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 15:05:33 UTC</tt>.

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

: The original revision id was <tt>~~209352342~~</tt>.

: The original revision id was <tt>209356328</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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==Definition==

To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space derived from a regular temperament, the octave equivalent ~~Teeny~~-Euclidean seminorm ~~ior~~ [[Tenney-Euclidean metrics#The OETES|OETES]]. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OE complexity for each scale step mapped to the integer i by the val v.

To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space derived from a regular temperament, the octave equivalent Tenney-Euclidean seminorm or [[Tenney-Euclidean metrics#The OETES|OETES]]. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer i by the val v.

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 < 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.

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, 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 < 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 [[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.

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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<h2 id="toc0"><a name="x-Definition"></a>Definition</h2>

To define the hobbit scale we first define a particular <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Seminorm.html" rel="nofollow">seminorm</a> on interval space derived from a regular temperament, the octave equivalent ~~Teeny~~-Euclidean seminorm ~~ior~~ <a class="wiki_link" href="/Tenney-Euclidean%20metrics#The OETES">OETES</a>. This seminorm applies to <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OE complexity for each scale step mapped to the integer i by the val v.

To define the hobbit scale we first define a particular <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Seminorm.html" rel="nofollow">seminorm</a> on interval space derived from a regular temperament, the octave equivalent Tenney-Euclidean seminorm or <a class="wiki_link" href="/Tenney-Euclidean%20metrics#The OETES">OETES</a>. This seminorm applies to <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer i by the val v.

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 &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.

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, 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 &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="/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.

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 15:05:33 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 15:14:18 UTC</tt>.<br>
-: The original revision id was <tt>209352342</tt>.<br>
+: The original revision id was <tt>209356328</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 11: / Line 11: @@
 ==Definition==
-To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space derived from a regular temperament, the octave equivalent Teeny-Euclidean seminorm ior [[Tenney-Euclidean metrics#The OETES|OETES]]. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OE complexity for each scale step mapped to the integer i by the val v.
+To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space derived from a regular temperament, the octave equivalent Tenney-Euclidean seminorm or [[Tenney-Euclidean metrics#The OETES|OETES]]. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer i by the val v.
-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 &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.
+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, 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 &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 [[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.
 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 31: / Line 31: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h2&gt;
-  To define the hobbit scale we first define a particular &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Seminorm.html" rel="nofollow"&gt;seminorm&lt;/a&gt; on interval space derived from a regular temperament, the octave equivalent Teeny-Euclidean seminorm ior &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics#The OETES"&gt;OETES&lt;/a&gt;. This seminorm applies to &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzos&lt;/a&gt; and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OE complexity for each scale step mapped to the integer i by the val v.&lt;br /&gt;
+  To define the hobbit scale we first define a particular &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Seminorm.html" rel="nofollow"&gt;seminorm&lt;/a&gt; on interval space derived from a regular temperament, the octave equivalent Tenney-Euclidean seminorm or &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics#The OETES"&gt;OETES&lt;/a&gt;. This seminorm applies to &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzos&lt;/a&gt; and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer i by the val v.&lt;br /&gt;
 &lt;br /&gt;
-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 &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;
+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, 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 &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="/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;
 &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;