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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''
|}
Retrieved from "https://en.xen.wiki/w/186zpi"