2-limit: Difference between revisions

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Since humans tend to perceive notes an octave apart as having the same pitch class, the 2-limit is said to be "easy to collapse", with this collapse being generally implemented in lattices. This will reduce the dimensionality of the lattice by one, allowing the [[5-limit]] (whose intervals are represented by 3 coordinates corresponding to each prime) to be drawn in 2 dimensions, forming the familiar classical [[Tonnetz]].

Counterintuitively, applying octave-equivalence to a lattice is mathematically equivalent to tempering out 2/1 - a single interval is used to stand for all intervals an octave apart, just as in meantone, a single interval stands in for all intervals an 81/80 apart.

Counterintuitively, applying octave-equivalence to a lattice is mathematically equivalent to [[tempering out]] 2/1 – a single interval is used to stand for all intervals an octave apart, just as in meantone, a single interval stands in for all intervals an 81/80 apart.

The 2-limit is equivalent to the [[1-odd-limit]], [[1edo]], and 1-''p''-fdo with arbitrary value of ''p'' (including [[AFDO|1afdo]] and [[IFDO|1ifdo]]).

~~The 2-limit~~ can be considered as ~~the~~ trivial rank-1 temperament ~~'''binary'''~~, equivalent to ~~1-ET~~ in the 2-limit, which is generated by only 2/1 and tempers out no commas. All equal temperaments with a mapping of 2/1 are weak extensions of ~~binary~~.

It can be considered as a trivial rank-1 temperament, equivalent to 1et in the 2-limit, which is generated by only 2/1 and tempers out no commas. All equal temperaments with a mapping of 2/1 are weak extensions of it.

== See also ==

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 Since humans tend to perceive notes an octave apart as having the same pitch class, the 2-limit is said to be "easy to collapse", with this collapse being generally implemented in lattices. This will reduce the dimensionality of the lattice by one, allowing the [[5-limit]] (whose intervals are represented by 3 coordinates corresponding to each prime) to be drawn in 2 dimensions, forming the familiar classical [[Tonnetz]].
-Counterintuitively, applying octave-equivalence to a lattice is mathematically equivalent to tempering out 2/1 - a single interval is used to stand for all intervals an octave apart, just as in meantone, a single interval stands in for all intervals an 81/80 apart.
+Counterintuitively, applying octave-equivalence to a lattice is mathematically equivalent to [[tempering out]] 2/1 – a single interval is used to stand for all intervals an octave apart, just as in meantone, a single interval stands in for all intervals an 81/80 apart.
 The 2-limit is equivalent to the [[1-odd-limit]], [[1edo]], and 1-''p''-fdo with arbitrary value of ''p'' (including [[AFDO|1afdo]] and [[IFDO|1ifdo]]).
-The 2-limit can be considered as the trivial rank-1 temperament '''binary''', equivalent to 1-ET in the 2-limit, which is generated by only 2/1 and tempers out no commas. All equal temperaments with a mapping of 2/1 are weak extensions of binary.
+It can be considered as a trivial rank-1 temperament, equivalent to 1et in the 2-limit, which is generated by only 2/1 and tempers out no commas. All equal temperaments with a mapping of 2/1 are weak extensions of it.
 == See also ==