Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 144121435 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 144125407 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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We may define an important function class(i) on the integers mod P which gives the //interval classes// of a periodic scale. This is defined by s[j] - s[i] mod O is in class(k) if j - i = k mod P. Here the modulo operation on real numbers may be defined in terms of the [[http://en.wikipedia.org/wiki/Floor_and_ceiling_functions|floor function]] by x mod O = x - O floor(x/O). Since s is quasiperiodic, class(0) consists only of {0}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have: | We may define an important function class(i) on the integers mod P which gives the //interval classes// of a periodic scale. This is defined by s[j] - s[i] mod O is in class(k) if j - i = k mod P. Here the modulo operation on real numbers may be defined in terms of the [[http://en.wikipedia.org/wiki/Floor_and_ceiling_functions|floor function]] by x mod O = x - O floor(x/O). Since s is quasiperiodic, class(0) consists only of {0}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have: | ||
**Constant Structure**: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by [[http://en.wikipedia.org/wiki/Erv_Wilson|Erv Wilson]]) means that i<>j implies class(i) intersect class(j) = {}. | **Constant Structure**: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by [[http://en.wikipedia.org/wiki/Erv_Wilson|Erv Wilson]]) means that i<>j implies class(i) intersect class(j) = {}. In academic music theory, this is called the //partitioning property//. | ||
**[[http://en.wikipedia.org/wiki/Rothenberg_propriety|Propriety]] **: If s is monotone, and if i <= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i < j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called //coherence//. Note that strict propriety implies constant structure. | **[[http://en.wikipedia.org/wiki/Rothenberg_propriety|Propriety]] **: If s is monotone, and if i <= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i < j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called //coherence//. Note that strict propriety implies constant structure. | ||
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We may define an important function class(i) on the integers mod P which gives the <em>interval classes</em> of a periodic scale. This is defined by s[j] - s[i] mod O is in class(k) if j - i = k mod P. Here the modulo operation on real numbers may be defined in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow">floor function</a> by x mod O = x - O floor(x/O). Since s is quasiperiodic, class(0) consists only of {0}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have:<br /> | We may define an important function class(i) on the integers mod P which gives the <em>interval classes</em> of a periodic scale. This is defined by s[j] - s[i] mod O is in class(k) if j - i = k mod P. Here the modulo operation on real numbers may be defined in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow">floor function</a> by x mod O = x - O floor(x/O). Since s is quasiperiodic, class(0) consists only of {0}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have:<br /> | ||
<br /> | <br /> | ||
<strong>Constant Structure</strong>: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow">Erv Wilson</a>) means that i&lt;&gt;j implies class(i) intersect class(j) = {}.<br /> | <strong>Constant Structure</strong>: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow">Erv Wilson</a>) means that i&lt;&gt;j implies class(i) intersect class(j) = {}. In academic music theory, this is called the <em>partitioning property</em>.<br /> | ||
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<strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Propriety</a> </strong>: If s is monotone, and if i &lt;= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i &lt; j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called <em>coherence</em>. Note that strict propriety implies constant structure. <br /> | <strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Propriety</a> </strong>: If s is monotone, and if i &lt;= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i &lt; j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called <em>coherence</em>. Note that strict propriety implies constant structure. <br /> | ||