Pentachords of 31edo: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 347714478 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 347804582 - 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-06- | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-06-25 07:28:02 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>347804582</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">The term "pentachord" may be used to refer to a scale segment in which a [[Perfect fourth|perfect fourth]] is divided into four steps. This follows the usage of [[Paul Erlich]] (see [[omnitetrachordality]]), and is a generalization of the classical "[[tetrachord]]," a division of the perfect fourth into three steps. [[31edo]]'s perfect fourth can be found at 13 degrees, and measures approximately 503.226¢. A complete list of the 66 tetrachords of 31edo can be found at [[Tricesimoprimal Tetrachordal Tesseract]]. In spite of | <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">The term "pentachord" may be used to refer to a scale segment in which a [[Perfect fourth|perfect fourth]] is divided into four steps. This follows the usage of [[Paul Erlich]] (see [[omnitetrachordality]]), and is a generalization of the classical "[[tetrachord]]," a division of the perfect fourth into three steps. [[31edo]]'s perfect fourth can be found at 13 degrees, and measures approximately 503.226¢. A complete list of the 66 tetrachords of 31edo can be found at [[Tricesimoprimal Tetrachordal Tesseract]]. In spite of that page's name, the organizing figure for the 31edo tetrachords is a triangle, not a tesseract. The 220 pentachords are analogously arranged here on a tetrahedron: | ||
<span style="font-family: Courier New,monospace;">T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1</span> | <span style="font-family: Courier New,monospace;">T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1</span> | ||
| Line 77: | Line 77: | ||
Note that the "T" in some of the pentachords above is short for "ten" and represents an interval of 10 degrees of 31edo (10\31). | Note that the "T" in some of the pentachords above is short for "ten" and represents an interval of 10 degrees of 31edo (10\31). | ||
The pentachords in boldface are the ones which exclude the 1-degree interval ([[diesis]]); this may be a desirable constraint for melodic considerations. There are 56 pentachords of this type. Further limiting the set to those pentachords that exclude both the diesis and the 2-degree interval gives us a list of four quasi-equal pentachords that form the core of this figure: 4333, 3433, 3343 and 3334. Those familiar with the [[31edo MOS scales|MOS scales of 31edo]] may recognize this pentachord as belonging to Miracle[10]. | The pentachords in boldface are the ones which exclude the 1-degree interval ([[diesis]]); this may be a desirable constraint for melodic considerations. There are 56 pentachords of this type. Further limiting the set to those pentachords that exclude both the diesis and the 2-degree interval gives us a list of four quasi-equal pentachords that form the core of this figure: 4333, 3433, 3343 and 3334. (Those familiar with the [[31edo MOS scales|MOS scales of 31edo]] may recognize this pentachord as belonging to Miracle[10].) | ||
A further subset of interest might include those pentachords with only two step sizes, as these can be found in moment of symmetry scales. (Of course, other pentachords may be found as subsets of MOS scales.) This relatively small list with 12 members includes: | |||
<span style="font-family: Courier New,monospace;">111T 11T1 1T11 T111</span> | |||
<span style="font-family: Courier New,monospace;">2227 2272 2722 7222</span> | |||
<span style="font-family: Courier New,monospace;">3334 3343 3433 4333</span> | |||
It should be mentioned that these divisions can be applied in any instance where a span of 13 units is divided into four groups. For instance, these may be interpreted as every possible tetrad within one octave of [[13edo]], or one [[tritave]] of [[BP]], etc.</pre></div> | It should be mentioned that these divisions can be applied in any instance where a span of 13 units is divided into four groups. For instance, these may be interpreted as every possible tetrad within one octave of [[13edo]], or one [[tritave]] of [[BP]], etc.</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>Pentachords of 31edo</title></head><body>The term &quot;pentachord&quot; may be used to refer to a scale segment in which a <a class="wiki_link" href="/Perfect%20fourth">perfect fourth</a> is divided into four steps. This follows the usage of <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> (see <a class="wiki_link" href="/omnitetrachordality">omnitetrachordality</a>), and is a generalization of the classical &quot;<a class="wiki_link" href="/tetrachord">tetrachord</a>,&quot; a division of the perfect fourth into three steps. <a class="wiki_link" href="/31edo">31edo</a>'s perfect fourth can be found at 13 degrees, and measures approximately 503.226¢. A complete list of the 66 tetrachords of 31edo can be found at <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a>. In spite of | <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>Pentachords of 31edo</title></head><body>The term &quot;pentachord&quot; may be used to refer to a scale segment in which a <a class="wiki_link" href="/Perfect%20fourth">perfect fourth</a> is divided into four steps. This follows the usage of <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> (see <a class="wiki_link" href="/omnitetrachordality">omnitetrachordality</a>), and is a generalization of the classical &quot;<a class="wiki_link" href="/tetrachord">tetrachord</a>,&quot; a division of the perfect fourth into three steps. <a class="wiki_link" href="/31edo">31edo</a>'s perfect fourth can be found at 13 degrees, and measures approximately 503.226¢. A complete list of the 66 tetrachords of 31edo can be found at <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a>. In spite of that page's name, the organizing figure for the 31edo tetrachords is a triangle, not a tesseract. The 220 pentachords are analogously arranged here on a tetrahedron:<br /> | ||
<br /> | <br /> | ||
<span style="font-family: Courier New,monospace;">T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1</span><br /> | <span style="font-family: Courier New,monospace;">T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1</span><br /> | ||
| Line 152: | Line 158: | ||
Note that the &quot;T&quot; in some of the pentachords above is short for &quot;ten&quot; and represents an interval of 10 degrees of 31edo (10\31).<br /> | Note that the &quot;T&quot; in some of the pentachords above is short for &quot;ten&quot; and represents an interval of 10 degrees of 31edo (10\31).<br /> | ||
<br /> | <br /> | ||
The pentachords in boldface are the ones which exclude the 1-degree interval (<a class="wiki_link" href="/diesis">diesis</a>); this may be a desirable constraint for melodic considerations. There are 56 pentachords of this type. Further limiting the set to those pentachords that exclude both the diesis and the 2-degree interval gives us a list of four quasi-equal pentachords that form the core of this figure: 4333, 3433, 3343 and 3334. Those familiar with the <a class="wiki_link" href="/31edo%20MOS%20scales">MOS scales of 31edo</a> may recognize this pentachord as belonging to Miracle[10].<br /> | The pentachords in boldface are the ones which exclude the 1-degree interval (<a class="wiki_link" href="/diesis">diesis</a>); this may be a desirable constraint for melodic considerations. There are 56 pentachords of this type. Further limiting the set to those pentachords that exclude both the diesis and the 2-degree interval gives us a list of four quasi-equal pentachords that form the core of this figure: 4333, 3433, 3343 and 3334. (Those familiar with the <a class="wiki_link" href="/31edo%20MOS%20scales">MOS scales of 31edo</a> may recognize this pentachord as belonging to Miracle[10].)<br /> | ||
<br /> | |||
A further subset of interest might include those pentachords with only two step sizes, as these can be found in moment of symmetry scales. (Of course, other pentachords may be found as subsets of MOS scales.) This relatively small list with 12 members includes:<br /> | |||
<br /> | |||
<span style="font-family: Courier New,monospace;">111T 11T1 1T11 T111</span><br /> | |||
<span style="font-family: Courier New,monospace;">2227 2272 2722 7222</span><br /> | |||
<span style="font-family: Courier New,monospace;">3334 3343 3433 4333</span><br /> | |||
<br /> | <br /> | ||
It should be mentioned that these divisions can be applied in any instance where a span of 13 units is divided into four groups. For instance, these may be interpreted as every possible tetrad within one octave of <a class="wiki_link" href="/13edo">13edo</a>, or one <a class="wiki_link" href="/tritave">tritave</a> of <a class="wiki_link" href="/BP">BP</a>, etc.</body></html></pre></div> | It should be mentioned that these divisions can be applied in any instance where a span of 13 units is divided into four groups. For instance, these may be interpreted as every possible tetrad within one octave of <a class="wiki_link" href="/13edo">13edo</a>, or one <a class="wiki_link" href="/tritave">tritave</a> of <a class="wiki_link" href="/BP">BP</a>, etc.</body></html></pre></div> | ||