User:BudjarnLambeth/Sandbox2

Title1

Octave stretch or compression

99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly compressing the octave is acceptable, using tunings such as 157edt or 256ed6. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.

What follows is a comparison of stretched- and compressed-octave 99edo tunings.

567zpi

• Step size: 12.138 ¢, octave size: 1201.66 ¢

Stretching the octave of 99edo by around 1.5 ¢ results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54 ¢. The tuning 567zpi does this.

Approximation of harmonics in 567zpi

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.66	+3.71	+3.32	+5.43	+5.37	+5.54	+4.99	-4.72	-5.05	-0.12	-5.10
	Relative (%)	+13.7	+30.6	+27.4	+44.7	+44.3	+45.6	+41.1	-38.9	-41.6	-1.0	-42.0
Step		99	157	198	230	256	278	297	313	328	342	354

Approximation of harmonics in 567zpi (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+1.98	-4.94	-3.00	-5.49	-1.20	-3.05	+0.45	-3.39	-2.89	+1.54	-2.59	-3.44
	Relative (%)	+16.3	-40.7	-24.7	-45.2	-9.9	-25.2	+3.7	-27.9	-23.8	+12.7	-21.3	-28.3
Step		366	376	386	395	404	412	420	427	434	441	447	453

99et, 13-limit WE tuning

• Step size: 12.123 ¢, octave size: 1200.18 ¢

Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 99et, 13-limit WE tuning

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.18	+1.36	+0.35	+1.98	+1.53	+1.37	+0.53	+2.71	+2.15	-5.25	+1.71
	Relative (%)	+1.5	+11.2	+2.9	+16.3	+12.6	+11.3	+4.4	+22.4	+17.8	-43.3	+14.1
Step		99	157	198	230	256	278	297	314	329	342	355

Approximation of harmonics in 99et, 13-limit WE tuning (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-3.51	+1.55	+3.33	+0.71	+4.86	+2.89	-5.85	+2.33	+2.72	-5.07	+2.83	+1.89
	Relative (%)	-29.0	+12.7	+27.5	+5.8	+40.1	+23.8	-48.3	+19.2	+22.5	-41.9	+23.3	+15.6
Step		366	377	387	396	405	413	420	428	435	441	448	454

99edo

• Step size: 12.121 ¢, octave size: 1200.00 ¢

Pure-octaves 99edo approximates all harmonics up to 16 within 5.86 ¢.

Approximation of harmonics in 99edo

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.00	+1.08	+0.00	+1.57	+1.08	+0.87	+0.00	+2.15	+1.57	-5.86	+1.08
	Relative (%)	+0.0	+8.9	+0.0	+12.9	+8.9	+7.2	+0.0	+17.7	+12.9	-48.4	+8.9
Steps (reduced)		99 (0)	157 (58)	198 (0)	230 (32)	256 (58)	278 (80)	297 (0)	314 (17)	329 (32)	342 (45)	355 (58)

Approximation of harmonics in 99edo (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-4.16	+0.87	+2.64	+0.00	+4.14	+2.15	+5.52	+1.57	+1.95	-5.86	+2.03	+1.08
	Relative (%)	-34.4	+7.2	+21.8	+0.0	+34.1	+17.7	+45.5	+12.9	+16.1	-48.4	+16.7	+8.9
Steps (reduced)		366 (69)	377 (80)	387 (90)	396 (0)	405 (9)	413 (17)	421 (25)	428 (32)	435 (39)	441 (45)	448 (52)	454 (58)

99et, 7-limit WE tuning / 256ed6

• Step size: 12.117 ¢, octave size: 1199.58 ¢

Compressing the octave of 99edo by around 0.6 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71 ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.

Approximation of harmonics in 99et, 7-limit WE tuning

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-0.42	+0.41	-0.83	+0.60	-0.00	-0.30	-1.25	+0.83	+0.18	+4.81	-0.42
	Relative (%)	-3.4	+3.4	-6.9	+4.9	-0.0	-2.5	-10.3	+6.8	+1.5	+39.7	-3.5
Step		99	157	198	230	256	278	297	314	329	343	355

Approximation of harmonics in 99et, 7-limit WE tuning (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-5.71	-0.72	+1.01	-1.67	+2.43	+0.41	+3.74	-0.24	+0.11	+4.40	+0.14	-0.84
	Relative (%)	-47.1	-5.9	+8.3	-13.8	+20.1	+3.4	+30.9	-2.0	+0.9	+36.3	+1.2	-6.9
Step		366	377	387	396	405	413	421	428	435	442	448	454

568zpi

• Step size: 12.115 ¢, octave size: 1199.39 ¢

Compressing the octave of 99edo by around 0.4 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68 ¢. The tuning 568zpi does this.

Approximation of harmonics in 568zpi

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-0.61	+0.10	-1.23	+0.14	-0.52	-0.86	-1.84	+0.20	-0.48	+4.13	-1.13
	Relative (%)	-5.1	+0.8	-10.2	+1.1	-4.3	-7.1	-15.2	+1.7	-4.0	+34.1	-9.3
Step		99	157	198	230	256	278	297	314	329	343	355

Approximation of harmonics in 568zpi (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+5.68	-1.47	+0.24	-2.46	+1.62	-0.42	+2.90	-1.09	-0.76	+3.51	-0.75	-1.75
	Relative (%)	+46.9	-12.1	+2.0	-20.3	+13.4	-3.4	+24.0	-9.0	-6.2	+29.0	-6.2	-14.4
Step		367	377	387	396	405	413	421	428	435	442	448	454

157edt / 230ed5

• Step size: 12.114 ¢, octave size: 1199.32 ¢

Compressing the octave of 99edo by around 0.3 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44 ¢. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.

Approximation of harmonics in 157edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-0.68	+0.00	-1.36	-0.01	-0.68	-1.03	-2.03	+0.00	-0.69	+3.91	-1.36
	Relative (%)	-5.6	+0.0	-11.2	-0.1	-5.6	-8.5	-16.8	+0.0	-5.7	+32.3	-11.2
Steps (reduced)		99 (99)	157 (0)	198 (41)	230 (73)	256 (99)	278 (121)	297 (140)	314 (0)	329 (15)	343 (29)	355 (41)

Approximation of harmonics in 157edt (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+5.44	-1.71	-0.01	-2.71	+1.36	-0.68	+2.63	-1.37	-1.03	+3.23	-1.04	-2.03
	Relative (%)	+44.9	-14.1	-0.1	-22.4	+11.2	-5.6	+21.7	-11.3	-8.5	+26.7	-8.6	-16.8
Steps (reduced)		367 (53)	377 (63)	387 (73)	396 (82)	405 (91)	413 (99)	421 (107)	428 (114)	435 (121)	442 (128)	448 (134)	454 (140)

Title2

Possible tunings to be used on each page

You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.

(Used https://x31eq.com/temper-pyscript/net.html, used WE instead of TE cause it kept defaulting to WE and I kept not remembering to switch it)

High-priority

23edo (narrow down edonoi & ZPIs)

• Main: "23edo and octave stretching"
• 36edt
• 59ed6
• 60ed6
• 68ed8
• 11ed7/5
• 1ed33/32
• 2.3.5.13 WE (52.447c)
• 2.7.11 WE (51.962c)
• 13-limit WE (52.237c)
• 83zpi (53.105c)
• 84zpi (52.615c)
• 85zpi (52.114c)
• 86zpi (51.653c)
• 87zpi (51.201c)

60edo (narrow down edonoi & ZPIs)

• 95edt
• 35edf
• 139ed5
• 155ed6
• 208ed11
• (???)ed12
• 255ed19
• 272ed23 (great for catnip temperament, maybe there's a similar but simpler tuning w similar benefits?)
• 13-limit WE (20.013c)
• 299zpi (20.128c)
• 300zpi (20.093c)
• 301zpi (20.027c)
• 302zpi (19.962c)
• 303zpi (19.913c)
• 304zpi (19.869c)

Medium priority

13edo

• Main: "13edo and optimal octave stretching"
• 2.5.11.13 WE (92.483c)
• 2.5.7.13 WE (92.804c)
• 2.3 WE (91.405c) (good for opposite 7 mapping)
• 38zpi (92.531c)

32edo (narrow down ZPIs)

• 90ed7
• 51edt
• 75ed5
• 1ed46/45
• 11-limit WE (37.453c)
• 13-limit WE (37.481c)
• 131zpi (37.862c)
• 132zpi (37.662c)
• 133zpi (37.418c)
• 134zpi (37.176c)

33edo (narrow down edonoi)

• 76ed5
• 92ed7
• 52edt
• 1ed47/46
• 114ed11
• 122ed13
• 93ed7
• 23edPhi
• 77ed5
• 123ed13
• 115ed11
• 11-limit WE (36.349c)
• 13-limit WE (36.357c)
• 137zpi (36.628c)
• 138zpi (36.394c)
• 139zpi (36.179c)

39edo (narrow down slightly)

• 62edt
• 101ed6
• 18ed11/8
• 2.3.5.11 WE (30.703c)
• 2.3.7.11.13 WE (30.787c)
• 13-limit WE (30.757c)
• 171zpi (30.973c)
• 172zpi (30.836c)
• 173zpi (30.672c)

42edo (narrow down slightly)

• 42ed257/128 (replace w something similar but simpler)
• AS123/121 (1ed123/121)
• 11ed6/5
• 34ed7/4
• 7-limit WE (28.484c)
• 13-limit WE (28.534c)
• 189zpi (28.689c)
• 190zpi (28.572c)
• 191zpi (28.444c)

45edo

• 126ed7
• 13ed11/9
• 7-limit WE (26.745c)
• 13-limit WE (26.695c)
• 207zpi (26.762)
• 208zpi (26.646)
• 209zpi (26.550)

54edo (narrow down slightly)

• 86edt
• 126ed5
• 152ed7
• 38ed5/3
• 40ed5/3
• 2.3.7.11.13 WE (22.180c)
• 13-limit WE (22.198c)
• 262zpi (22.313c)
• 263zpi (22.243c)
• 264zpi (22.175c)

59edo (narrow down ZPIs)

• 93edt
• 166ed7
• 203ed11
• 7-limit WE (20.301c)
• 11-limit WE (20.310c)
• 13-limit WE (20.320c)
• 293zpi (20.454c)
• 294zpi (20.399c)
• 295zpi (20.342c)
• 296zpi (20.282c)
• 297zpi (20.229c)

64edo (narrow down ZPIs)

• 149ed5
• 180ed7
• 222ed11
• 47ed5/3
• 11-limit WE (18.755c)
• 13-limit WE (18.752c)
• 325zpi (18.868c)
• 326zpi (18.816c)
• 327zpi (18.767c)
• 328zpi (18.721c)
• 329zpi (18.672c)
• 330zpi (18.630c)

103edo (narrow down edonoi, choose ZPIS)

• 163edt
• 239ed5
• (???)ed6
• 289ed7
• 356ed11
• (???)ed12
• 381ed13
• 421ed17
• 466ed23
• 13-limit WE (11.658c)
• Best nearby ZPI(s)

111edo (choose ZPIS)

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

118edo (choose ZPIS)

• 187edt
• 69edf
• 13-limit WE (10.171c)
• Best nearby ZPI(s)

Low priority

104edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

125edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

145edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

152edo

• 241edt
• 13-limit WE (7.894c)
• Best nearby ZPI(s)

159edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

166edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

182edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

198edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

212edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

243edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

247edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

Optional

25edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

26edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

29edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

30edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

34edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

35edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

36edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

37edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

5edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

6edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

9edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

10edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

11edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

15edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

18edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

48edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

20edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

24edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)

28edo

• Nearby edt, ed6, ed12 and/or edf
• Nearby ed5, ed10, ed7 and/or ed11 (optional)
• 1-2 WE tunings
• Best nearby ZPI(s)