Maximal evenness: Difference between revisions
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Wikispaces>keenanpepper **Imported revision 284372374 - Original comment: ** |
Wikispaces>hstraub **Imported revision 479367202 - 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: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2013-12-25 12:07:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>479367202</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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The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th). | The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th). | ||
The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.</pre></div> | The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo. | ||
Maximally even sets tend to be familiar and musically relevant scale collections. The maximally even heptatonic set of [[19edo]] is, like the one in 12edo, a diatonic scale. The maximally even heptatonic sets of [[17edo]] and [[24edo]], in contrary, are Maqamic[7]. The maximally even heptatonic set of [[22edo]] is Porcupine[7] (the diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is Porcupine[8], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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>Maximal evenness</title></head><body>Within every <a class="wiki_link" href="/edo">edo</a> one can specify a &quot;maximally even&quot; (ME) or &quot;quasi-equal&quot; scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo.<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>Maximal evenness</title></head><body>Within every <a class="wiki_link" href="/edo">edo</a> one can specify a &quot;maximally even&quot; (ME) or &quot;quasi-equal&quot; scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo.<br /> | ||
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The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th).<br /> | The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th).<br /> | ||
<br /> | <br /> | ||
The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.</body></html></pre></div> | The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.<br /> | ||
<br /> | |||
Maximally even sets tend to be familiar and musically relevant scale collections. The maximally even heptatonic set of <a class="wiki_link" href="/19edo">19edo</a> is, like the one in 12edo, a diatonic scale. The maximally even heptatonic sets of <a class="wiki_link" href="/17edo">17edo</a> and <a class="wiki_link" href="/24edo">24edo</a>, in contrary, are Maqamic[7]. The maximally even heptatonic set of <a class="wiki_link" href="/22edo">22edo</a> is Porcupine[7] (the diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is Porcupine[8], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.</body></html></pre></div> | |||
Revision as of 12:07, 25 December 2013
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author hstraub and made on 2013-12-25 12:07:40 UTC.
- The original revision id was 479367202.
- 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:
Within every [[edo]] one can specify a "maximally even" (ME) or "quasi-equal" scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo. The maximally even scale will be one: a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo). b. whose steps are distributed as evenly as possible. (a) and (b) above imply that the ME scale will be a [[MOSScales|moment of symmetry scale]]. For instance, here are all the ME scales available in [[31edo]]: <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 2 .. 15 16</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 10 10 11</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 8 8 8 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 6 6 6 6 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 5 5 5 5 5 6</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 5 4 5 4 5 4 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 4 4 4 4 4 4 4 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 4 3 4 3 4 3 4 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 3 3 3 3 3 3 3 3 3 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 3 3 3 3 2 3 3 3 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 3 3 2 3 2 3 3 2 3 2 3 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 13 . 3 2 3 2 2 3 2 3 2 2 3 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 14 . 2 2 2 2 3 2 2 2 2 3 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 15 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 16 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 17 . 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 18 . 2 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 19 . 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 20 . 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 21 . 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 22 . 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 23 . 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 24 . 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 25 . 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 26 . 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 27 . 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 28 . 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 29 . 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 30 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2</span> And here are all the ME scales available in [[13edo]]: <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;">2 .. 6 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 4 4 5</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 3 3 3 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 2 3 2 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 2 2 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 2 2 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 2 2 1 2 2 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 2 1 2 1 2 1 2 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 2 1 1 2 1 1 2 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 1 1 1 1 1 1 1 1 1 1 1 2 </span> The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th). The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo. Maximally even sets tend to be familiar and musically relevant scale collections. The maximally even heptatonic set of [[19edo]] is, like the one in 12edo, a diatonic scale. The maximally even heptatonic sets of [[17edo]] and [[24edo]], in contrary, are Maqamic[7]. The maximally even heptatonic set of [[22edo]] is Porcupine[7] (the diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is Porcupine[8], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.
Original HTML content:
<html><head><title>Maximal evenness</title></head><body>Within every <a class="wiki_link" href="/edo">edo</a> one can specify a "maximally even" (ME) or "quasi-equal" scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo.<br /> <br /> The maximally even scale will be one:<br /> a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo).<br /> b. whose steps are distributed as evenly as possible.<br /> (a) and (b) above imply that the ME scale will be a <a class="wiki_link" href="/MOSScales">moment of symmetry scale</a>.<br /> <br /> For instance, here are all the ME scales available in <a class="wiki_link" href="/31edo">31edo</a>:<br /> <br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 2 .. 15 16</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 10 10 11</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 8 8 8 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 6 6 6 6 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 5 5 5 5 5 6</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 5 4 5 4 5 4 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 4 4 4 4 4 4 4 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 4 3 4 3 4 3 4 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 3 3 3 3 3 3 3 3 3 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 3 3 3 3 2 3 3 3 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 3 3 2 3 2 3 3 2 3 2 3 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 13 . 3 2 3 2 2 3 2 3 2 2 3 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 14 . 2 2 2 2 3 2 2 2 2 3 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 15 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 16 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 17 . 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 18 . 2 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 19 . 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 20 . 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 21 . 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 22 . 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 23 . 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 24 . 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 25 . 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 26 . 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 27 . 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 28 . 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 29 . 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 30 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2</span><br /> <br /> And here are all the ME scales available in <a class="wiki_link" href="/13edo">13edo</a>:<br /> <br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;">2 .. 6 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 4 4 5</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 3 3 3 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 2 3 2 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 2 2 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 2 2 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 2 2 1 2 2 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 2 1 2 1 2 1 2 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 2 1 1 2 1 1 2 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 1 1 1 1 1 1 1 1 1 1 1 2 </span><br /> <br /> The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th).<br /> <br /> The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.<br /> <br /> Maximally even sets tend to be familiar and musically relevant scale collections. The maximally even heptatonic set of <a class="wiki_link" href="/19edo">19edo</a> is, like the one in 12edo, a diatonic scale. The maximally even heptatonic sets of <a class="wiki_link" href="/17edo">17edo</a> and <a class="wiki_link" href="/24edo">24edo</a>, in contrary, are Maqamic[7]. The maximally even heptatonic set of <a class="wiki_link" href="/22edo">22edo</a> is Porcupine[7] (the diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is Porcupine[8], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.</body></html>