7-limit symmetrical lattices: Difference between revisions

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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 [[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| {{vmp|''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 {{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 [[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| {{vmp|''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 {{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'' }}.