16808edo: Difference between revisions

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16808edo's step size is sometimes called a '''jinn''', a term proposed by [[Gene Ward Smith]]<ref>[https://www.huygens-fokker.org/docs/measures.html Stichting Huygens-Fokker: Logarithmic Interval Measures]</ref>, when used as an [[interval size unit]].

== Theory ==

16808edo is distinctly [[consistent]] and highly accurate through the 35-odd-limit, ~~and its step size can be used as an~~ [[~~interval size unit~~]] ~~(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 [[~~Tenney-Euclidean~~ temperament measures #TE simple badness|relative error]] up until [[148418edo|148418]]; in the 17- and 19-limit up until [[20203edo|20203]]; though in the 13-limit it is beaten out by smaller edos {{EDOs| 5585, 6079, 8269, 8539, 13112 and 14618 }}.

16808edo is distinctly [[consistent]] and highly accurate through the [[35-odd-limit]], being [[consistency #Generalization|consistent to distance 2]]. It is a very, very strong [[31-limit]] system, and a [[zeta peak edo|zeta peak]], [[zeta peak integer edo|zeta peak integer]] and [[zeta integral edo]]. In the [[23-limit|23-]], [[29-limit|29-]] and 31-limit it has the lowest [[Tenney–Euclidean temperament measures #TE simple badness|relative error]] up until [[148418edo|148418]]; in the [[17-limit|17-]] and [[19-limit]] up until [[20203edo|20203]]; though in the [[13-limit]] it is beaten out by smaller edos {{EDOs| 5585, 6079, 8269, 8539, 13112 and 14618 }}. As such, its step size can be used as an [[interval size unit]] (the jinn) for most intervals which occur in practice.

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.

Its [[3/2|perfect fifth]] ultimately comes from [[2101edo]], so it not only has two [[chain of fifths|circles of fifths]] ([[hemipyth]]), but ''eight'', giving itself another edge over similar systems.

Among the enormous list of 31-limit [[comma]]s it [[tempering out|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|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 16808edo (continued)}}

=== Subsets and supersets ===

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 subset edos 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. [[33616edo]], which doubles it, corrects its harmonics 37, [[43/1|43]], and [[47/1|47]] to near-just qualities.

== Intervals ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|16808}}
+{{ED intro}}
 edo's step size is sometimes called a '''jinn''', a term proposed by [[Gene Ward Smith]]<ref>[https://www.huygens-fokker.org/docs/measures.html Stichting Huygens-Fokker: Logarithmic Interval Measures]</ref>, when used as an [[interval size unit]].
 == Theory ==
-edo is distinctly [[consistent]] and highly accurate through the 35-odd-limit, and its step size can be used as an [[interval size unit]] (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 [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] up until [[148418edo|148418]]; in the 17- and 19-limit up until [[20203edo|20203]]; though in the 13-limit it is beaten out by smaller edos {{EDOs| 5585, 6079, 8269, 8539, 13112 and 14618 }}.
+edo is distinctly [[consistent]] and highly accurate through the [[35-odd-limit]], being [[consistency #Generalization|consistent to distance 2]]. It is a very, very strong [[31-limit]] system, and a [[zeta peak edo|zeta peak]], [[zeta peak integer edo|zeta peak integer]] and [[zeta integral edo]]. In the [[23-limit|23-]], [[29-limit|29-]] and 31-limit it has the lowest [[Tenney–Euclidean temperament measures #TE simple badness|relative error]] up until [[148418edo|148418]]; in the [[17-limit|17-]] and [[19-limit]] up until [[20203edo|20203]]; though in the [[13-limit]] it is beaten out by smaller edos {{EDOs| 5585, 6079, 8269, 8539, 13112 and 14618 }}. As such, its step size can be used as an [[interval size unit]] (the jinn) for most intervals which occur in practice.
-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.
+Its [[3/2|perfect fifth]] ultimately comes from [[2101edo]], so it not only has two [[chain of fifths|circles of fifths]] ([[hemipyth]]), but ''eight'', giving itself another edge over similar systems.
+Among the enormous list of 31-limit [[comma]]s it [[tempering out|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}}
+{{Harmonics in equal|16808|columns=11}}
+{{Harmonics in equal|16808|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 16808edo (continued)}}
 === Subsets and supersets ===
-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.
+has subset edos 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. [[33616edo]], which doubles it, corrects its harmonics 37, [[43/1|43]], and [[47/1|47]] to near-just qualities.
 == Intervals ==