Lattice: Difference between revisions

Line 1:

A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones. ~~It's a projection~~ of ~~a multi-dimensional structure onto a 2D screen~~.

A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[subgroups]] is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.

The '''Tonnetz''' is the lattice that represents ~~[[12edo]]~~'s mapping of [[5-limit]] intervals. '''Vogel's Tonnetz''' is a 7-limit ~~version~~ of Euler's 5-limit Tonnetz.

The '''Tonnetz''' is the lattice that represents a tuning system's mapping of [[5-limit]] intervals, conventionally arranged in a hexagonal fashion such that [[5/4]] and [[3/2]] are 60 degrees apart, with [[5/3]] being 60 degrees from 5/4 and 120 degrees from 3/2, so that the 5-limit [[4:5:6|major (4:5:6)]] and [[10:12:15|minor (10:12:15)]] chords form triangles. Such lattices can also be extended to other subgroups with 3 primes in them, where a respective fundamental chord such as [[4:5:7]] or [[8:11:14]] can be assigned to a triangle. '''Vogel's Tonnetz''' is a 7-limit extension of Euler's 5-limit Tonnetz.

It is also possible to create lattices for [[regular temperament|tempered]] systems, in particular [[rank-3 temperament|rank-3]] systems generated by two distinct intervals (aside from the [[equave]] or fraction thereof) and which thus function equivalently to three-prime JI subgroups.

== Examples ==

@@ Line 1: / Line 1: @@
 {{Wikipedia|Lattice (music)}}
-A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones. It's a projection of a multi-dimensional structure onto a 2D screen.
+A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[subgroups]] is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.
 {{Wikipedia|Tonnetz}}
 {{Wikipedia|Vogel's Tonnetz}}
-The '''Tonnetz''' is the lattice that represents [[12edo]]'s mapping of [[5-limit]] intervals. '''Vogel's Tonnetz''' is a 7-limit version of Euler's 5-limit Tonnetz.
+The '''Tonnetz''' is the lattice that represents a tuning system's mapping of [[5-limit]] intervals, conventionally arranged in a hexagonal fashion such that [[5/4]] and [[3/2]] are 60 degrees apart, with [[5/3]] being 60 degrees from 5/4 and 120 degrees from 3/2, so that the 5-limit [[4:5:6|major (4:5:6)]] and [[10:12:15|minor (10:12:15)]] chords form triangles. Such lattices can also be extended to other subgroups with 3 primes in them, where a respective fundamental chord such as [[4:5:7]] or [[8:11:14]] can be assigned to a triangle. '''Vogel's Tonnetz''' is a 7-limit extension of Euler's 5-limit Tonnetz.
+It is also possible to create lattices for [[regular temperament|tempered]] systems, in particular [[rank-3 temperament|rank-3]] systems generated by two distinct intervals (aside from the [[equave]] or fraction thereof) and which thus function equivalently to three-prime JI subgroups.
 == Examples ==