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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="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
! 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]]
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| ''+138.395''
|- style="background-color: #cccccc;"
| ''[[16/13]]''
| ''-40.199''
| ''-138.497''
|- style="background-color: #cccccc;"
| ''[[27/13]]''
| ''+40.782''
| ''+140.508''
|- style="background-color: #cccccc;"
| ''[[19/8]]''
| ''+40.806''
| ''+140.589''
|- style="background-color: #cccccc;"
| ''[[27/17]]''
| ''+40.812''
| ''+140.610''
|- style="background-color: #cccccc;"
| ''[[27/25]]''
| ''+40.912''
| ''+140.954''
|- style="background-color: #cccccc;"
| ''[[27/5]]''
| ''+40.987''
| ''+141.213''
|- style="background-color: #cccccc;"
| ''[[27/1]]''
| ''+41.062''
| ''+141.471''
|- style="background-color: #cccccc;"
| ''[[27/23]]''
| ''+41.683''
| ''+143.611''
|- style="background-color: #cccccc;"
| ''[[21/8]]''
| ''+41.687''
| ''+143.626''
|- style="background-color: #cccccc;"
| ''[[27/11]]''
| ''+41.822''
| ''+144.089''
|- style="background-color: #cccccc;"
| ''[[27/7]]''
| ''+43.001''
| ''+148.153''
|- style="background-color: #cccccc;"
| ''[[8/3]]''
| ''-43.627''
| ''-150.308''
|- style="background-color: #cccccc;"
| ''[[15/8]]''
| ''+43.702''
| ''+150.566''
|- style="background-color: #cccccc;"
| ''[[29/16]]''
| ''+44.343''
| ''+152.777''
|- style="background-color: #cccccc;"
| ''[[31/16]]''
| ''+44.985''
| ''+154.986''
|- style="background-color: #cccccc;"
| ''[[20/9]]''
| ''-47.259''
| ''-162.823''
|- style="background-color: #cccccc;"
| ''[[9/4]]''
| ''+47.334''
| ''+163.081''
|- style="background-color: #cccccc;"
| ''[[32/7]]''
| ''-47.959''
| ''-165.236''
|- style="background-color: #cccccc;"
| ''[[32/11]]''
| ''-49.139''
| ''-169.301''
|- style="background-color: #cccccc;"
| ''[[28/9]]''
| ''-49.274''
| ''-169.763''
|- style="background-color: #cccccc;"
| ''[[32/23]]''
| ''-49.278''
| ''-169.778''
|- style="background-color: #cccccc;"
| ''[[32/1]]''
| ''-49.899''
| ''-171.918''
|- style="background-color: #cccccc;"
| ''[[32/5]]''
| ''-49.974''
| ''-172.176''
|- style="background-color: #cccccc;"
| ''[[32/25]]''
| ''-50.049''
| ''-172.435''
|- style="background-color: #cccccc;"
| ''[[32/17]]''
| ''-50.149''
| ''-172.779''
|- style="background-color: #cccccc;"
| ''[[32/13]]''
| ''-50.178''
| ''-172.881''
|- style="background-color: #cccccc;"
| ''[[27/26]]''
| ''+50.762''
| ''+174.892''
|- style="background-color: #cccccc;"
| ''[[19/16]]''
| ''+50.786''
| ''+174.973''
|- style="background-color: #cccccc;"
| ''[[27/10]]''
| ''+50.967''
| ''+175.596''
|- style="background-color: #cccccc;"
| ''[[27/2]]''
| ''+51.042''
| ''+175.855''
|- style="background-color: #cccccc;"
| ''[[21/16]]''
| ''+51.667''
| ''+178.009''
|- style="background-color: #cccccc;"
| ''[[27/22]]''
| ''+51.801''
| ''+178.472''
|- style="background-color: #cccccc;"
| ''[[27/14]]''
| ''+52.981''
| ''+182.537''
|- style="background-color: #cccccc;"
| ''[[16/3]]''
| ''-53.606''
| ''-184.691''
|- style="background-color: #cccccc;"
| ''[[16/15]]''
| ''-53.682''
| ''-184.950''
|- style="background-color: #cccccc;"
| ''[[32/29]]''
| ''-54.323''
| ''-187.161''
|- style="background-color: #cccccc;"
| ''[[32/31]]''
| ''-54.964''
| ''-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''
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|- style="background-color: #cccccc;"
| ''[[32/27]]''
| ''-90.961''
| ''-313.389''
|}
[[Category:Zeta peak indexes]]

Latest revision as of 01:07, 20 August 2025

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 8ve 		Step size
(cents) 		Height Integral Gap Edo Octave (cents) Consistent Distinct
Size Stretch
186zpi 41.343835 29.024883 1.87659 0.241233 11.567493 41edo 1190.020215 −9.979785 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].

Intervals in 186zpi
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 %
⟨124 196] 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 +1 oct 120
41 1190.020 2/1 v1 +1 oct 123
42 1219.045 ^^1 +1 oct 126
43 1248.070 31/15, 29/14 vvvA1 +1 oct 129
44 1277.095 27/13, 25/12, 23/11, 21/10 v4m2 +1 oct 132
45 1306.120 19/9, 17/8, 32/15, 15/7 vm2 +1 oct 135
46 1335.145 28/13, 13/6 ^^m2 +1 oct 138
47 1364.170 24/11, 11/5, 31/14 vvvM2 +1 oct 141
48 1393.194 20/9, 29/13, 9/4 M2 +1 oct 144
49 1422.219 25/11, 16/7 ^^^M2 +1 oct 147
50 1451.244 23/10, 30/13 vvA2 +1 oct, ^^d3 +1 oct 150
51 1480.269 7/3, 26/11 vvvm3 +1 oct 153
52 1509.294 19/8, 31/13, 12/5 m3 +1 oct 156
53 1538.319 29/12, 17/7, 22/9 ^^^m3 +1 oct 159
54 1567.344 27/11, 32/13 vvM3 +1 oct 162
55 1596.369 5/2 ^M3 +1 oct 165
56 1625.393 28/11, 23/9, 18/7 ^4M3 +1 oct 168
57 1654.418 31/12, 13/5 ^^^d4 +1 oct 171
58 1683.443 21/8, 29/11 vv4 +1 oct 174
59 1712.468 8/3, 27/10 ^4 +1 oct 177
60 1741.493 19/7, 30/11, 11/4 ~4 +1 oct 180
61 1770.518 25/9, 14/5 vA4 +1 oct 183
62 1799.543 31/11, 17/6 ^^A4 +1 oct, vvd5 +1 oct 186
63 1828.568 20/7, 23/8, 26/9 ^d5 +1 oct 189
64 1857.593 29/10, 32/11 ~5 +1 oct 192
65 1886.617 v5 +1 oct 195
66 1915.642 3/1 ^^5 +1 oct 198
67 1944.667 31/10 vvvA5 +1 oct 201
68 1973.692 28/9, 25/8, 22/7 v4m6 +1 oct 204
69 2002.717 19/6, 16/5 vm6 +1 oct 207
70 2031.742 29/9, 13/4 ^^m6 +1 oct 210
71 2060.767 23/7 vvvM6 +1 oct 213
72 2089.792 10/3 M6 +1 oct 216
73 2118.816 27/8, 17/5, 24/7 ^^^M6 +1 oct 219
74 2147.841 31/9 vvA6 +1 oct, ^^d7 +1 oct 222
75 2176.866 7/2 vvvm7 +1 oct 225
76 2205.891 32/9, 25/7, 18/5 m7 +1 oct 228
77 2234.916 29/8, 11/3 ^^^m7 +1 oct 231
78 2263.941 26/7 vvM7 +1 oct 234
79 2292.966 15/4 ^M7 +1 oct 237
80 2321.991 19/5, 23/6 ^4M7 +1 oct 240
81 2351.016 27/7, 31/8 ^^^d1 +2 oct 243
82 2380.040 vv1 +2 oct 246
83 2409.065 4/1 ^1 +2 oct 249
84 2438.090 ^41 +2 oct 252
85 2467.115 29/7, 25/6 ^^^d2 +2 oct 255
86 2496.140 21/5, 17/4 vvm2 +2 oct 258
87 2525.165 30/7, 13/3 ^m2 +2 oct 261
88 2554.190 22/5 ~2 +2 oct 264
89 2583.215 31/7 vM2 +2 oct 267
90 2612.239 9/2 ^^M2 +2 oct 270
91 2641.264 32/7, 23/5 vvvA2 +2 oct 273
92 2670.289 14/3 v4m3 +2 oct 276
93 2699.314 19/4 vm3 +2 oct 279
94 2728.339 24/5, 29/6 ^^m3 +2 oct 282
95 2757.364 vvvM3 +2 oct 285
96 2786.389 5/1 M3 +2 oct 288
97 2815.414 ^^^M3 +2 oct 291
98 2844.439 31/6, 26/5 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 27/5 ^^^4 +2 oct 303
102 2960.538 11/2 vvA4 +2 oct 306
103 2989.563 28/5, 17/3 ^A4 +2 oct 309
104 3018.588 23/4 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 ^45 +2 oct 324
109 3163.712 31/5, 25/4 ^^^d6 +2 oct 327
110 3192.737 19/3 vvm6 +2 oct 330
111 3221.762 32/5 ^m6 +2 oct 333
112 3250.787 13/2 ~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 7/1 v4m7 +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 23/3 ^^^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 8/1 P1 +3 oct 372
125 3628.110 ^^^1 +3 oct 375
126 3657.135 25/3 vvA1 +3 oct, ^^d2 +3 oct 378
127 3686.160 vvvm2 +3 oct 381
128 3715.185 17/2 m2 +3 oct 384
129 3744.210 26/3 ^^^m2 +3 oct 387
130 3773.235 vvM2 +3 oct 390
131 3802.260 9/1 ^M2 +3 oct 393
132 3831.285 ^4M2 +3 oct 396
133 3860.309 28/3 ^^^d3 +3 oct 399
134 3889.334 19/2 vvm3 +3 oct 402
135 3918.359 29/3 ^m3 +3 oct 405
136 3947.384 ~3 +3 oct 408
137 3976.409 10/1 vM3 +3 oct 411
138 4005.434 ^^M3 +3 oct 414
139 4034.459 31/3 vvvA3 +3 oct 417
140 4063.484 21/2 v44 +3 oct 420
141 4092.509 32/3 v4 +3 oct 423
142 4121.533 ^^4 +3 oct 426
143 4150.558 11/1 vvvA4 +3 oct 429
144 4179.583 A4 +3 oct 432
145 4208.608 vd5 +3 oct 435
146 4237.633 23/2 ^^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 25/2 vvvm6 +3 oct 453
152 4411.782 m6 +3 oct 456
153 4440.807 13/1 ^^^m6 +3 oct 459
154 4469.832 vvM6 +3 oct 462
155 4498.857 27/2 ^M6 +3 oct 465
156 4527.882 ^4M6 +3 oct 468
157 4556.907 14/1 ^^^d7 +3 oct 471
158 4585.932 vvm7 +3 oct 474
159 4614.956 ^m7 +3 oct 477
160 4643.981 29/2 ~7 +3 oct 480
161 4673.006 vM7 +3 oct 483
162 4702.031 15/1 ^^M7 +3 oct 486
163 4731.056 31/2 vvvA7 +3 oct 489
164 4760.081 v41 +4 oct 492
165 4789.106 16/1 v1 +4 oct 495
166 4818.131 ^^1 +4 oct 498
167 4847.156 vvvA1 +4 oct 501
168 4876.180 v4m2 +4 oct 504
169 4905.205 17/1 vm2 +4 oct 507
170 4934.230 ^^m2 +4 oct 510
171 4963.255 vvvM2 +4 oct 513
172 4992.280 18/1 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 19/1 m3 +4 oct 528
177 5137.404 ^^^m3 +4 oct 531
178 5166.429 vvM3 +4 oct 534
179 5195.454 20/1 ^M3 +4 oct 537
180 5224.479 ^4M3 +4 oct 540
181 5253.504 ^^^d4 +4 oct 543
182 5282.529 21/1 vv4 +4 oct 546
183 5311.554 ^4 +4 oct 549
184 5340.579 22/1 ~4 +4 oct 552
185 5369.603 vA4 +4 oct 555
186 5398.628 ^^A4 +4 oct, vvd5 +4 oct 558
187 5427.653 23/1 ^d5 +4 oct 561
188 5456.678 ~5 +4 oct 564
189 5485.703 v5 +4 oct 567
190 5514.728 24/1 ^^5 +4 oct 570
191 5543.753 vvvA5 +4 oct 573
192 5572.778 25/1 v4m6 +4 oct 576
193 5601.802 vm6 +4 oct 579
194 5630.827 26/1 ^^m6 +4 oct 582
195 5659.852 vvvM6 +4 oct 585
196 5688.877 M6 +4 oct 588
197 5717.902 27/1 ^^^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 ^4M7 +4 oct 612
205 5950.101 31/1 ^^^d1 +5 oct 615
206 5979.126 vv1 +5 oct 618
207 6008.151 32/1 ^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.

32-integer-limit intervals in 186zpi (by direct approximation)
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
32-integer-limit intervals in 186zpi (by patent val mapping)
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
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
26/23 -9.079 -31.281
11/10 +9.145 +31.508
11/2 +9.220 +31.766
29/3 -9.263 -31.914
23/10 +9.284 +31.985
29/15 -9.338 -32.173
23/2 +9.359 +32.243
26/1 -9.700 -33.421
26/5 -9.775 -33.679
26/25 -9.850 -33.938
10/1 -9.905 -34.125
26/17 -9.950 -34.282
2/1 -9.980 -34.384
5/2 +10.055 +34.642
23/22 +10.118 +34.861
25/2 +10.130 +34.901
17/10 +10.155 +34.986
13/10 +10.184 +35.088
17/2 +10.230 +35.244
13/2 +10.259 +35.346
19/13 +10.587 +36.475
19/17 +10.617 +36.577
29/12 +10.697 +36.853
25/19 -10.716 -36.921
22/1 -10.739 -37.001
19/5 +10.791 +37.180
22/5 -10.814 -37.259
19/1 +10.866 +37.438
25/22 +10.890 +37.518
22/17 -10.989 -37.862
22/13 -11.019 -37.964
14/11 -11.160 -38.448
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
21/5 +11.673 +40.216
21/1 +11.748 +40.475
14/1 -11.919 -41.066
14/5 -11.994 -41.324
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
21/11 +12.507 +43.092
19/7 +12.806 +44.120
29/18 -12.970 -44.687
13/3 -13.408 -46.194
17/3 -13.437 -46.296
15/13 +13.483 +46.453
17/15 -13.513 -46.555
25/3 -13.537 -46.640
5/3 -13.612 -46.898
3/1 +13.687 +47.157
15/1 +13.762 +47.416
29/26 +14.125 +48.664
23/3 -14.308 -49.297
24/7 -14.313 -49.312
29/10 +14.329 +49.368
23/15 -14.384 -49.556
29/2 +14.404 +49.627
11/3 -14.447 -49.774
15/11 +14.522 +50.033
31/26 +14.766 +50.873
31/10 +14.970 +51.577
31/2 +15.045 +51.836
29/22 +15.164 +52.244
24/11 -15.492 -53.376
7/3 -15.627 -53.839
24/23 -15.631 -53.853
15/7 +15.702 +54.098
31/22 +15.805 +54.453
24/1 -16.252 -55.994
24/5 -16.327 -56.252
29/14 +16.344 +56.309
25/24 +16.402 +56.511
24/17 -16.502 -56.854
19/9 -16.508 -56.876
24/13 -16.532 -56.956
31/14 +16.985 +58.518
18/13 +17.115 +58.968
19/12 +17.139 +59.048
18/17 +17.145 +59.070
25/18 -17.245 -59.413
18/5 +17.320 +59.672
18/1 +17.395 +59.931
20/7 -17.945 -61.826
23/18 -18.016 -62.071
7/4 +18.020 +62.085
18/11 +18.154 +62.548
20/11 -19.125 -65.891
11/4 +19.200 +66.150
23/20 +19.263 +66.368
18/7 +19.334 +66.613
23/4 +19.338 +66.627
20/1 -19.884 -68.508
4/1 -19.960 -68.767
5/4 +20.035 +69.026
25/4 +20.110 +69.284
20/17 -20.134 -69.369
20/13 -20.164 -69.471
17/4 +20.209 +69.628
13/4 +20.239 +69.730
26/19 -20.567 -70.859
29/24 +20.676 +71.237
19/10 +20.771 +71.563
19/2 +20.846 +71.822
28/11 -21.139 -72.832
28/23 -21.278 -73.309
31/24 +21.318 +73.446
26/21 -21.448 -73.896
22/19 -21.606 -74.439
21/10 +21.653 +74.600
21/2 +21.728 +74.859
28/1 -21.899 -75.449
28/5 -21.974 -75.708
28/25 -22.049 -75.966
28/17 -22.149 -76.310
28/13 -22.178 -76.412
31/9 -22.309 -76.862
22/21 -22.487 -77.476
19/14 +22.786 +78.504
29/9 -22.950 -79.071
26/3 -23.388 -80.578
26/15 -23.463 -80.836
10/3 -23.592 -81.282
3/2 +23.667 +81.541
15/2 +23.742 +81.799
29/20 +24.309 +83.752
29/4 +24.384 +84.010
22/3 -24.427 -84.158
22/15 -24.502 -84.417
31/20 +24.950 +85.961
31/4 +25.025 +86.219
14/3 -25.607 -88.223
15/14 +25.682 +88.481
29/28 +26.323 +90.692
31/28 +26.965 +92.901
13/9 -27.095 -93.351
24/19 -27.119 -93.432
17/9 -27.125 -93.453
25/9 -27.224 -93.797
9/5 +27.300 +94.056
9/1 +27.375 +94.314
23/9 -27.996 -96.454
8/7 -28.000 -96.469
11/9 -28.134 -96.931
11/8 +29.180 +100.533
9/7 +29.314 +100.996
23/8 +29.318 +101.011
8/1 -29.939 -103.151
8/5 -30.014 -103.409
25/8 +30.090 +103.668
17/8 +30.189 +104.012
27/19 +30.195 +104.033
13/8 +30.219 +104.114
20/19 -30.751 -105.947
19/4 +30.826 +106.205
21/20 +31.632 +108.984
21/4 +31.707 +109.242
28/19 -32.765 -112.887
20/3 -33.572 -115.666
4/3 -33.647 -115.924
15/4 +33.722 +116.183
29/8 +34.364 +118.394
31/8 +35.005 +120.603
28/3 -35.586 -122.606
28/15 -35.661 -122.865
31/27 -35.996 -124.019
29/27 -36.638 -126.228
26/9 -37.075 -127.735
10/9 -37.279 -128.439
9/2 +37.354 +128.698
16/7 -37.980 -130.852
22/9 -38.114 -131.315
16/11 -39.160 -134.917
14/9 -39.294 -135.380
23/16 +39.298 +135.394
16/1 -39.919 -137.534
16/5 -39.994 -137.793
25/16 +40.069 +138.052
17/16 +40.169 +138.395
16/13 -40.199 -138.497
27/13 +40.782 +140.508
19/8 +40.806 +140.589
27/17 +40.812 +140.610
27/25 +40.912 +140.954
27/5 +40.987 +141.213
27/1 +41.062 +141.471
27/23 +41.683 +143.611
21/8 +41.687 +143.626
27/11 +41.822 +144.089
27/7 +43.001 +148.153
8/3 -43.627 -150.308
15/8 +43.702 +150.566
29/16 +44.343 +152.777
31/16 +44.985 +154.986
20/9 -47.259 -162.823
9/4 +47.334 +163.081
32/7 -47.959 -165.236
32/11 -49.139 -169.301
28/9 -49.274 -169.763
32/23 -49.278 -169.778
32/1 -49.899 -171.918
32/5 -49.974 -172.176
32/25 -50.049 -172.435
32/17 -50.149 -172.779
32/13 -50.178 -172.881
27/26 +50.762 +174.892
19/16 +50.786 +174.973
27/10 +50.967 +175.596
27/2 +51.042 +175.855
21/16 +51.667 +178.009
27/22 +51.801 +178.472
27/14 +52.981 +182.537
16/3 -53.606 -184.691
16/15 -53.682 -184.950
32/29 -54.323 -187.161
32/31 -54.964 -189.370
9/8 +57.314 +197.465
32/19 -60.765 -209.356
27/20 +60.946 +209.980
27/4 +61.021 +210.238
32/21 -61.647 -212.393
28/27 -62.961 -216.920
32/3 -63.586 -219.075
32/15 -63.661 -219.334
16/9 -67.294 -231.848
27/8 +71.001 +244.622
32/9 -77.274 -266.232
27/16 +80.981 +279.006
32/27 -90.961 -313.389
Retrieved from "https://en.xen.wiki/index.php?title=186zpi&oldid=207615"