Mapping to lattice: Difference between revisions

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Many temperaments documented on the wiki are accompanied with a '''mapping to lattice'''. These use the same [[extended bra-ket notation]] as temperament [[~~mappings~~]] but provide quite different information: they give the coordinates of the primes in a tempered lattice. The lattice for which they give coordinates does not necessarily correspond with the mapping otherwise provided.

Many temperaments documented on the wiki are accompanied with a '''mapping to lattice'''. These use the same [[extended bra-ket notation]] as temperament [[mapping]]s but provide quite different information: they give the coordinates of the primes in a tempered lattice. The lattice for which they give coordinates does not necessarily correspond with the mapping otherwise provided.

== Example ==

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Consider [[keemic]]. Its mapping to lattice is given as {{rket| {{map| 0 0 -1 3 }} {{map| 0 1 1 0 }} }}. We can read this in four vertical slices; the coordinate (0, 0) is for prime 2, (0, 1) is for prime 3, (-1, 1) is for prime 5, and (3, 0) is for prime 7.

Note that ~~supermagic~~ is a rank-3 temperament, meaning it has three generators. Or said another way, it has one period and two (other) generators. This corresponds with the fact that it has three generator maps – or rows – in its mapping matrix, which is given as {{rket| {{map| 1 0 0 5 }} {{map| 0 1 0 3 }} {{map| 0 0 1 -3 }} }}. The reason the mapping to lattice has only two rows while the mapping has three rows is because [[octave equivalence]] has been assumed in the lattice. So the first generator, the special one, the period, which in this case is equal to the octave, is not visualized on this lattice; we use one axis for the first generator, and the other axis of the second generator.

Note that keemic is a rank-3 temperament, meaning it has three generators. Or said another way, it has one period and two (other) generators. This corresponds with the fact that it has three generator maps – or rows – in its mapping matrix, which is given as {{rket| {{map| 1 0 0 5 }} {{map| 0 1 0 3 }} {{map| 0 0 1 -3 }} }}. The reason the mapping to lattice has only two rows while the mapping has three rows is because [[octave equivalence]] has been assumed in the lattice. So the first generator, the special one, the period, which in this case is equal to the octave, is not visualized on this lattice; we use one axis for the first generator, and the other axis of the second generator.

The octave equivalence also explains why prime 2 is found at (0, 0).

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[[Category:Regular temperament theory]]

[[Category:Mapping]]

[[Category:Lattice]]

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-Many temperaments documented on the wiki are accompanied with a '''mapping to lattice'''. These use the same [[extended bra-ket notation]] as temperament [[mappings]] but provide quite different information: they give the coordinates of the primes in a tempered lattice. The lattice for which they give coordinates does not necessarily correspond with the mapping otherwise provided.
+Many temperaments documented on the wiki are accompanied with a '''mapping to lattice'''. These use the same [[extended bra-ket notation]] as temperament [[mapping]]s but provide quite different information: they give the coordinates of the primes in a tempered lattice. The lattice for which they give coordinates does not necessarily correspond with the mapping otherwise provided.
 == Example ==
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 Consider [[keemic]]. Its mapping to lattice is given as {{rket| {{map| 0 0 -1 3 }} {{map| 0 1 1 0 }} }}. We can read this in four vertical slices; the coordinate (0, 0) is for prime 2, (0, 1) is for prime 3, (-1, 1) is for prime 5, and (3, 0) is for prime 7.
-Note that supermagic is a rank-3 temperament, meaning it has three generators. Or said another way, it has one period and two (other) generators. This corresponds with the fact that it has three generator maps – or rows – in its mapping matrix, which is given as {{rket| {{map| 1 0 0 5 }} {{map| 0 1 0 3 }} {{map| 0 0 1 -3 }} }}. The reason the mapping to lattice has only two rows while the mapping has three rows is because [[octave equivalence]] has been assumed in the lattice. So the first generator, the special one, the period, which in this case is equal to the octave, is not visualized on this lattice; we use one axis for the first generator, and the other axis of the second generator.
+Note that keemic is a rank-3 temperament, meaning it has three generators. Or said another way, it has one period and two (other) generators. This corresponds with the fact that it has three generator maps – or rows – in its mapping matrix, which is given as {{rket| {{map| 1 0 0 5 }} {{map| 0 1 0 3 }} {{map| 0 0 1 -3 }} }}. The reason the mapping to lattice has only two rows while the mapping has three rows is because [[octave equivalence]] has been assumed in the lattice. So the first generator, the special one, the period, which in this case is equal to the octave, is not visualized on this lattice; we use one axis for the first generator, and the other axis of the second generator.
 The octave equivalence also explains why prime 2 is found at (0, 0).
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 [[Category:Regular temperament theory]]
+[[Category:Mapping]]
+[[Category:Lattice]]