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>~~2010~~-12-~~12 01~~:21:37 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-15 23:23:45 UTC</tt>.

: The original revision id was <tt>~~187427749~~</tt>.

: The original revision id was <tt>193568886</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 OE seminorm 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that <v|m> = j, 0 less than <J|m> less than or equal to 1 where J is the JI mapping <log2(2) log2(3) ... log2(p)|, and T(m) is minimal.

If v[1] is even, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that <v|m> = j, 0 less than <J|m> less than or equal to 1, and where T(2m - u) is minimal.

If v[1] is even, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that <v|m> = j, 0 less than <J|m> less than or equal to 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 cannonical hobbit.

The intervals selected by this process are a [[transversal]] of te 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.

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.

An alternative and equivalent approach is to work directly with the notes of the temperament, using the [[Tenney-Euclidean metrics|temperamental norm]] defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament.

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Denoting the OE seminorm 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 1 where J is the JI mapping &lt;log2(2) log2(3) ... log2(p)|, and T(m) is minimal.

If v[1] is even, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 1, and where T(2m - u) is minimal.

If v[1] is even, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 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 cannonical hobbit.

The intervals selected by this process are a <a class="wiki_link" href="/transversal">transversal</a> of te 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.

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.

An alternative and equivalent approach is to work directly with the notes of the temperament, using the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">temperamental norm</a> defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament.

@@ 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>2010-12-12 01:21:37 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-15 23:23:45 UTC</tt>.<br>
-: The original revision id was <tt>187427749</tt>.<br>
+: The original revision id was <tt>193568886</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 13: / Line 13: @@
 Denoting the OE seminorm 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 1 where J is the JI mapping &lt;log2(2) log2(3) ... log2(p)|, and T(m) is minimal.
-If v[1] is even, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 1, and where T(2m - u) is minimal.
+If v[1] is even, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 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 cannonical hobbit.
-The intervals selected by this process are a [[transversal]] of te 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.
+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.
 An alternative and equivalent approach is to work directly with the notes of the temperament, using the [[Tenney-Euclidean metrics|temperamental norm]] defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament.
@@ Line 32: / Line 32: @@
 Denoting the OE seminorm 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &amp;lt;v|m&amp;gt; = j, 0 less than &amp;lt;J|m&amp;gt; less than or equal to 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, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &amp;lt;v|m&amp;gt; = j, 0 less than &amp;lt;J|m&amp;gt; less than or equal to 1, and where T(2m - u) is minimal.&lt;br /&gt;
+If v[1] is even, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that &amp;lt;v|m&amp;gt; = j, 0 less than &amp;lt;J|m&amp;gt; less than or equal to 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 cannonical 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 te 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;
+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;
 &lt;br /&gt;
 An alternative and equivalent approach is to work directly with the notes of the temperament, using the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;temperamental norm&lt;/a&gt; defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament.&lt;br /&gt;