3401edt: Difference between revisions
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3401edt is notable for being the denominator of a convergent to log<sub>3</sub>(7/3), after [[13edt]], [[35edt]], and [[153edt]], and the last before [[108985edt]], and therefore has an extremely accurate approximation to [[7/3]]. In fact, 3401edt demonstrates 16-strong 7-3 [[telicity]], even stronger than that of 153edt. | 3401edt is notable for being the denominator of a convergent to log<sub>3</sub>(7/3), after [[13edt]], [[35edt]], and [[153edt]], and the last before [[108985edt]], and therefore has an extremely accurate approximation to [[7/3]], only about 5 ''micro''cents flat. In fact, 3401edt demonstrates 16-strong 7-3 [[telicity]], even stronger than that of 153edt. It also has a very good approximation to [[15/13]]. | ||
== Harmonics == | |||
{{Harmonics in equal|3401|3|1|intervals=prime}} | |||
Latest revision as of 19:05, 25 December 2024
| ← 3400edt | 3401edt | 3402edt → |
3401 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 3401edt or 3401ed3), is a nonoctave tuning system that divides the interval of 3/1 into 3401 equal parts of about 0.559 ¢ each. Each step represents a frequency ratio of 31/3401, or the 3401st root of 3.
3401edt is notable for being the denominator of a convergent to log3(7/3), after 13edt, 35edt, and 153edt, and the last before 108985edt, and therefore has an extremely accurate approximation to 7/3, only about 5 microcents flat. In fact, 3401edt demonstrates 16-strong 7-3 telicity, even stronger than that of 153edt. It also has a very good approximation to 15/13.
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.116 | +0.000 | -0.210 | +0.000 | -0.124 | -0.209 | +0.086 | -0.095 | +0.211 | -0.121 | +0.182 |
| Relative (%) | +20.8 | +0.0 | -37.5 | +0.0 | -22.1 | -37.4 | +15.5 | -16.9 | +37.7 | -21.7 | +32.5 | |
| Steps (reduced) |
2146 (2146) |
3401 (0) |
4982 (1581) |
6024 (2623) |
7423 (621) |
7940 (1138) |
8771 (1969) |
9115 (2313) |
9707 (2905) |
10424 (221) |
10631 (428) | |