Hobbit: Difference between revisions
Wikispaces>genewardsmith **Imported revision 167479385 - 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:genewardsmith|genewardsmith]] and made on <tt>2010-10- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-04 01:05:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>167488473</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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==Definition== | ==Definition== | ||
To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows: | To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows: | ||
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where the norm on the right is the ordinary Euclidean norm. | where the norm on the right is the ordinary Euclidean norm. | ||
(8) If v[1] is odd then for each integer j, 0 < | (8) If v[1] is odd then for each integer j, 0 < j less than or equal to v[1], we choose a corresponding monzo mj such that <v|m> = j, 0 < <J|m> less than or equal to 1 where J is the JI mapping <log2(2) log2(3) ... log2(p)|, and ||m||_s is minimal. | ||
(9) If v[1] is even, we choose a monzo u such that ||u||_s > 0 and ||u||_s is minimal. Then for each integer j, where 0 < j less than or equal to v[1], we choose a corresponding monzo mj such that <v|m> = j, 0 < <J|m> less than or equal to 1, and where ||m - u/2||_s is minimal. | |||
( | (10) We now apply the chosen tuning to the monzos mj, obtaining values (in cents or fractional monzos) defining a scale. The monzos mj are defined only modulo the commas and the octave o, but since the commas are tempered out and mj is in the octave range from 0 < mj less than or equal to 1200 cents, this does not affect the definition of the scale. | ||
==Example== | |||
For an example, consider the 22 note hobbit for minerva temperament, the 11-limit temperament tempering out 99/98 and 176/175. Here the val is <22 35 51 62 76|, and an interval of minimal nonzero size for the temperament is 16/15, with monzo |4 -1 -1 0 0>. The fractional monzo for half of this, corresponding to the square root, is |4 -1/2 -1/2 0 0>, and intervals representing scale steps are 36/35, 15/14, 11/10, 8/7, 7/6, 40/33, 5/4, 9/7, 4/3, 48/35, 10/7, 22/15, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 64/35, 15/8, 64/33, 2/1. A tuning can be defined in various ways, for instance by approximating the above in [[53edo]], or by using the minimax tuning, which has eigenmonzsos 2, 3, and 11. | |||
If we use the minimax tuning, we find that </pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Hobbits</title></head><body>A <em>hobbit scale</em> is a generalization of <a class="wiki_link" href="/MOSScales">MOS</a> for arbitrary regular temperaments which is a sort of cousin to <a class="wiki_link" href="/Dwarves">dwarf scales</a>. Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.<br /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Hobbits</title></head><body>A <em>hobbit scale</em> is a generalization of <a class="wiki_link" href="/MOSScales">MOS</a> for arbitrary regular temperaments which is a sort of cousin to <a class="wiki_link" href="/Dwarves">dwarf scales</a>. Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->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. This seminorm applies to <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows:<br /> | 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. This seminorm applies to <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows:<br /> | ||
<br /> | <br /> | ||
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where the norm on the right is the ordinary Euclidean norm.<br /> | where the norm on the right is the ordinary Euclidean norm.<br /> | ||
<br /> | <br /> | ||
(8) If v[1] is odd then for each integer j, 0 &lt; | (8) If v[1] is odd then for each integer j, 0 &lt; j less than or equal to v[1], we choose a corresponding monzo mj such that &lt;v|m&gt; = j, 0 &lt; &lt;J|m&gt; less than or equal to 1 where J is the JI mapping &lt;log2(2) log2(3) ... log2(p)|, and ||m||_s is minimal.<br /> | ||
&lt; | <br /> | ||
(9) If v[1] is even, we choose a monzo u such that ||u||_s &gt; 0 and ||u||_s is minimal. Then for each integer j, where 0 &lt; j less than or equal to v[1], we choose a corresponding monzo mj such that &lt;v|m&gt; = j, 0 &lt; &lt;J|m&gt; less than or equal to 1, and where ||m - u/2||_s is minimal.<br /> | |||
<br /> | <br /> | ||
( | (10) We now apply the chosen tuning to the monzos mj, obtaining values (in cents or fractional monzos) defining a scale. The monzos mj are defined only modulo the commas and the octave o, but since the commas are tempered out and mj is in the octave range from 0 &lt; mj less than or equal to 1200 cents, this does not affect the definition of the scale.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Example"></a><!-- ws:end:WikiTextHeadingRule:2 -->Example</h2> | |||
For an example, consider the 22 note hobbit for minerva temperament, the 11-limit temperament tempering out 99/98 and 176/175. Here the val is &lt;22 35 51 62 76|, and an interval of minimal nonzero size for the temperament is 16/15, with monzo |4 -1 -1 0 0&gt;. The fractional monzo for half of this, corresponding to the square root, is |4 -1/2 -1/2 0 0&gt;, and intervals representing scale steps are 36/35, 15/14, 11/10, 8/7, 7/6, 40/33, 5/4, 9/7, 4/3, 48/35, 10/7, 22/15, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 64/35, 15/8, 64/33, 2/1. A tuning can be defined in various ways, for instance by approximating the above in <a class="wiki_link" href="/53edo">53edo</a>, or by using the minimax tuning, which has eigenmonzsos 2, 3, and 11.<br /> | |||
<br /> | <br /> | ||
If we use the minimax tuning, we find that</body></html></pre></div> | |||