Crystal Balls
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 abundently 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
[1, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]
Crystal ball 2 55 notes
[1, 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]
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
[1, 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]
Euclid 3 43 notes
[1, 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]