16808edo: Difference between revisions
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The '''16808 equal division''' divides the octave into 16808 steps of size 0.071395 [[cent]]s each. It is distinctly consistent and highly accurate through the 35 limit, and can be used as a [[interval size measure|measure of interval size]] (the [[jinn]]) for most intervals which occur in practice. It is a very, very strong 31-limit division, and a [[ | The '''16808 equal division''' divides the octave into 16808 steps of size 0.071395 [[cent]]s each. It is distinctly [[consistent]] and highly accurate through the 35-odd-limit, and can be used as a [[interval size measure|measure of interval size]] (the [[jinn]]) for most intervals which occur in practice. It is a very, very strong 31-limit division, and a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak]], [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak integer]], [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral]] and zeta gap tuning. In the [[23-limit|23-]], [[29-limit|29-]] and [[31-limit|31-limit]] it has the lowest logflat badness up until at least 200000; in the 19-limit it is beaten out by [[8539edo]], and in the 17-limit by [[72edo]], [[1506edo]], [[3395edo]] and [[7033edo]]. | ||
Among the enormous list of 31-limit commas it tempers out, the simplest are 43681/43680, 49011/49010, 52326/52325 and 53361/53360. In the 13-limit it tempers out 123201/123200 and 1990656/1990625; in the 17-limit 194481/194480 and 336141/336140; in the 19-limit 43681/43680,89376/89375 and104976/104975. Since 43681/43680 is both the simplest comma it tempers out and the limit is as low (in this context) as 19, it may be regarded as rather characteristic of 16808. | Among the enormous list of 31-limit commas it tempers out, the simplest are 43681/43680, 49011/49010, 52326/52325 and 53361/53360. In the 13-limit it tempers out 123201/123200 and 1990656/1990625; in the 17-limit 194481/194480 and 336141/336140; in the 19-limit 43681/43680,89376/89375 and104976/104975. Since 43681/43680 is both the simplest comma it tempers out and the limit is as low (in this context) as 19, it may be regarded as rather characteristic of 16808. | ||
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16808 has proper divisors 1, 2, 4, 8, 11, 22, 44, 88, 191, 382, 764, 1528, 2101, 4202 and 8404, among which [[22edo]] and [[764edo]] are particularly notable. One step of 22edo is 764 jinns, and one step of 764edo is 22 jinns. | 16808 has proper divisors 1, 2, 4, 8, 11, 22, 44, 88, 191, 382, 764, 1528, 2101, 4202 and 8404, among which [[22edo]] and [[764edo]] are particularly notable. One step of 22edo is 764 jinns, and one step of 764edo is 22 jinns. | ||
{{ | {{Harmonics in equal|16808|prec=5|columns=11}} | ||
[[Category:Equal divisions of the octave|#####]] <!-- 5-digit number --> | [[Category:Equal divisions of the octave|#####]] <!-- 5-digit number --> | ||
Revision as of 15:28, 14 November 2022
| ← 16807edo | 16808edo | 16809edo → |
The 16808 equal division divides the octave into 16808 steps of size 0.071395 cents each. It is distinctly consistent and highly accurate through the 35-odd-limit, and can be used as a measure of interval size (the jinn) for most intervals which occur in practice. It is a very, very strong 31-limit division, and a zeta peak, zeta peak integer, zeta integral and zeta gap tuning. In the 23-, 29- and 31-limit it has the lowest logflat badness up until at least 200000; in the 19-limit it is beaten out by 8539edo, and in the 17-limit by 72edo, 1506edo, 3395edo and 7033edo.
Among the enormous list of 31-limit commas it tempers out, the simplest are 43681/43680, 49011/49010, 52326/52325 and 53361/53360. In the 13-limit it tempers out 123201/123200 and 1990656/1990625; in the 17-limit 194481/194480 and 336141/336140; in the 19-limit 43681/43680,89376/89375 and104976/104975. Since 43681/43680 is both the simplest comma it tempers out and the limit is as low (in this context) as 19, it may be regarded as rather characteristic of 16808.
16808 has proper divisors 1, 2, 4, 8, 11, 22, 44, 88, 191, 382, 764, 1528, 2101, 4202 and 8404, among which 22edo and 764edo are particularly notable. One step of 22edo is 764 jinns, and one step of 764edo is 22 jinns.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | -0.00355 | +0.00233 | -0.00154 | -0.00904 | +0.00066 | -0.00539 | -0.01183 | -0.00210 | +0.00396 | -0.00939 |
| Relative (%) | +0.0 | -5.0 | +3.3 | -2.2 | -12.7 | +0.9 | -7.5 | -16.6 | -2.9 | +5.5 | -13.2 | |
| Steps (reduced) |
16808 (0) |
26640 (9832) |
39027 (5411) |
47186 (13570) |
58146 (7722) |
62197 (11773) |
68702 (1470) |
71399 (4167) |
76032 (8800) |
81653 (14421) |
83270 (16038) | |