187edt: Difference between revisions
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== Harmonics == | == Theory == | ||
{{Harmonics in equal | 187edt is nearly identical to [[118edo]], but with the [[3/1|perfect twelfth]] rather than the [[2/1|octave]] being just. The octave is [[stretched and compressed tuning|stretched]] by about 0.164 cents. Like 118edo, 187edt is [[consistent]] to the [[integer limit|12-integer-limit]]. It preserves the 5-limit [[microtemperament|microtempering]] quality of 118edo, and the approximated [[prime harmonic]]s [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]] are slighly improved. | ||
| | |||
| | === Harmonics === | ||
| | {{Harmonics in equal|187|3|1|intervals=integer|columns=11}} | ||
}} | {{Harmonics in equal|187|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 187edt (continued)}} | ||
{{Harmonics in equal | |||
| | === Subsets and supersets === | ||
| | Since 187 factors into primes as {{nowrap| 11 × 17 }}, 187edt contains [[11edt]] and [[17edt]] as subset edts. | ||
| | |||
| start = 12 | == See also == | ||
| collapsed = | * [[69edf]] – relative edf | ||
}} | * [[118edo]] – relative edo | ||
Latest revision as of 11:22, 15 April 2025
| ← 186edt | 187edt | 188edt → |
187 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 187edt or 187ed3), is a nonoctave tuning system that divides the interval of 3/1 into 187 equal parts of about 10.2 ¢ each. Each step represents a frequency ratio of 31/187, or the 187th root of 3.
Theory
187edt is nearly identical to 118edo, but with the perfect twelfth rather than the octave being just. The octave is stretched by about 0.164 cents. Like 118edo, 187edt is consistent to the 12-integer-limit. It preserves the 5-limit microtempering quality of 118edo, and the approximated prime harmonics 7, 11, 17, and 19 are slighly improved.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.16 | +0.00 | +0.33 | +0.51 | +0.16 | -2.26 | +0.49 | +0.00 | +0.67 | -1.60 | +0.33 |
| Relative (%) | +1.6 | +0.0 | +3.2 | +5.0 | +1.6 | -22.3 | +4.8 | +0.0 | +6.6 | -15.7 | +3.2 | |
| Steps (reduced) |
118 (118) |
187 (0) |
236 (49) |
274 (87) |
305 (118) |
331 (144) |
354 (167) |
374 (0) |
392 (18) |
408 (34) |
423 (49) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.15 | -2.10 | +0.51 | +0.66 | -2.59 | +0.16 | -1.90 | +0.84 | -2.26 | -1.43 | +2.98 | +0.49 |
| Relative (%) | +40.8 | -20.6 | +5.0 | +6.5 | -25.5 | +1.6 | -18.7 | +8.2 | -22.3 | -14.1 | +29.3 | +4.8 | |
| Steps (reduced) |
437 (63) |
449 (75) |
461 (87) |
472 (98) |
482 (108) |
492 (118) |
501 (127) |
510 (136) |
518 (144) |
526 (152) |
534 (160) |
541 (167) | |
Subsets and supersets
Since 187 factors into primes as 11 × 17, 187edt contains 11edt and 17edt as subset edts.