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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-04-20 15:40:26 UTC</tt>.<br>
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| : The original revision id was <tt>580722411</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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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.
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| 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.
| | == 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. |
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| 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. |
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| 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. |
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| 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. |
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| 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.</pre></div> | | 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. |
| <h4>Original HTML content:</h4>
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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 />
| | === Harmonics === |
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| | {{Harmonics in equal|49|6|1}} |
| 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 />
| | {{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 49ed6 (continued)}} |
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| Minor thirds are still excellent, only slightly less accurate than they are in standard 19edo.<br />
| | === Subsets and supersets === |
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| | Since 49 factors into primes as 7<sup>2</sup>, 49ed6 contains [[7ed6]] as its only nontrivial subset ed6. |
| 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 />
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| | == Intervals == |
| 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 />
| | {{Interval table}} |
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| 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></pre></div>
| | == See also == |
| | * [[11edf]] – relative edf |
| | * [[19edo]] – relative edo |
| | * [[30edt]] – relative edt |
| | * [[53ed7]] – relative ed7 |
| | * [[68ed12]] – relative ed12 |
| | * [[93ed30]] – relative ed30 |
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| | [[Category:19edo]] |
| | [[Category:Godzilla]] |
| | [[Category:Meantone]] |