44ed5 - Revision history

FloraC: Text replacement - "{{Infobox et}}↵{{ED intro}}" to "{{Infobox ET}} {{ED intro}}"

2025-06-26T12:00:20Z

Text replacement - "{{Infobox et}}↵{{ED intro}}" to "{{Infobox ET}} {{ED intro}}"

← Older revision		Revision as of 12:00, 26 June 2025
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	{{Infobox et}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}

FloraC: Text replacement - "{{infobox et}}↵{{ED intro}}" to "{{Infobox et}} {{ED intro}}"

2025-06-26T11:57:26Z

Text replacement - "{{infobox et}}↵{{ED intro}}" to "{{Infobox et}} {{ED intro}}"

← Older revision		Revision as of 11:57, 26 June 2025
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	{{~~infobox~~ et}}		{{Infobox et}}
	{{ED intro}}		{{ED intro}}

ArrowHead294: /* Theory */

2025-06-18T13:56:18Z

Theory

← Older revision		Revision as of 13:56, 18 June 2025
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	44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. The [[The Riemann zeta function and tuning\|local zeta peak]] around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}, and the octave of 44ed5 comes extremely close (differing 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.		44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. The [[The Riemann zeta function and tuning\|local zeta peak]] around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}, and the octave of 44ed5 comes extremely close (differing 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.		Other variants (which stretch the octave by different amounts, 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 allow 19edo to be a promising option for pianos and harpsichords with split sharps since they have stretched partials (especially on cheaper spinet pianos, since their short string lengths result in more inharmonicity compared to upright or grand pianos), and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203{{c}} 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 allow 19edo to be a promising option for pianos and harpsichords with split sharps since they have stretched partials (especially on cheaper spinet pianos, since their short string lengths result in more inharmonicity compared to upright or grand pianos), and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203{{c}} or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.

ArrowHead294: /* Theory */

2025-06-18T13:39:53Z

Theory

← Older revision		Revision as of 13:39, 18 June 2025
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	== Theory ==		== Theory ==
	44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. The [[The Riemann zeta function and tuning\|local zeta peak]] around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}, and the octave of 44ed5 comes extremely close (differing 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.		44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. The [[The Riemann zeta function and tuning\|local zeta peak]] around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}, and the octave of 44ed5 comes extremely close (differing 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.		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.

ArrowHead294 at 13:39, 18 June 2025

2025-06-18T13:39:44Z

← Older revision		Revision as of 13:39, 18 June 2025
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	== Theory ==		== 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}}, 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.		44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. The [[The Riemann zeta function and tuning\|local zeta peak]] around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}, and the octave of 44ed5 comes extremely close (differing 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.		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.		Tunings in this range allow 19edo to be a promising option for pianos and harpsichords with split sharps since they have stretched partials (especially on cheaper spinet pianos, since their short string lengths result in more inharmonicity compared to upright or grand pianos), and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203{{c}} or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.

	=== Harmonics ===		=== Harmonics ===

ArrowHead294: Substitute

2025-06-18T13:36:54Z

Substitute

← Older revision		Revision as of 13:36, 18 June 2025
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	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|44\|5\|1}}		{{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)}}		{{Harmonics in equal\|44\|5\|1\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 44ed5 (continued)}}

	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Zeta-optimized tunings]]		[[Category:Zeta-optimized tunings]]

ArrowHead294 at 13:36, 18 June 2025

2025-06-18T13:36:33Z

← Older revision		Revision as of 13:36, 18 June 2025
Line 3:		Line 3:

	== Theory ==		== 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}}
			{{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]]		[[Category:Zeta-optimized tunings]]

ArrowHead294 at 13:33, 18 June 2025

2025-06-18T13:33:55Z

← Older revision		Revision as of 13:33, 18 June 2025
Line 1:		Line 1:
	{{infobox et}}		{{infobox et}}
			{{ED intro}}

			== 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.

	[[Category:Zeta-optimized tunings]]		[[Category:Zeta-optimized tunings]]

ArrowHead294: Make page

2025-06-18T13:29:08Z

Make page

New page

{{infobox et}}
[[Category:Zeta-optimized tunings]]