49ed6: Difference between revisions

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'''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.

~~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 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.

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.

=== Harmonics ===

{{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 ==

== See also ==

* [[11edf]] – relative edf

* [[19edo]] – relative edo

* [[30edt]] – relative edt

* [[53ed7]] – relative ed7

* [[68ed12]] – relative ed12

* [[93ed30]] – relative ed30

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.

[[Category:19edo]]

[[Category:~~godzilla~~]]

[[Category:Godzilla]]

[[Category:~~meantone~~]]

[[Category:Meantone]]

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-'''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.
+{{Infobox ET}}
+{{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 ==
+ed6 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.
-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.
+=== 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)}}
+=== Subsets and supersets ===
+Since 49 factors into primes as 7<sup>2</sup>, 49ed6 contains [[7ed6]] as its only nontrivial subset ed6.
+== Intervals ==
+{{Interval table}}
+== See also ==
+* [[11edf]] – relative edf
+* [[19edo]] – relative edo
+* [[30edt]] – relative edt
+* [[53ed7]] – relative ed7
+* [[68ed12]] – relative ed12
+* [[93ed30]] – relative ed30
-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.
 [[Category:19edo]]
-[[Category:godzilla]]
+[[Category:Godzilla]]
-[[Category:meantone]]
+[[Category:Meantone]]