7-limit symmetrical lattices: Difference between revisions

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== Norms and lattices ==

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If ''T'' is the ~~[[vals and tuning space|val]]~~ {{nowrap| ~~''T'' {{=}}~~ {{val| 1 1 1 1 }} }} (~~note that this~~ is ~~''not'' the~~ [[~~just tuning map~~]]) then we may define a subspace of 7-limit interval space, '''symmetric interval class space''', as the subspace of all ~~vectors ''M'' in interval space such that~~ {{nowrap| {{~~vmprod~~|''T''|''M''}} {{=}} 0 }}, ~~which~~ has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.

If ''T'' is the equal temperament {{nowrap|{{val| 1 1 1 1 }} }} which maps 2, 3, 5, and 7 all to a single step (and is thus a val of [[1edo]], [[1edt]], etc), then we may define a subspace of 7-limit interval space, '''symmetric interval class space''', as the subspace of all intervals where {{nowrap| {{vmp|''T''|''M''}} {{=}} 0 }} - or in other words, the number of primes (including repetition) in the numerator and denominator is the same. This has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space (because only one power of two in the numerator or denominator satisfies the criterion for any odd fraction), and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.

If {{monzo| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }} is any element of symmetric interval class space, then by definition {{nowrap|‖{{monzo| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }}‖}} = {{nowrap|sqrt(2) sqrt(''x''2 + ''y''2 + ''z''2 + ''xy'' + ''yz'' + ''zx'')}} where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by {{monzo| <nowiki>*</nowiki> ''x'' ''y'' ''z'' }}.

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-If ''T'' is the [[vals and tuning space|val]] {{nowrap| ''T'' {{=}} {{val| 1 1 1 1 }} }} (note that this is ''not'' the [[just tuning map]]) then we may define a subspace of 7-limit interval space, '''symmetric interval class space''', as the subspace of all vectors ''M'' in interval space such that {{nowrap| {{vmprod|''T''|''M''}} {{=}} 0 }}, which has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.
+If ''T'' is the equal temperament {{nowrap|{{val| 1 1 1 1 }} }} which maps 2, 3, 5, and 7 all to a single step (and is thus a val of [[1edo]], [[1edt]], etc), then we may define a subspace of 7-limit interval space, '''symmetric interval class space''', as the subspace of all intervals where {{nowrap| {{vmp|''T''|''M''}} {{=}} 0 }} - or in other words, the number of primes (including repetition) in the numerator and denominator is the same. This has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space (because only one power of two in the numerator or denominator satisfies the criterion for any odd fraction), and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.
 If {{monzo| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }} is any element of symmetric interval class space, then by definition {{nowrap|‖{{monzo| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }}‖}} = {{nowrap|sqrt(2) sqrt(''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> + ''xy'' + ''yz'' + ''zx'')}} where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by {{monzo| <nowiki>*</nowiki> ''x'' ''y'' ''z'' }}.