User:BudjarnLambeth/Sandbox2
Title1
Octave stretch or compression
58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as 92edt or 150ed6.
What follows is a comparison of stretched- and compressed-octave 58edo tunings.
- Step size: 20.736 ¢, octave size: 1202.69 ¢
Stretching the octave of 58edo by around 2.5 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 288zpi does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.69 | +5.76 | +5.38 | -7.69 | +8.44 | -9.59 | +8.06 | -9.22 | -5.00 | -4.12 | -9.60 |
| Relative (%) | +13.0 | +27.8 | +25.9 | -37.1 | +40.7 | -46.3 | +38.9 | -44.5 | -24.1 | -19.9 | -46.3 | |
| Step | 58 | 92 | 116 | 134 | 150 | 162 | 174 | 183 | 192 | 200 | 207 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.02 | -6.91 | -1.93 | -9.98 | +9.48 | -6.53 | +3.54 | -2.31 | -3.84 | -1.43 | +4.56 | -6.92 |
| Relative (%) | -14.6 | -33.3 | -9.3 | -48.1 | +45.7 | -31.5 | +17.1 | -11.2 | -18.5 | -6.9 | +22.0 | -33.3 | |
| Step | 214 | 220 | 226 | 231 | 237 | 241 | 246 | 250 | 254 | 258 | 262 | 265 | |
- 58edo
- Step size: 20.690 ¢, octave size: 1200.00 ¢
Pure-octaves 58edo approximates all harmonics up to 16 within NNN ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.49 | +0.00 | +6.79 | +1.49 | +3.59 | +0.00 | +2.99 | +6.79 | +7.30 | +1.49 |
| Relative (%) | +0.0 | +7.2 | +0.0 | +32.8 | +7.2 | +17.3 | +0.0 | +14.4 | +32.8 | +35.3 | +7.2 | |
| Steps (reduced) |
58 (0) |
92 (34) |
116 (0) |
135 (19) |
150 (34) |
163 (47) |
174 (0) |
184 (10) |
193 (19) |
201 (27) |
208 (34) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.75 | +3.59 | +8.28 | +0.00 | -1.51 | +2.99 | -7.86 | +6.79 | +5.08 | +7.30 | -7.58 | +1.49 |
| Relative (%) | +37.4 | +17.3 | +40.0 | +0.0 | -7.3 | +14.4 | -38.0 | +32.8 | +24.6 | +35.3 | -36.7 | +7.2 | |
| Steps (reduced) |
215 (41) |
221 (47) |
227 (53) |
232 (0) |
237 (5) |
242 (10) |
246 (14) |
251 (19) |
255 (23) |
259 (27) |
262 (30) |
266 (34) | |
- Step size: 20.680 ¢, octave size: 1199.42 ¢
Compressing the octave of 58edo by around half a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 150ed6 does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.58 | +0.58 | -1.15 | +5.45 | +0.00 | +1.97 | -1.73 | +1.15 | +4.87 | +5.30 | -0.58 |
| Relative (%) | -2.8 | +2.8 | -5.6 | +26.3 | +0.0 | +9.5 | -8.4 | +5.6 | +23.5 | +25.6 | -2.8 | |
| Steps (reduced) |
58 (58) |
92 (92) |
116 (116) |
135 (135) |
150 (0) |
163 (13) |
174 (24) |
184 (34) |
193 (43) |
201 (51) |
208 (58) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.61 | +1.39 | +6.02 | -2.31 | -3.87 | +0.58 | -10.31 | +4.29 | +2.54 | +4.72 | -10.19 | -1.15 |
| Relative (%) | +27.1 | +6.7 | +29.1 | -11.2 | -18.7 | +2.8 | -49.8 | +20.7 | +12.3 | +22.8 | -49.3 | -5.6 | |
| Steps (reduced) |
215 (65) |
221 (71) |
227 (77) |
232 (82) |
237 (87) |
242 (92) |
246 (96) |
251 (101) |
255 (105) |
259 (109) |
262 (112) |
266 (116) | |
- Step size: 20.673 ¢, octave size: 1199.06 ¢
Compressing the octave of 58edo by around 1 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 92edt does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.94 | +0.00 | -1.88 | +4.60 | -0.94 | +0.94 | -2.82 | +0.00 | +3.66 | +4.04 | -1.88 |
| Relative (%) | -4.6 | +0.0 | -9.1 | +22.2 | -4.6 | +4.6 | -13.7 | +0.0 | +17.7 | +19.5 | -9.1 | |
| Steps (reduced) |
58 (58) |
92 (0) |
116 (24) |
135 (43) |
150 (58) |
163 (71) |
174 (82) |
184 (0) |
193 (9) |
201 (17) |
208 (24) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.26 | +0.00 | +4.60 | -3.77 | -5.35 | -0.94 | +8.82 | +2.72 | +0.94 | +3.10 | +8.84 | -2.82 |
| Relative (%) | +20.6 | +0.0 | +22.2 | -18.2 | -25.9 | -4.6 | +42.7 | +13.1 | +4.6 | +15.0 | +42.7 | -13.7 | |
| Steps (reduced) |
215 (31) |
221 (37) |
227 (43) |
232 (48) |
237 (53) |
242 (58) |
247 (63) |
251 (67) |
255 (71) |
259 (75) |
263 (79) |
266 (82) | |
- Step size: 20.666 ¢, octave size: 1198.63 ¢
Compressing the octave of 58edo by just under 1.5 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06 ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.37 | -0.68 | -2.74 | +3.60 | -2.06 | -0.27 | -4.12 | -1.37 | +2.22 | +2.55 | -3.43 |
| Relative (%) | -6.6 | -3.3 | -13.3 | +17.4 | -9.9 | -1.3 | -19.9 | -6.6 | +10.8 | +12.3 | -16.6 | |
| Step | 58 | 92 | 116 | 135 | 150 | 163 | 174 | 184 | 193 | 201 | 208 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.66 | -1.64 | +2.91 | -5.49 | -7.11 | -2.74 | +6.99 | +0.85 | -0.95 | +1.18 | +6.88 | -4.80 |
| Relative (%) | +12.9 | -7.9 | +14.1 | -26.6 | -34.4 | -13.2 | +33.8 | +4.1 | -4.6 | +5.7 | +33.3 | -23.2 | |
| Step | 215 | 221 | 227 | 232 | 237 | 242 | 247 | 251 | 255 | 259 | 263 | 266 | |
- Step size: 20.663 ¢, octave size: 1198.45 ¢
Compressing the octave of 58edo by just over 1.5 ¢ 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 (¢) | -1.55 | -0.96 | -3.09 | +3.19 | -2.51 | -0.76 | -4.64 | -1.92 | +1.65 | +1.95 | -4.05 |
| Relative (%) | -7.5 | -4.6 | -15.0 | +15.4 | -12.1 | -3.7 | -22.4 | -9.3 | +8.0 | +9.4 | -19.6 | |
| Step | 58 | 92 | 116 | 135 | 150 | 163 | 174 | 184 | 193 | 201 | 208 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.02 | -2.30 | +2.23 | -6.18 | -7.82 | -3.46 | +6.25 | +0.10 | -1.72 | +0.40 | +6.09 | -5.60 |
| Relative (%) | +9.8 | -11.1 | +10.8 | -29.9 | -37.9 | -16.8 | +30.2 | +0.5 | -8.3 | +1.9 | +29.5 | -27.1 | |
| Step | 215 | 221 | 227 | 232 | 237 | 242 | 247 | 251 | 255 | 259 | 263 | 266 | |
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)