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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{inacc}}
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>2011-11-22 21:10:00 UTC</tt>.<br>
A '''crystal ball''' is an interval set generated from a [[tonality diamond]] and defined by [[Hahn distance]].  
: The original revision id was <tt>278346330</tt>.<br>
 
: The revision comment was: <tt></tt><br>
We may define the ''n''th ''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'' &gt; 0) 6''n'', and in the 7-limit Hahn shell ''n'' has 10''n''^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 ''n''th 5-limit crystal ball is 3''n''^2 + 3''n'' + 1 and in the ''n''th 7-limit crystal ball is (2''n'' + 1)(5''n''^2 + 5''n'' + 3)/3. An alternative definition, not employing Hahn distance, is that the ''n''th 5- and 7- limit crystal balls are the ''n''th [[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''.
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>
Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.
<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&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. Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.
Here are the first few 5-limit Hahn shells:
Here are the first few 5-limit Hahn shells:


Shell 0  
Shell 0
 
[1]
[1]


Shell 1 -- the 5-limit consonances
Shell 1 -- the 5-limit consonances
[6/5, 5/4, 4/3, 3/2, 8/5, 5/3]
[6/5, 5/4, 4/3, 3/2, 8/5, 5/3]


Shell 2
Shell 2
[25/24, 16/15, 10/9, 9/8, 32/25, 25/18, 36/25, 25/16, 16/9, 9/5, 15/8, 48/25]
[25/24, 16/15, 10/9, 9/8, 32/25, 25/18, 36/25, 25/16, 16/9, 9/5, 15/8, 48/25]


Shell 3
Shell 3
[128/125, 27/25, 144/125, 125/108, 75/64, 32/27, 125/96, 27/20, 45/32, 64/45, 40/27, 192/125, 27/16, 128/75, 216/125, 125/72, 50/27, 125/64]
[128/125, 27/25, 144/125, 125/108, 75/64, 32/27, 125/96, 27/20, 45/32, 64/45, 40/27, 192/125, 27/16, 128/75, 216/125, 125/72, 50/27, 125/64]


Shell 4
Shell 4
[81/80, 648/625, 135/128, 625/576, 256/225, 625/512, 768/625, 100/81, 81/64, 162/125, 512/375, 864/625, 625/432, 375/256, 125/81, 128/81, 81/50, 625/384, 1024/625, 225/128, 1152/625, 256/135, 625/324, 160/81]
[81/80, 648/625, 135/128, 625/576, 256/225, 625/512, 768/625, 100/81, 81/64, 162/125, 512/375, 864/625, 625/432, 375/256, 125/81, 128/81, 81/50, 625/384, 1024/625, 225/128, 1152/625, 256/135, 625/324, 160/81]


Here are the first three 7-limit Hahn shells:
Here are the first three 7-limit Hahn shells:


Shell 0  
Shell 0
 
[1]
[1]


Shell 1 -- the 7-limit consonances
Shell 1 -- the 7-limit consonances
[8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]
[8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]


Shell 2
Shell 2
[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 9/8, 25/21, 60/49, 49/40, 32/25, 9/7, 64/49, 21/16, 49/36, 48/35, 25/18, 36/25,35/24, 72/49, 32/21, 49/32, 14/9, 25/16, 80/49, 49/30, 42/25, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25]
 
[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 9/8, 25/21, 60/49, 49/40, 32/25, 9/7, 64/49, 21/16, 49/36, 48/35, 25/18, 36/25, 35/24, 72/49, 32/21, 49/32, 14/9, 25/16, 80/49, 49/30, 42/25, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25]


Here are the first two 7-limit crystal ball scales:
Here are the first two 7-limit crystal ball scales:


**Crystal ball 1 13 notes -- the 7-limit Tonality Diamond**
'''Crystal ball 1 13 notes -- the 7-limit Tonality Diamond'''
 
[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]


**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,
 
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]
[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]


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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The first two crystal balls can also equally well be described as Euclidean ball scales; they began to diverge with the third crystal ball. If we take everything within a radius of one of the unison, we get crystal ball one; if we take everything within a radius of two, we get crystal ball two. This means we also have two intermediate scales, Euclidean balls of radius √2 and √3.
The first two crystal balls can also equally well be described as Euclidean ball scales; they began to diverge with the third crystal ball. If we take everything within a radius of one of the unison, we get crystal ball one; if we take everything within a radius of two, we get crystal ball two. This means we also have two intermediate scales, Euclidean balls of radius √2 and √3.


**Euclid 2 19 notes**
'''Euclid 2 19 notes'''
 
[21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21, 2]
[21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21, 2]


**Euclid 3 43 notes**
'''Euclid 3 43 notes'''
 
[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|Crystal2]]
[[crystal2breed]]</pre></div>
* [[crystal2breed|Crystal2breed]]
<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;Crystal balls&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;br /&gt;
[[Category:math]]
We may define the nth q-limit &lt;em&gt;Hahn shell&lt;/em&gt; as the octave classes at exactly &lt;a class="wiki_link" href="/Hahn%20distance"&gt;Hahn distance&lt;/a&gt; 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&amp;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. Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.&lt;br /&gt;
Here are the first few 5-limit Hahn shells:&lt;br /&gt;
&lt;br /&gt;
Shell 0 &lt;br /&gt;
[1]&lt;br /&gt;
&lt;br /&gt;
Shell 1 -- the 5-limit consonances&lt;br /&gt;
[6/5, 5/4, 4/3, 3/2, 8/5, 5/3]&lt;br /&gt;
&lt;br /&gt;
Shell 2&lt;br /&gt;
[25/24, 16/15, 10/9, 9/8, 32/25, 25/18, 36/25, 25/16, 16/9, 9/5, 15/8, 48/25]&lt;br /&gt;
&lt;br /&gt;
Shell 3&lt;br /&gt;
[128/125, 27/25, 144/125, 125/108, 75/64, 32/27, 125/96, 27/20, 45/32, 64/45, 40/27, 192/125, 27/16, 128/75, 216/125, 125/72, 50/27, 125/64]&lt;br /&gt;
&lt;br /&gt;
Shell 4&lt;br /&gt;
[81/80, 648/625, 135/128, 625/576, 256/225, 625/512, 768/625, 100/81, 81/64, 162/125, 512/375, 864/625, 625/432, 375/256, 125/81, 128/81, 81/50, 625/384, 1024/625, 225/128, 1152/625, 256/135, 625/324, 160/81]&lt;br /&gt;
&lt;br /&gt;
Here are the first three 7-limit Hahn shells:&lt;br /&gt;
&lt;br /&gt;
Shell 0 &lt;br /&gt;
[1]&lt;br /&gt;
&lt;br /&gt;
Shell 1 -- the 7-limit consonances&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]&lt;br /&gt;
&lt;br /&gt;
Shell 2&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, 25/21, 60/49, 49/40, 32/25, 9/7, 64/49, 21/16, 49/36, 48/35, 25/18, 36/25,35/24, 72/49, 32/21, 49/32, 14/9, 25/16, 80/49, 49/30, 42/25, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25]&lt;br /&gt;
&lt;br /&gt;
Here are the first two 7-limit crystal ball scales:&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Crystal ball 1 13 notes -- the 7-limit Tonality Diamond&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;Crystal ball 2 55 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,&lt;br /&gt;
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]&lt;br /&gt;
&lt;br /&gt;
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].&lt;br /&gt;
&lt;br /&gt;
The first two crystal balls can also equally well be described as Euclidean ball scales; they began to diverge with the third crystal ball. If we take everything within a radius of one of the unison, we get crystal ball one; if we take everything within a radius of two, we get crystal ball two. This means we also have two intermediate scales, Euclidean balls of radius √2 and √3.&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Euclid 2 19 notes&lt;/strong&gt;&lt;br /&gt;
[21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21, 2]&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Euclid 3 43 notes&lt;/strong&gt;&lt;br /&gt;
[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]&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="/crystal2"&gt;crystal2&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/crystal2breed"&gt;crystal2breed&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/Crystal_ball"