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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | A cube scale is a 7-limit scale (we might also consider the 9 odd limit) whose notes are derived from a cube in the [[The_Seven_Limit_Symmetrical_Lattices|7-limit lattice of chords]]. For odd n, we will call the octave-reduced set of notes deriving from all chords of the form [i, j, k], (1-n)/2 ≤ i, j, k ≤ (n-1)/2, Cube(n). If n is even, we will use Cube(n) to refer to the notes of [i, j, k] with (2-n)/2 ≤ i, j, k < n/2. If n is odd, Cube(n) has (n+1)^3/2 notes to it; if n is even, its growth is more complicated but still approximately cubic. There are however two types of chord cubes for each n; for even n, we define the alternative cube Alt(n) via -n/2 ≤ i, j, k ≤ (n-2)/2, and for odd n, Alt(n), (1-n)/2 ≤ i+1, j, k ≤ (n-1)/2. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-28 17:52:59 UTC</tt>.<br>
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| : The original revision id was <tt>269726358</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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| <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">
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| A cube scale is a 7-limit scale (we might also consider the 9 odd limit) whose notes are derived from a cube in the [[The Seven Limit Symmetrical Lattices|7-limit lattice of chords]]. For odd n, we will call the octave-reduced set of notes deriving from all chords of the form [i, j, k], (1-n)/2 <= i, j, k <= (n-1)/2, Cube(n). If n is even, we will use Cube(n) to refer to the notes of [i, j, k] with (2-n)/2 <= i, j, k < n/2. If n is odd, Cube(n) has (n+1)^3/2 notes to it; if n is even, its growth is more complicated but still approximately cubic. There are however two types of chord cubes for each n; for even n, we define the alternative cube Alt(n) via -n/2 <= i, j, k <= (n-2)/2, and for odd n, Alt(n) (1-n)/2 <= i+1, j, k <= (n-1)/2. | |
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| Here are the smaller cube scales: | | Here are the smaller cube scales: |
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| **Cube[2] -- the stellated hexany, 14 notes**
| | '''Cube(2) -- the stellated hexany, 14 notes''' |
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| [21/20, 15/14, 35/32, 9/8, 5/4, 21/16, 35/24, 3/2, 49/32, 25/16, 105/64, 7/4, 15/8, 2] | | [21/20, 15/14, 35/32, 9/8, 5/4, 21/16, 35/24, 3/2, 49/32, 25/16, 105/64, 7/4, 15/8, 2] |
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| **Alt[2] -- the 7-limit tonality diamond, 13 notes**
| | '''Alt(2) -- the 7-limit tonality diamond, 13 notes''' |
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| [8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 2] | | [8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 2] |
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| **Cube[3] 32 notes**
| | '''Cube(3) 32 notes''' |
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| [49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5, 105/64, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 15/8, 35/18, 2] | | [49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5, 105/64, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 15/8, 35/18, 2] |
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| **Alt[3] 32 notes**
| | '''Alt(3) 32 notes''' |
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| [36/35, 21/20, 15/14, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 36/25, 72/49, 3/2, 54/35, 8/5, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35, 15/8, 48/25, 96/49, 2] | | [36/35, 21/20, 15/14, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 36/25, 72/49, 3/2, 54/35, 8/5, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35, 15/8, 48/25, 96/49, 2] |
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| **Cube[4] 62 notes**
| | '''Cube(4) 62 notes''' |
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| [49/48, 525/512, 25/24, 21/20, 15/14, 343/320, 35/32, 125/112, 9/8, 8/7, 147/128, 7/6, 75/64, 6/5, 175/144, 60/49, 49/40, 315/256, 5/4, 63/50, 245/192, 9/7, 125/96, 21/16, 4/3, 75/56, 343/256, 27/20, 175/128, 7/5, 45/32, 10/7, 735/512, 35/24, 147/100, 3/2, 75/49, 49/32, 25/16, 63/40, 8/5, 45/28, 105/64, 5/3, 42/25, 27/16, 245/144, 12/7, 7/4, 25/14, 343/192, 9/5, 175/96, 90/49, 147/80, 15/8, 245/128, 27/14, 35/18, 125/64, 63/32, 2] | | [49/48, 525/512, 25/24, 21/20, 15/14, 343/320, 35/32, 125/112, 9/8, 8/7, 147/128, 7/6, 75/64, 6/5, 175/144, 60/49, 49/40, 315/256, 5/4, 63/50, 245/192, 9/7, 125/96, 21/16, 4/3, 75/56, 343/256, 27/20, 175/128, 7/5, 45/32, 10/7, 735/512, 35/24, 147/100, 3/2, 75/49, 49/32, 25/16, 63/40, 8/5, 45/28, 105/64, 5/3, 42/25, 27/16, 245/144, 12/7, 7/4, 25/14, 343/192, 9/5, 175/96, 90/49, 147/80, 15/8, 245/128, 27/14, 35/18, 125/64, 63/32, 2] |
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| **Alt[4] 63 notes**
| | '''Alt(4) 63 notes''' |
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| [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, 128/105, 60/49, 49/40, 5/4, 32/25, 9/7, 35/27, 64/49, 21/16, 4/3, 168/125, 49/36, 48/35, 25/18, 480/343, 7/5, 10/7, 343/240, 36/25, 35/24, 72/49, 125/84, 3/2, 32/21, 49/32, 54/35, 14/9, 25/16, 8/5, 80/49, 49/30, 105/64, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, | | [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, 128/105, 60/49, 49/40, 5/4, 32/25, 9/7, 35/27, 64/49, 21/16, 4/3, 168/125, 49/36, 48/35, 25/18, 480/343, 7/5, 10/7, 343/240, 36/25, 35/24, 72/49, 125/84, 3/2, 32/21, 49/32, 54/35, 14/9, 25/16, 8/5, 80/49, 49/30, 105/64, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, |
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| 48/25, 35/18, 96/49, 49/25, 2] | | 48/25, 35/18, 96/49, 49/25, 2] |
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| =Scales= | | =Scales= |
| [[cube3]] | | [[cube3|cube3]] |
| [[cube4]] | | |
| | [[cube4|cube4]] |
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| Scales tempered in 3600et | | Scales tempered in 3600et |
| [[cube3enn]] | | |
| [[cube4enn]]</pre></div> | | [[cube3enn|cube3enn]] |
| <h4>Original HTML content:</h4>
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| <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>Chord cubes</title></head><body><br />
| | [[cube4enn|cube4enn]] [[Category:math]] |
| A cube scale is a 7-limit scale (we might also consider the 9 odd limit) whose notes are derived from a cube in the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">7-limit lattice of chords</a>. For odd n, we will call the octave-reduced set of notes deriving from all chords of the form [i, j, k], (1-n)/2 &lt;= i, j, k &lt;= (n-1)/2, Cube(n). If n is even, we will use Cube(n) to refer to the notes of [i, j, k] with (2-n)/2 &lt;= i, j, k &lt; n/2. If n is odd, Cube(n) has (n+1)^3/2 notes to it; if n is even, its growth is more complicated but still approximately cubic. There are however two types of chord cubes for each n; for even n, we define the alternative cube Alt(n) via -n/2 &lt;= i, j, k &lt;= (n-2)/2, and for odd n, Alt(n) (1-n)/2 &lt;= i+1, j, k &lt;= (n-1)/2.<br />
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| <br />
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| Here are the smaller cube scales:<br />
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| <br />
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| <strong>Cube[2] -- the stellated hexany, 14 notes</strong><br />
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| [21/20, 15/14, 35/32, 9/8, 5/4, 21/16, 35/24, 3/2, 49/32, 25/16, 105/64, 7/4, 15/8, 2]<br />
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| <br />
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| <strong>Alt[2] -- the 7-limit tonality diamond, 13 notes</strong><br />
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| [8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 2]<br />
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| <br />
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| <strong>Cube[3] 32 notes</strong><br />
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| [49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5, 105/64, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 15/8, 35/18, 2]<br />
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| <br />
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| <strong>Alt[3] 32 notes</strong><br />
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| [36/35, 21/20, 15/14, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 36/25, 72/49, 3/2, 54/35, 8/5, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35, 15/8, 48/25, 96/49, 2]<br />
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| <br />
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| <strong>Cube[4] 62 notes</strong><br />
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| [49/48, 525/512, 25/24, 21/20, 15/14, 343/320, 35/32, 125/112, 9/8, 8/7, 147/128, 7/6, 75/64, 6/5, 175/144, 60/49, 49/40, 315/256, 5/4, 63/50, 245/192, 9/7, 125/96, 21/16, 4/3, 75/56, 343/256, 27/20, 175/128, 7/5, 45/32, 10/7, 735/512, 35/24, 147/100, 3/2, 75/49, 49/32, 25/16, 63/40, 8/5, 45/28, 105/64, 5/3, 42/25, 27/16, 245/144, 12/7, 7/4, 25/14, 343/192, 9/5, 175/96, 90/49, 147/80, 15/8, 245/128, 27/14, 35/18, 125/64, 63/32, 2]<br />
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| <br />
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| <strong>Alt[4] 63 notes</strong><br />
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| [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, 128/105, 60/49, 49/40, 5/4, 32/25, 9/7, 35/27, 64/49, 21/16, 4/3, 168/125, 49/36, 48/35, 25/18, 480/343, 7/5, 10/7, 343/240, 36/25, 35/24, 72/49, 125/84, 3/2, 32/21, 49/32, 54/35, 14/9, 25/16, 8/5, 80/49, 49/30, 105/64, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21,<br />
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| 48/25, 35/18, 96/49, 49/25, 2]<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h1>
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| <a class="wiki_link" href="/cube3">cube3</a><br />
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| <a class="wiki_link" href="/cube4">cube4</a><br />
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| <br />
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| Scales tempered in 3600et<br />
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| <a class="wiki_link" href="/cube3enn">cube3enn</a><br />
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| <a class="wiki_link" href="/cube4enn">cube4enn</a></body></html></pre></div>
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