153edt: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
153edt is notable for being the denominator of a convergent to log<sub>3</sub>(7/3), after [[9edt]], [[13edt]] and [[35edt]], and the last before [[3401edt]], and therefore has an extremely accurate approximation to [[7/3]]. 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. | 153edt is notable for being the denominator of a convergent to log<sub>3</sub>(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. | 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 == | |||
{{Harmonics in equal|153|3|1|intervals = prime|columns = 9}} | |||
{{Harmonics in equal|153|3|1|start = 12|collapsed = 1|intervals = odd}} | |||
Latest revision as of 12:10, 5 October 2024
| ← 152edt | 153edt | 154edt → |
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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.81 | +0.00 | -1.75 | -0.00 | +0.66 | -2.63 | +5.32 | -0.77 | +4.11 |
| Relative (%) | +46.8 | +0.0 | -14.1 | -0.0 | +5.3 | -21.2 | +42.8 | -6.2 | +33.0 | |
| Steps (reduced) |
97 (97) |
153 (0) |
224 (71) |
271 (118) |
334 (28) |
357 (51) |
395 (89) |
410 (104) |
437 (131) | |
| Harmonic | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.50 | +0.00 | +0.60 | -2.98 | +0.66 | -1.76 | +1.49 | -2.63 | -2.19 | +2.37 | -1.75 |
| Relative (%) | -28.2 | +0.0 | +4.8 | -24.0 | +5.3 | -14.1 | +12.0 | -21.2 | -17.7 | +19.0 | -14.1 | |
| Steps (reduced) |
448 (142) |
459 (0) |
469 (10) |
478 (19) |
487 (28) |
495 (36) |
503 (44) |
510 (51) |
517 (58) |
524 (65) |
530 (71) | |