Chord cubes - Revision history

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
Line 1:		Line 1:
	~~<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>~~
	~~: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2013-10-31 02:03:05 UTC</tt>.<br>~~
	~~: The original revision id was <tt>464713570</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 ≤ 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:		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"><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 ≤ 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.<br />
	~~<br />~~
	~~Here are the smaller cube scales~~:~~<br />~~
	~~<br />~~
	~~<strong>Cube(2) -- the stellated hexany, 14 notes</strong><br />~~
	~~[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 />~~
	~~<br />~~
	~~<strong>Alt(2) -- the 7-limit tonality diamond, 13 notes</strong><br />~~
	~~[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 />~~
	~~<br />~~
	~~<strong>Cube(3) 32 notes</strong><br />~~
	[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 />
	~~<br />~~
	~~<strong>Alt(3) 32 notes</strong><br />~~
	[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 />
	~~<br />~~
	~~<strong>Cube(4) 62 notes</strong><br />~~
	[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 />~~
	~~<br />~~
	~~<strong>Alt(4) 63 notes</strong><br />~~
	[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 />
	~~48/25, 35/18, 96/49, 49/25, 2~~]~~<br />~~
	~~<br />~~
	~~<!-- 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="/cube3">cube3</a><br />~~
	~~<a class="wiki_link" href="/cube4">cube4</a><br />~~
	~~<br />~~
	~~Scales tempered in 3600et<br />~~
	~~<a class="wiki_link" href="/cube3enn">cube3enn</a><br />~~
	~~<a class="wiki_link" href="/cube4enn">cube4enn</a></body></html></pre></div>~~

Wikispaces>genewardsmith: Imported revision 464713570 - Original comment:

2013-10-31T02:03:05Z

**Imported revision 464713570 - Original comment: **

Wikispaces>genewardsmith: Imported revision 269727848 - Original comment:

2011-10-28T18:00:43Z

**Imported revision 269727848 - Original comment: **

Wikispaces>genewardsmith: Imported revision 269727664 - Original comment:

2011-10-28T17:59:36Z

**Imported revision 269727664 - Original comment: **

← Older revision		Revision as of 17:59, 28 October 2011
Line 1:		Line 1:
	<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>2011-10-28 17:52:59 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-10-28 17:59:36 UTC</tt>.<br>
	: The original revision id was <tt>~~269726358~~</tt>.<br>		: The original revision id was <tt>269727664</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>
Line 11:		Line 11:
	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]
Line 43:		Line 43:
	Here are the smaller cube scales:<br />		Here are the smaller cube scales:<br />
	<br />		<br />
	<strong>Cube[2] -- the stellated hexany, 14 notes</strong><br />		<strong>Cube(2) -- the stellated hexany, 14 notes</strong><br />
	[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 />		[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 />
	<br />		<br />
	<strong>Alt[2] -- the 7-limit tonality diamond, 13 notes</strong><br />		<strong>Alt(2) -- the 7-limit tonality diamond, 13 notes</strong><br />
	[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 />		[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 />
	<br />		<br />
	<strong>Cube[3] 32 notes</strong><br />		<strong>Cube(3) 32 notes</strong><br />
	[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 />		[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 />
	<br />		<br />
	<strong>Alt[3] 32 notes</strong><br />		<strong>Alt(3) 32 notes</strong><br />
	[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 />		[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 />
	<br />		<br />
	<strong>Cube[4] 62 notes</strong><br />		<strong>Cube(4) 62 notes</strong><br />
	[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 />		[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 />
	<br />		<br />
	<strong>Alt[4] 63 notes</strong><br />		<strong>Alt(4) 63 notes</strong><br />
	[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 />		[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 />
	48/25, 35/18, 96/49, 49/25, 2]<br />		48/25, 35/18, 96/49, 49/25, 2]<br />

Wikispaces>genewardsmith: Imported revision 269726358 - Original comment:

2011-10-28T17:52:59Z

**Imported revision 269726358 - Original comment: **

← Older revision		Revision as of 17:52, 28 October 2011
Line 1:		Line 1:
	<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>2011-10-28 14:02:18 UTC</tt>.<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>~~269661578~~</tt>.<br>		: The original revision id was <tt>269726358</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">
	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 1-n/2 <= i, j, k < n/2~~, but an alternative cube is derived from -n/2 < i, j, k < n/2-1~~. 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. ~~Note that in the case of even n, the~~ two types of cubes ~~are distinctly different~~; for ~~instance~~ the alternative ~~2x2x2~~ cube ~~is the 7~~-~~limit [[tonality diamond]]~~, ~~which has only 13 notes. For~~ odd n, ~~the inversion of the scale gives another scale~~, ~~centered around a minor rather than a major tetrad~~.		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.

	Here are the ~~first three~~ 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
			[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]

	Cube[4] 63 notes		Alt[3] 32 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]		[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=		=Scales=
Line 29:		Line 39:
	<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>Chord cubes</title></head><body><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>Chord cubes</title></head><body><br />
	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 1-n/2 &lt;= i, j, k &lt; n/2~~, but an alternative cube is derived from -n/2 &lt; i, j, k &lt; n/2-1~~. 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. ~~Note that in the case of even n, the~~ two types of cubes ~~are distinctly different~~; for ~~instance~~ the alternative ~~2x2x2~~ cube ~~is the 7~~-~~limit~~ <~~a class~~=~~"wiki_link" href="/tonality%20diamond"~~>~~tonality diamond&~~lt;/~~a>~~, ~~which has only 13 notes. For~~ odd n, ~~the inversion of the scale gives another scale~~, ~~centered around a minor rather than a major tetrad~~. <br />		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 />
	<br />		<br />
	Here are the ~~first three~~ cube scales:<br />		Here are the smaller cube scales:<br />
	<br />		<br />
	<strong>Cube[2] -- the stellated hexany, 14 notes</strong><br />		<strong>Cube[2] -- the stellated hexany, 14 notes</strong><br />
	[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 />		[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 />
			<br />
			<strong>Alt[2] -- the 7-limit tonality diamond, 13 notes</strong><br />
			[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 />
	<br />		<br />
	<strong>Cube[3] 32 notes</strong><br />		<strong>Cube[3] 32 notes</strong><br />
	[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 />		[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 />
	<br />		<br />
	<strong>Cube[4] 63 notes</strong><br />		<strong>Alt[3] 32 notes</strong><br />
	[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]<br />		[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 />
			<br />
			<strong>Cube[4] 62 notes</strong><br />
			[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 />
			<br />
			<strong>Alt[4] 63 notes</strong><br />
			[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 />
			48/25, 35/18, 96/49, 49/25, 2]<br />
	<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>

Wikispaces>genewardsmith: Imported revision 269661578 - Original comment:

2011-10-28T14:02:18Z

**Imported revision 269661578 - Original comment: **

Wikispaces>genewardsmith: Imported revision 240321807 - Original comment:

2011-07-07T03:11:24Z

**Imported revision 240321807 - Original comment: **

← Older revision		Revision as of 03:11, 7 July 2011
Line 1:		Line 1:
	<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~~-07-05 03:17:43 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-07-07 03:11:24 UTC</tt>.<br>
	: The original revision id was <tt>~~151546173~~</tt>.<br>		: The original revision id was <tt>240321807</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>
Line 17:		Line 17:
	Cube[4] 63 notes		Cube[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]		[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]
	</pre></div>
			=Scales=
			[[cube3]]
			[[cube4]]

			Scales tempered in 3600et
			[[cube3enn]]
			[[cube4enn]]</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>Chord cubes</title></head><body><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>Chord cubes</title></head><body><br />
Line 29:		Line 36:
	<br />		<br />
	<strong>Cube[4] 63 notes</strong><br />		<strong>Cube[4] 63 notes</strong><br />
	[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]</body></html></pre></div>		[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]<br />
			<br />
			<!-- 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="/cube3">cube3</a><br />
			<a class="wiki_link" href="/cube4">cube4</a><br />
			<br />
			Scales tempered in 3600et<br />
			<a class="wiki_link" href="/cube3enn">cube3enn</a><br />
			<a class="wiki_link" href="/cube4enn">cube4enn</a></body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 151546173 - Original comment:

2010-07-05T03:17:43Z

**Imported revision 151546173 - Original comment: **

New page

<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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>2010-07-05 03:17:43 UTC</tt>.<br>
: The original revision id was <tt>151546173</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 <= 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 1-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. For odd n, the inversion of the scale gives another scale, centered around a minor rather than a major tetrad. Here are the first three 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]

**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]

**Cube[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]
</pre></div>
<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>Chord cubes</title></head><body><br />
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 1-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. For odd n, the inversion of the scale gives another scale, centered around a minor rather than a major tetrad. Here are the first three cube scales:<br />
<br />
<strong>Cube[2] -- the stellated hexany, 14 notes</strong><br />
[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 />
<br />
<strong>Cube[3] 32 notes</strong><br />
[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 />
<br />
<strong>Cube[4] 63 notes</strong><br />
[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]</body></html></pre></div>

← Older revision		Revision as of 02:03, 31 October 2013
Line 1:		Line 1:
	<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>~~2011~~-10-~~28 18~~:00:43 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2013-10-31 02:03:05 UTC</tt>.<br>
	: The original revision id was <tt>~~269727848~~</tt>.<br>		: The original revision id was <tt>464713570</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">
	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.		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.

	Here are the smaller cube scales:		Here are the smaller cube scales:
Line 39:		Line 39:
	<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>Chord cubes</title></head><body><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>Chord cubes</title></head><body><br />
	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 />		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 ≤ 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.<br />
	<br />		<br />
	Here are the smaller cube scales:<br />		Here are the smaller cube scales:<br />

← Older revision		Revision as of 18:00, 28 October 2011
Line 1:		Line 1:
	<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>2011-10-28 17:59:36 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-10-28 18:00:43 UTC</tt>.<br>
	: The original revision id was <tt>~~269727664~~</tt>.<br>		: The original revision id was <tt>269727848</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">
	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.		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.

	Here are the smaller cube scales:		Here are the smaller cube scales:
Line 39:		Line 39:
	<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>Chord cubes</title></head><body><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>Chord cubes</title></head><body><br />
	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 />		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 />
	<br />		<br />
	Here are the smaller cube scales:<br />		Here are the smaller cube scales:<br />

← Older revision		Revision as of 14:02, 28 October 2011
Line 1:		Line 1:
	<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>2011-07-~~07 03~~:11:24 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-10-28 14:02:18 UTC</tt>.<br>
	: The original revision id was <tt>~~240321807~~</tt>.<br>		: The original revision id was <tt>269661578</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">
	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 1-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. For odd n, the inversion of the scale gives another scale, centered around a minor rather than a major tetrad. Here are the first three cube scales:		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 1-n/2 <= i, j, k < n/2, but an alternative cube is derived from -n/2 < i, j, k < n/2-1. 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. Note that in the case of even n, the two types of cubes are distinctly different; for instance the alternative 2x2x2 cube is the 7-limit [[tonality diamond]], which has only 13 notes. For odd n, the inversion of the scale gives another scale, centered around a minor rather than a major tetrad.

			Here are the first three cube scales:

	Cube[2] -- the stellated hexany, 14 notes		Cube[2] -- the stellated hexany, 14 notes
Line 27:		Line 29:
	<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>Chord cubes</title></head><body><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>Chord cubes</title></head><body><br />
	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 1-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. For odd n, the inversion of the scale gives another scale, centered around a minor rather than a major tetrad. Here are the first three cube scales:<br />		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 1-n/2 &lt;= i, j, k &lt; n/2, but an alternative cube is derived from -n/2 &lt; i, j, k &lt; n/2-1. 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. Note that in the case of even n, the two types of cubes are distinctly different; for instance the alternative 2x2x2 cube is the 7-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, which has only 13 notes. For odd n, the inversion of the scale gives another scale, centered around a minor rather than a major tetrad. <br />
			<br />
			Here are the first three cube scales:<br />
	<br />		<br />
	<strong>Cube[2] -- the stellated hexany, 14 notes</strong><br />		<strong>Cube[2] -- the stellated hexany, 14 notes</strong><br />

Chord cubes - Revision history

Wikispaces>FREEZE at 00:00, 17 July 2018

Wikispaces>genewardsmith: **Imported revision 464713570 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 269727848 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 269727664 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 269726358 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 269661578 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 240321807 - Original comment: **