16edf: Difference between revisions
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Cleanup; note its lack of similarity to 27edo |
→Theory: +subsets and supersets |
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== Theory == | == Theory == | ||
16edf corresponds to 27.3522…[[edo]]. It is not quite similar to [[27edo]], but it is similar to every third step of [[82edo]]. It contains good approximations of the [[7/1|7th]] and [[13/1|13th]] [[ | 16edf corresponds to 27.3522…[[edo]]. It is not quite similar to [[27edo]], but it is similar to every third step of [[82edo]]. It contains good approximations of the [[7/1|7th]] and [[13/1|13th]] [[harmonic]]s. | ||
It serves as a good approximation to [[halftone]] temperament, containing the [[~]][[7/5]] generator at 13 steps. | It serves as a good approximation to [[halftone]] temperament, containing the [[~]][[7/5]] generator at 13 steps. | ||
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{{Harmonics in equal|16|3|2}} | {{Harmonics in equal|16|3|2}} | ||
{{Harmonics in equal|16|3|2|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 16edf (continued)}} | {{Harmonics in equal|16|3|2|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 16edf (continued)}} | ||
=== Subsets and supersets === | |||
Since 16 factors into primes as 2<sup>4</sup>, 16edf contains subset edfs {{EDs|equave=f| 2, 4, and 8 }}. | |||
== Intervals == | == Intervals == | ||
Revision as of 08:29, 4 March 2025
| ← 15edf | 16edf | 17edf → |
16 equal divisions of the perfect fifth (abbreviated 16edf or 16ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 16 equal parts of about 43.9 ¢ each. Each step represents a frequency ratio of (3/2)1/16, or the 16th root of 3/2.
Theory
16edf corresponds to 27.3522…edo. It is not quite similar to 27edo, but it is similar to every third step of 82edo. It contains good approximations of the 7th and 13th harmonics.
It serves as a good approximation to halftone temperament, containing the ~7/5 generator at 13 steps.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -15.5 | -15.5 | +13.0 | +21.5 | +13.0 | +9.3 | -2.5 | +13.0 | +6.1 | +16.5 | -2.5 |
| Relative (%) | -35.2 | -35.2 | +29.6 | +49.0 | +29.6 | +21.3 | -5.7 | +29.6 | +13.8 | +37.7 | -5.7 | |
| Steps (reduced) |
27 (11) |
43 (11) |
55 (7) |
64 (0) |
71 (7) |
77 (13) |
82 (2) |
87 (7) |
91 (11) |
95 (15) |
98 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.4 | -6.1 | +6.1 | -17.9 | +8.7 | -2.5 | -8.3 | -9.4 | -6.1 | +1.1 | +11.9 | -17.9 |
| Relative (%) | -21.5 | -13.9 | +13.8 | -40.9 | +19.9 | -5.7 | -19.0 | -21.4 | -13.9 | +2.5 | +27.1 | -40.9 | |
| Steps (reduced) |
101 (5) |
104 (8) |
107 (11) |
109 (13) |
112 (0) |
114 (2) |
116 (4) |
118 (6) |
120 (8) |
122 (10) |
124 (12) |
125 (13) | |
Subsets and supersets
Since 16 factors into primes as 24, 16edf contains subset edfs 2, 4, and 8.
Intervals
| # | Cents | Approximate ratios | Halftone[6] notation (using ups and downs) |
Comments |
|---|---|---|---|---|
| 0 | 0.0 | 1/1 | C | |
| 1 | 43.9 | 40/39, 39/38 | ^C | |
| 2 | 87.7 | 20/19 | Db | |
| 3 | 131.6 | 55/51, (27/25) | vD | |
| 4 | 175.5 | (21/19) | D | |
| 5 | 219.4 | vE | ||
| 6 | 263.2 | (7/6) | E | |
| 7 | 307.1 | Fb | ||
| 8 | 351.0 | 60/49, 49/40 | vF | |
| 9 | 394.8 | (44/35) | F | |
| 10 | 438.7 | (9/7) | Ab | |
| 11 | 482.6 | vA | ||
| 12 | 526.5 | (19/14) | A | |
| 13 | 570.3 | (25/18), 153/110, 112/81 | B | |
| 14 | 614.2 | (10/7) | Cb | |
| 15 | 658.1 | 19/13 | vC | |
| 16 | 702.0 | 3/2 | C | Just perfect fifth |
| 17 | 745.8 | 20/13 | ||
| 18 | 789.7 | 30/19 | ||
| 19 | 833.6 | 55/34 | ||
| 20 | 877.4 | |||
| 21 | 921.3 | |||
| 22 | 965.2 | 7/4 | ||
| 23 | 1009.0 | |||
| 24 | 1052.9 | 90/49, (11/6) | ||
| 25 | 1096.8 | (66/35) | ||
| 26 | 1140.7 | |||
| 27 | 1184.5 | |||
| 28 | 1228.4 | 128/63 | ||
| 29 | 1272.3 | 25/12 | ||
| 30 | 1316.2 | 15/7 | ||
| 31 | 1360.0 | 57/26 | ||
| 32 | 1403.9 | 9/4 | Pythagorean major ninth |