49ed6 - Revision history

FloraC: /* Theory */ "very nearly identical" was an overstatement

2025-03-30T13:03:14Z

Theory: "very nearly identical" was an overstatement

← Older revision		Revision as of 13:03, 30 March 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	49ed6 is very ~~nearly identical~~ to [[19edo]], but with the [[6/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning\|zeta peak]] near 19, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.		49ed6 is very similar to [[19edo]], but with the [[6/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning\|zeta peak]] near 19, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.

	The fifth is ~696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of [[31edo]]. The fourth is less accurate than in 19edo, and is close in size to a [[flattone]] fourth. Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.		The fifth is ~696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of [[31edo]]. The fourth is less accurate than in 19edo, and is close in size to a [[flattone]] fourth. Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.
Line 18:		Line 18:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 49 factors into primes as 7<sup>2</sup>, 49ed6 contains [[7ed6]] as its only nontrivial subset ed6.		Since 49 factors into primes as 7<sup>2</sup>, 49ed6 contains [[7ed6]] as its only nontrivial subset ed6.

	== Intervals ==		== Intervals ==

FloraC: +see also

2025-03-30T13:01:49Z

+see also

← Older revision		Revision as of 13:01, 30 March 2025
Line 22:		Line 22:
	== Intervals ==		== Intervals ==
	{{Interval table}}		{{Interval table}}

			== See also ==
			* [[11edf]] – relative edf
			* [[19edo]] – relative edo
			* [[30edt]] – relative edt
			* [[53ed7]] – relative ed7
			* [[68ed12]] – relative ed12
			* [[93ed30]] – relative ed30

	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Godzilla]]		[[Category:Godzilla]]
	[[Category:Meantone]]		[[Category:Meantone]]

FloraC: +subsets and supersets

2025-01-23T12:08:13Z

+subsets and supersets

← Older revision		Revision as of 12:08, 23 January 2025
Line 16:		Line 16:
	{{Harmonics in equal\|49\|6\|1}}		{{Harmonics in equal\|49\|6\|1}}
	{{Harmonics in equal\|49\|6\|1\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 49ed6 (continued)}}		{{Harmonics in equal\|49\|6\|1\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 49ed6 (continued)}}

			=== Subsets and supersets ===
			Since 49 factors into primes as 7<sup>2</sup>, 49ed6 contains [[7ed6]] as its only nontrivial subset ed6.

	== Intervals ==		== Intervals ==

FloraC: Cleanup

2025-01-23T12:05:32Z

Cleanup

← Older revision		Revision as of 12:05, 23 January 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	'''[[Ed6\|Division of the sixth harmonic]] into 49 equal parts''' (49ED6) is very nearly identical to [[19edo\|19 EDO]], but with the [[6/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning\|zeta peak]] near 19, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.		{{ED intro}}

	~~The fifth~~ is ~~~ 696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone~~, ~~or half a cent flatter than~~ the ~~fifth of~~ [[~~31edo\|31edo~~]]. ~~The fourth is less accurate than in 19edo, and~~ is close ~~in size~~ to a [[~~Flattone~~\|~~flattone~~]] ~~fourth~~.		== Theory ==
			49ed6 is very nearly identical to [[19edo]], but with the [[6/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning\|zeta peak]] near 19, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.

	Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.		The fifth is ~696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of [[31edo]]. The fourth is less accurate than in 19edo, and is close in size to a [[flattone]] fourth. Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.

	Usable prime harmonics include the 3:1 (about 3 cents flat), the 5:1 (about a cent flat), ~~and~~ the 7:1 and 13:1 (~~around 12 and~~ 9 cents flat~~, respectively~~). The 7:1 and 13:1 in particular are much improved; with pure octaves they are too far out of tune to be usable for most, but the situation changes with the stretched version.		Usable prime harmonics include the [[3/1\|3]] (about 3 cents flat), the [[5/1\|5]] (about a cent flat), the [[7/1\|7]] (about 14 cents flat) and the [[13/1\|13]] (about 9 cents flat). The 7 and 13 in particular are much improved; with pure octaves they are too far out of tune to be usable for most, but the situation changes with the stretched version.

	Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.		Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are [[44ed5]] and [[93ed30]]. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.

	Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.		Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.

			=== Harmonics ===
			{{Harmonics in equal\|49\|6\|1}}
			{{Harmonics in equal\|49\|6\|1\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 49ed6 (continued)}}

	== Intervals ==		== Intervals ==
	{{Interval table}}		{{Interval table}}

	~~== Harmonics ==~~
	~~{{Harmonics in equal~~
	~~\| steps = 49~~
	~~\| num = 6~~
	~~\| denom = 1~~
	}}
	~~{{Harmonics in equal~~
	~~\| steps = 49~~
	~~\| num = 6~~
	~~\| denom = 1~~
	~~\| start = 12~~
	~~\| collapsed = 1~~
	}}

	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Godzilla]]		[[Category:Godzilla]]
	[[Category:Meantone]]		[[Category:Meantone]]

BudjarnLambeth at 02:01, 30 November 2024

2024-11-30T02:01:22Z

← Older revision		Revision as of 02:01, 30 November 2024
Line 10:		Line 10:
	Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.		Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.

	Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.		Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.

			== Intervals ==
			{{Interval table}}

	== Harmonics ==		== Harmonics ==
	{{Harmonics in equal\|49\|6\|1\|~~prec~~=2\|~~columns~~=15}}		{{Harmonics in equal
			\| steps = 49
			\| num = 6
			\| denom = 1
			}}
			{{Harmonics in equal
			\| steps = 49
			\| num = 6
			\| denom = 1
			\| start = 12
			\| collapsed = 1
			}}

	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Godzilla]]		[[Category:Godzilla]]
	[[Category:Meantone]]		[[Category:Meantone]]

BudjarnLambeth: Undo revision 167930 by BudjarnLambeth (talk)

2024-11-30T01:58:33Z

Undo revision 167930 by BudjarnLambeth (talk)

← Older revision		Revision as of 01:58, 30 November 2024
Line 10:		Line 10:
	Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.		Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.

	Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has. ~~octave~~		Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.

			== Harmonics ==
			{{Harmonics in equal\|49\|6\|1\|prec=2\|columns=15}}

	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Godzilla]]		[[Category:Godzilla]]
	[[Category:Meantone]]		[[Category:Meantone]]

BudjarnLambeth: Intervals harmonics

2024-11-30T01:58:18Z

Intervals harmonics

← Older revision		Revision as of 01:58, 30 November 2024
Line 10:		Line 10:
	Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.		Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 44ed5 and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.

	Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.		Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has. octave

	~~== Harmonics ==~~
	~~{{Harmonics in equal\|49\|6\|1\|prec=2\|columns=15}}~~

	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Godzilla]]		[[Category:Godzilla]]
	[[Category:Meantone]]		[[Category:Meantone]]

ArrowHead294 at 14:10, 10 September 2024

2024-09-10T14:10:08Z

← Older revision		Revision as of 14:10, 10 September 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	'''[[Ed6\|Division of the sixth harmonic]] into 49 equal parts''' (49ED6) is very nearly identical to [[19edo\|19 EDO]], but with the [[6/1]] rather than the 2/1 being just. It is ~~a stretched version of [[19edo\|19edo]] and~~ extremely close to the [[~~The_Riemann_Zeta_Function_and_Tuning~~\|zeta peak]], thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.		'''[[Ed6\|Division of the sixth harmonic]] into 49 equal parts''' (49ED6) is very nearly identical to [[19edo\|19 EDO]], but with the [[6/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning\|zeta peak]] near 19, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.

	The fifth is ~ 696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of [[31edo\|31edo]]. The fourth is less accurate than in 19edo, and is close in size to a [[Flattone\|flattone]] fourth.		The fifth is ~ 696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of [[31edo\|31edo]]. The fourth is less accurate than in 19edo, and is close in size to a [[Flattone\|flattone]] fourth.

ArrowHead294 at 14:09, 10 September 2024

2024-09-10T14:09:12Z

← Older revision		Revision as of 14:09, 10 September 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	'''~~49ed6~~''' ~~divides~~ the ~~just~~ 6:1 ~~into 49 equal parts, resulting in a step size of about 63.3053 cents and an octave approximately 3 cents sharp~~. It is a stretched version of [[19edo\|19edo]] and extremely close to the [[The_Riemann_Zeta_Function_and_Tuning\|zeta peak]], thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave ~~by this much~~ improves the overall tuning accuracy.		'''[[Ed6\|Division of the sixth harmonic]] into 49 equal parts''' (49ED6) is very nearly identical to [[19edo\|19 EDO]], but with the [[6/1]] rather than the 2/1 being just. It is a stretched version of [[19edo\|19edo]] and extremely close to the [[The_Riemann_Zeta_Function_and_Tuning\|zeta peak]], thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.

	The fifth is ~ 696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of [[31edo\|31edo]]. The fourth is less accurate than in 19edo, and is close in size to a [[Flattone\|flattone]] fourth.		The fifth is ~ 696.36 cents; about 1/4 of a cent flatter than the fifth of quarter-comma meantone, or half a cent flatter than the fifth of [[31edo\|31edo]]. The fourth is less accurate than in 19edo, and is close in size to a [[Flattone\|flattone]] fourth.
Line 12:		Line 12:
	Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.		Tunings in this range are a promising option for stiff-stringed instruments since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.

	==Harmonics==		== Harmonics ==
	{{Harmonics in equal\|49\|6\|1\|prec=2\|columns=15}}		{{Harmonics in equal\|49\|6\|1\|prec=2\|columns=15}}

	[[Category:19edo]]		[[Category:19edo]]
	[[Category:~~godzilla~~]]		[[Category:Godzilla]]
	[[Category:~~meantone~~]]		[[Category:Meantone]]

ArrowHead294 at 01:17, 12 May 2024

2024-05-12T01:17:00Z

← Older revision		Revision as of 01:17, 12 May 2024
Line 1:		Line 1:
			{{Infobox ET}}
	'''49ed6''' divides the just 6:1 into 49 equal parts, resulting in a step size of about 63.3053 cents and an octave approximately 3 cents sharp. It is a stretched version of [[19edo\|19edo]] and extremely close to the [[The_Riemann_Zeta_Function_and_Tuning\|zeta peak]], thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave by this much improves the overall tuning accuracy.		'''49ed6''' divides the just 6:1 into 49 equal parts, resulting in a step size of about 63.3053 cents and an octave approximately 3 cents sharp. It is a stretched version of [[19edo\|19edo]] and extremely close to the [[The_Riemann_Zeta_Function_and_Tuning\|zeta peak]], thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave by this much improves the overall tuning accuracy.

Line 12:		Line 13:

	==Harmonics==		==Harmonics==
	{{Harmonics in equal\|49\|6\|1\|prec=2}}		{{Harmonics in equal\|49\|6\|1\|prec=2\|columns=15}}

	[[Category:19edo]]		[[Category:19edo]]
	[[Category:godzilla]]		[[Category:godzilla]]
	[[Category:meantone]]		[[Category:meantone]]