16808edo - Revision history

Overthink: Undo revision 230160 by Domin (talk) please give a better reason

2026-05-13T00:14:55Z

Undo revision 230160 by Domin (talk) please give a better reason

← Older revision		Revision as of 00:14, 13 May 2026
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	~~{{delete\|written under an influence of substances}}~~
	{{Infobox ET}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}

Domin at 00:08, 13 May 2026

2026-05-13T00:08:38Z

← Older revision		Revision as of 00:08, 13 May 2026
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			{{delete\|written under an influence of substances}}
	{{Infobox ET}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}

Overthink: /* Theory */ Isn't that every edo of this size?

2026-01-02T05:14:08Z

Theory: Isn't that every edo of this size?

← Older revision		Revision as of 05:14, 2 January 2026
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	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.		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 ===		=== Prime harmonics ===

FloraC: Use standard precision in the harmonics table so that we can expand it a little

2025-10-02T15:50:51Z

Use standard precision in the harmonics table so that we can expand it a little

← Older revision		Revision as of 15:50, 2 October 2025
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	=== Prime harmonics ===		=== 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 ===		=== Subsets and supersets ===

FloraC: /* Theory */ the implications of the perfect fifth

2025-02-22T09:50:55Z

Theory: the implications of the perfect fifth

← Older revision		Revision as of 09:50, 22 February 2025
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	== Theory ==		== 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.		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.

FloraC: /* Theory */ note the perfect fifth from 2101edo

2025-02-22T09:34:18Z

Theory: note the perfect fifth from 2101edo

← Older revision		Revision as of 09:34, 22 February 2025
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	== Theory ==		== Theory ==
	16808edo is distinctly [[consistent]] and highly accurate through the [[35-odd-limit]], being [[consistency #Generalization\|consistent to distance 2]]~~. 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 [[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 }}.		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.

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

	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 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.

ArrowHead294 at 13:38, 21 February 2025

2025-02-21T13:38:34Z

← Older revision		Revision as of 13:38, 21 February 2025
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	== Theory ==		== Theory ==
	16808edo is distinctly [[consistent]] and highly accurate through the [[35-odd-limit]], ~~in fact~~ [[consistency #Generalization\|consistent to distance 2]]. 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 [[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 }}.		16808edo is distinctly [[consistent]] and highly accurate through the [[35-odd-limit]], being [[consistency #Generalization\|consistent to distance 2]]. 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 [[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 }}.

	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 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.

FloraC: /* Theory */ primes 43 and 47

2025-02-21T12:53:32Z

Theory: primes 43 and 47

← Older revision		Revision as of 12:53, 21 February 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	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 ~~harmonic~~ 37 to near-just ~~quality~~.		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 ==		== Intervals ==

FloraC: /* Theory */ write a little about prime 37

2025-02-21T12:20:24Z

Theory: write a little about prime 37

← Older revision		Revision as of 12:20, 21 February 2025
Line 8:		Line 8:

	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 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.

			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 ===		=== Prime harmonics ===
	{{Harmonics in equal\|16808\|prec=5\|columns=11}}		{{Harmonics in equal\|16808\|prec=5\|columns=9}}
			{{Harmonics in equal\|16808\|prec=5\|columns=9\|start=10\|collapsed=true\|title=Approximation of prime harmonics in 16808edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	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.		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 harmonic 37 to near-just quality.

	== Intervals ==		== Intervals ==

FloraC: /* Theory */ note consistency to distance 2 (stronger than purely consistent)

2025-02-20T15:37:31Z

Theory: note consistency to distance 2 (stronger than purely consistent)

← Older revision		Revision as of 15:37, 20 February 2025
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	== Theory ==		== 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]], in fact [[consistency #Generalization\|consistent to distance 2]]. 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 [[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 }}.

	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 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.

	~~It is also notable for being the smallest EDO after [[311edo\|311]] that is [[purely consistent]] in the 31-odd-limit (approximates all odd harmonics from 1 to 31 with less than 25% relative error)~~.

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== 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.

	== Intervals ==		== Intervals ==