Crystal ball: Difference between revisions
Wikispaces>genewardsmith **Imported revision 297268894 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 598609118 - 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: | : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-11-06 14:35:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>598609118</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html"> | <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;white-space: pre-wrap ! important" class="old-revision-html">We may define the nth q-limit //Hahn shell// as the octave classes at exactly [[Hahn distance]] n from the unison in terms of the q-odd-limit Hahn norm. The number of notes in the 5-limit Hahn shell is (for n>0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. An alternative definition, not employing Hahn distance, is that the nth 5- and 7- limit crystal balls are the nth [[Scale products and scale powers|scale powers]] of the 5- and 7-limit tonality diamonds, respectively. This easily generalizes to the scale power of the q-limit tonality diamond for any odd number q. | ||
We may define the nth q-limit //Hahn shell// as the octave classes at exactly [[Hahn distance]] n from the unison in terms of the q-odd-limit Hahn norm. The number of notes in the 5-limit Hahn shell is (for n>0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. An alternative definition, not employing Hahn distance, is that the nth 5- and 7- limit crystal balls are the nth [[Scale products and scale powers|scale powers]] of the 5- and 7-limit tonality diamonds, respectively. This easily generalizes to the scale power of the q-limit tonality diamond for any odd number q. | |||
Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords. | Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords. | ||
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Here are the first few 5-limit Hahn shells: | Here are the first few 5-limit Hahn shells: | ||
Shell 0 | Shell 0 | ||
[1] | [1] | ||
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Here are the first three 7-limit Hahn shells: | Here are the first three 7-limit Hahn shells: | ||
Shell 0 | Shell 0 | ||
[1] | [1] | ||
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**Crystal ball 2 55 notes** | **Crystal ball 2 55 notes** | ||
[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, | [50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, | ||
28/25,9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49, 21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25, 2] | 28/25, 9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49, 21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25, 2] | ||
Crystal ball one can also be described as Cube[2], the 2x2x2 cube scale, which consists of the notes of the eight chords [i, j, k] with i, j, and k either -1 or 0. Crystal ball two consists of Cube[4], the 4x4x4 cube with i, j, and k from -2 to 1, minus the eight chords [-2 -2 1], [-2 1 -2], [-2 1 1], [1 -2 -2], [1 -2 1], [-2 -2 -2], [1 1 -2], [1 1 1]. | Crystal ball one can also be described as Cube[2], the 2x2x2 cube scale, which consists of the notes of the eight chords [i, j, k] with i, j, and k either -1 or 0. Crystal ball two consists of Cube[4], the 4x4x4 cube with i, j, and k from -2 to 1, minus the eight chords [-2 -2 1], [-2 1 -2], [-2 1 1], [1 -2 -2], [1 -2 1], [-2 -2 -2], [1 1 -2], [1 1 1]. | ||
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[49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 32/21, 14/9, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 2] | [49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 32/21, 14/9, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 2] | ||
=Scales= | =Scales= | ||
[[crystal2]] | [[crystal2]] | ||
[[crystal2breed]]</pre></div> | [[crystal2breed]]</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>Crystal balls</title></head><body> | <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>Crystal balls</title></head><body>We may define the nth q-limit <em>Hahn shell</em> as the octave classes at exactly <a class="wiki_link" href="/Hahn%20distance">Hahn distance</a> n from the unison in terms of the q-odd-limit Hahn norm. The number of notes in the 5-limit Hahn shell is (for n&gt;0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. An alternative definition, not employing Hahn distance, is that the nth 5- and 7- limit crystal balls are the nth <a class="wiki_link" href="/Scale%20products%20and%20scale%20powers">scale powers</a> of the 5- and 7-limit tonality diamonds, respectively. This easily generalizes to the scale power of the q-limit tonality diamond for any odd number q.<br /> | ||
We may define the nth q-limit <em>Hahn shell</em> as the octave classes at exactly <a class="wiki_link" href="/Hahn%20distance">Hahn distance</a> n from the unison in terms of the q-odd-limit Hahn norm. The number of notes in the 5-limit Hahn shell is (for n&gt;0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. An alternative definition, not employing Hahn distance, is that the nth 5- and 7- limit crystal balls are the nth <a class="wiki_link" href="/Scale%20products%20and%20scale%20powers">scale powers</a> of the 5- and 7-limit tonality diamonds, respectively. This easily generalizes to the scale power of the q-limit tonality diamond for any odd number q.<br /> | |||
<br /> | <br /> | ||
Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.<br /> | Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.<br /> | ||
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Here are the first few 5-limit Hahn shells:<br /> | Here are the first few 5-limit Hahn shells:<br /> | ||
<br /> | <br /> | ||
Shell 0 <br /> | Shell 0<br /> | ||
[1]<br /> | [1]<br /> | ||
<br /> | <br /> | ||
| Line 86: | Line 84: | ||
Here are the first three 7-limit Hahn shells:<br /> | Here are the first three 7-limit Hahn shells:<br /> | ||
<br /> | <br /> | ||
Shell 0 <br /> | Shell 0<br /> | ||
[1]<br /> | [1]<br /> | ||
<br /> | <br /> | ||
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<strong>Crystal ball 2 55 notes</strong><br /> | <strong>Crystal ball 2 55 notes</strong><br /> | ||
[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9,<br /> | [50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9,<br /> | ||
28/25,9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49, 21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25, 2]<br /> | 28/25, 9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49, 21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25, 2]<br /> | ||
<br /> | <br /> | ||
Crystal ball one can also be described as Cube[2], the 2x2x2 cube scale, which consists of the notes of the eight chords [i, j, k] with i, j, and k either -1 or 0. Crystal ball two consists of Cube[4], the 4x4x4 cube with i, j, and k from -2 to 1, minus the eight chords [-2 -2 1], [-2 1 -2], [-2 1 1], [1 -2 -2], [1 -2 1], [-2 -2 -2], [1 1 -2], [1 1 1].<br /> | Crystal ball one can also be described as Cube[2], the 2x2x2 cube scale, which consists of the notes of the eight chords [i, j, k] with i, j, and k either -1 or 0. Crystal ball two consists of Cube[4], the 4x4x4 cube with i, j, and k from -2 to 1, minus the eight chords [-2 -2 1], [-2 1 -2], [-2 1 1], [1 -2 -2], [1 -2 1], [-2 -2 -2], [1 1 -2], [1 1 1].<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h1> | ||
<a class="wiki_link" href="/crystal2">crystal2</a><br /> | <a class="wiki_link" href="/crystal2">crystal2</a><br /> | ||
<a class="wiki_link" href="/crystal2breed">crystal2breed</a></body></html></pre></div> | <a class="wiki_link" href="/crystal2breed">crystal2breed</a></body></html></pre></div> | ||