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-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]] and [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]]. 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]].

==Theory==

16808edo 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]] and [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]]. 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.

=== Divisors ===

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.

=== Prime harmonics ===

[[Category:Equal divisions of the octave|#####]]

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 {{Infobox ET}}
-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]] and [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]]. 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]].
+{{EDO intro|16808}}
+==Theory==
+edo 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]] and [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]]. 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.
+=== Divisors ===
 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.
+=== Prime harmonics ===
 {{Harmonics in equal|16808|prec=5|columns=11}}
 [[Category:Equal divisions of the octave|#####]] <!-- 5-digit number -->