Hobbit: Difference between revisions
Wikispaces>genewardsmith **Imported revision 167493227 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 167493297 - 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>2010-10-04 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-04 01:57:31 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>167493297</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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After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank two temperament; by adding both, we obtain the rank one temperament with val <65 103 151 183 225|, giving a scale with steps 2433333242432424233333. This sort of thing seems to happen fairly often with hobbit scales. | After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank two temperament; by adding both, we obtain the rank one temperament with val <65 103 151 183 225|, giving a scale with steps 2433333242432424233333. This sort of thing seems to happen fairly often with hobbit scales. | ||
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<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 /> | ||
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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 /> | 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 /> | ||
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After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank two temperament; by adding both, we obtain the rank one temperament with val &lt;65 103 151 183 225|, giving a scale with steps 2433333242432424233333. This sort of thing seems to happen fairly often with hobbit scales. | After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank two temperament; by adding both, we obtain the rank one temperament with val &lt;65 103 151 183 225|, giving a scale with steps 2433333242432424233333. This sort of thing seems to happen fairly often with hobbit scales.</body></html></pre></div> | ||