User:BudjarnLambeth/Sandbox2
Title1
Octave stretch or compression
Having a flat tendency, 16et is best tuned with stretched octaves, which improve the accuracy of wide-voiced JI chords and rooted harmonics especially on inharmonic timbres such as bells and gamelans, with 25edt, 41ed6, and 57ed12 being good options.
What follows is a comparison of stretched- and compressed-octave 16edo tunings.
- 16edo
- Step size: 75.000 ¢, octave size: 1200.0 ¢
Pure-octaves 16edo approximates all harmonics up to 16 within NNN ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -27.0 | +0.0 | -11.3 | -27.0 | +6.2 | +0.0 | +21.1 | -11.3 | -26.3 | -27.0 |
| Relative (%) | +0.0 | -35.9 | +0.0 | -15.1 | -35.9 | +8.2 | +0.0 | +28.1 | -15.1 | -35.1 | -35.9 | |
| Steps (reduced) |
16 (0) |
25 (9) |
32 (0) |
37 (5) |
41 (9) |
45 (13) |
48 (0) |
51 (3) |
53 (5) |
55 (7) |
57 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -15.5 | +6.2 | +36.7 | +0.0 | -30.0 | +21.1 | +2.5 | -11.3 | -20.8 | -26.3 | -28.3 | -27.0 |
| Relative (%) | -20.7 | +8.2 | +49.0 | +0.0 | -39.9 | +28.1 | +3.3 | -15.1 | -27.7 | -35.1 | -37.7 | -35.9 | |
| Steps (reduced) |
59 (11) |
61 (13) |
63 (15) |
64 (0) |
65 (1) |
67 (3) |
68 (4) |
69 (5) |
70 (6) |
71 (7) |
72 (8) |
73 (9) | |
- Step size: 75.105 ¢, octave size: NNN ¢
Stretching the octave of 16edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 2.5.7.13 WE tuning and 2.5.7.13 TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.7 | -24.3 | +3.4 | -7.4 | -22.7 | +10.9 | +5.0 | +26.4 | -5.7 | -20.5 | -21.0 |
| Relative (%) | +2.2 | -32.4 | +4.5 | -9.9 | -30.2 | +14.5 | +6.7 | +35.2 | -7.7 | -27.4 | -27.9 | |
| Step | 16 | 25 | 32 | 37 | 41 | 45 | 48 | 51 | 53 | 55 | 57 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.3 | +12.6 | -31.8 | +6.7 | -23.1 | +28.1 | +9.6 | -4.1 | -13.4 | -18.9 | -20.7 | -19.3 |
| Relative (%) | -12.4 | +16.7 | -42.3 | +8.9 | -30.8 | +37.4 | +12.8 | -5.4 | -17.9 | -25.1 | -27.6 | -25.7 | |
| Step | 59 | 61 | 62 | 64 | 65 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | |
- Step size: 75.262 ¢, octave size: NNN ¢
Stretching the octave of 16edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 15zpi does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.2 | -20.4 | +8.4 | -1.6 | -16.2 | +18.0 | +12.6 | +34.5 | +2.6 | -11.9 | -12.0 |
| Relative (%) | +5.6 | -27.1 | +11.1 | -2.2 | -21.5 | +23.9 | +16.7 | +45.8 | +3.4 | -15.8 | -16.0 | |
| Step | 16 | 25 | 32 | 37 | 41 | 45 | 48 | 51 | 53 | 55 | 57 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.1 | +22.2 | -22.0 | +16.8 | -12.9 | -36.6 | +20.3 | +6.8 | -2.4 | -7.7 | -9.4 | -7.8 |
| Relative (%) | -0.1 | +29.4 | -29.3 | +22.3 | -17.2 | -48.7 | +27.0 | +9.0 | -3.2 | -10.3 | -12.5 | -10.4 | |
| Step | 59 | 61 | 62 | 64 | 65 | 66 | 68 | 69 | 70 | 71 | 72 | 73 | |
- Step size: 75.315 ¢, octave size: NNN ¢
Stretching the octave of 16edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.0 | -19.1 | +10.1 | +0.3 | -14.0 | +20.3 | +15.1 | +37.2 | +5.4 | -9.0 | -9.0 |
| Relative (%) | +6.7 | -25.3 | +13.4 | +0.5 | -18.6 | +27.0 | +20.1 | +49.3 | +7.1 | -11.9 | -11.9 | |
| Step | 16 | 25 | 32 | 37 | 41 | 45 | 48 | 51 | 53 | 55 | 57 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.1 | +25.4 | -18.7 | +20.2 | -9.5 | -33.1 | +23.9 | +10.4 | +1.3 | -4.0 | -5.6 | -4.0 |
| Relative (%) | +4.1 | +33.7 | -24.9 | +26.8 | -12.6 | -44.0 | +31.7 | +13.8 | +1.7 | -5.2 | -7.4 | -5.3 | |
| Step | 59 | 61 | 62 | 64 | 65 | 66 | 68 | 69 | 70 | 71 | 72 | 73 | |
- Step size: NNN ¢, octave size: NNN ¢
Stretching the octave of 16edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 57ed12 does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.6 | -15.1 | +15.1 | +6.2 | -7.6 | +27.5 | +22.7 | -30.3 | +13.7 | -0.3 | +0.0 |
| Relative (%) | +10.0 | -20.1 | +20.1 | +8.2 | -10.0 | +36.4 | +30.1 | -40.1 | +18.2 | -0.4 | +0.0 | |
| Steps (reduced) |
16 (16) |
25 (25) |
32 (32) |
37 (37) |
41 (41) |
45 (45) |
48 (48) |
50 (50) |
53 (53) |
55 (55) |
57 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +12.4 | +35.0 | -8.9 | +30.3 | +0.8 | -22.7 | +34.6 | +21.3 | +12.3 | +7.3 | +5.8 | +7.6 |
| Relative (%) | +16.4 | +46.4 | -11.9 | +40.1 | +1.0 | -30.1 | +45.9 | +28.2 | +16.3 | +9.6 | +7.7 | +10.0 | |
| Steps (reduced) |
59 (2) |
61 (4) |
62 (5) |
64 (7) |
65 (8) |
66 (9) |
68 (11) |
69 (12) |
70 (13) |
71 (14) |
72 (15) |
73 (16) | |
- Step size: NNN ¢, octave size: NNN ¢
Stretching the octave of 16edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 41ed6 does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.5 | -10.5 | +21.0 | +13.0 | +0.0 | +35.8 | +31.6 | -21.0 | +23.5 | +9.8 | +10.5 |
| Relative (%) | +13.9 | -13.9 | +27.8 | +17.2 | +0.0 | +47.3 | +41.7 | -27.8 | +31.1 | +13.0 | +13.9 | |
| Steps (reduced) |
16 (16) |
25 (25) |
32 (32) |
37 (37) |
41 (0) |
45 (4) |
48 (7) |
50 (9) |
53 (12) |
55 (14) |
57 (16) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +23.3 | -29.4 | +2.5 | -33.6 | +12.8 | -10.5 | -28.5 | +34.0 | +25.2 | +20.4 | +19.1 | +21.0 |
| Relative (%) | +30.7 | -38.8 | +3.3 | -44.4 | +16.9 | -13.9 | -37.6 | +45.0 | +33.4 | +26.9 | +25.2 | +27.8 | |
| Steps (reduced) |
59 (18) |
60 (19) |
62 (21) |
63 (22) |
65 (24) |
66 (25) |
67 (26) |
69 (28) |
70 (29) |
71 (30) |
72 (31) |
73 (32) | |
- Step size: NNN ¢, octave size: NNN ¢
Stretching the octave of 16edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 25edt does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +17.3 | +0.0 | +34.5 | +28.6 | +17.3 | -21.4 | -24.3 | +0.0 | -30.2 | +33.0 | +34.5 |
| Relative (%) | +22.7 | +0.0 | +45.4 | +37.6 | +22.7 | -28.1 | -32.0 | +0.0 | -39.8 | +43.4 | +45.4 | |
| Steps (reduced) |
16 (16) |
25 (0) |
32 (7) |
37 (12) |
41 (16) |
44 (19) |
47 (22) |
50 (0) |
52 (2) |
55 (5) |
57 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -28.0 | -4.1 | +28.6 | -7.1 | -36.0 | +17.3 | -0.3 | -13.0 | -21.4 | -25.8 | -26.7 | -24.3 |
| Relative (%) | -36.8 | -5.4 | +37.6 | -9.3 | -47.3 | +22.7 | -0.4 | -17.1 | -28.1 | -34.0 | -35.1 | -32.0 | |
| Steps (reduced) |
58 (8) |
60 (10) |
62 (12) |
63 (13) |
64 (14) |
66 (16) |
67 (17) |
68 (18) |
69 (19) |
70 (20) |
71 (21) |
72 (22) | |
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
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)
14edo
- 22edt
- 36ed6
- 11-limit WE (85.842c)
- 13-limit WE (85.759c)
- 42zpi (86.329c)
16edo
- 25edt
- 41ed6
- 57ed12
- 2.5.7.13 WE (75.105c)
- 13-limit WE (75.315c)
- 15zpi (75.262c)
99edo
- 157edt
- 256ed6
- 7-limit WE (12.117c)
- 13-limit WE (12.123c)
- 567zpi (12.138c)
- 568zpi (12.115c)
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
- 139ed5
- 155ed6
- 208ed11
- 255ed19
- 272ed23 (great for catnip temperament)
- 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
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
- 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
- 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
- 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
- 289ed7
- 356ed11
- 381ed13
- 421ed17
- 466ed23
- 13-limit WE (11.658c)
- Best nearby ZPI(s)
118edo (choose ZPIS)
- 187edt
- 69edf
- 13-limit WE (10.171c)
- Best nearby ZPI(s)
152edo (choose ZPIS)
- 241edt
- 13-limit WE (7.894c)
- Best nearby ZPI(s)
- Low priority
111edo
- 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)
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)