8539edo: Difference between revisions
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While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists|strict zeta]] tuning, and is also the first [[trivial temperament|non-trivial]] edo to be consistent in the [[odd prime sum limit|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo|20203]]. | While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists|strict zeta]] tuning, and is also the first [[trivial temperament|non-trivial]] edo to be consistent in the [[odd prime sum limit|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo|20203]]. | ||
Some of the simpler commas it [[tempering out|tempers out]] include the [[senior|senior comma]] in the [[5-limit]]; [[very high accuracy temperaments #Sepbizo-asegu (171 & 3566 & 4973)|{{monzo| -3 2 -17 14 }}]] in the [[7-limit]]; [[3294225/3294172]] in the [[11-limit]]; [[123201/123200]] in the [[13-limit]]; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the [[17-limit]]; 27456/27455, 43681/43680, 89376/89375 in the [[19-limit]]; 12168/12167, 16929/16928, 19551/19550, 21736/21735, [[25025/25024]], 43264/43263 among others in the [[23-limit]]. | |||
Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]]. | Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]]. | ||
Revision as of 10:19, 5 January 2026
| ← 8538edo | 8539edo | 8540edo → |
8539 equal divisions of the octave (abbreviated 8539edo or 8539ed2), also called 8539-tone equal temperament (8539tet) or 8539 equal temperament (8539et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8539 equal parts of about 0.141 ¢ each. Each step represents a frequency ratio of 21/8539, or the 8539th root of 2.
Theory
While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a unit of interval measure, the tina. This is because it is a very strong higher-limit system, distinctly consistent through the 27-odd-limit. It is a strict zeta tuning, and is also the first non-trivial edo to be consistent in the 27-odd-prime-sum-limit. In the 13-limit, the only smaller systems with a lower logflat badness are 72, 270, 494, 5585 and 6079; in the 17-limit, that becomes 72, 494, 1506, 3395 and 7033. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until 20203.
Some of the simpler commas it tempers out include the senior comma in the 5-limit; [-3 2 -17 14⟩ in the 7-limit; 3294225/3294172 in the 11-limit; 123201/123200 in the 13-limit; 28561/28560, 31213/31212, 37180/37179 in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.
Since it tempers out 12168/12167, it allows vicetertismic chords in the 23-odd-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0007 | +0.0079 | -0.0005 | -0.0122 | -0.0077 | +0.0218 | -0.0075 | +0.0428 | -0.0421 | +0.0165 |
| Relative (%) | +0.0 | +0.5 | +5.6 | -0.4 | -8.7 | -5.5 | +15.5 | -5.3 | +30.4 | -30.0 | +11.8 | |
| Steps (reduced) |
8539 (0) |
13534 (4995) |
19827 (2749) |
23972 (6894) |
29540 (3923) |
31598 (5981) |
34903 (747) |
36273 (2117) |
38627 (4471) |
41482 (7326) |
42304 (8148) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0671 | -0.0192 | +0.0176 | +0.0514 | +0.0404 | +0.0156 | +0.0610 | -0.0472 | +0.0435 | +0.0124 | +0.0023 |
| Relative (%) | +47.8 | -13.7 | +12.5 | +36.6 | +28.7 | +11.1 | +43.4 | -33.6 | +30.9 | +8.8 | +1.6 | |
| Steps (reduced) |
44484 (1789) |
45748 (3053) |
46335 (3640) |
47431 (4736) |
48911 (6216) |
50232 (7537) |
50643 (7948) |
51798 (564) |
52513 (1279) |
52855 (1621) |
53828 (2594) | |
Subsets and supersets
8539edo is the 1065th prime edo. On that basis, the tina as a unit of measure could be criticized; however, some people prefer primes for this sort of job, as they do not imply a preference for one smaller edo over another.