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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | We may define the nth q-limit ''Hahn shell'' as the octave classes at exactly [[Hahn_distance|Hahn distance]] n from the unison in terms of the q-odd-limit Hahn norm. The number of notes in the 5-limit Hahn shell is (for n>0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. An alternative definition, not employing Hahn distance, is that the nth 5- and 7- limit crystal balls are the nth [[Scale_products_and_scale_powers|scale powers]] of the 5- and 7-limit tonality diamonds, respectively. This easily generalizes to the scale power of the q-limit tonality diamond for any odd number q. |
| 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:JosephRuhf|JosephRuhf]] and made on <tt>2016-11-06 14:35:37 UTC</tt>.<br>
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| : The original revision id was <tt>598609118</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">We may define the nth q-limit //Hahn shell// as the octave classes at exactly [[Hahn distance]] n from the unison in terms of the q-odd-limit Hahn norm. The number of notes in the 5-limit Hahn shell is (for n>0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. An alternative definition, not employing Hahn distance, is that the nth 5- and 7- limit crystal balls are the nth [[Scale products and scale powers|scale powers]] of the 5- and 7-limit tonality diamonds, respectively. This easily generalizes to the scale power of the q-limit tonality diamond for any odd number q.
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| Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords. | | Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords. |
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| Shell 0 | | Shell 0 |
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| [1] | | [1] |
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| Shell 1 -- the 5-limit consonances | | Shell 1 -- the 5-limit consonances |
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| [6/5, 5/4, 4/3, 3/2, 8/5, 5/3] | | [6/5, 5/4, 4/3, 3/2, 8/5, 5/3] |
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| Shell 2 | | Shell 2 |
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| [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] |
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| Shell 3 | | Shell 3 |
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| [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] |
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| Shell 4 | | Shell 4 |
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| [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] |
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| Shell 0 | | Shell 0 |
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| [1] | | [1] |
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| Shell 1 -- the 7-limit consonances | | Shell 1 -- the 7-limit consonances |
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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] | | [8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4] |
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| Shell 2 | | Shell 2 |
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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, 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] |
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| Here are the first two 7-limit crystal ball scales: | | Here are the first two 7-limit crystal ball scales: |
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| **Crystal ball 1 13 notes -- the 7-limit Tonality Diamond**
| | '''Crystal ball 1 13 notes -- the 7-limit Tonality Diamond''' |
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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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| **Crystal ball 2 55 notes**
| | '''Crystal ball 2 55 notes''' |
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| [50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, | | [50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, |
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| 28/25, 9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49, 21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25, 2] | | 28/25, 9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49, 21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25, 2] |
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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. |
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| **Euclid 2 19 notes**
| | '''Euclid 2 19 notes''' |
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| [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] |
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| **Euclid 3 43 notes**
| | '''Euclid 3 43 notes''' |
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| [49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 32/21, 14/9, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 2] | | [49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 32/21, 14/9, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 2] |
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| =Scales= | | =Scales= |
| [[crystal2]] | | [[crystal2|crystal2]] |
| [[crystal2breed]]</pre></div> | | |
| <h4>Original HTML content:</h4>
| | [[crystal2breed|crystal2breed]] [[Category:math]] |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Crystal balls</title></head><body>We may define the nth q-limit <em>Hahn shell</em> as the octave classes at exactly <a class="wiki_link" href="/Hahn%20distance">Hahn distance</a> n from the unison in terms of the q-odd-limit Hahn norm. The number of notes in the 5-limit Hahn shell is (for n&gt;0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. An alternative definition, not employing Hahn distance, is that the nth 5- and 7- limit crystal balls are the nth <a class="wiki_link" href="/Scale%20products%20and%20scale%20powers">scale powers</a> of the 5- and 7-limit tonality diamonds, respectively. This easily generalizes to the scale power of the q-limit tonality diamond for any odd number q.<br />
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| <br />
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| Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.<br />
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| <br />
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| Here are the first few 5-limit Hahn shells:<br />
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| <br />
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| Shell 0<br />
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| [1]<br />
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| <br />
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| Shell 1 -- the 5-limit consonances<br />
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| [6/5, 5/4, 4/3, 3/2, 8/5, 5/3]<br /> | |
| <br />
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| Shell 2<br />
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| [25/24, 16/15, 10/9, 9/8, 32/25, 25/18, 36/25, 25/16, 16/9, 9/5, 15/8, 48/25]<br />
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| <br />
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| Shell 3<br />
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| [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]<br />
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| <br />
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| Shell 4<br />
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| [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]<br />
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| <br />
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| Here are the first three 7-limit Hahn shells:<br />
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| <br />
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| Shell 0<br />
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| [1]<br />
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| <br />
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| Shell 1 -- the 7-limit consonances<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]<br />
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| <br />
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| Shell 2<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, 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]<br /> | |
| <br />
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| Here are the first two 7-limit crystal ball scales:<br />
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| <br />
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| <strong>Crystal ball 1 13 notes -- the 7-limit Tonality Diamond</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>Crystal ball 2 55 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,<br />
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| 28/25, 9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49, 21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25, 2]<br />
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| <br />
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| Crystal ball one can also be described as Cube[2], the 2x2x2 cube scale, which consists of the notes of the eight chords [i, j, k] with i, j, and k either -1 or 0. Crystal ball two consists of Cube[4], the 4x4x4 cube with i, j, and k from -2 to 1, minus the eight chords [-2 -2 1], [-2 1 -2], [-2 1 1], [1 -2 -2], [1 -2 1], [-2 -2 -2], [1 1 -2], [1 1 1].<br />
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| <br />
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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.<br />
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| <br />
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| <strong>Euclid 2 19 notes</strong><br />
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| [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]<br />
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| <br />
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| <strong>Euclid 3 43 notes</strong><br />
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| [49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 32/21, 14/9, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 2]<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="/crystal2">crystal2</a><br />
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| <a class="wiki_link" href="/crystal2breed">crystal2breed</a></body></html></pre></div>
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