16808edo: Difference between revisions

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== Theory ==

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- 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.

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.

Its [[3/2|perfect fifth]] ultimately comes from [[2101edo]].

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 ~~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.

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.

It is not as excellent, but certainly usable beyond the 31-limit, as at this level little can be complained about inaccuracy, even though the [[37/1|37th]] [[harmonic]] is about halfway between its steps.

=== Prime harmonics ===

{{Harmonics in equal|16808~~|prec=5~~|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 16808edo (continued)}}

{{Harmonics in equal|16808|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 16808edo (continued)}}

=== Subsets and supersets ===

@@ Line 5: / Line 5: @@
 == Theory ==
-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- 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.
+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.
-Its [[3/2|perfect fifth]] ultimately comes from [[2101edo]].
+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 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.
+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.
-It is not as excellent, but certainly usable beyond the 31-limit, as at this level little can be complained about inaccuracy, even though the [[37/1|37th]] [[harmonic]] is about halfway between its steps.
 === Prime harmonics ===
-{{Harmonics in equal|16808|prec=5|columns=9}}
+{{Harmonics in equal|16808|columns=11}}
-{{Harmonics in equal|16808|prec=5|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 16808edo (continued)}}
+{{Harmonics in equal|16808|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 16808edo (continued)}}
 === Subsets and supersets ===