16808edo: Difference between revisions

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~~==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.

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]] 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]].

~~=== 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.

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 and 104976/104975. 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.

=== Prime harmonics ===

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

=== 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.

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 {{Infobox ET}}
 {{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.
+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]] 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]].
-=== 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.
+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 and 104976/104975. 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.
 === Prime harmonics ===
 {{Harmonics in equal|16808|prec=5|columns=11}}
-[[Category:Equal divisions of the octave|#####]] <!-- 5-digit number -->
+=== 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.