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'''186 zeta peak index''' (abbreviated '''186zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 186st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].
'''186 zeta peak index''' (abbreviated '''186zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 186st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].


{| class="wikitable"
{{ZPI
! colspan="3" | Tuning
| zpi = 186
! colspan="3" | Strength
| steps = 41.3438354846780
! colspan="2" | Closest EDO
| step size = 29.0248832971658
! colspan="2" | Integer limit
| height = 1.876590
|-
| integral = 0.241233
! ZPI
| gap = 11.567493
! Steps per octave
| edo = 41edo
! Step size (cents)
| octave = 1190.02021518380
! Height
| consistent = 2
! Integral
| distinct = 2
! Gap
}}
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[186zpi]]
| 41.3438354846780
| 29.0248832971658
| 1.876590
| 0.241233
| 11.567493
| [[41edo]]
| 1190.02021518380
| 2
| 2
|}


== Theory ==
== Theory ==
=== Record on the Riemann zeta function with primes 2 and 3 removed ===
=== 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.
'''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.


{| class="wikitable"
{| class="wikitable"
! colspan="6" |Unmodified Riemann zeta function
|-
! colspan="5" |Riemann zeta function with primes 2 and 3 removed
! colspan="6" | Unmodified Riemann zeta function
! colspan="5" | Riemann zeta function with primes 2 and 3 removed
|-
|-
! colspan="3" | Tuning
! colspan="3" | Tuning
Line 61: Line 45:
| 30.6006474885974
| 30.6006474885974
| 39.2148564976330
| 39.2148564976330
|1.468164
| 1.468164
| [[31edo]]
| [[31edo]]
| 1215.66055142662
| 1215.66055142662
| 30.5974484926723
| 30.5974484926723
| 39.2189564527704
| 39.2189564527704
|3.769318
| 3.769318
| [[31edo]]
| [[31edo]]
| 1215.78765003588
| 1215.78765003588
|-
|-
|[[186zpi]]
| [[186zpi]]
|41.3438354846780
| 41.3438354846780
|29.0248832971658
| 29.0248832971658
|1.876590
| 1.876590
|[[41edo]]
| [[41edo]]
|1190.02021518380
| 1190.02021518380
|41.3477989230936
| 41.3477989230936
|29.0221010852836
| 29.0221010852836
|4.469823
| 4.469823
|[[41edo]]
| [[41edo]]
|1189.90614449663
| 1189.90614449663
|-
|-
|[[565zpi]]
| [[565zpi]]
|98.6209462564991
| 98.6209462564991
|12.1678005084130
| 12.1678005084130
|2.305330
| 2.305330
|[[99edo]]
| [[99edo]]
|1204.61225033289
| 1204.61225033289
|98.6257548378926
| 98.6257548378926
|12.1672072570942
| 12.1672072570942
|4.883729
| 4.883729
|[[99edo]]
| [[99edo]]
|1204.55351845233
| 1204.55351845233
|}
|}


Line 107: Line 91:
It is important to note that [[124edo]] provides two possible [[3/2|fifths (3/2)]]. The closest one, from the [[val]] <124 197] (i.e. the [[patent val]]), is the [[3/2|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 [[3/2|fifth]], which appears in [[124ed8]], also corresponds to the [[3/2|fifth]] of [[31edo]]. Therefore, we choose to use the [[ups and downs notation]] of the 124b temperament, denoted as <124 196].
It is important to note that [[124edo]] provides two possible [[3/2|fifths (3/2)]]. The closest one, from the [[val]] <124 197] (i.e. the [[patent val]]), is the [[3/2|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 [[3/2|fifth]], which appears in [[124ed8]], also corresponds to the [[3/2|fifth]] of [[31edo]]. Therefore, we choose to use the [[ups and downs notation]] of the 124b temperament, denoted as <124 196].


{| class="wikitable center-all left-1 right-2 left-3 center-4 left-5"
{| class="wikitable center-1 right-2 left-3 center-4 center-5"
|+ style="font-size: 105%; white-space: nowrap;" | Intervals in 186zpi
|-
|-
| colspan="3" | JI ratios are comprised of 32-integer limit ratios,<br>and are stylized as follows to indicate their accuracy:
| colspan="3" style="text-align:left;" | JI ratios are comprised of [[32-integer-limit]] ratios,<br>and are stylized as follows to indicate their accuracy:
* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
* '''Bold:''' relative error < 16.667 %
* '''Bold:''' relative error < 16.667 %
Line 116: Line 101:
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
| colspan="2" | <center>'''124b at every 3 steps'''</center><br>[[9/8|Whole tone]] = 20 steps<br>[[256/243|Limma]] = 12 steps<br>[[2187/2048|Apotome]] = 8 steps
| colspan="2" style="text-align:right;" | <center>'''⟨124 196] at every 3 steps'''</center><br>[[9/8|Whole tone]] = 20 steps<br>[[256/243|Limma]] = 12 steps<br>[[2187/2048|Apotome]] = 8 steps
|-
|-
! Degree
! Degree
! Cents
! Cents
! Ratios
! Ratios
! Ups and Downs Notation
! Ups and downs notation
! Step
! Step
|-
|-
Line 1,373: Line 1,358:
|}
|}


===Approximation to JI===
== 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''.
=== Interval mappings ===


{| class="wikitable center-all mw-collapsible mw-collapsed"
The following tables show how [[32-integer-limit]] intervals are represented in 186zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italics''.
|+ style="white-space: nowrap;" | Intervals by direct approximation (even if inconsistent)
 
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 32-integer-limit intervals in 186zpi (by direct approximation)
|-
|-
! Ratio
! Ratio
Line 1,385: Line 1,372:
|-
|-
| [[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
|-
|- style="background-color: #cccccc;"
| ''[[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'''
|-
|- style="background-color: #cccccc;"
| ''[[9/7]]''
| ''[[9/7]]''
| ''0.289''
| ''+0.289''
| ''0.996''
| ''+0.996''
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[8/1]]''
| ''[[8/1]]''
| ''0.914''
| ''-0.914''
| ''3.151''
| ''-3.151''
|-
|- style="background-color: #cccccc;"
| ''[[8/5]]''
| ''[[8/5]]''
| ''0.990''
| ''-0.990''
| ''3.409''
| ''-3.409''
|-
|-
| [[17/11]]
| [[17/11]]
| 1.009
| +1.009
| 3.478
| +3.478
|-
|- style="background-color: #cccccc;"
| ''[[8/7]]''
| ''[[8/7]]''
| ''1.025''
| ''+1.025''
| ''3.531''
| ''+3.531''
|-
|- style="background-color: #cccccc;"
| ''[[23/9]]''
| ''[[23/9]]''
| ''1.029''
| ''+1.029''
| ''3.546''
| ''+3.546''
|-
|-
| [[13/11]]
| [[13/11]]
| 1.039
| +1.039
| 3.580
| +3.580
|-
|- style="background-color: #cccccc;"
| ''[[25/8]]''
| ''[[25/8]]''
| ''1.065''
| ''+1.065''
| ''3.668''
| ''+3.668''
|-
|- style="background-color: #cccccc;"
| ''[[17/8]]''
| ''[[17/8]]''
| ''1.164''
| ''+1.164''
| ''4.012''
| ''+4.012''
|-
|- style="background-color: #cccccc;"
| ''[[27/19]]''
| ''[[27/19]]''
| ''1.171''
| ''+1.171''
| ''4.033''
| ''+4.033''
|-
|-
| [[11/7]]
| [[11/7]]
| 1.180
| +1.180
| 4.065
| +4.065
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[9/1]]''
| ''[[9/1]]''
| ''1.650''
| ''-1.650''
| ''5.686''
| ''-5.686''
|-
|- style="background-color: #cccccc;"
| ''[[9/5]]''
| ''[[9/5]]''
| ''1.725''
| ''-1.725''
| ''5.944''
| ''-5.944''
|-
|- style="background-color: #cccccc;"
| ''[[20/19]]''
| ''[[20/19]]''
| ''1.726''
| ''-1.726''
| ''5.947''
| ''-5.947''
|-
|- style="background-color: #cccccc;"
| ''[[25/9]]''
| ''[[25/9]]''
| ''1.800''
| ''+1.800''
| ''6.203''
| ''+6.203''
|-
|- style="background-color: #cccccc;"
| ''[[19/4]]''
| ''[[19/4]]''
| ''1.801''
| ''+1.801''
| ''6.205''
| ''+6.205''
|-
|- style="background-color: #cccccc;"
| ''[[17/9]]''
| ''[[17/9]]''
| ''1.900''
| ''+1.900''
| ''6.547''
| ''+6.547''
|-
|- style="background-color: #cccccc;"
| ''[[24/19]]''
| ''[[24/19]]''
| ''1.906''
| ''+1.906''
| ''6.568''
| ''+6.568''
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[21/20]]''
| ''[[21/20]]''
| ''2.607''
| ''+2.607''
| ''8.984''
| ''+8.984''
|-
|- style="background-color: #cccccc;"
| ''[[21/4]]''
| ''[[21/4]]''
| ''2.683''
| ''+2.683''
| ''9.242''
| ''+9.242''
|-
|- style="background-color: #cccccc;"
| ''[[29/28]]''
| ''[[29/28]]''
| ''2.702''
| ''-2.702''
| ''9.308''
| ''-9.308''
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[27/20]]''
| ''[[27/20]]''
| ''2.897''
| ''+2.897''
| ''9.980''
| ''+9.980''
|-
|- style="background-color: #cccccc;"
| ''[[27/4]]''
| ''[[27/4]]''
| ''2.972''
| ''+2.972''
| ''10.238''
| ''+10.238''
|-
|- style="background-color: #cccccc;"
| ''[[32/31]]''
| ''[[32/31]]''
| ''3.085''
| ''+3.085''
| ''10.630''
| ''+10.630''
|-
|- style="background-color: #cccccc;"
| ''[[15/14]]''
| ''[[15/14]]''
| ''3.343''
| ''-3.343''
| ''11.519''
| ''-11.519''
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[32/29]]''
| ''[[32/29]]''
| ''3.726''
| ''+3.726''
| ''12.839''
| ''+12.839''
|-
|- style="background-color: #cccccc;"
| ''[[28/19]]''
| ''[[28/19]]''
| ''3.741''
| ''-3.741''
| ''12.887''
| ''-12.887''
|-
|-
| [[30/1]]
| [[30/1]]
| 3.783
| +3.783
| 13.032
| +13.032
|-
|- style="background-color: #cccccc;"
| ''[[32/27]]''
| ''[[32/27]]''
| ''3.886''
| ''-3.886''
| ''13.389''
| ''-13.389''
|-
|- style="background-color: #cccccc;"
| ''[[31/4]]''
| ''[[31/4]]''
| ''4.000''
| ''-4.000''
| ''13.781''
| ''-13.781''
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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'''
|-
|- style="background-color: #cccccc;"
| ''[[16/3]]''
| ''[[16/3]]''
| ''4.443''
| ''+4.443''
| ''15.309''
| ''+15.309''
|-
|-
| [[11/6]]
| [[11/6]]
| 4.467
| -4.467
| 15.391
| -15.391
|-
|- style="background-color: #cccccc;"
| ''[[22/15]]''
| ''[[22/15]]''
| ''4.523''
| ''+4.523''
| ''15.583''
| ''+15.583''
|-
|-
| [[30/11]]
| [[30/11]]
| 4.542
| +4.542
| 15.649
| +15.649
|-
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| [[31/17]]
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| [[31/25]]
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| +16.935
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| [[31/5]]
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| [[29/23]]
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| '''5.066'''
| '''+5.066'''
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| '''+17.452'''
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| [[29/11]]
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| +17.860
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| ''+19.422''
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| -19.456
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| +19.470
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| [[31/23]]
| [[31/23]]
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| +19.592
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| [[30/7]]
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| +19.714
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| [[31/19]]
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| [[31/11]]
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| +20.069
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| [[29/7]]
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| +21.925
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|-
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| +22.127
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| [[17/12]]
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|-
| [[19/18]]
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| -22.492
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| [[13/12]]
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| [[31/21]]
| [[31/21]]
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| ''[[31/27]]''
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| [[31/7]]
| [[31/7]]
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| [[30/19]]
| [[30/19]]
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| -24.406
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|-
| [[19/6]]
| [[19/6]]
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| +24.665
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| [[29/21]]
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| -25.232
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|- style="background-color: #cccccc;"
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| -26.739
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| -27.443
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|-
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| +27.702
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| ''+29.141''
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| -29.705
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|-
| [[31/15]]
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| -8.697
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| -29.964
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| ''[[32/23]]''
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| ''+30.237''
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| ''-30.270''
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| [[22/7]]
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| -30.319
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| ''[[32/11]]''
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| -30.803
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| ''+31.233''
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|-
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| [[26/23]]
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| -9.079
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| -31.281
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| ''+31.492''
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| [[11/10]]
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| +9.145
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| +31.508
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|-
| [[11/2]]
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| +9.220
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| +31.766
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| ''-31.848''
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|-
| [[29/3]]
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| -9.263
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| -31.914
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|-
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| +31.985
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| -32.173
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|-
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| +32.243
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|-
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| ''-33.387''
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|-
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| [[26/1]]
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| -9.700
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| -33.421
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| ''-33.632''
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|-
| [[26/5]]
| [[26/5]]
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| -9.775
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| -33.679
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| ''[[32/9]]''
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| ''9.801''
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| ''+33.768''
|-
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| ''[[11/4]]''
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| ''9.825''
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|-
| [[26/25]]
| [[26/25]]
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| -9.850
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| -33.938
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| ''[[20/11]]''
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| ''9.900''
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| ''+34.109''
|-
|-
| [[10/1]]
| [[10/1]]
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| -9.905
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| -34.125
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|-
| [[26/17]]
| [[26/17]]
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| -9.950
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| -34.282
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|-
| '''[[2/1]]'''
| '''[[2/1]]'''
| '''9.980'''
| '''-9.980'''
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| '''-34.384'''
|-
|-
| [[5/2]]
| [[5/2]]
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| +10.055
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| +34.642
|-
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| ''[[32/7]]''
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| ''10.090''
| ''+10.090''
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| ''+34.764''
|-
|-
| [[23/22]]
| [[23/22]]
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| +10.118
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| +34.861
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|-
| [[25/2]]
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| +10.130
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| +34.901
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| ''-34.917''
|-
|-
| [[17/10]]
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| +34.986
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|-
| [[13/10]]
| [[13/10]]
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| +10.184
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| +35.088
|-
|-
| [[17/2]]
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| +10.230
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| +35.244
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|-
| [[13/2]]
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| +10.259
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| +35.346
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| ''[[14/9]]''
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| ''10.269''
| ''-10.269''
| ''35.380''
| ''-35.380''
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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'''
|-
|- style="background-color: #cccccc;"
| ''[[18/11]]''
| ''[[18/11]]''
| ''10.870''
| ''-10.870''
| ''37.452''
| ''-37.452''
|-
|-
| [[25/22]]
| [[25/22]]
| 10.890
| +10.890
| 37.518
| +37.518
|-
|- style="background-color: #cccccc;"
| ''[[16/1]]''
| ''[[16/1]]''
| ''10.894''
| ''-10.894''
| ''37.534''
| ''-37.534''
|-
|- style="background-color: #cccccc;"
| ''[[16/5]]''
| ''[[16/5]]''
| ''10.969''
| ''-10.969''
| ''37.793''
| ''-37.793''
|-
|-
| [[22/17]]
| [[22/17]]
| 10.989
| -10.989
| 37.862
| -37.862
|-
|- style="background-color: #cccccc;"
| ''[[7/4]]''
| ''[[7/4]]''
| ''11.005''
| ''-11.005''
| ''37.915''
| ''-37.915''
|-
|- style="background-color: #cccccc;"
| ''[[23/18]]''
| ''[[23/18]]''
| ''11.009''
| ''+11.009''
| ''37.929''
| ''+37.929''
|-
|-
| [[22/13]]
| [[22/13]]
| 11.019
| -11.019
| 37.964
| -37.964
|-
|- style="background-color: #cccccc;"
| ''[[25/16]]''
| ''[[25/16]]''
| ''11.044''
| ''+11.044''
| ''38.052''
| ''+38.052''
|-
|- style="background-color: #cccccc;"
| ''[[20/7]]''
| ''[[20/7]]''
| ''11.080''
| ''+11.080''
| ''38.174''
| ''+38.174''
|-
|- style="background-color: #cccccc;"
| ''[[17/16]]''
| ''[[17/16]]''
| ''11.144''
| ''+11.144''
| ''38.395''
| ''+38.395''
|-
|-
| [[14/11]]
| [[14/11]]
| 11.160
| -11.160
| 38.448
| -38.448
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[18/1]]''
| ''[[18/1]]''
| ''11.630''
| ''-11.630''
| ''40.069''
| ''-40.069''
|-
|-
| [[21/5]]
| [[21/5]]
| 11.673
| +11.673
| 40.216
| +40.216
|-
|- style="background-color: #cccccc;"
| ''[[18/5]]''
| ''[[18/5]]''
| ''11.705''
| ''-11.705''
| ''40.328''
| ''-40.328''
|-
|-
| [[21/1]]
| [[21/1]]
| 11.748
| +11.748
| 40.475
| +40.475
|-
|- style="background-color: #cccccc;"
| ''[[27/13]]''
| ''[[27/13]]''
| ''11.758''
| ''+11.758''
| ''40.508''
| ''+40.508''
|-
|- style="background-color: #cccccc;"
| ''[[25/18]]''
| ''[[25/18]]''
| ''11.780''
| ''+11.780''
| ''40.587''
| ''+40.587''
|-
|- style="background-color: #cccccc;"
| ''[[19/8]]''
| ''[[19/8]]''
| ''11.781''
| ''+11.781''
| ''40.589''
| ''+40.589''
|-
|- style="background-color: #cccccc;"
| ''[[27/17]]''
| ''[[27/17]]''
| ''11.787''
| ''+11.787''
| ''40.610''
| ''+40.610''
|-
|- style="background-color: #cccccc;"
| ''[[18/17]]''
| ''[[18/17]]''
| ''11.880''
| ''-11.880''
| ''40.930''
| ''-40.930''
|-
|- style="background-color: #cccccc;"
| ''[[19/12]]''
| ''[[19/12]]''
| ''11.886''
| ''-11.886''
| ''40.952''
| ''-40.952''
|-
|- style="background-color: #cccccc;"
| ''[[27/25]]''
| ''[[27/25]]''
| ''11.887''
| ''+11.887''
| ''40.954''
| ''+40.954''
|-
|- style="background-color: #cccccc;"
| ''[[18/13]]''
| ''[[18/13]]''
| ''11.910''
| ''-11.910''
| ''41.032''
| ''-41.032''
|-
|-
| [[14/1]]
| [[14/1]]
| 11.919
| -11.919
| 41.066
| -41.066
|-
|- style="background-color: #cccccc;"
| ''[[27/5]]''
| ''[[27/5]]''
| ''11.962''
| ''+11.962''
| ''41.213''
| ''+41.213''
|-
|-
| [[14/5]]
| [[14/5]]
| 11.994
| -11.994
| 41.324
| -41.324
|-
|- style="background-color: #cccccc;"
| ''[[27/1]]''
| ''[[27/1]]''
| ''12.037''
| ''+12.037''
| ''41.471''
| ''+41.471''
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[24/13]]''
| ''[[24/13]]''
| ''12.493''
| ''+12.493''
| ''43.044''
| ''+43.044''
|-
|-
| [[21/11]]
| [[21/11]]
| 12.507
| +12.507
| 43.092
| +43.092
|-
|- style="background-color: #cccccc;"
| ''[[19/9]]''
| ''[[19/9]]''
| ''12.517''
| ''+12.517''
| ''43.124''
| ''+43.124''
|-
|- style="background-color: #cccccc;"
| ''[[24/17]]''
| ''[[24/17]]''
| ''12.523''
| ''+12.523''
| ''43.146''
| ''+43.146''
|-
|- style="background-color: #cccccc;"
| ''[[25/24]]''
| ''[[25/24]]''
| ''12.623''
| ''-12.623''
| ''43.489''
| ''-43.489''
|-
|- style="background-color: #cccccc;"
| ''[[27/23]]''
| ''[[27/23]]''
| ''12.658''
| ''+12.658''
| ''43.611''
| ''+43.611''
|-
|- style="background-color: #cccccc;"
| ''[[21/8]]''
| ''[[21/8]]''
| ''12.662''
| ''+12.662''
| ''43.626''
| ''+43.626''
|-
|- style="background-color: #cccccc;"
| ''[[29/14]]''
| ''[[29/14]]''
| ''12.681''
| ''-12.681''
| ''43.691''
| ''-43.691''
|-
|- style="background-color: #cccccc;"
| ''[[24/5]]''
| ''[[24/5]]''
| ''12.698''
| ''+12.698''
| ''43.748''
| ''+43.748''
|-
|- style="background-color: #cccccc;"
| ''[[24/1]]''
| ''[[24/1]]''
| ''12.773''
| ''+12.773''
| ''44.006''
| ''+44.006''
|-
|- style="background-color: #cccccc;"
| ''[[27/11]]''
| ''[[27/11]]''
| ''12.797''
| ''+12.797''
| ''44.089''
| ''+44.089''
|-
|-
| [[19/7]]
| [[19/7]]
| 12.806
| +12.806
| 44.120
| +44.120
|-
|- style="background-color: #cccccc;"
| ''[[27/8]]''
| ''[[27/8]]''
| ''12.951''
| ''+12.951''
| ''44.622''
| ''+44.622''
|-
|-
| [[29/18]]
| [[29/18]]
| 12.970
| -12.970
| 44.687
| -44.687
|-
|- style="background-color: #cccccc;"
| ''[[31/16]]''
| ''[[31/16]]''
| ''13.065''
| ''-13.065''
| ''45.014''
| ''-45.014''
|-
|- style="background-color: #cccccc;"
| ''[[31/22]]''
| ''[[31/22]]''
| ''13.220''
| ''-13.220''
| ''45.547''
| ''-45.547''
|-
|- style="background-color: #cccccc;"
| ''[[15/7]]''
| ''[[15/7]]''
| ''13.323''
| ''-13.323''
| ''45.902''
| ''-45.902''
|-
|- style="background-color: #cccccc;"
| ''[[24/23]]''
| ''[[24/23]]''
| ''13.394''
| ''+13.394''
| ''46.147''
| ''+46.147''
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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'''
|-
|- style="background-color: #cccccc;"
| ''[[29/16]]''
| ''[[29/16]]''
| ''13.706''
| ''-13.706''
| ''47.223''
| ''-47.223''
|-
|-
| [[15/1]]
| [[15/1]]
| 13.762
| +13.762
| 47.416
| +47.416
|-
|- style="background-color: #cccccc;"
| ''[[29/22]]''
| ''[[29/22]]''
| ''13.861''
| ''-13.861''
| ''47.756''
| ''-47.756''
|-
|- style="background-color: #cccccc;"
| ''[[27/7]]''
| ''[[27/7]]''
| ''13.976''
| ''+13.976''
| ''48.153''
| ''+48.153''
|-
|- style="background-color: #cccccc;"
| ''[[31/2]]''
| ''[[31/2]]''
| ''13.980''
| ''-13.980''
| ''48.164''
| ''-48.164''
|-
|- style="background-color: #cccccc;"
| ''[[31/10]]''
| ''[[31/10]]''
| ''14.055''
| ''-14.055''
| ''48.423''
| ''-48.423''
|-
|-
| [[29/26]]
| [[29/26]]
| 14.125
| +14.125
| 48.664
| +48.664
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[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
|-
|- style="background-color: #cccccc;"
| ''[[8/3]]''
| ''[[8/3]]''
| ''14.423''
| ''+14.423''
| ''49.692''
| ''+49.692''
|-
|-
| [[11/3]]
| [[11/3]]
| 14.447
| -14.447
| 49.774
| -49.774
|- style="background-color: #cccccc;"
| ''[[15/11]]''
| ''-14.503''
| ''-49.967''
|}
 
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 32-integer-limit intervals in 186zpi (by patent val mapping)
|-
|-
! Ratio
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[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
|-
| [[17/5]]
| +0.175
| +0.602
|-
| [[13/5]]
| +0.204
| +0.704
|-
| '''[[17/1]]'''
| '''+0.250'''
| '''+0.861'''
|-
| '''[[13/1]]'''
| '''+0.279'''
| '''+0.963'''
|-
| '''[[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
|-
| '''[[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
|-
| [[23/13]]
| -0.901
| -3.103
|-
| [[25/11]]
| +0.910
| +3.135
|-
| [[17/11]]
| +1.009
| +3.478
|-
| [[13/11]]
| +1.039
| +3.580
|-
| [[11/7]]
| +1.180
| +4.065
|-
| [[31/30]]
| +1.283
| +4.420
|-
| [[23/7]]
| +1.318
| +4.542
|-
| [[31/6]]
| +1.358
| +4.679
|-
| '''[[7/1]]'''
| '''-1.939'''
| '''-6.682'''
|-
| [[7/5]]
| -2.015
| -6.941
|-
| [[25/7]]
| +2.090
| +7.199
|-
| [[17/7]]
| +2.189
| +7.543
|-
| [[13/7]]
| +2.219
| +7.645
|-
| [[19/3]]
| -2.821
| -9.719
|-
| [[19/15]]
| -2.896
| -9.977
|-
| [[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
|-
| [[6/5]]
| +3.632
| +12.515
|-
| [[6/1]]
| +3.708
| +12.774
|-
| [[30/1]]
| +3.783
| +13.032
|-
| [[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
|-
| [[30/23]]
| +4.404
| +15.172
|-
| '''[[29/1]]'''
| '''+4.424'''
| '''+15.243'''
|-
| [[11/6]]
| -4.467
| -15.391
|-
| [[30/11]]
| +4.542
| +15.649
|-
| [[31/13]]
| +4.786
| +16.489
|-
| [[31/17]]
| +4.816
| +16.591
|-
| [[31/25]]
| +4.915
| +16.935
|-
| [[31/5]]
| +4.990
| +17.194
|-
| [[29/23]]
| +5.046
| +17.383
|-
| '''[[31/1]]'''
| '''+5.066'''
| '''+17.452'''
|-
| [[29/11]]
| +5.184
| +17.860
|-
| [[12/11]]
| -5.513
| -18.993
|-
| [[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
|-
| [[12/1]]
| -6.272
| -21.610
|-
| [[12/5]]
| -6.347
| -21.869
|-
| [[29/7]]
| +6.364
| +21.925
|-
| [[25/12]]
| +6.422
| +22.127
|-
| [[29/19]]
| -6.442
| -22.195
|-
| [[17/12]]
| +6.522
| +22.471
|-
| [[19/18]]
| -6.528
| -22.492
|-
| [[13/12]]
| +6.552
| +22.573
|-
| [[31/21]]
| -6.682
| -23.023
|-
| [[31/7]]
| +7.005
| +24.134
|-
| [[30/19]]
| -7.084
| -24.406
|-
| [[19/6]]
| +7.159
| +24.665
|-
| [[29/21]]
| -7.324
| -25.232
|-
| [[26/7]]
| -7.761
| -26.739
|-
| [[10/7]]
| -7.965
| -27.443
|-
| [[7/2]]
| +8.040
| +27.702
|-
| [[31/3]]
| -8.622
| -29.705
|-
| [[31/15]]
| -8.697
| -29.964
|-
| [[22/7]]
| -8.800
| -30.319
|-
| [[26/11]]
| -8.941
| -30.803
|-
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| ''-189.370''
|- style="background-color: #cccccc;"
| ''[[9/8]]''
| ''+57.314''
| ''+197.465''
|- style="background-color: #cccccc;"
| ''[[32/19]]''
| ''-60.765''
| ''-209.356''
|- style="background-color: #cccccc;"
| ''[[27/20]]''
| ''+60.946''
| ''+209.980''
|- style="background-color: #cccccc;"
| ''[[27/4]]''
| ''+61.021''
| ''+210.238''
|- style="background-color: #cccccc;"
| ''[[32/21]]''
| ''-61.647''
| ''-212.393''
|- style="background-color: #cccccc;"
| ''[[28/27]]''
| ''-62.961''
| ''-216.920''
|- style="background-color: #cccccc;"
| ''[[32/3]]''
| ''-63.586''
| ''-219.075''
|- style="background-color: #cccccc;"
| ''[[32/15]]''
| ''-63.661''
| ''-219.334''
|- style="background-color: #cccccc;"
| ''[[16/9]]''
| ''-67.294''
| ''-231.848''
|- style="background-color: #cccccc;"
| ''[[27/8]]''
| ''+71.001''
| ''+244.622''
|- style="background-color: #cccccc;"
| ''[[32/9]]''
| ''-77.274''
| ''-266.232''
|- style="background-color: #cccccc;"
| ''[[27/16]]''
| ''+80.981''
| ''+279.006''
|- style="background-color: #cccccc;"
| ''[[32/27]]''
| ''-90.961''
| ''-313.389''
|}
|}


[[Category:Zeta peak indexes]]
[[Category:Zeta peak indexes]]
Retrieved from "https://en.xen.wiki/w/186zpi"