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.

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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.		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.