 38.925
31/12 11.338 39.062
21/13 11.468 39.512
23/19 11.488 39.579
21/17 11.498 39.614
25/21 11.598 39.958
19/11 11.626 40.056
18/1 11.630 40.069
21/5 11.673 40.216
18/5 11.705 40.328
21/1 11.748 40.475
27/13 11.758 40.508
25/18 11.780 40.587
19/8 11.781 40.589
27/17 11.787 40.610
18/17 11.880 40.930
19/12 11.886 40.952
27/25 11.887 40.954
18/13 11.910 41.032
14/1 11.919 41.066
27/5 11.962 41.213
14/5 11.994 41.324
27/1 12.037 41.471
31/14 12.040 41.482
25/14 12.069 41.583
17/14 12.169 41.926
14/13 12.199 42.028
31/18 12.329 42.478
23/21 12.369 42.615
24/13 12.493 43.044
21/11 12.507 43.092
19/9 12.517 43.124
24/17 12.523 43.146
25/24 12.623 43.489
27/23 12.658 43.611
21/8 12.662 43.626
29/14 12.681 43.691
24/5 12.698 43.748
24/1 12.773 44.006
27/11 12.797 44.089
19/7 12.806 44.120
27/8 12.951 44.622
29/18 12.970 44.687
31/16 13.065 45.014
31/22 13.220 45.547
15/7 13.323 45.902
24/23 13.394 46.147
7/3 13.398 46.161
13/3 13.408 46.194
17/3 13.437 46.296
15/13 13.483 46.453
17/15 13.513 46.555
24/11 13.532 46.624
25/3 13.537 46.640
5/3 13.612 46.898
3/1 13.687 47.157
29/16 13.706 47.223
15/1 13.762 47.416
29/22 13.861 47.756
27/7 13.976 48.153
31/2 13.980 48.164
31/10 14.055 48.423
29/26 14.125 48.664
31/26 14.259 49.127
23/3 14.308 49.297
24/7 14.313 49.312
29/10 14.329 49.368
15/8 14.348 49.434
23/15 14.384 49.556
29/2 14.404 49.627
8/3 14.423 49.692
11/3 14.447 49.774
15/11 14.503 49.967
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