165edt: Difference between revisions
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== | == Theory == | ||
165edt is related to [[104edo]], but with the [[3/1|perfect twelfth]] instead of the [[2/1|octave]] tuned just. The octave is about 1.19 cents compressed. Unlike 104edo, which is only [[consistent]] to the [[integer limit|4-integer-limit]], 165edt is consistent to the 6-integer-limit. It may be said to have a flat tuning tendency, as within [[harmonic]]s 1–16, only multiples of [[5/1|5]] are tuned sharp. | |||
== Harmonics == | === Harmonics === | ||
{{Harmonics in equal | {{Harmonics in equal|165|3|1|intervals=integer|columns=11}} | ||
| | {{Harmonics in equal|165|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 165edt (continued)}} | ||
| | |||
| | === Subsets and supersets === | ||
}} | Since 165 factors into primes as {{nowrap| 3 × 5 × 11 }}, 165edt has subset edts {{EDs|equave=t| 3, 5, 11, 15, 33, and 55 }}. | ||
{{Harmonics in equal | |||
| | == See also == | ||
| | * [[104edo]] – relative edo | ||
| | * [[269ed6]] – relative ed6 | ||
| start = 12 | |||
| collapsed = | |||
}} | |||
Latest revision as of 12:10, 9 April 2025
| ← 164edt | 165edt | 166edt → |
165 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 165edt or 165ed3), is a nonoctave tuning system that divides the interval of 3/1 into 165 equal parts of about 11.5 ¢ each. Each step represents a frequency ratio of 31/165, or the 165th root of 3.
Theory
165edt is related to 104edo, but with the perfect twelfth instead of the octave tuned just. The octave is about 1.19 cents compressed. Unlike 104edo, which is only consistent to the 4-integer-limit, 165edt is consistent to the 6-integer-limit. It may be said to have a flat tuning tendency, as within harmonics 1–16, only multiples of 5 are tuned sharp.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.19 | +0.00 | -2.38 | +3.22 | -1.19 | -2.94 | -3.58 | +0.00 | +2.03 | -1.60 | -2.38 |
| Relative (%) | -10.3 | +0.0 | -20.7 | +27.9 | -10.3 | -25.5 | -31.0 | +0.0 | +17.6 | -13.9 | -20.7 | |
| Steps (reduced) |
104 (104) |
165 (0) |
208 (43) |
242 (77) |
269 (104) |
292 (127) |
312 (147) |
330 (0) |
346 (16) |
360 (30) |
373 (43) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | -4.13 | +3.22 | -4.77 | +5.55 | -1.19 | -2.58 | +0.84 | -2.94 | -2.79 | +0.94 | -3.58 |
| Relative (%) | -22.8 | -35.9 | +27.9 | -41.4 | +48.1 | -10.3 | -22.4 | +7.3 | -25.5 | -24.2 | +8.2 | -31.0 | |
| Steps (reduced) |
385 (55) |
396 (66) |
407 (77) |
416 (86) |
426 (96) |
434 (104) |
442 (112) |
450 (120) |
457 (127) |
464 (134) |
471 (141) |
477 (147) | |
Subsets and supersets
Since 165 factors into primes as 3 × 5 × 11, 165edt has subset edts 3, 5, 11, 15, 33, and 55.