Chord cubes: Difference between revisions

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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 &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 ≤ 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>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-28 17:52:59 UTC</tt>.<br>
: The original revision id was <tt>269726358</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>
<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">
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 &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.


Here are the smaller cube scales:
Here are the smaller cube scales:


**Cube[2] -- the stellated hexany, 14 notes**
'''Cube(2) -- the stellated hexany, 14 notes'''
 
[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]


**Alt[2] -- the 7-limit tonality diamond, 13 notes**
'''Alt(2) -- the 7-limit tonality diamond, 13 notes'''
 
[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]


**Cube[3] 32 notes**
'''Cube(3) 32 notes'''
 
[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]


**Alt[3] 32 notes**
'''Alt(3) 32 notes'''
 
[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]


**Cube[4] 62 notes**
'''Cube(4) 62 notes'''
 
[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]


**Alt[4] 63 notes**
'''Alt(4) 63 notes'''
 
[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,
48/25, 35/18, 96/49, 49/25, 2]
48/25, 35/18, 96/49, 49/25, 2]


=Scales=
=Scales=
[[cube3]]
[[cube3|cube3]]
[[cube4]]
 
[[cube4|cube4]]


Scales tempered in 3600et
Scales tempered in 3600et
[[cube3enn]]
 
[[cube4enn]]</pre></div>
[[cube3enn|cube3enn]]
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Chord cubes&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;br /&gt;
[[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 &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;7-limit lattice of chords&lt;/a&gt;. 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 &amp;lt;= i, j, k &amp;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 &amp;lt;= i, j, k &amp;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 &amp;lt;= i, j, k &amp;lt;= (n-2)/2, and for odd n, Alt(n) (1-n)/2 &amp;lt;= i+1, j, k &amp;lt;= (n-1)/2.&lt;br /&gt;
&lt;br /&gt;
Here are the smaller cube scales:&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Cube[2] -- the stellated hexany, 14 notes&lt;/strong&gt;&lt;br /&gt;
[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]&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Alt[2] -- the 7-limit tonality diamond, 13 notes&lt;/strong&gt;&lt;br /&gt;
[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]&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Cube[3] 32 notes&lt;/strong&gt;&lt;br /&gt;
[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]&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Alt[3] 32 notes&lt;/strong&gt;&lt;br /&gt;
[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]&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Cube[4] 62 notes&lt;/strong&gt;&lt;br /&gt;
[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]&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Alt[4] 63 notes&lt;/strong&gt;&lt;br /&gt;
[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,&lt;br /&gt;
48/25, 35/18, 96/49, 49/25, 2]&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Scales&lt;/h1&gt;
&lt;a class="wiki_link" href="/cube3"&gt;cube3&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/cube4"&gt;cube4&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
Scales tempered in 3600et&lt;br /&gt;
&lt;a class="wiki_link" href="/cube3enn"&gt;cube3enn&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/cube4enn"&gt;cube4enn&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 00:00, 17 July 2018

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 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.

Here are the smaller cube scales:

Cube(2) -- the stellated hexany, 14 notes

[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]

Alt(2) -- the 7-limit tonality diamond, 13 notes

[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]

Cube(3) 32 notes

[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]

Alt(3) 32 notes

[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]

Cube(4) 62 notes

[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]

Alt(4) 63 notes

[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,

48/25, 35/18, 96/49, 49/25, 2]

Scales

cube3

cube4

Scales tempered in 3600et

cube3enn

cube4enn

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