Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 401004732 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 401005304 - 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:genewardsmith|genewardsmith]] and made on <tt>2013-01-24 04: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-24 04:33:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>401005304</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> | ||
| Line 21: | Line 21: | ||
(3)\ i < j\text{ implies }s[i] < s[j] | (3)\ i < j\text{ implies }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: | ||
| Line 30: | Line 28: | ||
**[[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. | ||
The set {s[i] | i∈ℤ} generates a group G, the **group of the scale**; this is a free, finitely generated subgroup of the reals ℝ. | The set {s[i] | i∈ℤ} generates a group G, the **group of the scale**; this is a free, finitely generated subgroup of the reals ℝ. The **rank of the scale** is the rank of G. | ||
**Epimorphic**: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by [[Yves Hellegouarch]]. | **Epimorphic**: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by [[Yves Hellegouarch]]. | ||
| Line 54: | Line 52: | ||
(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 /> | ||
<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 /> | ||
| Line 63: | Line 59: | ||
<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 ≤ 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 ≤ 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 /> | ||
<br /> | <br /> | ||
The set {s[i] | i∈ℤ} generates a group G, the <strong>group of the scale</strong>; this is a free, finitely generated subgroup of the reals ℝ.<br /> | The set {s[i] | i∈ℤ} generates a group G, the <strong>group of the scale</strong>; this is a free, finitely generated subgroup of the reals ℝ. The <strong>rank of the scale</strong> is the rank of G.<br /> | ||
<br /> | <br /> | ||
<strong>Epimorphic</strong>: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a>.<br /> | <strong>Epimorphic</strong>: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a>.<br /> | ||
<br /> | <br /> | ||
<strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a></strong> : A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the <strong>trivalence property</strong>. If every class has less than three elements, it has the property of distributional evenness.</body></html></pre></div> | <strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a></strong> : A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the <strong>trivalence property</strong>. If every class has less than three elements, it has the property of distributional evenness.</body></html></pre></div> | ||