44ed5: Difference between revisions

Line 3:

== Theory ==

44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning|zeta peak]] near 19; the local zeta peak around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}; the octave of 44ed5 is only {{sfrac|1|9}}{{c}} ~~off~~, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.

44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning|zeta peak]] near 19; the local zeta peak around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}, and the octave of 44ed5 differs by only {{sfrac|1|9}}{{c}}, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.

Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are [[30edt]], [[49ed6]], 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 pianos and harpsichords 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|44|5|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in {{susbt:PAGENAME}} (continued)}}

[[Category:19edo]]

[[Category:Zeta-optimized tunings]]

@@ Line 3: / Line 3: @@
 == Theory ==
-ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning|zeta peak]] near 19; the  local zeta peak around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}; the octave of 44ed5 is only {{sfrac|1|9}}{{c}} off, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.
+ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning|zeta peak]] near 19; the local zeta peak around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}, and the octave of 44ed5 differs by only {{sfrac|1|9}}{{c}}, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.
+Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are [[30edt]], [[49ed6]], 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 pianos and harpsichords 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|44|5|1}}
+{{Harmonics in equal|44|5|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in {{susbt:PAGENAME}} (continued)}}
+[[Category:19edo]]
 [[Category:Zeta-optimized tunings]]