Crystal ball: Difference between revisions

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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.

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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:

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'''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].

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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]

=Scales=

== Scales==

[[crystal2|~~crystal2~~]]

* [[crystal2|Crystal2]]

* [[crystal2breed|Crystal2breed]]

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

[[Category:math]]

@@ Line 1: / Line 1: @@
 {{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&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 [[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'' &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''.
 Because of the way they are formed crystal balls are not especially regular as scales, but they are abundantly supplied with chords.
@@ Line 39: / Line 41: @@
 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:
@@ Line 49: / Line 51: @@
 '''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]
-/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].
@@ Line 65: / Line 65: @@
 [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]]