Table of zeta-stretched edos

This table lists tuning instructions for equal divisions of the octave which have been stretched or compressed using optimal octave stretch based on zeta peaks, as described here: the Riemann zeta function and tuning.

No such table could possibly be complete (as there are so many possible edos), so please add tunings of interest as you see fit.

All of the tunings listed on this page are zeta peak index tunings, e.g. 1zpi, 2zpi, 3zpi... However, not all zeta peak index tunings are listed here - only those with intervals close to the octave. For a more complete table see: zeta peak index.

Calculation instructions

How to calculate the third column using the free version of Wolfram Cloud:

1. Copy-paste Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}] into a cell.
2. Change "11.9" and "12.1" to whatever values you want, e.g. to view the curve around 15edo you might use the values "14.9" and "15.1".
3. Ensure that cell is still selected
4. In the menu select Evaluation > Evaluate Cells

Table

This is a list of zeta peak-based octave tunings, which includes all EDOs up to 100 and certain noteworthy ones above 100.

Zeta-optimal tunings for selected EDOs

Tuning	Associated edo	No. of steps per 1200 cents	Step size (cents)	Tuning of 2/1 (cents)	Gram point index
1zpi	1edo	1.127	1065.177	1065.177	−1
2zpi	2edo	1.973	608.283	1216.565	0
4zpi	3edo	3.060	392.187	1176.562	2
6zpi	4edo	3.904	307.342	1229.367	4
9zpi	5edo	5.034	238.357	1191.783	7
12zpi	6edo	6.035	198.843	1193.056	10
15zpi	7edo	6.957	172.496	1207.471	13
19zpi	8edo	8.137	147.467	1179.734	17
22zpi	9edo	8.950	134.078	1206.705	20
26zpi	10edo	10.008	119.899	1198.986	24
30zpi	11edo	11.037	108.722	1195.938	28
34zpi	12edo	12.023	99.807	1197.686	32
38zpi	13edo	12.969	92.531	1202.900	36
42zpi	14edo	13.900	86.329	1208.611	40
47zpi	15edo	15.053	79.716	1195.736	45
51zpi	16edo	15.944	75.262	1204.187	49
56zpi	17edo	17.045	70.404	1196.861	54
61zpi	18edo	18.119	66.228	1192.113	59
65zpi	19edo	18.948	63.331	1203.288	63
70zpi	20edo	19.982	60.054	1201.087	68
75zpi	21edo	21.028	57.067	1198.406	73
80zpi	22edo	22.025	54.483	1198.630	78
84zpi	23edo	22.807	52.615	1210.148	82
90zpi	24edo	24.006	49.988	1199.713	88
95zpi	25edo	24.965	48.067	1201.678	93
100zpi	26edo	25.936	46.268	1202.975	98
106zpi	27edo	27.087	44.302	1196.163	104
111zpi	28edo	28.032	42.808	1198.629	109
116zpi	29edo	28.940	41.465	1202.489	114
122zpi	30edo	30.061	39.918	1197.555	120
127zpi	31edo	30.978	38.737	1200.837	125
133zpi	32edo	32.070	37.418	1197.375	131
138zpi	33edo	32.972	36.394	1201.009	136
144zpi	34edo	34.045	35.248	1198.419	142
149zpi	35edo	34.925	34.359	1202.564	147
155zpi	36edo	35.982	33.350	1200.587	153
161zpi	37edo	37.028	32.408	1199.108	159
166zpi	38edo	37.890	31.671	1203.480	164
173zpi	39edo	39.124	30.672	1196.204	171
178zpi	40edo	39.968	30.024	1200.965	176
184zpi	41edo	40.988	29.277	1200.349	182
190zpi	42edo	41.999	28.572	1200.032	188
196zpi	43edo	43.026	27.890	1199.261	194
202zpi	44edo	44.015	27.263	1199.579	200
207zpi	45edo	44.840	26.762	1204.289	205
214zpi	46edo	46.009	26.082	1199.766	212
220zpi	47edo	47.006	25.529	1199.846	218
226zpi	48edo	47.988	25.006	1200.292	224
233zpi	49edo	49.141	24.419	1196.552	231
238zpi	50edo	49.939	24.030	1201.477	236
245zpi	51edo	51.080	23.493	1198.128	243
251zpi	52edo	52.043	23.058	1199.018	249
257zpi	53edo	52.997	22.643	1200.072	255
264zpi	54edo	54.116	22.175	1197.430	262
269zpi	55edo	54.894	21.860	1202.325	267
276zpi	56edo	56.008	21.425	1199.821	274
282zpi	57edo	56.968	21.064	1200.668	280
289zpi	58edo	58.067	20.666	1198.621	287
295zpi	59edo	58.992	20.342	1200.157	293
301zpi	60edo	59.920	20.027	1201.599	299
308zpi	61edo	61.003	19.671	1199.937	306
314zpi	62edo	61.938	19.374	1201.200	312
321zpi	63edo	63.019	19.042	1199.633	319
328zpi	64edo	64.099	18.721	1198.140	326
334zpi	65edo	65.016	18.457	1199.708	332
340zpi	66edo	65.916	18.205	1201.533	338
347zpi	67edo	66.998	17.911	1200.029	345
354zpi	68edo	68.049	17.634	1199.131	352
360zpi	69edo	68.960	17.401	1200.696	358
367zpi	70edo	70.004	17.142	1199.931	365
374zpi	71edo	71.059	16.887	1198.998	372
380zpi	72edo	71.951	16.678	1200.824	378
387zpi	73edo	72.983	16.442	1200.273	385
394zpi	74edo	74.052	16.205	1199.155	392
401zpi	75edo	75.091	15.981	1198.544	399
407zpi	76edo	75.968	15.796	1200.503	405
414zpi	77edo	76.992	15.586	1200.127	412
420zpi	78edo	77.851	15.414	1202.292	418
427zpi	79edo	78.892	15.211	1201.637	425
435zpi	80edo	80.073	14.986	1198.904	433
441zpi	81edo	80.948	14.824	1200.777	439
448zpi	82edo	81.954	14.642	1200.671	446
455zpi	83edo	82.967	14.464	1200.484	453
462zpi	84edo	83.997	14.286	1200.040	460
469zpi	85edo	84.991	14.119	1200.131	467
476zpi	86edo	86.019	13.950	1199.741	474
483zpi	87edo	87.014	13.791	1199.808	481
490zpi	88edo	88.027	13.632	1199.635	488
497zpi	89edo	89.023	13.480	1199.691	495
504zpi	90edo	90.006	13.332	1199.917	502
510zpi	91edo	90.852	13.208	1201.956	508
518zpi	92edo	91.993	13.044	1200.089	516
525zpi	93edo	93.002	12.903	1199.969	523
532zpi	94edo	93.984	12.768	1200.208	530
540zpi	95edo	95.117	12.616	1198.526	538
546zpi	96edo	95.954	12.506	1200.570	544
553zpi	97edo	96.925	12.381	1200.927	551
560zpi	98edo	97.923	12.254	1200.941	558
568zpi	99edo	99.047	12.115	1199.427	566
575zpi	100edo	99.869	12.016	1201.577	573
1936zpi	270edo	270.018	4.444	1199.920	1934
2293zpi	311edo	311.004	3.858	1199.985	2291
2568zpi	342edo	341.975	3.509	1200.088	2566
3971zpi	494edo	494.014	2.429	1199.966	3969
5818zpi	684edo	683.939	1.755	1200.107	5816

Record zeta peaks

    0.00000
    1.12657
    1.97277
    3.05976
    3.90445
    5.03448
    6.95669
   10.00846
   12.02318
   18.94809
   22.02515
   27.08661
   30.97838
   40.98808
   52.99683
   71.95061
   99.04733
  117.96951
  130.00391
  152.05285
  170.99589
  217.02470
  224.00255
  270.01779
  341.97485
  422.05570
  441.01827
  494.01377
  742.01093
  764.01938
  935.03297
  953.94128
 1012.02423
 1105.99972
 1177.96567
 1236.02355
 1394.98350
 1447.97300
 1577.98315
 2459.98488
 2683.99168
 3395.02659
 5585.00172
 6079.01642
 7032.96529
 8268.98378
 8539.00834
11664.01488
14347.99444
16807.99325
28742.01019
34691.00191
36268.98775
57578.00854
58972.99326
95524.04578
102557.01877
112984.99531
148418.01630
212146.99129
241199.99851

See also

• Zeta peak index
• The Riemann zeta function and tuning