Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 312095674 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 314424870 - 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>2012-03- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-25 17:51:00 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>314424870</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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Hence S satisfies the conditions for being a [[Periodic scale|periodic scale]], and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block. | Hence S satisfies the conditions for being a [[Periodic scale|periodic scale]], and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block. | ||
Note that because any ak can be chosen satisfying 0 | Note that because any ak can be chosen satisfying 0 ≤ ak < P, each Fokker block comes associated not only with various modes, but also [[Dome|domes]], which are variations on the same basic theme. | ||
=Second definition of a Fokker block= | =Second definition of a Fokker block= | ||
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The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val <22 35 51 62|. Using a basis for the Fokker group, for instance the one listed [[Minkowski reduced bases for Fokker groups of certain vals|here]], pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS. | The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val <22 35 51 62|. Using a basis for the Fokker group, for instance the one listed [[Minkowski reduced bases for Fokker groups of certain vals|here]], pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS. | ||
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Hence S satisfies the conditions for being a <a class="wiki_link" href="/Periodic%20scale">periodic scale</a>, and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.<br /> | Hence S satisfies the conditions for being a <a class="wiki_link" href="/Periodic%20scale">periodic scale</a>, and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.<br /> | ||
<br /> | <br /> | ||
Note that because any ak can be chosen satisfying 0 | Note that because any ak can be chosen satisfying 0 ≤ ak &lt; P, each Fokker block comes associated not only with various modes, but also <a class="wiki_link" href="/Dome">domes</a>, which are variations on the same basic theme. <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Second definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:4 -->Second definition of a Fokker block</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Second definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:4 -->Second definition of a Fokker block</h1> | ||