186ed6: Difference between revisions
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{{Harmonics in equal|186|6|1|intervals=integer|columns=11}} | {{Harmonics in equal|186|6|1|intervals=integer|columns=11}} | ||
{{Harmonics in equal|186|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 186ed6 (continued)}} | {{Harmonics in equal|186|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 186ed6 (continued)}} | ||
=== Subsets and supersets === | |||
Since 186 factors into primes as {{nowrap| 2 × 3 × 31 }}, 186ed6 contains subset ed6's {{EDs|equave=6| 2, 3, 6, 31, 62, and 93 }}. | |||
== See also == | == See also == | ||
* [[72edo]] – relative edo | * [[72edo]] – relative edo | ||
* [[114edt]] – relative edt | * [[114edt]] – relative edt | ||
Revision as of 19:57, 10 April 2025
| ← 185ed6 | 186ed6 | 187ed6 → |
186 equal divisions of the 6th harmonic (abbreviated 186ed6) is a nonoctave tuning system that divides the interval of 6/1 into 186 equal parts of about 16.7 ¢ each. Each step represents a frequency ratio of 61/186, or the 186th root of 6.
Theory
186ed6 is closely related to 72edo, but with the 6th harmonic rather than the octave being just, which results in the octaves being stretched by about 0.757 ¢. Like 72edo, 186ed6 is consistent to the 18-integer-limit, and significantly improves on 72edo's approximation to 13.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.76 | -0.76 | +1.51 | -1.23 | +0.00 | -0.04 | +2.27 | -1.51 | -0.47 | +1.30 | +0.76 |
| Relative (%) | +4.5 | -4.5 | +9.1 | -7.3 | +0.0 | -0.2 | +13.6 | -9.1 | -2.8 | +7.8 | +4.5 | |
| Steps (reduced) |
72 (72) |
114 (114) |
144 (144) |
167 (167) |
186 (0) |
202 (16) |
216 (30) |
228 (42) |
239 (53) |
249 (63) |
258 (72) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.40 | +0.72 | -1.98 | +3.03 | -1.87 | -0.76 | +5.70 | +0.29 | -0.79 | +2.06 | -8.19 | +1.51 |
| Relative (%) | -26.4 | +4.3 | -11.9 | +18.2 | -11.2 | -4.5 | +34.2 | +1.7 | -4.8 | +12.3 | -49.1 | +9.1 | |
| Steps (reduced) |
266 (80) |
274 (88) |
281 (95) |
288 (102) |
294 (108) |
300 (114) |
306 (120) |
311 (125) |
316 (130) |
321 (135) |
325 (139) |
330 (144) | |
Subsets and supersets
Since 186 factors into primes as 2 × 3 × 31, 186ed6 contains subset ed6's 2, 3, 6, 31, 62, and 93.