49ed6: Difference between revisions
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Wikispaces>MasonGreen1 **Imported revision 580364841 - Original comment: ** |
Wikispaces>MasonGreen1 **Imported revision 580722411 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-04- | : This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-04-20 15:40:26 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>580722411</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**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]] 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. | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**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]] 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]]. 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 (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. | ||
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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>49ed6</title></head><body><strong>49ed6</strong> 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 <a class="wiki_link" href="/19edo">19edo</a> and extremely close to the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">zeta peak</a>, 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.<br /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>49ed6</title></head><body><strong>49ed6</strong> 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 <a class="wiki_link" href="/19edo">19edo</a> and extremely close to the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">zeta peak</a>, 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.<br /> | ||
<br /> | <br /> | ||
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 <a class="wiki_link" href="/31edo">31edo</a>. Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.<br /> | 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 <a class="wiki_link" href="/31edo">31edo</a>. The fourth is less accurate than in 19edo, and is close in size to a <a class="wiki_link" href="/flattone">flattone</a> fourth.<br /> | ||
<br /> | |||
Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.<br /> | |||
<br /> | <br /> | ||
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.<br /> | 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.<br /> | ||
Revision as of 15:40, 20 April 2016
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author MasonGreen1 and made on 2016-04-20 15:40:26 UTC.
- The original revision id was 580722411.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
**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]] 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]]. 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. 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.
Original HTML content:
<html><head><title>49ed6</title></head><body><strong>49ed6</strong> 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 <a class="wiki_link" href="/19edo">19edo</a> and extremely close to the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">zeta peak</a>, 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.<br /> <br /> 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 <a class="wiki_link" href="/31edo">31edo</a>. The fourth is less accurate than in 19edo, and is close in size to a <a class="wiki_link" href="/flattone">flattone</a> fourth.<br /> <br /> Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.<br /> <br /> 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.<br /> <br /> 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.<br /> <br /> 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.</body></html>