16ed5: Difference between revisions
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== Theory == | |||
16ed5 is a strong no-2s, no-13s, no-29s [[41-limit]] tuning. Alternatively, it could include 2s, but the [[equave]] might need to be [[Octave stretch|compressed]] to make the [[2/1]] and [[4/1]] more in tune. | |||
This tuning tempers out [[25/24]] in the [[5-limit]]; [[21/20]], [[35/32]], and [[28/27]] in the [[7-limit]]; [[33/32]] and [[35/33]] in the [[11-limit]]; [[13/12]] and [[26/25]] in the [[13-limit]]; [[17/16]], [[34/33]], and [[35/34]] in the [[17-limit]]; [[19/18]] and [[39/38]] in the [[19-limit]]; [[23/22]] in the [[23-limit]]; [[29/28]] and [[29/27]] in the [[29-limit]]; [[31/30]] in the [[31-limit]]; [[37/36]], [[39/37]], and [[38/37]] in the [[37-limit]]; [[41/40]] and [[42/41]] in the [[41-limit]]; [[43/40]], [[43/42]] and [[43/41]] in the [[43-limit]]; and [[47/44]] in the [[47-limit]]. | |||
== Intervals == | == Intervals == | ||
Latest revision as of 22:25, 10 January 2025
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| ← 15ed5 | 16ed5 | 17ed5 → |
(semiconvergent)
(semiconvergent)
16 equal divisions of the 5th harmonic (abbreviated 16ed5) is a nonoctave tuning system that divides the interval of 5/1 into 16 equal parts of about 174 ¢ each. Each step represents a frequency ratio of 51/16, or the 16th root of 5.
Theory
16ed5 is a strong no-2s, no-13s, no-29s 41-limit tuning. Alternatively, it could include 2s, but the equave might need to be compressed to make the 2/1 and 4/1 more in tune.
This tuning tempers out 25/24 in the 5-limit; 21/20, 35/32, and 28/27 in the 7-limit; 33/32 and 35/33 in the 11-limit; 13/12 and 26/25 in the 13-limit; 17/16, 34/33, and 35/34 in the 17-limit; 19/18 and 39/38 in the 19-limit; 23/22 in the 23-limit; 29/28 and 29/27 in the 29-limit; 31/30 in the 31-limit; 37/36, 39/37, and 38/37 in the 37-limit; 41/40 and 42/41 in the 41-limit; 43/40, 43/42 and 43/41 in the 43-limit; and 47/44 in the 47-limit.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 174.1 | 10/9, 11/10, 19/17, 21/19, 23/21 |
| 2 | 348.3 | 11/9, 17/14, 21/17, 23/19 |
| 3 | 522.4 | 15/11, 19/14, 23/17 |
| 4 | 696.6 | 3/2 |
| 5 | 870.7 | 5/3, 18/11, 23/14 |
| 6 | 1044.9 | 11/6, 20/11 |
| 7 | 1219 | 2/1 |
| 8 | 1393.2 | 9/4, 20/9 |
| 9 | 1567.3 | 5/2, 22/9 |
| 10 | 1741.4 | 11/4, 19/7 |
| 11 | 1915.6 | 3/1 |
| 12 | 2089.7 | 10/3 |
| 13 | 2263.9 | 11/3 |
| 14 | 2438 | |
| 15 | 2612.2 | 9/2 |
| 16 | 2786.3 | 5/1 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +19.0 | +13.6 | +0.0 | -60.1 | +28.2 | -86.9 | -28.9 | -47.3 | -29.8 | -82.8 | -24.1 |
| Relative (%) | +10.9 | +7.8 | +0.0 | -34.5 | +16.2 | -49.9 | -16.6 | -27.2 | -17.1 | -47.5 | -13.8 | |
| Steps (reduced) |
7 (7) |
11 (11) |
16 (0) |
19 (3) |
24 (8) |
25 (9) |
28 (12) |
29 (13) |
31 (15) |
33 (1) |
34 (2) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +17.9 | +14.3 | -68.2 | -48.0 | -81.9 | +80.8 | +23.0 | +34.8 | -65.6 | +60.4 | -76.3 |
| Relative (%) | +10.3 | +8.2 | -39.1 | -27.6 | -47.0 | +46.4 | +13.2 | +20.0 | -37.7 | +34.7 | -43.8 | |
| Steps (reduced) |
36 (4) |
37 (5) |
37 (5) |
38 (6) |
39 (7) |
41 (9) |
41 (9) |
42 (10) |
42 (10) |
43 (11) |
43 (11) | |