User:Contribution/Collection of tunings

Equal-step tunings

About this list

The table that follows is not a “best-of” roster but a modest snapshot of equal-step tunings that happen to score highly under a few specific mathematical lenses. In particular, it gathers:

• Prominent peak counts from the classic Riemann zeta function
• Prominent peaks after removing the prime 2 from the zeta product
• Prominent peaks after removing the prime 3
• Prominent peaks after simultaneously removing the primes 2 and 3
• The α–β–γ family, with an equave sliding from 3/1 down to 4/3

These tunings earn the label “optimized” only relative to the limited set of zeta-derived functions explored here. When you layer many differently pruned zeta functions in a tool such as Wolfram Mathematica, striking peaks emerge almost everywhere; the peaks simply shift as each combination of omitted primes reshapes the landscape. That ubiquity means there is no absolute “good” or “bad” equal-step tuning, only different alignments of primes that reveal different musical affordances.

Consequently, the list below is inherently biased toward a handful of functions and can only hint at the boundless diversity of xenharmonic equal-step systems. Treat it as a useful starting palette, not a definitive canon.

Todo: use sigma 1.0 instead sigma 1/2 is not a good sigma choice for musical applications

Notable Local Maxima of the Riemann Zeta Function

Notable Local Maxima of the Riemann Zeta Function

Tuning			Strength			Closest EDO		Integer limit
ZPI	Steps per octave	Cents	Height	Integral	Gap	EDO	Octave	Consistent	Distinct
34zpi	12.0231830072926	99.8071807833375	5.193290	1.269599	15.899282	12edo	1197.68616940005	10	6
42zpi	13.9002525327005	86.3293668353859	4.592177	0.984037	14.097244	14edo	1208.61113569540	7	5
47zpi	15.0534898676781	79.7157343943591	5.050324	1.104057	14.918297	15edo	1195.73601591539	8	7
56zpi	17.0445886606675	70.4035764012981	5.056957	1.032175	14.269437	17edo	1196.86079882207	4	4
65zpi	18.9480867166984	63.3309324546460	5.980169	1.313799	16.699651	19edo	1203.28771663827	10	7
80zpi	22.0251467420146	54.4831784348982	6.062600	1.258178	16.213941	22edo	1198.62992556776	12	8
90zpi	24.0057421830853	49.9880399800983	5.721613	1.092055	14.821136	24edo	1199.71295952236	6	6
100zpi	25.9356996537225	46.2682717652372	5.545073	1.031155	14.793013	26edo	1202.97506589617	14	9
106zpi	27.0866140827635	44.3023257293579	6.069233	1.185939	16.215619	27edo	1196.16279469266	10	8
116zpi	28.9399661541990	41.4651487014917	5.566209	1.000619	14.904418	29edo	1202.48931234326	8	7
127zpi	30.9783816349790	38.7366910944446	7.003472	1.403777	17.739476	31edo	1200.83742392778	12	9
144zpi	34.0448410043159	35.2476312005063	6.685147	1.241437	16.236989	34edo	1198.41946081721	6	6
155zpi	35.9823877000425	33.3496490006021	6.027497	1.028887	14.706508	36edo	1200.58736402167	8	8
184zpi	40.9880783925993	29.2768055263764	7.570230	1.423937	17.722623	41edo	1200.34902658143	16	10
214zpi	46.0089748051542	26.0818678330031	7.495674	1.356067	17.747832	46edo	1199.76592031814	14	11
238zpi	49.9385162652878	24.0295485277387	6.655352	1.111229	15.942083	50edo	1201.47742638693	10	9
257zpi	52.9968291550147	22.6428640945673	8.249774	1.486620	18.069918	53edo	1200.07179701207	10	10
289zpi	58.0667185533159	20.6658827964969	7.814035	1.358357	18.056292	58edo	1198.62120219682	16	12
301zpi	59.9201656607655	20.0266469020418	7.046396	1.131000	15.932359	60edo	1201.59881412251	10	10
334zpi	65.0158450885860	18.4570391781413	7.813349	1.269821	16.514861	65edo	1199.70754657919	6	6
354zpi	68.0493056282519	17.6342725163943	7.666604	1.254592	17.034505	68edo	1199.13053111481	10	10
380zpi	71.9506065993786	16.6781081733140	9.157547	1.625363	19.964746	72edo	1200.82378847861	18	13
414zpi	76.9918536925042	15.5860645308353	8.194847	1.311364	17.029289	77edo	1200.12696887432	10	10
435zpi	80.0731374302484	14.9862992572924	7.873146	1.247325	17.087322	80edo	1198.90394058339	12	12
462zpi	83.9972142607288	14.2861880666087	8.020965	1.241945	16.733121	84edo	1200.03979759513	10	10
483zpi	87.0139255957575	13.7908960178956	8.869041	1.439474	18.061741	87edo	1199.80795355692	16	14
532zpi	93.9836761074943	12.7681747480009	8.806201	1.394050	17.832744	94edo	1200.20842631208	24	15
568zpi	99.0473345956631	12.1154194093028	9.406495	1.510412	18.536483	99edo	1199.42652152097	12	12
596zpi	102.936629522070	11.6576577800491	8.543510	1.340775	18.270998	103edo	1200.73875134506	15	15
655zpi	111.059577998833	10.8050113427643	9.038544	1.394739	18.041165	111edo	1199.35625904684	22	16
706zpi	117.969513574257	10.1721195895637	9.850823	1.544280	18.861062	118edo	1200.31011156852	12	12
796zpi	130.003910460506	9.23049157328654	10.355108	1.634018	19.594551	130edo	1199.96390452725	16	16
872zpi	139.990541024216	8.57200773152536	10.076688	1.548424	19.514765	140edo	1200.08108241355	10	10
965zpi	152.052848107925	7.89199291517551	10.468420	1.593855	19.487224	152edo	1199.58292310668	15	15
1114zpi	170.995891689006	7.01771246166817	11.076998	1.652856	19.091741	171edo	1200.02883094526	14	14

Notable Local Maxima of the Riemann Zeta Function after removing the prime 2 from the zeta product

Tuning			Strength			Closest EDT		No-2 Integer limit
No-2 ZPI analog	Steps per octave	Cents	Height	Integral	Gap	EDT	Tritave	Consistent	Distinct
no-2 93zpi analog	24.5738316304204	48.8324335434323	4.665720	0.766618	13.261693	39edt	1904.46490819386	15	15
no-2 151zpi analog	35.3059427335609	33.9886123153798	4.738265	0.709543	13.081926	56edt	1903.36228966127	15	15
no-2 199zpi analog	43.5176229677494	27.5750355411028	4.824506	0.678480	12.871286	69edt	1902.67745233609	9	9
no-2 207zpi analog	44.8152489207676	26.7766001282638	4.819120	0.732965	14.719415	71edt	1901.13860910673	17	17
no-2 222zpi analog	47.3521317910583	25.3420480686067	5.059485	0.721113	13.412098	75edt	1900.65360514550	15	15
no-2 233zpi analog	49.1685275266548	24.4058559481869	4.790248	0.736865	15.624024	78edt	1903.65676395858	21	21
no-2 249zpi analog	51.6860577447882	23.2170928168922	4.848916	0.664134	13.043858	82edt	1903.80161098516	17	17
no-2 273zpi analog	55.5353711835277	21.6078505360910	5.441186	0.771944	14.061502	88edt	1901.49084717601	11	11
no-2 317zpi analog	62.4092182976906	19.2279287055965	5.154539	0.705887	14.235540	99edt	1903.56494185405	25	23
no-2 326zpi analog	63.7619933650274	18.8199887843874	4.961196	0.662970	13.437518	101edt	1900.81886722313	9	9
no-2 342zpi analog	66.2581615380500	18.1109764011620	5.073625	0.677884	13.529076	105edt	1901.65252212201	17	17
no-2 363zpi analog	69.4221749409126	17.2855431426825	5.247825	0.705262	14.276498	110edt	1901.40974569508	23	23
no-2 453zpi analog	82.6705208991009	14.5154522670130	6.410342	0.925687	16.646686	131edt	1901.52424697870	27	27
no-2 492zpi analog	88.3242305963095	13.5863057271867	5.480169	0.696272	13.636687	140edt	1902.08280180614	9	9
no-2 510zpi analog	90.8297848520406	13.2115252937654	5.712975	0.810755	16.378662	144edt	1902.45964230221	39	27
no-2 550zpi analog	96.5193707902430	12.4327374927449	6.047703	0.795582	14.790729	153edt	1902.20883638997	15	15
no-2 627zpi analog	107.244707551072	11.1893633485693	6.217266	0.828658	15.375247	170edt	1902.19176925679	15	15
no-2 687zpi analog	115.410497106759	10.3976677172610	5.985004	0.754232	14.631506	183edt	1902.77319225877	15	15
no-2 697zpi analog	116.733331758968	10.2798402300191	5.835644	0.746180	15.041001	185edt	1901.77044255353	29	29
no-2 777zpi analog	127.487421022497	9.41269334947362	6.134922	0.758067	14.474624	202edt	1901.36405659367	17	17
no-2 810zpi analog	131.820548689719	9.10328482112888	6.140639	0.820704	16.484428	209edt	1902.58652761594	21	21
no-2 829zpi analog	134.375301622234	8.93021251311149	5.870928	0.707721	14.252150	213edt	1902.13526529275	29	29
no-2 839zpi analog	135.657235331861	8.84582379306507	5.733350	0.672634	13.637550	215edt	1901.85211550899	15	15
no-2 858zpi analog	138.196733558228	8.68327325185579	5.998270	0.762777	15.383590	219edt	1901.63684215642	11	11
no-2 985zpi analog	154.604938100947	7.76171844664157	7.104335	0.924588	16.674411	245edt	1901.62101942718	21	21

Notable Local Maxima of the Riemann Zeta Function after removing the prime 3 from the zeta product

Tuning			Strength			Closest EDO		No-3 Integer limit
No-3 ZPI analog	Steps per octave	Cents	Height	Integral	Gap	EDO	Octave	Consistent	Distinct
no-3 51zpi analog	15.9698898591818	75.1414073973756	5.367776	0.953376	13.070433	16edo	1202.26251835801	26	8
no-3 75zpi analog	21.0437746046821	57.0239903507143	5.752828	0.956754	12.853639	21edo	1197.50379736500	17	10
no-3 95zpi analog	24.9596545948521	48.0775883912872	6.060198	0.954994	12.605015	25edo	1201.93970978218	14	11
no-3 111zpi analog	28.0369867749215	42.8006051304121	5.701943	0.838390	11.937782	28edo	1198.41694365154	16	8
no-3 149zpi analog	34.9357059709719	34.3488121006365	6.001080	0.875916	12.775820	35edo	1202.20842352228	14	11
no-3 161zpi analog	37.0117501336435	32.4221360964286	7.215934	1.160421	15.095854	37edo	1199.61903556786	22	16
no-3 196zpi analog	43.0546167485686	27.8715754690789	6.495142	1.018487	15.545919	43edo	1198.47774517039	22	19
no-3 220zpi analog	47.0058691719873	25.5287269683150	6.758393	0.939366	13.012654	47edo	1199.85016751081	10	10
no-3 251zpi analog	52.0433965143593	23.0576803277801	6.442846	0.856289	12.619985	52edo	1198.99937704456	11	11
no-3 276zpi analog	55.9872265526305	21.4334603424577	6.932381	1.003267	14.804703	56edo	1200.27377917763	20	19
no-3 340zpi analog	65.9172827630736	18.2046338941664	7.029648	0.948492	13.998526	66edo	1201.50583701498	16	16
no-3 394zpi analog	74.0597618189548	16.2031306950932	7.464214	1.007842	14.386154	74edo	1199.03167143690	16	16
no-3 421zpi analog	78.0110209886063	15.3824419267024	7.592394	1.008960	14.204322	78edo	1199.83047028279	17	16
no-3 525zpi analog	93.0076810773635	12.9021601882735	8.466134	1.133255	15.018535	93edo	1199.90089750944	35	19
no-3 640zpi analog	108.976082315502	11.0115905665045	8.633826	1.182085	16.319873	109edo	1200.26337174899	16	16
no-3 751zpi analog	124.014367753602	9.67629817203298	9.498846	1.276085	16.564895	124edo	1199.86097333209	28	26

Notable Local Maxima of the Riemann Zeta Function after removing the primes 2 and 3 from the zeta product

Tuning			Strength			Closest ED5		No-2 No-3 Integer limit
No-2 No-3 ZPI analog	Steps per octave	Cents	Height	Integral	Gap	ED5	Pentave	Consistent	Distinct
no-2 no-3 55zpi analog	16.7630030425585	71.5862185882446	3.480299	0.477759	9.649416	39ed5	2791.86252494154	13	13
no-2 no-3 125zpi analog	30.5974484926723	39.2189564527704	3.769318	0.448541	9.828199	71ed5	2784.54590814670	19	19
no-2 no-3 176zpi analog	39.5828667040955	30.3161468564337	3.603524	0.421674	10.452207	92ed5	2789.08551079190	11	11
no-2 no-3 186zpi analog	41.3477989230936	29.0221010852836	4.469823	0.556068	11.567493	96ed5	2786.12170418722	35	23
no-2 no-3 212zpi analog	45.6783815054539	26.2706330752267	3.818225	0.433470	10.611042	106ed5	2784.68710597403	13	13
no-2 no-3 235zpi analog	49.4631517377883	24.2604839732289	3.853032	0.428042	10.508697	115ed5	2789.95565692132	25	25
no-2 no-3 284zpi analog	57.2705618247184	20.9531731794898	3.913350	0.465932	11.922515	133ed5	2786.77203287214	17	17
no-2 no-3 298zpi analog	59.4923782274424	20.1706510271339	4.083075	0.465782	11.463643	138ed5	2783.54984174448	23	23
no-2 no-3 312zpi analog	61.6047959566046	19.4790029147292	4.416896	0.501431	11.339301	143ed5	2785.49741680628	25	23
no-2 no-3 340zpi analog	65.8904943328257	18.2120351676004	4.092923	0.526694	13.998526	153ed5	2786.44138064287	13	13
no-2 no-3 368zpi analog	70.2158409653819	17.0901606176251	4.382540	0.518334	12.481351	163ed5	2785.69618067290	19	19
no-2 no-3 423zpi analog	78.3601842342727	15.3138996765548	4.270381	0.502072	12.963711	182ed5	2787.12974113297	19	19
no-2 no-3 438zpi analog	80.4944089071946	14.9078677176639	4.243838	0.450422	11.371118	187ed5	2787.77126320314	7	7
no-2 no-3 465zpi analog	84.4075187897342	14.2167429774745	4.301350	0.486089	12.332303	196ed5	2786.48162358500	17	17
no-2 no-3 477zpi analog	86.1814871554687	13.9241041157161	4.459348	0.505570	12.446285	200ed5	2784.82082314323	25	25
no-2 no-3 565zpi analog	98.6257548378926	12.1672072570942	4.883729	0.545550	12.639964	229ed5	2786.29046187457	29	29
no-2 no-3 581zpi analog	100.797128599965	11.9051010347969	4.579796	0.536282	13.693791	234ed5	2785.79364214247	25	25
no-2 no-3 671zpi analog	113.256639862217	10.5954052800778	5.104294	0.563708	12.937931	263ed5	2786.59158866045	19	19
no-2 no-3 764zpi analog	125.745930952370	9.54305233506547	5.001815	0.548008	12.976730	292ed5	2786.57128183912	37	37
no-2 no-3 905zpi analog	144.300058486204	8.31600494545005	5.030210	0.539592	13.254432	335ed5	2785.86165672577	43	41
no-2 no-3 938zpi analog	148.561761173834	8.07744866861039	5.510552	0.600083	13.846076	345ed5	2786.71979067058	25	25

The α–β–γ family

α–β–γ family

Optimization				Equal division of a ratio
Proposed name	Steps per octave	Cents	Optimization method
Alpha 3/1	1.90739592696007	629.130000247254	Dave Benson	3ed3/1
Beta 3/1	3.14186231690763	381.939079106782	Dave Benson	5ed3/1
Alpha 2/1	5.00991270509077	239.525131601721	Dave Benson	5ed2/1
Gamma 3/1	5.04255621376059	237.974540913462	Dave Benson	8ed3/1
Beta 2/1	6.99104980248710	171.648040552235	Dave Benson	7ed2/1
Alpha 5/3	9.50583353877785	126.238272015258	Dave Benson	7ed5/3
Gamma 2/1	11.9978480914311	100.017935787756	Dave Benson	12ed2/1
Beta 5/3	12.2053823008782	98.3172808862904	Dave Benson	9ed5/3
Alpha 3/2	15.3915238996928	77.9649895501219	Dave Benson	9ed3/2
Beta 3/2	18.7990736394111	63.8329325698408	Dave Benson	11ed3/2
Gamma 5/3	21.7094399215509	55.2754932571412	Dave Benson	16ed5/3
Alpha 7/5	22.6653911133366	52.9441558718088	Dave Benson	11ed7/5
Beta 7/5	26.7758951088566	44.8164289231577	Dave Benson	13ed7/5
Alpha 4/3	31.3266790320926	38.3060074376432	Dave Benson	13ed4/3
Gamma 3/2	34.1894540921914	35.0985422804417	Dave Benson	20ed3/2
Beta 4/3	36.1372975038827	33.2066890135065	Dave Benson	15ed4/3
Gamma 7/5	49.4404896216012	24.2716042900130	Dave Benson	24ed7/5
Gamma 4/3	67.4633901646646	17.7874251067289	Dave Benson	28ed4/3

Unequal-step tunings

Unequal-step tunings from equal divisions of a ratio

Tuning	Period	Mode	Why it matters
Stretched hemififth	94\93<2/1>	16 11 16 12 16 11 12
833 Cent Acoustic Golden Scale [11]	25\36<2/1>	3 1 3 3 1 3 1 3 3 1 3
833 Cent Logarithmic Golden Scale [8]	ϕ	ϕ 1 ϕ ϕ 1 ϕ 1 ϕ