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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 ==
'''186zpi''' sets a height record on the Riemann zeta function with primes 2 and 3 removed. The last record is [[125zpi]] and the next 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.
=== 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.


{| class="wikitable"
{| class="wikitable"
! colspan="5" |Unmodified Riemann zeta function
|-
! colspan="4" |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
! colspan="1" | Height
! colspan="1" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Closest EDO
! colspan="2" | Tuning
! colspan="2" | Tuning
! colspan="1" | Height
! colspan="1" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Closest EDO
|-
|-
Line 47: Line 33:
! Steps per octave
! Steps per octave
! Step size (cents)
! Step size (cents)
!  
! colspan="1" | Height
! EDO
! EDO
! Octave (cents)
! Octave (cents)
! Steps per octave
! Steps per octave
! Step size (cents)
! Step size (cents)
!
! colspan="1" |Height
! EDO
! EDO
! Octave (cents)
! Octave (cents)
|-
| [[125zpi]]
| 30.6006474885974
| 39.2148564976330
| 1.468164
| [[31edo]]
| 1215.66055142662
| 30.5974484926723
| 39.2189564527704
| 3.769318
| [[31edo]]
| 1215.78765003588
|-
|-
| [[186zpi]]
| [[186zpi]]
| 41.3438354846780
| 41.3438354846780
| 29.0248832971658
| 29.0248832971658
|
| 1.876590
| [[41edo]]
| [[41edo]]
| 1190.02021518380
| 1190.02021518380
| 41.3477989230936
| 41.3477989230936
| 29.0221010852836
| 29.0221010852836
|
| 4.469823
| [[41edo]]
| [[41edo]]
| 1189.90614449663
| 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.{{Harmonics in cet|29.0248832971658|columns=15|title=Approximation of harmonics in 186zpi}}
{{Harmonics in cet|29.0248832971658|columns=16|start=16|title=Approximation of harmonics in 186zpi}}
=== Approximation of [[Edonoi|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 [[List of music software|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 [[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-1 right-2 left-3 center-4 center-5"
|+ style="font-size: 105%; white-space: nowrap;" | Intervals in 186zpi
|-
| 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 %
* '''Bold:''' relative error < 16.667 %
* Normal: relative error < 25 %
* <small>Small:</small> relative error < 33.333 %
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
| 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
! Cents
! Ratios
! Ups and downs notation
! Step
|-
| 0
| 0.000
|
| P1
| 0
|-
| 1
| 29.025
|
| ^^^1
| 3
|-
| 2
| 58.050
| '''[[32/31]]''', '''<u>[[31/30]]'''</u>, '''<u>[[30/29]]'''</u>, '''[[29/28]]''', [[28/27]], <small>[[27/26]]</small>, <small><small>[[26/25]]</small></small>, <small><small><small>[[25/24]]</small></small></small>
| vvA1, ^^d2
| 6
|-
| 3
| 87.075
| <small><small><small>[[24/23]]</small></small></small>, <small><small>[[23/22]]</small></small>, [[22/21]], '''[[21/20]]''', '''<u>[[20/19]]'''</u>, [[19/18]], <small><small>[[18/17]]</small></small>
| vvvm2
| 9
|-
| 4
| 116.100
| <small><small>[[17/16]]</small></small>, '''[[16/15]]''', '''<u>[[31/29]]'''</u>, '''[[15/14]]''', <small>[[29/27]]</small>, <small><small><small>[[14/13]]</small></small></small>
| m2
| 12
|-
| 5
| 145.124
| <small><small>[[27/25]]</small></small>, [[13/12]], '''<u>[[25/23]]'''</u>, [[12/11]], <small><small><small>[[23/21]]</small></small></small>
| ^^^m2
| 15
|-
| 6
| 174.149
| <small>[[11/10]]</small>, '''[[32/29]]''', '''<u>[[21/19]]'''</u>, '''<u>[[31/28]]'''</u>, <small>[[10/9]]</small>
| vvM2
| 18
|-
| 7
| 203.174
| <small><small><small>[[29/26]]</small></small></small>, <small><small>[[19/17]]</small></small>, [[28/25]], '''<u>[[9/8]]'''</u>, <small>[[26/23]]</small>, <small><small><small>[[17/15]]</small></small></small>
| ^M2
| 21
|-
| 8
| 232.199
| <small><small>[[25/22]]</small></small>, '''<u>[[8/7]]'''</u>, [[31/27]], <small><small>[[23/20]]</small></small>
| ^<sup>4</sup>M2
| 24
|-
| 9
| 261.224
| <small><small><small>[[15/13]]</small></small></small>, <small>[[22/19]]</small>, '''[[29/25]]''', [[7/6]]
| ^^^d3
| 27
|-
| 10
| 290.249
| <small><small><small>[[27/23]]</small></small></small>, <small>[[20/17]]</small>, '''<u>[[13/11]]'''</u>, '''[[32/27]]''', <small>[[19/16]]</small>, <small><small>[[25/21]]</small></small>, <small><small><small>[[31/26]]</small></small></small>
| vvm3
| 30
|-
| 11
| 319.274
| '''[[6/5]]''', <small>[[29/24]]</small>, <small><small>[[23/19]]</small></small>
| ^m3
| 33
|-
| 12
| 348.299
| <small><small><small>[[17/14]]</small></small></small>, <small>[[28/23]]</small>, '''<u>[[11/9]]'''</u>, [[27/22]], <small><small>[[16/13]]</small></small>
| ~3
| 36
|-
| 13
| 377.323
| <small><small>[[21/17]]</small></small>, <small>[[26/21]]</small>, [[31/25]], <small>[[5/4]]</small>
| vM3
| 39
|-
| 14
| 406.348
| [[29/23]], '''<u>[[24/19]]'''</u>, '''[[19/15]]''', <small><small>[[14/11]]</small></small>
| ^^M3
| 42
|-
| 15
| 435.373
| <small><small>[[23/18]]</small></small>, <small>[[32/25]]</small>, '''<u>[[9/7]]'''</u>, <small>[[31/24]]</small>, <small><small>[[22/17]]</small></small>
| vvvA3
| 45
|-
| 16
| 464.398
| <small><small>[[13/10]]</small></small>, '''[[30/23]]''', '''<u>[[17/13]]'''</u>, [[21/16]], <small><small>[[25/19]]</small></small>, <small><small><small>[[29/22]]</small></small></small>
| v<sup>4</sup>4
| 48
|-
| 17
| 493.423
| '''[[4/3]]'''
| v4
| 51
|-
| 18
| 522.448
| [[31/23]], '''[[27/20]]''', '''<u>[[23/17]]'''</u>, [[19/14]], <small><small><small>[[15/11]]</small></small></small>
| ^^4
| 54
|-
| 19
| 551.473
| <small>[[26/19]]</small>, '''<u>[[11/8]]'''</u>, <small>[[29/21]]</small>, <small><small>[[18/13]]</small></small>
| vvvA4
| 57
|-
| 20
| 580.498
| <small><small>[[25/18]]</small></small>, <small>[[32/23]]</small>, '''<u>[[7/5]]'''</u>, <small><small><small>[[31/22]]</small></small></small>
| A4
| 60
|-
| 21
| 609.523
| <small><small><small>[[24/17]]</small></small></small>, [[17/12]], '''<u>[[27/19]]'''</u>, <small>[[10/7]]</small>
| vd5
| 63
|-
| 22
| 638.547
| <small><small>[[23/16]]</small></small>, '''<u>[[13/9]]'''</u>, '''[[29/20]]''', <small><small>[[16/11]]</small></small>
| ^^d5
| 66
|-
| 23
| 667.572
| <small><small>[[19/13]]</small></small>, '''[[22/15]]''', '''<u>[[25/17]]'''</u>, '''[[28/19]]''', [[31/21]]
| vvv5
| 69
|-
| 24
| 696.597
| [[3/2]]
| P5
| 72
|-
| 25
| 725.622
| '''[[32/21]]''', [[29/19]], <small><small>[[26/17]]</small></small>, <small><small><small>[[23/15]]</small></small></small>
| ^^^5
| 75
|-
| 26
| 754.647
| <small>[[20/13]]</small>, '''<u>[[17/11]]'''</u>, '''[[31/20]]''', <small><small>[[14/9]]</small></small>
| vvA5, ^^d6
| 78
|-
| 27
| 783.672
| <small><small>[[25/16]]</small></small>, '''<u>[[11/7]]'''</u>, [[30/19]], <small><small>[[19/12]]</small></small>
| vvvm6
| 81
|-
| 28
| 812.697
| <small><small>[[27/17]]</small></small>, '''<u>[[8/5]]'''</u>, <small><small><small>[[29/18]]</small></small></small>
| m6
| 84
|-
| 29
| 841.722
| <small><small>[[21/13]]</small></small>, '''<u>[[13/8]]'''</u>, [[31/19]], <small><small>[[18/11]]</small></small>
| ^^^m6
| 87
|-
| 30
| 870.746
| <small><small>[[23/14]]</small></small>, [[28/17]], <small><small><small>[[5/3]]</small></small></small>
| vvM6
| 90
|-
| 31
| 899.771
| '''[[32/19]]''', [[27/16]], <small><small>[[22/13]]</small></small>
| ^M6
| 93
|-
| 32
| 928.796
| <small><small>[[17/10]]</small></small>, '''[[29/17]]''', '''[[12/7]]''', <small><small><small>[[31/18]]</small></small></small>
| ^<sup>4</sup>M6
| 96
|-
| 33
| 957.821
| <small><small>[[19/11]]</small></small>, [[26/15]], <small><small>[[7/4]]</small></small>
| ^^^d7
| 99
|-
| 34
| 986.846
| '''[[30/17]]''', '''<u>[[23/13]]'''</u>, <small>[[16/9]]</small>
| vvm7
| 102
|-
| 35
| 1015.871
| <small><small>[[25/14]]</small></small>, '''<u>[[9/5]]'''</u>, <small><small><small>[[29/16]]</small></small></small>
| ^m7
| 105
|-
| 36
| 1044.896
| <small><small>[[20/11]]</small></small>, '''[[31/17]]''', '''[[11/6]]'''
| ~7
| 108
|-
| 37
| 1073.921
| <small><small><small>[[24/13]]</small></small></small>, '''<u>[[13/7]]'''</u>, [[28/15]], <small><small><small>[[15/8]]</small></small></small>
| vM7
| 111
|-
| 38
| 1102.946
| <small>[[32/17]]</small>, '''<u>[[17/9]]'''</u>, <small>[[19/10]]</small>
| ^^M7
| 114
|-
| 39
| 1131.970
| <small><small><small>[[21/11]]</small></small></small>, [[23/12]], '''<u>[[25/13]]'''</u>, [[27/14]], <small>[[29/15]]</small>, <small><small><small>[[31/16]]</small></small></small>
| vvvA7
| 117
|-
| 40
| 1160.995
|
| v<sup>4</sup>1 +1 oct
| 120
|-
| 41
| 1190.020
| <small><small>[[2/1]]</small></small>
| v1 +1 oct
| 123
|-
| 42
| 1219.045
|
| ^^1 +1 oct
| 126
|-
| 43
| 1248.070
| <small>[[31/15]]</small>, <small><small><small>[[29/14]]</small></small></small>
| vvvA1 +1 oct
| 129
|-
| 44
| 1277.095
| <small><small>[[27/13]]</small></small>, [[25/12]], '''<u>[[23/11]]'''</u>, <small>[[21/10]]</small>
| v<sup>4</sup>m2 +1 oct
| 132
|-
| 45
| 1306.120
| <small><small><small>[[19/9]]</small></small></small>, '''<u>[[17/8]]'''</u>, [[32/15]], <small><small><small>[[15/7]]</small></small></small>
| vm2 +1 oct
| 135
|-
| 46
| 1335.145
| [[28/13]], '''[[13/6]]'''
| ^^m2 +1 oct
| 138
|-
| 47
| 1364.170
| <small><small><small>[[24/11]]</small></small></small>, '''<u>[[11/5]]'''</u>, <small><small>[[31/14]]</small></small>
| vvvM2 +1 oct
| 141
|-
| 48
| 1393.194
| <small><small>[[20/9]]</small></small>, '''[[29/13]]''', <small><small>[[9/4]]</small></small>
| M2 +1 oct
| 144
|-
| 49
| 1422.219
| '''<u>[[25/11]]'''</u>, <small>[[16/7]]</small>
| ^^^M2 +1 oct
| 147
|-
| 50
| 1451.244
| <small>[[23/10]]</small>, '''[[30/13]]'''
| vvA2 +1 oct, ^^d3 +1 oct
| 150
|-
| 51
| 1480.269
| <small><small><small>[[7/3]]</small></small></small>, <small>[[26/11]]</small>
| vvvm3 +1 oct
| 153
|-
| 52
| 1509.294
| <small><small>[[19/8]]</small></small>, '''[[31/13]]''', [[12/5]]
| m3 +1 oct
| 156
|-
| 53
| 1538.319
| <small><small>[[29/12]]</small></small>, '''<u>[[17/7]]'''</u>, <small>[[22/9]]</small>
| ^^^m3 +1 oct
| 159
|-
| 54
| 1567.344
| <small><small><small>[[27/11]]</small></small></small>, <small>[[32/13]]</small>
| vvM3 +1 oct
| 162
|-
| 55
| 1596.369
| <small><small>[[5/2]]</small></small>
| ^M3 +1 oct
| 165
|-
| 56
| 1625.393
| <small>[[28/11]]</small>, '''<u>[[23/9]]'''</u>, <small><small>[[18/7]]</small></small>
| ^<sup>4</sup>M3 +1 oct
| 168
|-
| 57
| 1654.418
| <small><small>[[31/12]]</small></small>, '''<u>[[13/5]]'''</u>
| ^^^d4 +1 oct
| 171
|-
| 58
| 1683.443
| <small><small><small>[[21/8]]</small></small></small>, [[29/11]]
| vv4 +1 oct
| 174
|-
| 59
| 1712.468
| <small><small><small>[[8/3]]</small></small></small>, [[27/10]]
| ^4 +1 oct
| 177
|-
| 60
| 1741.493
| <small><small><small>[[19/7]]</small></small></small>, '''[[30/11]]''', <small><small>[[11/4]]</small></small>
| ~4 +1 oct
| 180
|-
| 61
| 1770.518
| '''<u>[[25/9]]'''</u>, <small><small>[[14/5]]</small></small>
| vA4 +1 oct
| 183
|-
| 62
| 1799.543
| [[31/11]], '''[[17/6]]'''
| ^^A4 +1 oct, vvd5 +1 oct
| 186
|-
| 63
| 1828.568
| <small><small>[[20/7]]</small></small>, '''<u>[[23/8]]'''</u>, <small>[[26/9]]</small>
| ^d5 +1 oct
| 189
|-
| 64
| 1857.593
| <small><small><small>[[29/10]]</small></small></small>, <small>[[32/11]]</small>
| ~5 +1 oct
| 192
|-
| 65
| 1886.617
|
| v5 +1 oct
| 195
|-
| 66
| 1915.642
| <small><small><small>[[3/1]]</small></small></small>
| ^^5 +1 oct
| 198
|-
| 67
| 1944.667
| <small><small><small>[[31/10]]</small></small></small>
| vvvA5 +1 oct
| 201
|-
| 68
| 1973.692
| <small>[[28/9]]</small>, '''<u>[[25/8]]'''</u>, <small>[[22/7]]</small>
| v<sup>4</sup>m6 +1 oct
| 204
|-
| 69
| 2002.717
| [[19/6]], <small><small>[[16/5]]</small></small>
| vm6 +1 oct
| 207
|-
| 70
| 2031.742
| [[29/9]], <small>[[13/4]]</small>
| ^^m6 +1 oct
| 210
|-
| 71
| 2060.767
| '''<u>[[23/7]]'''</u>
| vvvM6 +1 oct
| 213
|-
| 72
| 2089.792
| [[10/3]]
| M6 +1 oct
| 216
|-
| 73
| 2118.816
| <small><small><small>[[27/8]]</small></small></small>, '''<u>[[17/5]]'''</u>, <small><small><small>[[24/7]]</small></small></small>
| ^^^M6 +1 oct
| 219
|-
| 74
| 2147.841
| [[31/9]]
| vvA6 +1 oct, ^^d7 +1 oct
| 222
|-
| 75
| 2176.866
| <small>[[7/2]]</small>
| vvvm7 +1 oct
| 225
|-
| 76
| 2205.891
| <small><small>[[32/9]]</small></small>, '''<u>[[25/7]]'''</u>, <small><small>[[18/5]]</small></small>
| m7 +1 oct
| 228
|-
| 77
| 2234.916
| [[29/8]], <small><small><small>[[11/3]]</small></small></small>
| ^^^m7 +1 oct
| 231
|-
| 78
| 2263.941
| <small>[[26/7]]</small>
| vvM7 +1 oct
| 234
|-
| 79
| 2292.966
| '''[[15/4]]'''
| ^M7 +1 oct
| 237
|-
| 80
| 2321.991
| <small><small>[[19/5]]</small></small>, '''[[23/6]]'''
| ^<sup>4</sup>M7 +1 oct
| 240
|-
| 81
| 2351.016
| <small><small><small>[[27/7]]</small></small></small>, [[31/8]]
| ^^^d1 +2 oct
| 243
|-
| 82
| 2380.040
|
| vv1 +2 oct
| 246
|-
| 83
| 2409.065
| <small>[[4/1]]</small>
| ^1 +2 oct
| 249
|-
| 84
| 2438.090
|
| ^<sup>4</sup>1 +2 oct
| 252
|-
| 85
| 2467.115
| [[29/7]], '''[[25/6]]'''
| ^^^d2 +2 oct
| 255
|-
| 86
| 2496.140
| <small><small>[[21/5]]</small></small>, <small>[[17/4]]</small>
| vvm2 +2 oct
| 258
|-
| 87
| 2525.165
| [[30/7]], <small><small><small>[[13/3]]</small></small></small>
| ^m2 +2 oct
| 261
|-
| 88
| 2554.190
| <small><small>[[22/5]]</small></small>
| ~2 +2 oct
| 264
|-
| 89
| 2583.215
| [[31/7]]
| vM2 +2 oct
| 267
|-
| 90
| 2612.239
| <small>[[9/2]]</small>
| ^^M2 +2 oct
| 270
|-
| 91
| 2641.264
| <small><small>[[32/7]]</small></small>, '''<u>[[23/5]]'''</u>
| vvvA2 +2 oct
| 273
|-
| 92
| 2670.289
| '''[[14/3]]'''
| v<sup>4</sup>m3 +2 oct
| 276
|-
| 93
| 2699.314
| '''<u>[[19/4]]'''</u>
| vm3 +2 oct
| 279
|-
| 94
| 2728.339
| <small><small><small>[[24/5]]</small></small></small>, '''<u>[[29/6]]'''</u>
| ^^m3 +2 oct
| 282
|-
| 95
| 2757.364
|
| vvvM3 +2 oct
| 285
|-
| 96
| 2786.389
| '''<u>[[5/1]]'''</u>
| M3 +2 oct
| 288
|-
| 97
| 2815.414
|
| ^^^M3 +2 oct
| 291
|-
| 98
| 2844.439
| '''<u>[[31/6]]'''</u>, <small><small>[[26/5]]</small></small>
| vvA3 +2 oct, ^^d4 +2 oct
| 294
|-
| 99
| 2873.463
| '''[[21/4]]'''
| vvv4 +2 oct
| 297
|-
| 100
| 2902.488
| '''[[16/3]]'''
| P4 +2 oct
| 300
|-
| 101
| 2931.513
| <small><small>[[27/5]]</small></small>
| ^^^4 +2 oct
| 303
|-
| 102
| 2960.538
| <small>[[11/2]]</small>
| vvA4 +2 oct
| 306
|-
| 103
| 2989.563
| [[28/5]], <small><small><small>[[17/3]]</small></small></small>
| ^A4 +2 oct
| 309
|-
| 104
| 3018.588
| <small><small>[[23/4]]</small></small>
| d5 +2 oct
| 312
|-
| 105
| 3047.613
| '''[[29/5]]'''
| ^^^d5 +2 oct
| 315
|-
| 106
| 3076.638
|
| vv5 +2 oct
| 318
|-
| 107
| 3105.663
| '''[[6/1]]'''
| ^5 +2 oct
| 321
|-
| 108
| 3134.687
|
| ^<sup>4</sup>5 +2 oct
| 324
|-
| 109
| 3163.712
| [[31/5]], <small>[[25/4]]</small>
| ^^^d6 +2 oct
| 327
|-
| 110
| 3192.737
| '''[[19/3]]'''
| vvm6 +2 oct
| 330
|-
| 111
| 3221.762
| <small>[[32/5]]</small>
| ^m6 +2 oct
| 333
|-
| 112
| 3250.787
| <small><small>[[13/2]]</small></small>
| ~6 +2 oct
| 336
|-
| 113
| 3279.812
| '''[[20/3]]'''
| vM6 +2 oct
| 339
|-
| 114
| 3308.837
| '''[[27/4]]'''
| ^^M6 +2 oct
| 342
|-
| 115
| 3337.862
|
| vvvA6 +2 oct
| 345
|-
| 116
| 3366.886
| '''<u>[[7/1]]'''</u>
| v<sup>4</sup>m7 +2 oct
| 348
|-
| 117
| 3395.911
|
| vm7 +2 oct
| 351
|-
| 118
| 3424.936
| '''[[29/4]]'''
| ^^m7 +2 oct
| 354
|-
| 119
| 3453.961
| '''[[22/3]]'''
| vvvM7 +2 oct
| 357
|-
| 120
| 3482.986
| [[15/2]]
| M7 +2 oct
| 360
|-
| 121
| 3512.011
| <small><small><small>[[23/3]]</small></small></small>
| ^^^M7 +2 oct
| 363
|-
| 122
| 3541.036
| '''[[31/4]]'''
| vvA7 +2 oct, ^^d1 +3 oct
| 366
|-
| 123
| 3570.061
|
| vvv1 +3 oct
| 369
|-
| 124
| 3599.086
| '''<u>[[8/1]]'''</u>
| P1 +3 oct
| 372
|-
| 125
| 3628.110
|
| ^^^1 +3 oct
| 375
|-
| 126
| 3657.135
| <small><small><small>[[25/3]]</small></small></small>
| vvA1 +3 oct, ^^d2 +3 oct
| 378
|-
| 127
| 3686.160
|
| vvvm2 +3 oct
| 381
|-
| 128
| 3715.185
| <small><small>[[17/2]]</small></small>
| m2 +3 oct
| 384
|-
| 129
| 3744.210
| [[26/3]]
| ^^^m2 +3 oct
| 387
|-
| 130
| 3773.235
|
| vvM2 +3 oct
| 390
|-
| 131
| 3802.260
| '''<u>[[9/1]]'''</u>
| ^M2 +3 oct
| 393
|-
| 132
| 3831.285
|
| ^<sup>4</sup>M2 +3 oct
| 396
|-
| 133
| 3860.309
| [[28/3]]
| ^^^d3 +3 oct
| 399
|-
| 134
| 3889.334
| <small>[[19/2]]</small>
| vvm3 +3 oct
| 402
|-
| 135
| 3918.359
| <small>[[29/3]]</small>
| ^m3 +3 oct
| 405
|-
| 136
| 3947.384
|
| ~3 +3 oct
| 408
|-
| 137
| 3976.409
| <small><small>[[10/1]]</small></small>
| vM3 +3 oct
| 411
|-
| 138
| 4005.434
|
| ^^M3 +3 oct
| 414
|-
| 139
| 4034.459
| <small>[[31/3]]</small>
| vvvA3 +3 oct
| 417
|-
| 140
| 4063.484
| <small>[[21/2]]</small>
| v<sup>4</sup>4 +3 oct
| 420
|-
| 141
| 4092.509
| [[32/3]]
| v4 +3 oct
| 423
|-
| 142
| 4121.533
|
| ^^4 +3 oct
| 426
|-
| 143
| 4150.558
| '''<u>[[11/1]]'''</u>
| vvvA4 +3 oct
| 429
|-
| 144
| 4179.583
|
| A4 +3 oct
| 432
|-
| 145
| 4208.608
|
| vd5 +3 oct
| 435
|-
| 146
| 4237.633
| <small>[[23/2]]</small>
| ^^d5 +3 oct
| 438
|-
| 147
| 4266.658
|
| vvv5 +3 oct
| 441
|-
| 148
| 4295.683
| [[12/1]]
| P5 +3 oct
| 444
|-
| 149
| 4324.708
|
| ^^^5 +3 oct
| 447
|-
| 150
| 4353.732
|
| vvA5 +3 oct, ^^d6 +3 oct
| 450
|-
| 151
| 4382.757
| <small><small>[[25/2]]</small></small>
| vvvm6 +3 oct
| 453
|-
| 152
| 4411.782
|
| m6 +3 oct
| 456
|-
| 153
| 4440.807
| '''<u>[[13/1]]'''</u>
| ^^^m6 +3 oct
| 459
|-
| 154
| 4469.832
|
| vvM6 +3 oct
| 462
|-
| 155
| 4498.857
| [[27/2]]
| ^M6 +3 oct
| 465
|-
| 156
| 4527.882
|
| ^<sup>4</sup>M6 +3 oct
| 468
|-
| 157
| 4556.907
| <small><small>[[14/1]]</small></small>
| ^^^d7 +3 oct
| 471
|-
| 158
| 4585.932
|
| vvm7 +3 oct
| 474
|-
| 159
| 4614.956
|
| ^m7 +3 oct
| 477
|-
| 160
| 4643.981
| <small><small><small>[[29/2]]</small></small></small>
| ~7 +3 oct
| 480
|-
| 161
| 4673.006
|
| vM7 +3 oct
| 483
|-
| 162
| 4702.031
| <small><small><small>[[15/1]]</small></small></small>
| ^^M7 +3 oct
| 486
|-
| 163
| 4731.056
| <small><small><small>[[31/2]]</small></small></small>
| vvvA7 +3 oct
| 489
|-
| 164
| 4760.081
|
| v<sup>4</sup>1 +4 oct
| 492
|-
| 165
| 4789.106
| <small><small>[[16/1]]</small></small>
| v1 +4 oct
| 495
|-
| 166
| 4818.131
|
| ^^1 +4 oct
| 498
|-
| 167
| 4847.156
|
| vvvA1 +4 oct
| 501
|-
| 168
| 4876.180
|
| v<sup>4</sup>m2 +4 oct
| 504
|-
| 169
| 4905.205
| '''<u>[[17/1]]'''</u>
| vm2 +4 oct
| 507
|-
| 170
| 4934.230
|
| ^^m2 +4 oct
| 510
|-
| 171
| 4963.255
|
| vvvM2 +4 oct
| 513
|-
| 172
| 4992.280
| <small><small>[[18/1]]</small></small>
| M2 +4 oct
| 516
|-
| 173
| 5021.305
|
| ^^^M2 +4 oct
| 519
|-
| 174
| 5050.330
|
| vvA2 +4 oct, ^^d3 +4 oct
| 522
|-
| 175
| 5079.355
|
| vvvm3 +4 oct
| 525
|-
| 176
| 5108.379
| <small><small>[[19/1]]</small></small>
| m3 +4 oct
| 528
|-
| 177
| 5137.404
|
| ^^^m3 +4 oct
| 531
|-
| 178
| 5166.429
|
| vvM3 +4 oct
| 534
|-
| 179
| 5195.454
| <small>[[20/1]]</small>
| ^M3 +4 oct
| 537
|-
| 180
| 5224.479
|
| ^<sup>4</sup>M3 +4 oct
| 540
|-
| 181
| 5253.504
|
| ^^^d4 +4 oct
| 543
|-
| 182
| 5282.529
| <small><small>[[21/1]]</small></small>
| vv4 +4 oct
| 546
|-
| 183
| 5311.554
|
| ^4 +4 oct
| 549
|-
| 184
| 5340.579
| <small><small>[[22/1]]</small></small>
| ~4 +4 oct
| 552
|-
| 185
| 5369.603
|
| vA4 +4 oct
| 555
|-
| 186
| 5398.628
|
| ^^A4 +4 oct, vvd5 +4 oct
| 558
|-
| 187
| 5427.653
| '''<u>[[23/1]]'''</u>
| ^d5 +4 oct
| 561
|-
| 188
| 5456.678
|
| ~5 +4 oct
| 564
|-
| 189
| 5485.703
|
| v5 +4 oct
| 567
|-
| 190
| 5514.728
| <small><small><small>[[24/1]]</small></small></small>
| ^^5 +4 oct
| 570
|-
| 191
| 5543.753
|
| vvvA5 +4 oct
| 573
|-
| 192
| 5572.778
| '''<u>[[25/1]]'''</u>
| v<sup>4</sup>m6 +4 oct
| 576
|-
| 193
| 5601.802
|
| vm6 +4 oct
| 579
|-
| 194
| 5630.827
| <small><small>[[26/1]]</small></small>
| ^^m6 +4 oct
| 582
|-
| 195
| 5659.852
|
| vvvM6 +4 oct
| 585
|-
| 196
| 5688.877
|
| M6 +4 oct
| 588
|-
| 197
| 5717.902
| <small><small>[[27/1]]</small></small>
| ^^^M6 +4 oct
| 591
|-
| 198
| 5746.927
|
| vvA6 +4 oct, ^^d7 +4 oct
| 594
|-
| 199
| 5775.952
| [[28/1]]
| vvvm7 +4 oct
| 597
|-
| 200
| 5804.977
|
| m7 +4 oct
| 600
|-
| 201
| 5834.002
| '''[[29/1]]'''
| ^^^m7 +4 oct
| 603
|-
| 202
| 5863.026
|
| vvM7 +4 oct
| 606
|-
| 203
| 5892.051
| '''[[30/1]]'''
| ^M7 +4 oct
| 609
|-
| 204
| 5921.076
|
| ^<sup>4</sup>M7 +4 oct
| 612
|-
| 205
| 5950.101
| [[31/1]]
| ^^^d1 +5 oct
| 615
|-
| 206
| 5979.126
|
| vv1 +5 oct
| 618
|-
| 207
| 6008.151
| <small>[[32/1]]</small>
| ^1 +5 oct
| 621
|}
|}
== Approximation to JI ==
=== Interval mappings ===
The following tables show how [[32-integer-limit]] intervals are represented in 186zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italics''.
{| 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
! 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
|- style="background-color: #cccccc;"
| ''[[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'''
|- style="background-color: #cccccc;"
| ''[[9/7]]''
| ''+0.289''
| ''+0.996''
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[11/9]]''
| ''+0.891''
| ''+3.069''
|-
| [[23/13]]
| -0.901
| -3.103
|-
| [[25/11]]
| +0.910
| +3.135
|- style="background-color: #cccccc;"
| ''[[8/1]]''
| ''-0.914''
| ''-3.151''
|- style="background-color: #cccccc;"
| ''[[8/5]]''
| ''-0.990''
| ''-3.409''
|-
| [[17/11]]
| +1.009
| +3.478
|- style="background-color: #cccccc;"
| ''[[8/7]]''
| ''+1.025''
| ''+3.531''
|- style="background-color: #cccccc;"
| ''[[23/9]]''
| ''+1.029''
| ''+3.546''
|-
| [[13/11]]
| +1.039
| +3.580
|- style="background-color: #cccccc;"
| ''[[25/8]]''
| ''+1.065''
| ''+3.668''
|- style="background-color: #cccccc;"
| ''[[17/8]]''
| ''+1.164''
| ''+4.012''
|- style="background-color: #cccccc;"
| ''[[27/19]]''
| ''+1.171''
| ''+4.033''
|-
| [[11/7]]
| +1.180
| +4.065
|- style="background-color: #cccccc;"
| ''[[13/8]]''
| ''+1.194''
| ''+4.114''
|-
| [[31/30]]
| +1.283
| +4.420
|-
| [[23/7]]
| +1.318
| +4.542
|-
| [[31/6]]
| +1.358
| +4.679
|- style="background-color: #cccccc;"
| ''[[9/1]]''
| ''-1.650''
| ''-5.686''
|- style="background-color: #cccccc;"
| ''[[9/5]]''
| ''-1.725''
| ''-5.944''
|- style="background-color: #cccccc;"
| ''[[20/19]]''
| ''-1.726''
| ''-5.947''
|- style="background-color: #cccccc;"
| ''[[25/9]]''
| ''+1.800''
| ''+6.203''
|- style="background-color: #cccccc;"
| ''[[19/4]]''
| ''+1.801''
| ''+6.205''
|- style="background-color: #cccccc;"
| ''[[17/9]]''
| ''+1.900''
| ''+6.547''
|- style="background-color: #cccccc;"
| ''[[24/19]]''
| ''+1.906''
| ''+6.568''
|- style="background-color: #cccccc;"
| ''[[13/9]]''
| ''+1.930''
| ''+6.649''
|-
| '''[[7/1]]'''
| '''-1.939'''
| '''-6.682'''
|-
| [[7/5]]
| -2.015
| -6.941
|- style="background-color: #cccccc;"
| ''[[31/28]]''
| ''-2.060''
| ''-7.099''
|-
| [[25/7]]
| +2.090
| +7.199
|-
| [[17/7]]
| +2.189
| +7.543
|-
| [[13/7]]
| +2.219
| +7.645
|- style="background-color: #cccccc;"
| ''[[21/20]]''
| ''+2.607''
| ''+8.984''
|- style="background-color: #cccccc;"
| ''[[21/4]]''
| ''+2.683''
| ''+9.242''
|- style="background-color: #cccccc;"
| ''[[29/28]]''
| ''-2.702''
| ''-9.308''
|- style="background-color: #cccccc;"
| ''[[32/19]]''
| ''-2.716''
| ''-9.356''
|-
| [[19/3]]
| -2.821
| -9.719
|-
| [[19/15]]
| -2.896
| -9.977
|- style="background-color: #cccccc;"
| ''[[27/20]]''
| ''+2.897''
| ''+9.980''
|- style="background-color: #cccccc;"
| ''[[27/4]]''
| ''+2.972''
| ''+10.238''
|- style="background-color: #cccccc;"
| ''[[32/31]]''
| ''+3.085''
| ''+10.630''
|- style="background-color: #cccccc;"
| ''[[15/14]]''
| ''-3.343''
| ''-11.519''
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[32/21]]''
| ''-3.597''
| ''-12.393''
|-
| [[6/5]]
| +3.632
| +12.515
|-
| [[6/1]]
| +3.708
| +12.774
|- style="background-color: #cccccc;"
| ''[[32/29]]''
| ''+3.726''
| ''+12.839''
|- style="background-color: #cccccc;"
| ''[[28/19]]''
| ''-3.741''
| ''-12.887''
|-
| [[30/1]]
| +3.783
| +13.032
|- style="background-color: #cccccc;"
| ''[[32/27]]''
| ''-3.886''
| ''-13.389''
|- style="background-color: #cccccc;"
| ''[[31/4]]''
| ''-4.000''
| ''-13.781''
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[16/15]]''
| ''+4.368''
| ''+15.050''
|-
| [[30/23]]
| +4.404
| +15.172
|-
| '''[[29/1]]'''
| '''+4.424'''
| '''+15.243'''
|- style="background-color: #cccccc;"
| ''[[16/3]]''
| ''+4.443''
| ''+15.309''
|-
| [[11/6]]
| -4.467
| -15.391
|- style="background-color: #cccccc;"
| ''[[22/15]]''
| ''+4.523''
| ''+15.583''
|-
| [[30/11]]
| +4.542
| +15.649
|- style="background-color: #cccccc;"
| ''[[20/3]]''
| ''-4.547''
| ''-15.666''
|- style="background-color: #cccccc;"
| ''[[22/3]]''
| ''+4.598''
| ''+15.842''
|- style="background-color: #cccccc;"
| ''[[4/3]]''
| ''-4.622''
| ''-15.924''
|- style="background-color: #cccccc;"
| ''[[29/4]]''
| ''-4.641''
| ''-15.990''
|- style="background-color: #cccccc;"
| ''[[15/4]]''
| ''+4.697''
| ''+16.183''
|- style="background-color: #cccccc;"
| ''[[29/20]]''
| ''-4.716''
| ''-16.248''
|-
| [[31/13]]
| +4.786
| +16.489
|-
| [[31/17]]
| +4.816
| +16.591
|- style="background-color: #cccccc;"
| ''[[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'''
|- style="background-color: #cccccc;"
| ''[[27/14]]''
| ''-5.069''
| ''-17.463''
|-
| [[29/11]]
| +5.184
| +17.860
|- style="background-color: #cccccc;"
| ''[[15/2]]''
| ''-5.283''
| ''-18.201''
|- style="background-color: #cccccc;"
| ''[[29/8]]''
| ''+5.339''
| ''+18.394''
|- style="background-color: #cccccc;"
| ''[[3/2]]''
| ''-5.358''
| ''-18.459''
|- style="background-color: #cccccc;"
| ''[[10/3]]''
| ''+5.433''
| ''+18.718''
|-
| [[12/11]]
| -5.513
| -18.993
|- style="background-color: #cccccc;"
| ''[[32/3]]''
| ''-5.536''
| ''-19.075''
|- style="background-color: #cccccc;"
| ''[[26/15]]''
| ''+5.562''
| ''+19.164''
|- style="background-color: #cccccc;"
| ''[[32/15]]''
| ''-5.612''
| ''-19.334''
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[31/8]]''
| ''+5.980''
| ''+20.603''
|- style="background-color: #cccccc;"
| ''[[29/9]]''
| ''+6.075''
| ''+20.929''
|- style="background-color: #cccccc;"
| ''[[27/16]]''
| ''-6.094''
| ''-20.994''
|- style="background-color: #cccccc;"
| ''[[19/14]]''
| ''-6.239''
| ''-21.496''
|- style="background-color: #cccccc;"
| ''[[27/22]]''
| ''-6.248''
| ''-21.528''
|-
| [[12/1]]
| -6.272
| -21.610
|-
| [[12/5]]
| -6.347
| -21.869
|-
| [[29/7]]
| +6.364
| +21.925
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[22/21]]''
| ''+6.538''
| ''+22.524''
|-
| [[13/12]]
| +6.552
| +22.573
|- style="background-color: #cccccc;"
| ''[[28/3]]''
| ''-6.561''
| ''-22.606''
|- style="background-color: #cccccc;"
| ''[[28/15]]''
| ''-6.637''
| ''-22.865''
|-
| [[31/21]]
| -6.682
| -23.023
|- style="background-color: #cccccc;"
| ''[[31/9]]''
| ''+6.716''
| ''+23.138''
|- style="background-color: #cccccc;"
| ''[[28/13]]''
| ''+6.846''
| ''+23.588''
|- style="background-color: #cccccc;"
| ''[[28/17]]''
| ''+6.876''
| ''+23.690''
|- style="background-color: #cccccc;"
| ''[[31/27]]''
| ''-6.972''
| ''-24.019''
|- style="background-color: #cccccc;"
| ''[[28/25]]''
| ''+6.976''
| ''+24.034''
|-
| [[31/7]]
| +7.005
| +24.134
|- style="background-color: #cccccc;"
| ''[[27/2]]''
| ''-7.008''
| ''-24.145''
|- style="background-color: #cccccc;"
| ''[[28/5]]''
| ''+7.051''
| ''+24.292''
|- style="background-color: #cccccc;"
| ''[[27/10]]''
| ''-7.083''
| ''-24.404''
|-
| [[30/19]]
| -7.084
| -24.406
|- style="background-color: #cccccc;"
| ''[[28/1]]''
| ''+7.126''
| ''+24.551''
|-
| [[19/6]]
| +7.159
| +24.665
|- style="background-color: #cccccc;"
| ''[[19/16]]''
| ''-7.264''
| ''-25.027''
|- style="background-color: #cccccc;"
| ''[[27/26]]''
| ''-7.288''
| ''-25.108''
|- style="background-color: #cccccc;"
| ''[[21/2]]''
| ''-7.297''
| ''-25.141''
|-
| [[29/21]]
| -7.324
| -25.232
|- style="background-color: #cccccc;"
| ''[[21/10]]''
| ''-7.372''
| ''-25.400''
|- style="background-color: #cccccc;"
| ''[[22/19]]''
| ''+7.419''
| ''+25.561''
|- style="background-color: #cccccc;"
| ''[[26/21]]''
| ''+7.577''
| ''+26.104''
|- style="background-color: #cccccc;"
| ''[[29/27]]''
| ''-7.613''
| ''-26.228''
|- style="background-color: #cccccc;"
| ''[[31/24]]''
| ''-7.707''
| ''-26.554''
|- style="background-color: #cccccc;"
| ''[[28/23]]''
| ''+7.747''
| ''+26.691''
|-
| [[26/7]]
| -7.761
| -26.739
|- style="background-color: #cccccc;"
| ''[[32/13]]''
| ''+7.871''
| ''+27.119''
|- style="background-color: #cccccc;"
| ''[[28/11]]''
| ''+7.886''
| ''+27.168''
|- style="background-color: #cccccc;"
| ''[[32/17]]''
| ''+7.901''
| ''+27.221''
|-
| [[10/7]]
| -7.965
| -27.443
|- style="background-color: #cccccc;"
| ''[[32/25]]''
| ''+8.001''
| ''+27.565''
|-
| [[7/2]]
| +8.040
| +27.702
|- style="background-color: #cccccc;"
| ''[[26/9]]''
| ''-8.050''
| ''-27.735''
|- style="background-color: #cccccc;"
| ''[[32/5]]''
| ''+8.076''
| ''+27.824''
|- style="background-color: #cccccc;"
| ''[[32/1]]''
| ''+8.151''
| ''+28.082''
|- style="background-color: #cccccc;"
| ''[[19/2]]''
| ''-8.179''
| ''-28.178''
|- style="background-color: #cccccc;"
| ''[[19/10]]''
| ''-8.254''
| ''-28.437''
|- style="background-color: #cccccc;"
| ''[[10/9]]''
| ''-8.254''
| ''-28.439''
|- style="background-color: #cccccc;"
| ''[[9/2]]''
| ''+8.329''
| ''+28.698''
|- style="background-color: #cccccc;"
| ''[[29/24]]''
| ''-8.348''
| ''-28.763''
|- style="background-color: #cccccc;"
| ''[[26/19]]''
| ''+8.458''
| ''+29.141''
|-
| [[31/3]]
| -8.622
| -29.705
|-
| [[31/15]]
| -8.697
| -29.964
|- style="background-color: #cccccc;"
| ''[[32/23]]''
| ''+8.772''
| ''+30.222''
|- style="background-color: #cccccc;"
| ''[[28/9]]''
| ''+8.776''
| ''+30.237''
|- style="background-color: #cccccc;"
| ''[[13/4]]''
| ''-8.786''
| ''-30.270''
|-
| [[22/7]]
| -8.800
| -30.319
|- style="background-color: #cccccc;"
| ''[[17/4]]''
| ''-8.815''
| ''-30.372''
|- style="background-color: #cccccc;"
| ''[[20/13]]''
| ''+8.861''
| ''+30.529''
|- style="background-color: #cccccc;"
| ''[[20/17]]''
| ''+8.891''
| ''+30.631''
|- style="background-color: #cccccc;"
| ''[[32/11]]''
| ''+8.910''
| ''+30.699''
|- style="background-color: #cccccc;"
| ''[[25/4]]''
| ''-8.915''
| ''-30.716''
|-
| [[26/11]]
| -8.941
| -30.803
|- style="background-color: #cccccc;"
| ''[[16/7]]''
| ''-8.955''
| ''-30.852''
|- style="background-color: #cccccc;"
| ''[[5/4]]''
| ''-8.990''
| ''-30.974''
|- style="background-color: #cccccc;"
| ''[[4/1]]''
| ''+9.065''
| ''+31.233''
|-
| [[26/23]]
| -9.079
| -31.281
|- style="background-color: #cccccc;"
| ''[[22/9]]''
| ''-9.089''
| ''-31.315''
|- style="background-color: #cccccc;"
| ''[[20/1]]''
| ''+9.140''
| ''+31.492''
|-
| [[11/10]]
| +9.145
| +31.508
|-
| [[11/2]]
| +9.220
| +31.766
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[23/4]]''
| ''-9.686''
| ''-33.373''
|- style="background-color: #cccccc;"
| ''[[18/7]]''
| ''-9.691''
| ''-33.387''
|-
| [[26/1]]
| -9.700
| -33.421
|- style="background-color: #cccccc;"
| ''[[23/20]]''
| ''-9.762''
| ''-33.632''
|-
| [[26/5]]
| -9.775
| -33.679
|- style="background-color: #cccccc;"
| ''[[32/9]]''
| ''+9.801''
| ''+33.768''
|- style="background-color: #cccccc;"
| ''[[11/4]]''
| ''-9.825''
| ''-33.850''
|-
| [[26/25]]
| -9.850
| -33.938
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[32/7]]''
| ''+10.090''
| ''+34.764''
|-
| [[23/22]]
| +10.118
| +34.861
|-
| [[25/2]]
| +10.130
| +34.901
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[14/9]]''
| ''-10.269''
| ''-35.380''
|- style="background-color: #cccccc;"
| ''[[23/16]]''
| ''+10.273''
| ''+35.394''
|-
| [[19/13]]
| +10.587
| +36.475
|-
| [[19/17]]
| +10.617
| +36.577
|-
| [[29/12]]
| +10.697
| +36.853
|- style="background-color: #cccccc;"
| ''[[9/4]]''
| ''-10.716''
| ''-36.919''
|-
| [[25/19]]
| -10.716
| -36.921
|-
| [[22/1]]
| -10.739
| -37.001
|- style="background-color: #cccccc;"
| ''[[20/9]]''
| ''+10.791''
| ''+37.177''
|-
| [[19/5]]
| +10.791
| +37.180
|-
| [[22/5]]
| -10.814
| -37.259
|-
| '''[[19/1]]'''
| '''+10.866'''
| '''+37.438'''
|- style="background-color: #cccccc;"
| ''[[18/11]]''
| ''-10.870''
| ''-37.452''
|-
| [[25/22]]
| +10.890
| +37.518
|- style="background-color: #cccccc;"
| ''[[16/1]]''
| ''-10.894''
| ''-37.534''
|- style="background-color: #cccccc;"
| ''[[16/5]]''
| ''-10.969''
| ''-37.793''
|-
| [[22/17]]
| -10.989
| -37.862
|- style="background-color: #cccccc;"
| ''[[7/4]]''
| ''-11.005''
| ''-37.915''
|- style="background-color: #cccccc;"
| ''[[23/18]]''
| ''+11.009''
| ''+37.929''
|-
| [[22/13]]
| -11.019
| -37.964
|- style="background-color: #cccccc;"
| ''[[25/16]]''
| ''+11.044''
| ''+38.052''
|- style="background-color: #cccccc;"
| ''[[20/7]]''
| ''+11.080''
| ''+38.174''
|- style="background-color: #cccccc;"
| ''[[17/16]]''
| ''+11.144''
| ''+38.395''
|-
| [[14/11]]
| -11.160
| -38.448
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[18/1]]''
| ''-11.630''
| ''-40.069''
|-
| [[21/5]]
| +11.673
| +40.216
|- style="background-color: #cccccc;"
| ''[[18/5]]''
| ''-11.705''
| ''-40.328''
|-
| [[21/1]]
| +11.748
| +40.475
|- style="background-color: #cccccc;"
| ''[[27/13]]''
| ''+11.758''
| ''+40.508''
|- style="background-color: #cccccc;"
| ''[[25/18]]''
| ''+11.780''
| ''+40.587''
|- style="background-color: #cccccc;"
| ''[[19/8]]''
| ''+11.781''
| ''+40.589''
|- style="background-color: #cccccc;"
| ''[[27/17]]''
| ''+11.787''
| ''+40.610''
|- style="background-color: #cccccc;"
| ''[[18/17]]''
| ''-11.880''
| ''-40.930''
|- style="background-color: #cccccc;"
| ''[[19/12]]''
| ''-11.886''
| ''-40.952''
|- style="background-color: #cccccc;"
| ''[[27/25]]''
| ''+11.887''
| ''+40.954''
|- style="background-color: #cccccc;"
| ''[[18/13]]''
| ''-11.910''
| ''-41.032''
|-
| [[14/1]]
| -11.919
| -41.066
|- style="background-color: #cccccc;"
| ''[[27/5]]''
| ''+11.962''
| ''+41.213''
|-
| [[14/5]]
| -11.994
| -41.324
|- style="background-color: #cccccc;"
| ''[[27/1]]''
| ''+12.037''
| ''+41.471''
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[24/13]]''
| ''+12.493''
| ''+43.044''
|-
| [[21/11]]
| +12.507
| +43.092
|- style="background-color: #cccccc;"
| ''[[19/9]]''
| ''+12.517''
| ''+43.124''
|- style="background-color: #cccccc;"
| ''[[24/17]]''
| ''+12.523''
| ''+43.146''
|- style="background-color: #cccccc;"
| ''[[25/24]]''
| ''-12.623''
| ''-43.489''
|- style="background-color: #cccccc;"
| ''[[27/23]]''
| ''+12.658''
| ''+43.611''
|- style="background-color: #cccccc;"
| ''[[21/8]]''
| ''+12.662''
| ''+43.626''
|- style="background-color: #cccccc;"
| ''[[29/14]]''
| ''-12.681''
| ''-43.691''
|- style="background-color: #cccccc;"
| ''[[24/5]]''
| ''+12.698''
| ''+43.748''
|- style="background-color: #cccccc;"
| ''[[24/1]]''
| ''+12.773''
| ''+44.006''
|- style="background-color: #cccccc;"
| ''[[27/11]]''
| ''+12.797''
| ''+44.089''
|-
| [[19/7]]
| +12.806
| +44.120
|- style="background-color: #cccccc;"
| ''[[27/8]]''
| ''+12.951''
| ''+44.622''
|-
| [[29/18]]
| -12.970
| -44.687
|- style="background-color: #cccccc;"
| ''[[31/16]]''
| ''-13.065''
| ''-45.014''
|- style="background-color: #cccccc;"
| ''[[31/22]]''
| ''-13.220''
| ''-45.547''
|- style="background-color: #cccccc;"
| ''[[15/7]]''
| ''-13.323''
| ''-45.902''
|- style="background-color: #cccccc;"
| ''[[24/23]]''
| ''+13.394''
| ''+46.147''
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[24/11]]''
| ''+13.532''
| ''+46.624''
|-
| [[25/3]]
| -13.537
| -46.640
|-
| [[5/3]]
| -13.612
| -46.898
|-
| '''[[3/1]]'''
| '''+13.687'''
| '''+47.157'''
|- style="background-color: #cccccc;"
| ''[[29/16]]''
| ''-13.706''
| ''-47.223''
|-
| [[15/1]]
| +13.762
| +47.416
|- style="background-color: #cccccc;"
| ''[[29/22]]''
| ''-13.861''
| ''-47.756''
|- style="background-color: #cccccc;"
| ''[[27/7]]''
| ''+13.976''
| ''+48.153''
|- style="background-color: #cccccc;"
| ''[[31/2]]''
| ''-13.980''
| ''-48.164''
|- style="background-color: #cccccc;"
| ''[[31/10]]''
| ''-14.055''
| ''-48.423''
|-
| [[29/26]]
| +14.125
| +48.664
|- style="background-color: #cccccc;"
| ''[[31/26]]''
| ''-14.259''
| ''-49.127''
|-
| [[23/3]]
| -14.308
| -49.297
|-
| [[24/7]]
| -14.313
| -49.312
|-
| [[29/10]]
| +14.329
| +49.368
|- style="background-color: #cccccc;"
| ''[[15/8]]''
| ''-14.348''
| ''-49.434''
|-
| [[23/15]]
| -14.384
| -49.556
|-
| [[29/2]]
| +14.404
| +49.627
|- style="background-color: #cccccc;"
| ''[[8/3]]''
| ''+14.423''
| ''+49.692''
|-
| [[11/3]]
| -14.447
| -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
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| ''[[32/29]]''
| ''-54.323''
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| ''[[32/27]]''
| ''-90.961''
| ''-313.389''
|}
[[Category:Zeta peak indexes]]
Retrieved from "https://en.xen.wiki/w/186zpi"