153edt: Difference between revisions
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However, 153edt's approximation of [[2/1]] is close to maximally bad, meaning that it is as far from an octave-equivalent tuning that an [[EDT]] of this size can be (though by this point, it is only 6 or so cents off). | However, 153edt's approximation of [[2/1]] is close to maximally bad, meaning that it is as far from an octave-equivalent tuning that an [[EDT]] of this size can be (though by this point, it is only 6 or so cents off). | ||
== Harmonics == | == Harmonics == | ||
Revision as of 10:18, 5 October 2024
| ← 152edt | 153edt | 154edt → |
153 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 153edt or 153ed3), is a nonoctave tuning system that divides the interval of 3/1 into 153 equal parts of about 12.4 ¢ each. Each step represents a frequency ratio of 31/153, or the 153rd root of 3.
153edt is notable for being the denominator of a convergent to log3(7/3), after 9edt, 13edt and 35edt, and the last before 3401edt, and therefore has an extremely accurate approximation to 7/3, a mere 0.0036 cents flat. In fact, 153edt demonstrates 11-strong 7-3 telicity, due to the next term in the continued fraction expansion being large (note how much larger 3401 is than 153), although 3401edt in fact surpasses it, demonstrating 16-strong 7-3 telicity.
In the no-twos 7-limit, 153edt supports canopus temperament, which gives it a rather accurate approximation of the 5th harmonic; and it additionally is accurate in the 11-limit, tempering out the comma 387420489/386683451 in the 3.7.11 subgroup. Harmonics 19 and 29 are also notably good.
However, 153edt's approximation of 2/1 is close to maximally bad, meaning that it is as far from an octave-equivalent tuning that an EDT of this size can be (though by this point, it is only 6 or so cents off).
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.81 | +0.00 | -0.80 | -1.75 | +5.81 | -0.00 | +5.01 | +0.00 | +4.06 | +0.66 | -0.80 |
| Relative (%) | +46.8 | +0.0 | -6.5 | -14.1 | +46.8 | -0.0 | +40.3 | +0.0 | +32.7 | +5.3 | -6.5 | |
| Steps (reduced) |
97 (97) |
153 (0) |
193 (40) |
224 (71) |
250 (97) |
271 (118) |
290 (137) |
306 (0) |
321 (15) |
334 (28) |
346 (40) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | +5.81 | -1.75 | -1.60 | +5.32 | +5.81 | -0.77 | -2.55 | -0.00 | -5.95 | +4.11 |
| Relative (%) | -21.2 | +46.7 | -14.1 | -12.9 | +42.8 | +46.8 | -6.2 | -20.5 | -0.0 | -47.9 | +33.0 | |
| Steps (reduced) |
357 (51) |
368 (62) |
377 (71) |
386 (80) |
395 (89) |
403 (97) |
410 (104) |
417 (111) |
424 (118) |
430 (124) |
437 (131) | |