Periodic scale: Difference between revisions
Wikispaces>guest **Imported revision 199010844 - Original comment: ** |
Wikispaces>guest **Imported revision 199010910 - 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:guest|guest]] and made on <tt>2011-02-05 21:28: | : This revision was by author [[User:guest|guest]] and made on <tt>2011-02-05 21:28:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>199010910</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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[[math]] | [[math]] | ||
(1) s[0] = 0 | (1) s[0] = 0 | ||
[[math]] | [[math]] | ||
[[math]] | [[math]] | ||
(2) s[i + P] = s[i] + O | (2) s[i + P] = s[i] + O | ||
[[math]] | [[math]] | ||
Scales written in the widely used [[http://www.huygens-fokker.org/scala/scl_format.html|Scala format]] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Neither Scala nor the above definition assumes that the scales are [[http://en.wikipedia.org/wiki/Monotonic_function|monotonically strictly increasing]], but this condition, giving a **monotone periodic scale**, is often important to add: | Scales written in the widely used [[http://www.huygens-fokker.org/scala/scl_format.html|Scala format]] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Neither Scala nor the above definition assumes that the scales are [[http://en.wikipedia.org/wiki/Monotonic_function|monotonically strictly increasing]], but this condition, giving a **monotone periodic scale**, is often important to add: | ||
[[math]] | [[math]] (3) i < j[[math]] implies [[math]]s[i] < s[j] [[math]] | ||
[[math]] | |||
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: | ||
**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//. | **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//. | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
(1) s[0] = 0&lt;br/&gt;[[math]] | (1) s[0] = 0&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> (1) s[0] = 0</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex"> (1) s[0] = 0</script><!-- ws:end:WikiTextMathRule:0 --> <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextMathRule:1: | <!-- ws:start:WikiTextMathRule:1: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
(2) s[i + P] = s[i] + O&lt;br/&gt;[[math]] | (2) s[i + P] = s[i] + O&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> (2) s[i + P] = s[i] + O</script><!-- ws:end:WikiTextMathRule:1 --><br /> | --><script type="math/tex"> (2) s[i + P] = s[i] + O</script><!-- ws:end:WikiTextMathRule:1 --> <br /> | ||
<br /> | <br /> | ||
Scales written in the widely used <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/scl_format.html" rel="nofollow">Scala format</a> are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Neither Scala nor the above definition assumes that the scales are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotonically strictly increasing</a>, but this condition, giving a <strong>monotone periodic scale</strong>, is often important to add:<br /> | Scales written in the widely used <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/scl_format.html" rel="nofollow">Scala format</a> are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Neither Scala nor the above definition assumes that the scales are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotonically strictly increasing</a>, but this condition, giving a <strong>monotone periodic scale</strong>, is often important to add:<br /> | ||
<br /> | <br /> | ||
< | <a class="wiki_link" href="/math">math</a> (3) i &lt; j<a class="wiki_link" href="/math">math</a> implies <a class="wiki_link" href="/math">math</a>s[i] &lt; s[j] <a class="wiki_link" href="/math">math</a><br /> | ||
<br /> | <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 /> | 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 /> | ||
<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) = {}. In academic music theory, this is called the <em>partitioning property</em>.<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 /> | ||