Maximal evenness: Difference between revisions
Wikispaces>hstraub **Imported revision 479367202 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 479373534 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2013-12-25 13:48:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>479373534</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">Within every [[edo]] one can specify a "maximally even" (ME) | <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">Within every [[edo]] one can specify a "maximally even" (ME) 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: | The maximally even scale will be one: | ||
| Line 63: | Line 63: | ||
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. | 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> | 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. | ||
Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel Mandelbaum]]'s 1961 thesis [[http://www.anaphoria.com/mandelbaum.html|Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.</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) | <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) scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo.<br /> | ||
<br /> | <br /> | ||
The maximally even scale will be one:<br /> | The maximally even scale will be one:<br /> | ||
| Line 122: | Line 124: | ||
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 /> | 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 /> | <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> | 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.<br /> | ||
<br /> | |||
Note that &quot;maximally even&quot; is equivalent to &quot;quasi-equal-interval-symmetrical&quot; in <a class="wiki_link" href="/Joel%20Mandelbaum">Joel Mandelbaum</a>'s 1961 thesis <a class="wiki_link_ext" href="http://www.anaphoria.com/mandelbaum.html" rel="nofollow">Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament</a>. Previous versions of this article have conflated &quot;quasi-equal&quot; with &quot;quasi-equal-interval symmetrical&quot;. In fact, &quot;quasi-equal&quot; scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.</body></html></pre></div> | |||