Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 385523468 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 400700856 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2013-01-23 07:08:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>400700856</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> | 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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(3)\ i < j\text{ implies }s[i] < s[j] | (3)\ i < j\text{ implies }s[i] < s[j] | ||
[[math]] | [[math]] | ||
For any periodic scale p, the **rank** of that scale is the cardinality of the smallest set S of real numbers which has the property that everything in Im(p) is a Z-linear combination of elements in S. | |||
We may define an important function **class(i)** on the integers which gives the //generic intervals// of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, 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 which gives the //generic intervals// of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have: | ||
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(3)\ i &lt; j\text{ implies }s[i] &lt; s[j]&lt;br/&gt;[[math]] | (3)\ i &lt; j\text{ implies }s[i] &lt; s[j]&lt;br/&gt;[[math]] | ||
--><script type="math/tex">(3)\ i < j\text{ implies }s[i] < s[j]</script><!-- ws:end:WikiTextMathRule:2 --><br /> | --><script type="math/tex">(3)\ i < j\text{ implies }s[i] < s[j]</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
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For any periodic scale p, the <strong>rank</strong> of that scale is the cardinality of the smallest set S of real numbers which has the property that everything in Im(p) is a Z-linear combination of elements in S.<br /> | |||
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We may define an important function <strong>class(i)</strong> on the integers which gives the <em>generic intervals</em> of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, 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 <strong>class(i)</strong> on the integers which gives the <em>generic intervals</em> of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have:<br /> | ||