User:BudjarnLambeth/Sandbox2: Difference between revisions
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| Line 6: | Line 6: | ||
= Title2 = | = Title2 = | ||
== Octave | == Octave compression == | ||
What follows is a comparison of | What follows is a comparison of compressed-octave 17edo tunings. | ||
; [[ | ; 17edo | ||
* Step size: | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
_ing the octave of 17edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning | Pure-octaves 17edo approximates all harmonics up to 16 within NNN{{c}}. | ||
{{Harmonics in | {{Harmonics in equal|17|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo}} | ||
{{Harmonics in | {{Harmonics in equal|17|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo (continued)}} | ||
; [[44ed6]] | |||
* Step size: NNN{{c}}, octave size: NNN{{c}} | |||
_ing the octave of 17edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 44ed6 does this. | |||
{{Harmonics in equal|44|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6}} | |||
{{Harmonics in equal|44|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6 (continued)}} | |||
; [[27edt]] | ; [[27edt]] | ||
| Line 21: | Line 27: | ||
{{Harmonics in equal|27|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt (continued)}} | {{Harmonics in equal|27|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt (continued)}} | ||
; [[ | ; [[zpi|56zpi]] | ||
* Step size: | * Step size: 70.403{{c}}, octave size: NNN{{c}} | ||
_ing the octave of 17edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning | _ing the octave of 17edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 56zpi does this. | ||
{{Harmonics in | {{Harmonics in cet|70.403|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi}} | ||
{{Harmonics in | {{Harmonics in cet|70.403|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi (continued)}} | ||
; [[WE|17et, 2.3.7.11.13 WE tuning]] | |||
* Step size: 70.410{{c}}, octave size: NNN{{c}} | |||
_ing the octave of 17edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.7.11.13 WE tuning and 2.3.7.11.13 [[TE]] tuning both do this. | |||
{{Harmonics in cet|70.410|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11.13 WE tuning}} | |||
{{Harmonics in cet|70.410|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11.13 WE tuning (continued)}} | |||
; [[WE|17et, 2.3.7.11 WE tuning]] | ; [[WE|17et, 2.3.7.11 WE tuning]] | ||
| Line 32: | Line 44: | ||
{{Harmonics in cet|70.392|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning}} | {{Harmonics in cet|70.392|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning}} | ||
{{Harmonics in cet|70.392|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning (continued)}} | {{Harmonics in cet|70.392|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning (continued)}} | ||
Revision as of 03:50, 22 August 2025
Title1
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.1 | -8.5 | -8.2 | +4.1 | -12.6 | +19.5 | -12.3 | -16.9 | +0.0 | +34.3 | -16.7 |
| Relative (%) | -4.1 | -8.5 | -8.2 | +4.1 | -12.6 | +19.6 | -12.4 | -17.0 | +0.0 | +34.4 | -16.7 | |
| Steps (reduced) |
12 (12) |
19 (19) |
24 (24) |
28 (28) |
31 (31) |
34 (34) |
36 (36) |
38 (38) |
40 (0) |
42 (2) |
43 (3) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.4 | +3.4 | +6.7 | +21.5 | +6.7 | +40.7 | +10.1 | +6.7 | +24.9 | -39.9 | +10.1 |
| Relative (%) | +3.3 | +3.3 | +6.7 | +21.4 | +6.7 | +40.6 | +10.0 | +6.7 | +24.8 | -39.8 | +10.0 | |
| Steps (reduced) |
12 (5) |
19 (5) |
24 (3) |
28 (0) |
31 (3) |
34 (6) |
36 (1) |
38 (3) |
40 (5) |
41 (6) |
43 (1) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.2 | +0.0 | +2.5 | +16.6 | +1.2 | +34.7 | +3.7 | +0.0 | +17.8 | -47.1 | +2.5 |
| Relative (%) | +1.2 | +0.0 | +2.5 | +16.6 | +1.2 | +34.6 | +3.7 | +0.0 | +17.8 | -47.1 | +2.5 | |
| Steps (reduced) |
12 (12) |
19 (0) |
24 (5) |
28 (9) |
31 (12) |
34 (15) |
36 (17) |
38 (0) |
40 (2) |
41 (3) |
43 (5) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.8 | -0.8 | +1.5 | +15.5 | +0.0 | +33.3 | +2.3 | -1.5 | +16.2 | -48.7 | +0.8 |
| Relative (%) | +0.8 | -0.8 | +1.5 | +15.4 | +0.0 | +33.3 | +2.3 | -1.5 | +16.2 | -48.7 | +0.8 | |
| Steps (reduced) |
12 (12) |
19 (19) |
24 (24) |
28 (28) |
31 (0) |
34 (3) |
36 (5) |
38 (7) |
40 (9) |
41 (10) |
43 (12) | |
Title2
Octave compression
What follows is a comparison of compressed-octave 17edo tunings.
- 17edo
- Step size: NNN ¢, octave size: NNN ¢
Pure-octaves 17edo approximates all harmonics up to 16 within NNN ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | +3.9 | +0.0 | -33.4 | +3.9 | +19.4 | +0.0 | +7.9 | -33.4 | +13.4 | +3.9 |
| Relative (%) | +0.0 | +5.6 | +0.0 | -47.3 | +5.6 | +27.5 | +0.0 | +11.1 | -47.3 | +19.0 | +5.6 | |
| Steps (reduced) |
17 (0) |
27 (10) |
34 (0) |
39 (5) |
44 (10) |
48 (14) |
51 (0) |
54 (3) |
56 (5) |
59 (8) |
61 (10) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.5 | +19.4 | -29.4 | +0.0 | -34.4 | +7.9 | -15.2 | -33.4 | +23.3 | +13.4 | +7.0 | +3.9 |
| Relative (%) | +9.3 | +27.5 | -41.7 | +0.0 | -48.7 | +11.1 | -21.5 | -47.3 | +33.1 | +19.0 | +9.9 | +5.6 | |
| Steps (reduced) |
63 (12) |
65 (14) |
66 (15) |
68 (0) |
69 (1) |
71 (3) |
72 (4) |
73 (5) |
75 (7) |
76 (8) |
77 (9) |
78 (10) | |
- Step size: NNN ¢, octave size: NNN ¢
_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 44ed6 does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.5 | +1.5 | -3.0 | +33.6 | +0.0 | +15.1 | -4.6 | +3.0 | +32.1 | +8.1 | -1.5 |
| Relative (%) | -2.2 | +2.2 | -4.3 | +47.7 | +0.0 | +21.5 | -6.5 | +4.3 | +45.6 | +11.5 | -2.2 | |
| Steps (reduced) |
17 (17) |
27 (27) |
34 (34) |
40 (40) |
44 (0) |
48 (4) |
51 (7) |
54 (10) |
57 (13) |
59 (15) |
61 (17) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.9 | +13.6 | +35.2 | -6.1 | +30.0 | +1.5 | -21.6 | +30.6 | +16.6 | +6.6 | +0.1 | -3.0 |
| Relative (%) | +1.3 | +19.3 | +49.9 | -8.6 | +42.5 | +2.2 | -30.6 | +43.4 | +23.6 | +9.4 | +0.2 | -4.3 | |
| Steps (reduced) |
63 (19) |
65 (21) |
67 (23) |
68 (24) |
70 (26) |
71 (27) |
72 (28) |
74 (30) |
75 (31) |
76 (32) |
77 (33) |
78 (34) | |
- Step size: NNN ¢, octave size: NNN ¢
_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 27edt does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.5 | +0.0 | -4.9 | +31.4 | -2.5 | +12.4 | -7.4 | +0.0 | +28.9 | +4.8 | -4.9 |
| Relative (%) | -3.5 | +0.0 | -7.0 | +44.6 | -3.5 | +17.6 | -10.5 | +0.0 | +41.1 | +6.8 | -7.0 | |
| Steps (reduced) |
17 (17) |
27 (0) |
34 (7) |
40 (13) |
44 (17) |
48 (21) |
51 (24) |
54 (0) |
57 (3) |
59 (5) |
61 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | +10.0 | +31.4 | -9.9 | +26.0 | -2.5 | -25.6 | +26.5 | +12.4 | +2.3 | -4.2 | -7.4 |
| Relative (%) | -3.7 | +14.1 | +44.6 | -14.0 | +37.0 | -3.5 | -36.4 | +37.6 | +17.6 | +3.3 | -5.9 | -10.5 | |
| Steps (reduced) |
63 (9) |
65 (11) |
67 (13) |
68 (14) |
70 (16) |
71 (17) |
72 (18) |
74 (20) |
75 (21) |
76 (22) |
77 (23) |
78 (24) | |
- Step size: 70.403 ¢, octave size: NNN ¢
_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 56zpi does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.1 | -1.1 | -6.3 | +29.8 | -4.2 | +10.5 | -9.4 | -2.1 | +26.7 | +2.5 | -7.4 |
| Relative (%) | -4.5 | -1.5 | -8.9 | +42.3 | -6.0 | +14.9 | -13.4 | -3.1 | +37.9 | +3.5 | -10.5 | |
| Step | 17 | 27 | 34 | 40 | 44 | 48 | 51 | 54 | 57 | 59 | 61 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.1 | +7.4 | +28.7 | -12.6 | +23.3 | -5.3 | -28.5 | +23.5 | +9.4 | -0.7 | -7.2 | -10.5 |
| Relative (%) | -7.3 | +10.5 | +40.8 | -17.9 | +33.0 | -7.5 | -40.5 | +33.4 | +13.4 | -1.0 | -10.3 | -14.9 | |
| Step | 63 | 65 | 67 | 68 | 70 | 71 | 72 | 74 | 75 | 76 | 77 | 78 | |
- Step size: 70.410 ¢, octave size: NNN ¢
_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 2.3.7.11.13 WE tuning and 2.3.7.11.13 TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.0 | -0.9 | -6.1 | +30.1 | -3.9 | +10.9 | -9.1 | -1.8 | +27.1 | +2.9 | -6.9 |
| Relative (%) | -4.3 | -1.3 | -8.6 | +42.7 | -5.6 | +15.4 | -12.9 | -2.5 | +38.4 | +4.1 | -9.9 | |
| Step | 17 | 27 | 34 | 40 | 44 | 48 | 51 | 54 | 57 | 59 | 61 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.7 | +7.8 | +29.2 | -12.1 | +23.7 | -4.8 | -28.0 | +24.0 | +10.0 | -0.2 | -6.7 | -10.0 |
| Relative (%) | -6.7 | +11.1 | +41.5 | -17.2 | +33.7 | -6.8 | -39.8 | +34.1 | +14.2 | -0.2 | -9.5 | -14.2 | |
| Step | 63 | 65 | 67 | 68 | 70 | 71 | 72 | 74 | 75 | 76 | 77 | 78 | |
- Step size: 70.392 ¢, octave size: NNN ¢
_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 2.3.7.11 WE tuning and 2.3.7.11 TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.3 | -1.4 | -6.7 | +29.4 | -4.7 | +10.0 | -10.0 | -2.7 | +26.0 | +1.8 | -8.0 |
| Relative (%) | -4.7 | -1.9 | -9.5 | +41.7 | -6.7 | +14.2 | -14.2 | -3.9 | +37.0 | +2.6 | -11.4 | |
| Step | 17 | 27 | 34 | 40 | 44 | 48 | 51 | 54 | 57 | 59 | 61 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.8 | +6.7 | +28.0 | -13.3 | +22.5 | -6.1 | -29.3 | +22.7 | +8.6 | -1.5 | -8.1 | -11.4 |
| Relative (%) | -8.3 | +9.5 | +39.8 | -19.0 | +31.9 | -8.6 | -41.6 | +32.2 | +12.2 | -2.2 | -11.5 | -16.2 | |
| Step | 63 | 65 | 67 | 68 | 70 | 71 | 72 | 74 | 75 | 76 | 77 | 78 | |