Crystal ball - Revision history

Xenllium at 22:15, 27 January 2026

2026-01-27T22:15:09Z

← Older revision		Revision as of 22:15, 27 January 2026
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	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.		A '''crystal ball''' is an interval set generated from a [[tonality diamond]] and defined by [[Hahn distance]].

			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'' > 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''.

	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 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:
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	'''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,		[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]

	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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	[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\|Crystal2]]
			* [[crystal2breed\|Crystal2breed]]

	~~[[crystal2breed\|crystal2breed]]~~ [[Category:math]]		[[Category:math]]

Lériendil: Lériendil moved page Crystal balls to Crystal ball: plurals

2025-04-24T22:00:12Z

Lériendil moved page Crystal balls to Crystal ball: plurals

← Older revision	Revision as of 22:00, 24 April 2025
(No difference)

Sintel: Inaccessible

2025-03-19T20:08:07Z

Inaccessible

← Older revision		Revision as of 20:08, 19 March 2025
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			{{inacc}}

	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.		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.

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>JosephRuhf: Imported revision 598609118 - Original comment:

2016-11-06T14:35:37Z

**Imported revision 598609118 - Original comment: **

← Older revision		Revision as of 14:35, 6 November 2016
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	<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>~~2012~~-01-~~31 23~~:39:40 UTC</tt>.<br>		: This revision was by author [[User:JosephRuhf\|JosephRuhf]] and made on <tt>2016-11-06 14:35:37 UTC</tt>.<br>
	: The original revision id was <tt>~~297268894~~</tt>.<br>		: The original revision id was <tt>598609118</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">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.
	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.

	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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	Here are the first few 5-limit Hahn shells:		Here are the first few 5-limit Hahn shells:

	Shell 0		Shell 0
	[1]		[1]

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

	Shell 0		Shell 0
	[1]		[1]

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	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,		[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]		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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	[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]]
	[[crystal2breed]]</pre></div>		[[crystal2breed]]</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>Crystal balls</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>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 />
	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 />
	<br />		<br />
	Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.<br />		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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	Here are the first few 5-limit Hahn shells:<br />		Here are the first few 5-limit Hahn shells:<br />
	<br />		<br />
	Shell 0 <br />		Shell 0<br />
	[1]<br />		[1]<br />
	<br />		<br />
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	Here are the first three 7-limit Hahn shells:<br />		Here are the first three 7-limit Hahn shells:<br />
	<br />		<br />
	Shell 0 <br />		Shell 0<br />
	[1]<br />		[1]<br />
	<br />		<br />
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	<strong>Crystal ball 2 55 notes</strong><br />		<strong>Crystal ball 2 55 notes</strong><br />
	[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9,<br />		[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9,<br />
	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 />		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 />
	<br />		<br />
	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 />		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 />		<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>
	<a class="wiki_link" href="/crystal2">crystal2</a><br />		<a class="wiki_link" href="/crystal2">crystal2</a><br />
	<a class="wiki_link" href="/crystal2breed">crystal2breed</a></body></html></pre></div>		<a class="wiki_link" href="/crystal2breed">crystal2breed</a></body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 297268894 - Original comment:

2012-01-31T23:39:40Z

**Imported revision 297268894 - Original comment: **

← Older revision		Revision as of 23:39, 31 January 2012
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	<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~~-11-~~22 21~~:10:00 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2012-01-31 23:39:40 UTC</tt>.<br>
	: The original revision id was <tt>~~278346330~~</tt>.<br>		: The original revision id was <tt>297268894</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">
	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. Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.		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.

			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:

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	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Crystal balls</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>Crystal balls</title></head><body><br />
	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. Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.<br />		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 />
			<br />
			Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.<br />
			<br />
	Here are the first few 5-limit Hahn shells:<br />		Here are the first few 5-limit Hahn shells:<br />
	<br />		<br />

Wikispaces>genewardsmith: Imported revision 278346330 - Original comment:

2011-11-22T21:10:00Z

**Imported revision 278346330 - Original comment: **

← Older revision		Revision as of 21:10, 22 November 2011
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	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-~~06 20~~:20:10 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-11-22 21:10:00 UTC</tt>.<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].		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].

	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 ~~sqrt(2)~~ and ~~sqrt(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
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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 />		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 />
	<br />		<br />
	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 ~~sqrt(2)~~ and ~~sqrt(3)~~.<br />		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 />
	<br />		<br />
	<strong>Euclid 2 19 notes</strong><br />		<strong>Euclid 2 19 notes</strong><br />

Wikispaces>genewardsmith: Imported revision 240282537 - Original comment:

2011-07-06T20:20:10Z

**Imported revision 240282537 - Original comment: **

← Older revision		Revision as of 20:20, 6 July 2011
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	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 02~~:50:48 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-07-06 20:20:10 UTC</tt>.<br>
	: The original revision id was <tt>~~151545195~~</tt>.<br>		: The original revision id was <tt>240282537</tt>.<br>
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	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]
	</pre></div>
			=Scales=
			[[crystal2]]
			[[crystal2breed]]</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>Crystal balls</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>Crystal balls</title></head><body><br />
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	<br />		<br />
	<strong>Euclid 3 43 notes</strong><br />		<strong>Euclid 3 43 notes</strong><br />
	[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]</body></html></pre></div>		[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 />
			<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="/crystal2">crystal2</a><br />
			<a class="wiki_link" href="/crystal2breed">crystal2breed</a></body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 151545195 - Original comment:

2010-07-05T02:50:48Z

**Imported revision 151545195 - Original comment: **

New page

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This is an imported revision from Wikispaces. The revision metadata is included below for reference: 
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<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">
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. 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:

Shell 0
[1]

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

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]

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]

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]

Here are the first three 7-limit Hahn shells:

Shell 0
[1]

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]

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]

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

**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]

**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]

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].

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 sqrt(2) and sqrt(3).

**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]

**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]
</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>Crystal balls</title></head><body> 
We may define the nth q-limit Hahn shell 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. 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: 
 
Shell 0 
[1] 
 
Shell 1 -- the 5-limit consonances 
[6/5, 5/4, 4/3, 3/2, 8/5, 5/3] 
 
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] 
 
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] 
 
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] 
 
Here are the first three 7-limit Hahn shells: 
 
Shell 0 
[1] 
 
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] 
 
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] 
 
Here are the first two 7-limit crystal ball scales: 
 
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] 
 
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] 
 
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]. 
 
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 sqrt(2) and sqrt(3). 
 
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] 
 
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]</body></html></pre></div>

Crystal ball - Revision history

Xenllium at 22:15, 27 January 2026

Lériendil: Lériendil moved page Crystal balls to Crystal ball: plurals

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Wikispaces>FREEZE at 00:00, 17 July 2018

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