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.

Notable Local Maxima of the Riemann Zeta Function

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.5 and cents ≥ 40.0) or (height ≥ 2.8 and cents ≥ 12.0) or (height ≥ 3.25 and cents ≥ 10.0) or (height ≥ 3.6 and cents ≥ 6.0)

Tuning			Strength	Closest EDO		Integer limit
ZPI (σ = 1)	Steps per octave	Step size (cents)	Height	EDO	Octave (cents)	Consistent	Distinct
15zpi (σ = 1)	6.95688550773	172.490980147	2.55384	7edo	1207.43686103	6	5
26zpi (σ = 1)	10.0089746115	119.892401228	2.57426	10edo	1198.92401228	8	5
34zpi (σ = 1)	12.0220488259	99.8165967700	2.85866	12edo	1197.79916124	10	6
42zpi (σ = 1)	13.9020220557	86.3183783764	2.50514	14edo	1208.45729727	7	5
47zpi (σ = 1)	15.0534708836	79.7158349246	2.69313	15edo	1195.73752387	8	7
56zpi (σ = 1)	17.0432556931	70.4090827252	2.65741	17edo	1196.95440633	4	4
65zpi (σ = 1)	18.9489976130	63.3278880767	3.02387	19edo	1203.22987346	10	7
80zpi (σ = 1)	22.0251749360	54.4831086920	2.99601	22edo	1198.62839122	12	8
90zpi (σ = 1)	24.0053572889	49.9888414723	2.82476	24edo	1199.73219533	6	6
100zpi (σ = 1)	25.9356337472	46.2683893402	2.71167	26edo	1202.97812285	14	9
106zpi (σ = 1)	27.0853383248	44.3044124320	2.90524	27edo	1196.21913566	10	8
116zpi (σ = 1)	28.9431579907	41.4605759463	2.68561	29edo	1202.35670244	8	7
127zpi (σ = 1)	30.9779815456	38.7371913897	3.23190	31edo	1200.85293308	12	9
144zpi (σ = 1)	34.0437506778	35.2487600839	3.07414	34edo	1198.45784285	6	6
155zpi (σ = 1)	35.9827898689	33.3492762616	2.80355	36edo	1200.57394542	8	8
184zpi (σ = 1)	40.9880790756	29.2768050385	3.32966	41edo	1200.34900658	16	10
214zpi (σ = 1)	46.0106419996	26.0809227572	3.25119	46edo	1199.72244683	14	11
238zpi (σ = 1)	49.9382924730	24.0296562132	2.90274	50edo	1201.48281066	10	9
257zpi (σ = 1)	52.9969882711	22.6427961125	3.46399	53edo	1200.06819396	10	10
289zpi (σ = 1)	58.0645692462	20.6666477609	3.25823	58edo	1198.66557013	16	12
301zpi (σ = 1)	59.9223835273	20.0259056693	2.98826	60edo	1201.55434016	10	10
321zpi (σ = 1)	63.0197888699	19.0416378969	2.87513	63edo	1199.62318750	8	8
334zpi (σ = 1)	65.0145858034	18.4573966776	3.23462	65edo	1199.73078404	6	6
354zpi (σ = 1)	68.0496579343	17.6341812204	3.14200	68edo	1199.12432299	10	10
380zpi (σ = 1)	71.9512656175	16.6779554147	3.61665	72edo	1200.81278986	18	13
414zpi (σ = 1)	76.9924672555	15.5859403235	3.28825	77edo	1200.11740491	10	10
435zpi (σ = 1)	80.0733926855	14.9862514845	3.14833	80edo	1198.90011876	12	12
462zpi (σ = 1)	83.9950884037	14.2865496400	3.19687	84edo	1200.07016976	10	10
483zpi (σ = 1)	87.0139579095	13.7908908965	3.44872	87edo	1199.80750799	16	14
497zpi (σ = 1)	89.0215260329	13.4798857476	3.02681	89edo	1199.70983154	12	12
532zpi (σ = 1)	93.9843698073	12.7680805059	3.39762	94edo	1200.19956756	24	15
546zpi (σ = 1)	95.9558568688	12.5057504477	2.93099	96edo	1200.55204298	6	6
568zpi (σ = 1)	99.0456175574	12.1156294402	3.56676	99edo	1199.44731458	12	12
596zpi (σ = 1)	102.936325452	11.6576922163	3.25007	103edo	1200.74229828	15	15
655zpi (σ = 1)	111.058159333	10.8051493669	3.39509	111edo	1199.37157972	22	16
706zpi (σ = 1)	117.971388652	10.1719579104	3.62695	118edo	1200.29103343	12	12
796zpi (σ = 1)	130.004267285	9.23046623824	3.72487	130edo	1199.96061097	16	16
872zpi (σ = 1)	139.992781938	8.57187051639	3.60746	140edo	1200.06187229	10	10
965zpi (σ = 1)	152.050659206	7.89210652729	3.68901	152edo	1199.60019215	15	15
1114zpi (σ = 1)	170.995049914	7.01774700849	3.82285	171edo	1200.03473845	14	14
1210zpi (σ = 1)	183.000273182	6.55736726036	3.76064	183edo	1199.99820865	18	18

Todo: use sigma 1.0 instead of sigma 1/2

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 19zpi analog	8.18712929074	146.571521883	1.87661	13ed3	1905.42978449	15	11
no-2 29zpi analog	10.7334869381	111.799642271	1.95394	17ed3	1900.59391860	17	11
no-2 53zpi analog	16.4033618519	73.1557354420	2.01896	26ed3	1902.04912149	21	15
no-2 71zpi analog	20.2433432017	59.2787460076	2.00269	32ed3	1896.91987224	21	15
no-2 84zpi analog	22.7835155508	52.6696592247	1.89685	36ed3	1896.10773209	17	13
no-2 93zpi analog	24.5747239922	48.8306603314	2.12985	39ed3	1904.39575293	15	15
no-2 106zpi analog	27.1258094838	44.2383111448	1.97822	43ed3	1902.24737923	11	11
no-2 113zpi analog	28.4085507996	42.2408030759	1.96399	45ed3	1900.83613842	9	9
no-2 137zpi analog	32.7488975372	36.6424548685	2.02055	52ed3	1905.40765316	25	15
no-2 151zpi analog	35.3061077059	33.9884534992	2.08576	56ed3	1903.35339595	15	15
no-2 166zpi analog	37.8594891129	31.6961487891	1.97021	60ed3	1901.76892734	15	15
no-2 173zpi analog	39.1519961740	30.6497782301	1.99822	62ed3	1900.28625027	9	9
no-2 199zpi analog	43.5167998698	27.5755571088	2.05686	69ed3	1902.71344050	9	9
no-2 207zpi analog	44.8164999984	26.7758526445	2.10342	71ed3	1901.08553776	17	17
no-2 222zpi analog	47.3516876312	25.3422857776	2.11876	75ed3	1900.67143332	15	15
no-2 233zpi analog	49.1657210129	24.4072491012	2.07714	78ed3	1903.76542989	21	21
no-2 249zpi analog	51.6879877530	23.2162259002	2.03774	82ed3	1903.73052382	17	17
no-2 273zpi analog	55.5359583782	21.6076220712	2.19450	88ed3	1901.47074227	11	11
no-2 289zpi analog	58.0976839265	20.6548681272	1.99993	92ed3	1900.24786771	15	15
no-2 301zpi analog	59.8907003349	20.0364997118	1.93131	95ed3	1903.46747262	11	11
no-2 309zpi analog	61.2052267978	19.6061686686	1.96785	97ed3	1901.79836086	11	11
no-2 317zpi analog	62.4122030931	19.2270091509	2.07392	99ed3	1903.47390594	25	23
no-2 326zpi analog	63.7602215687	18.8205117623	2.05280	101ed3	1900.87168799	9	9
no-2 342zpi analog	66.2583876236	18.1109146033	2.06825	105ed3	1901.64603334	17	17
no-2 363zpi analog	69.4191721809	17.2862908372	2.08043	110ed3	1901.49199210	23	23
no-2 380zpi analog	71.9200195089	16.6852012582	2.07565	114ed3	1902.11294344	17	17
no-2 397zpi analog	74.4867252346	16.1102531521	1.92629	118ed3	1901.00987195	15	15
no-2 409zpi analog	76.2807590080	15.7313589378	1.97954	121ed3	1903.49443147	25	23
no-2 418zpi analog	77.5713604064	15.4696268534	1.90376	123ed3	1902.76410297	9	9
no-2 435zpi analog	80.1032694573	14.9806619396	1.99098	127ed3	1902.54406634	11	11
no-2 453zpi analog	82.6700405439	14.5155366092	2.38406	131ed3	1901.53529581	27	27
no-2 492zpi analog	88.3238806401	13.5863595587	2.12238	140ed3	1902.09033822	9	9
no-2 510zpi analog	90.8334979880	13.2109852266	2.23067	144ed3	1902.38187263	39	27
no-2 519zpi analog	92.1840749628	13.0174327885	1.99259	146ed3	1900.54518712	17	17
no-2 550zpi analog	96.5187261015	12.4328205362	2.24293	153ed3	1902.22154203	15	15
no-2 568zpi analog	99.0730275901	12.1122774704	2.00937	157ed3	1901.62756285	11	11
no-2 577zpi analog	100.316260311	11.9621684090	1.98584	159ed3	1901.98477703	11	11
no-2 596zpi analog	102.908364024	11.6608597502	1.96654	163ed3	1900.72013927	15	15
no-2 609zpi analog	104.713326539	11.4598594053	2.00635	166ed3	1902.33666128	11	11
no-2 614zpi analog	105.436045548	11.3813069692	1.92595	167ed3	1900.67826385	23	23
no-2 627zpi analog	107.244021785	11.1894348983	2.29774	170ed3	1902.20393272	15	15
no-2 646zpi analog	109.793603482	10.9295984642	1.96998	174ed3	1901.75013278	15	15
no-2 655zpi analog	111.085500608	10.8024899148	2.00672	176ed3	1901.23822501	21	21
no-2 659zpi analog	111.586744725	10.7539654729	1.88303	177ed3	1903.45188870	7	7
no-2 687zpi analog	115.412802617	10.3974600113	2.18983	183ed3	1902.73518207	15	15
no-2 697zpi analog	116.734850378	10.2797064983	2.15793	185ed3	1901.74570218	29	29
no-2 706zpi analog	117.949591604	10.1738376851	1.91643	187ed3	1902.50764711	11	11
no-2 725zpi analog	120.530724507	9.95596769960	1.89765	191ed3	1901.58983062	5	5
no-2 729zpi analog	121.102378223	9.90897138117	2.05767	192ed3	1902.52250518	17	17
no-2 748zpi analog	123.601895646	9.70858896401	1.91762	196ed3	1902.88343695	11	11
no-2 753zpi analog	124.304838560	9.65368696748	1.91680	197ed3	1901.77633259	21	21
no-2 767zpi analog	126.183698594	9.50994473428	2.05769	200ed3	1901.98894686	9	9
no-2 777zpi analog	127.486291223	9.41277676594	2.21095	202ed3	1901.38090672	17	17
no-2 810zpi analog	131.822840677	9.10312654342	2.25360	209ed3	1902.55344758	21	21
no-2 829zpi analog	134.373782790	8.93031345169	2.13475	213ed3	1902.15676521	29	29
no-2 839zpi analog	135.657892938	8.84578091263	2.11125	215ed3	1901.84289622	15	15
no-2 858zpi analog	138.196070465	8.68331491602	2.20051	219ed3	1901.64596661	11	11
no-2 878zpi analog	140.756053126	8.52538823977	1.91894	223ed3	1901.16157747	15	15
no-2 882zpi analog	141.320264620	8.49135121014	1.94097	224ed3	1902.06267107	17	17
no-2 902zpi analog	143.873905513	8.34063686336	2.09948	228ed3	1901.66520485	11	11
no-2 911zpi analog	145.102065664	8.27004077793	1.96452	230ed3	1902.10937892	23	23
no-2 921zpi analog	146.379932964	8.19784498941	1.96989	232ed3	1901.90003754	9	9
no-2 945zpi analog	149.470277594	8.02835198621	1.92855	237ed3	1902.71942073	19	19
no-2 965zpi analog	152.075713777	7.89080629768	2.10893	241ed3	1901.68431774	15	15
no-2 985zpi analog	154.604034485	7.76176381166	2.40811	245ed3	1901.63213386	21	21
no-2 995zpi analog	155.863142206	7.69906202978	1.88900	247ed3	1901.66832135	7	7
no-2 1019zpi analog	158.932236585	7.55038767329	1.94652	252ed3	1902.69769367	15	15
no-2 1029zpi analog	160.260260060	7.48782012177	2.17192	254ed3	1901.90631093	9	9
no-2 1049zpi analog	162.750022676	7.37327086209	2.14738	258ed3	1902.30388242	17	17
no-2 1069zpi analog	165.332187903	7.25811480039	2.19607	262ed3	1901.62607770	17	17
no-2 1083zpi analog	167.112289634	7.18080042243	1.93984	265ed3	1902.91211194	11	11
no-2 1104zpi analog	169.714157484	7.07071241310	1.92771	269ed3	1902.02163912	15	15
no-2 1114zpi analog	170.990381058	7.01793862657	1.91502	271ed3	1901.86136780	9	9
no-2 1134zpi analog	173.506549648	6.91616542681	2.26764	275ed3	1901.94549237	29	29
no-2 1145zpi analog	174.860916353	6.86259700012	1.98752	277ed3	1900.93936903	15	15
no-2 1159zpi analog	176.625850825	6.79402247404	2.14379	280ed3	1902.32629273	11	11
no-2 1179zpi analog	179.167803205	6.69763193238	2.29964	284ed3	1902.12746880	15	15
no-2 1200zpi analog	181.734924328	6.60302363146	1.98334	288ed3	1901.67080586	11	11
no-2 1210zpi analog	183.000523023	6.55735830793	1.88033	290ed3	1901.63390930	17	17
no-2 1225zpi analog	184.832854856	6.49235224405	1.92540	293ed3	1902.25920751	9	9
no-2 1245zpi analog	187.354933401	6.40495544056	2.28021	297ed3	1902.27176585	21	21
no-2 1266zpi analog	189.909845446	6.31878772364	2.17116	301ed3	1901.95510482	17	17
no-2 1297zpi analog	193.736743714	6.19397217583	2.12380	307ed3	1901.54945798	21	21
no-2 1301zpi analog	194.272130007	6.17690247159	1.87710	308ed3	1902.48596125	7	7
no-2 1312zpi analog	195.595668163	6.13510519569	1.92538	310ed3	1901.88261066	9	9
no-2 1332zpi analog	198.083101013	6.05806347873	2.07112	314ed3	1902.23193232	15	15
no-2 1343zpi analog	199.415414525	6.01758897555	2.36503	316ed3	1901.55811627	39	39

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

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.5 and cents ≥ 40.0) or (height ≥ 2.6 and cents ≥ 15.0) or (height ≥ 2.8 and cents ≥ 12.0) or (height ≥ 3.1 and cents ≥ 6.0)

Tuning			Strength	Closest EDO		No-3 Integer limit
No-3 ZPI analog	Steps per octave	Cents	Height	EDO	Octave	Consistent	Distinct
no-3 51zpi (σ = 1)	15.9687074547	75.1469712502	2.56677	16edo	1202.35154000	26	8
no-3 75zpi (σ = 1)	21.0417134383	57.0295762045	2.60042	21edo	1197.62110029	17	10
no-3 95zpi (σ = 1)	24.9617781085	48.0734984016	2.64675	25edo	1201.83746004	14	11
no-3 127zpi (σ = 1)	31.0146799866	38.6913552073	2.60405	31edo	1199.43201143	11	11
no-3 161zpi (σ = 1)	37.0135086000	32.4205957606	2.92705	37edo	1199.56204314	22	16
no-3 196zpi (σ = 1)	43.0494972034	27.8748900209	2.71380	43edo	1198.62027090	22	19
no-3 220zpi (σ = 1)	47.0043385196	25.5295582875	2.69328	47edo	1199.88923951	10	10
no-3 276zpi (σ = 1)	55.9891415481	21.4327272543	2.76321	56edo	1200.23272624	20	19
no-3 340zpi (σ = 1)	65.9204029312	18.2037722259	2.65263	66edo	1201.44896691	16	16
no-3 354zpi (σ = 1)	68.0229453080	17.6411061674	2.76285	68edo	1199.59521939	11	11
no-3 394zpi (σ = 1)	74.0566473758	16.2038121158	2.76672	74edo	1199.08209657	16	16
no-3 421zpi (σ = 1)	78.0097604150	15.3826904943	2.81219	78edo	1199.84985856	17	16
no-3 525zpi (σ = 1)	93.0066513531	12.9023030347	2.97919	93edo	1199.91418223	35	19
no-3 751zpi (σ = 1)	124.013627761	9.67635591079	3.13747	124edo	1199.86813294	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