Hobbit: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-~~10 15~~:14:18 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-25 16:19:51 UTC</tt>.

: The original revision id was <tt>~~209356328~~</tt>.

: The original revision id was <tt>242784719</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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Denoting the OETES for any element x of interval space by T(x), we first define the hobbit of an odd-numbered scale; that is, a scale for which v[1] is an odd number. If v[1] is odd then for each integer j, 0 < j __<__ v[1], we choose a corresponding monzo m such that <v|m> = j, 0 < <J|m> __<__ 1 where J is the JI mapping <log2(2) log2(3) ... log2(p)|, and T(m) is minimal.

If v[1] is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least [[~~Tenney~~ height]]. Another approach adopted here is to choose a monzo u such that T(u) is minimal under the condition that T(u) > 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 < j __<__ to v[1], we choose a corresponding monzo m such that <v|m> = j, 0 < <J|m> __<__ 1, and where T(2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is full equal as a scale to the canonical hobbit.

If v[1] is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least [[Benedetti height]]. Another approach adopted here is to choose a monzo u such that T(u) is minimal under the condition that T(u) > 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 < j __<__ to v[1], we choose a corresponding monzo m such that <v|m> = j, 0 < <J|m> __<__ 1, and where T(2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is full equal as a scale to the canonical hobbit.

The intervals selected by this process are a [[transversal]] of the scale, and we may now apply the chosen tuning to the monzos in the transversal, obtaining values (in cents or fractional monzos) defining a scale. The monzos in the transversal are defined only modulo the commas of the temperament, but since these are tempered out this does not affect the definition of the scale.

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Denoting the OETES for any element x of interval space by T(x), we first define the hobbit of an odd-numbered scale; that is, a scale for which v[1] is an odd number. If v[1] is odd then for each integer j, 0 &lt; j &lt; v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 &lt; &lt;J|m&gt; &lt; 1 where J is the JI mapping &lt;log2(2) log2(3) ... log2(p)|, and T(m) is minimal.

If v[1] is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least <a class="wiki_link" href="/~~Tenney~~%20height">~~Tenney~~ height</a>. Another approach adopted here is to choose a monzo u such that T(u) is minimal under the condition that T(u) &gt; 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 &lt; j &lt; to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 &lt; &lt;J|m&gt; &lt; 1, and where T(2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is full equal as a scale to the canonical hobbit.

If v[1] is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a>. Another approach adopted here is to choose a monzo u such that T(u) is minimal under the condition that T(u) &gt; 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 &lt; j &lt; to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 &lt; &lt;J|m&gt; &lt; 1, and where T(2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is full equal as a scale to the canonical hobbit.

The intervals selected by this process are a <a class="wiki_link" href="/transversal">transversal</a> of the scale, and we may now apply the chosen tuning to the monzos in the transversal, obtaining values (in cents or fractional monzos) defining a scale. The monzos in the transversal are defined only modulo the commas of the temperament, but since these are tempered out this does not affect the definition of the scale.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-03-10 15:14:18 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-25 16:19:51 UTC</tt>.<br>
-: The original revision id was <tt>209356328</tt>.<br>
+: The original revision id was <tt>242784719</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>
@@ Line 15: / Line 15: @@
 Denoting the OETES for any element x of interval space by T(x), we first define the hobbit of an odd-numbered scale; that is, a scale for which v[1] is an odd number. If v[1] is odd then for each integer j, 0 &lt; j __&lt;__ v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 &lt; &lt;J|m&gt; __&lt;__ 1 where J is the JI mapping &lt;log2(2) log2(3) ... log2(p)|, and T(m) is minimal.
-If v[1] is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least [[Tenney height]]. Another approach adopted here is to choose a monzo u such that T(u) is minimal under the condition that T(u) &gt; 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 &lt; j __&lt;__ to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 &lt; &lt;J|m&gt; __&lt;__ 1, and where T(2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is full equal as a scale to the canonical hobbit.
+If v[1] is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least [[Benedetti height]]. Another approach adopted here is to choose a monzo u such that T(u) is minimal under the condition that T(u) &gt; 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 &lt; j __&lt;__ to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 &lt; &lt;J|m&gt; __&lt;__ 1, and where T(2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is full equal as a scale to the canonical hobbit.
 The intervals selected by this process are a [[transversal]] of the scale, and we may now apply the chosen tuning to the monzos in the transversal, obtaining values (in cents or fractional monzos) defining a scale. The monzos in the transversal are defined only modulo the commas of the temperament, but since these are tempered out this does not affect the definition of the scale.
@@ Line 35: / Line 35: @@
 Denoting the OETES for any element x of interval space by T(x), we first define the hobbit of an odd-numbered scale; that is, a scale for which v[1] is an odd number. If v[1] is odd then for each integer j, 0 &amp;lt; j &lt;u&gt;&amp;lt;&lt;/u&gt; v[1], we choose a corresponding monzo m such that &amp;lt;v|m&amp;gt; = j, 0 &amp;lt; &amp;lt;J|m&amp;gt; &lt;u&gt;&amp;lt;&lt;/u&gt; 1 where J is the JI mapping &amp;lt;log2(2) log2(3) ... log2(p)|, and T(m) is minimal.&lt;br /&gt;
 &lt;br /&gt;
-If v[1] is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least &lt;a class="wiki_link" href="/Tenney%20height"&gt;Tenney height&lt;/a&gt;. Another approach adopted here is to choose a monzo u such that T(u) is minimal under the condition that T(u) &amp;gt; 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 &amp;lt; j &lt;u&gt;&amp;lt;&lt;/u&gt; to v[1], we choose a corresponding monzo m such that &amp;lt;v|m&amp;gt; = j, 0 &amp;lt; &amp;lt;J|m&amp;gt; &lt;u&gt;&amp;lt;&lt;/u&gt; 1, and where T(2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is full equal as a scale to the canonical hobbit.&lt;br /&gt;
+If v[1] is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt;. Another approach adopted here is to choose a monzo u such that T(u) is minimal under the condition that T(u) &amp;gt; 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 &amp;lt; j &lt;u&gt;&amp;lt;&lt;/u&gt; to v[1], we choose a corresponding monzo m such that &amp;lt;v|m&amp;gt; = j, 0 &amp;lt; &amp;lt;J|m&amp;gt; &lt;u&gt;&amp;lt;&lt;/u&gt; 1, and where T(2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is full equal as a scale to the canonical hobbit.&lt;br /&gt;
 &lt;br /&gt;
 The intervals selected by this process are a &lt;a class="wiki_link" href="/transversal"&gt;transversal&lt;/a&gt; of the scale, and we may now apply the chosen tuning to the monzos in the transversal, obtaining values (in cents or fractional monzos) defining a scale. The monzos in the transversal are defined only modulo the commas of the temperament, but since these are tempered out this does not affect the definition of the scale.&lt;br /&gt;