176edt: Difference between revisions
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== Harmonics == | == Theory == | ||
{{Harmonics in equal | 176edt is closely related to [[111edo]], but with the [[3/1|perfect twelfth]] tuned just instead of the [[2/1|octave]]. The octave is [[stretched and compressed tuning|compressed]] by about 0.472 cents. Like 111edo, 176edt is [[consistent]] to the [[integer limit|22-integer-limit]]. While it tunes 2 and [[11/1|11]] flat, the [[5/1|5]], [[7/1|7]], [[13/1|13]], [[17/1|17]], and [[19/1|19]] remain sharp as in 111edo but significantly less so. The [[23/1|23]], which is flat to begin with, becomes worse. | ||
| | |||
| | === Harmonics === | ||
| | {{Harmonics in equal|176|3|1|interval=integer|columns=11}} | ||
}} | {{Harmonics in equal|176|3|1|interval=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 176edt (continued)}} | ||
{{Harmonics in equal | |||
| | === Subsets and supersets === | ||
| | Since 176 factors into primes as {{nowrap| 2<sup>4</sup> × 11 }}, 176edt contains subset edts {{EDs|equave=t| 2, 4, 8, 11, 16, 22, 44, and 88 }}. | ||
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| start = 12 | == See also == | ||
| collapsed = | * [[111edo]] – relative edo | ||
}} | * [[287ed6]] – relative ed6 | ||
Latest revision as of 15:32, 12 April 2025
| ← 175edt | 176edt | 177edt → |
176 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 176edt or 176ed3), is a nonoctave tuning system that divides the interval of 3/1 into 176 equal parts of about 10.8 ¢ each. Each step represents a frequency ratio of 31/176, or the 176th root of 3.
Theory
176edt is closely related to 111edo, but with the perfect twelfth tuned just instead of the octave. The octave is compressed by about 0.472 cents. Like 111edo, 176edt is consistent to the 22-integer-limit. While it tunes 2 and 11 flat, the 5, 7, 13, 17, and 19 remain sharp as in 111edo but significantly less so. The 23, which is flat to begin with, becomes worse.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.47 | +0.00 | -0.94 | +1.78 | -0.47 | +2.82 | -1.41 | +0.00 | +1.31 | -1.60 | -0.94 |
| Relative (%) | -4.4 | +0.0 | -8.7 | +16.5 | -4.4 | +26.1 | -13.1 | +0.0 | +12.1 | -14.8 | -8.7 | |
| Steps (reduced) |
111 (111) |
176 (0) |
222 (46) |
258 (82) |
287 (111) |
312 (136) |
333 (157) |
352 (0) |
369 (17) |
384 (32) |
398 (46) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.97 | +2.35 | +1.78 | -1.89 | +1.22 | -0.47 | +3.18 | +0.84 | +2.82 | -2.07 | -3.38 | -1.41 |
| Relative (%) | +9.0 | +21.7 | +16.5 | -17.5 | +11.3 | -4.4 | +29.5 | +7.7 | +26.1 | -19.2 | -31.3 | -13.1 | |
| Steps (reduced) |
411 (59) |
423 (71) |
434 (82) |
444 (92) |
454 (102) |
463 (111) |
472 (120) |
480 (128) |
488 (136) |
495 (143) |
502 (150) |
509 (157) | |
Subsets and supersets
Since 176 factors into primes as 24 × 11, 176edt contains subset edts 2, 4, 8, 11, 16, 22, 44, and 88.