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== Intervals and notation ==
== Intervals and notation ==
There are multiple ways to approach notation. The simplest method is to use the notations from [[41edo]]. However, this approach will not preserve octave compression when the audio is rendered by notation software. To address this, consider using the ups and downs notation from [[124edo]] at every 3-degree step (i.e., the [[edonoi]] [[124ed8]]).
There are multiple ways to approach notation. The simplest method is to use the notations from [[41edo]]. However, this approach will not preserve octave compression when the audio is rendered by notation software. To address this, consider using the ups and downs notation from [[124edo]] at every 3-degree step (i.e., the [[edonoi]] [[124ed8]]).
=== Approximation to JI ===
The following table illustrates the representation of the 32-integer limit intervals in 186zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Intervals by direct approximation (even if inconsistent)
|-
! 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
|-
| ''[[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''
|}

Revision as of 17:47, 28 June 2024

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

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

Theory

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

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

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

Harmonic series

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

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

Approximation of EDONOIs

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

Intervals and notation

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

Approximation to JI

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

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