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]]. | 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 | 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 is equivalent to the [[1-odd-limit]], [[1edo]], and 1-''p''-fdo with arbitrary value of ''p'' (including [[AFDO|1afdo]] and [[IFDO|1ifdo]]). | ||
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 == | == See also == | ||
Revision as of 11:50, 20 April 2025
The 2-limit consists of intervals that are either an integer whose only prime factor is 2, or the reciprocal of such an integer. Naturally, since 2/1 is the octave, this limits us to unisons, octaves and stacks of octaves. The 2-limit can be represented by any edo.
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.
The 2-limit is equivalent to the 1-odd-limit, 1edo, and 1-p-fdo with arbitrary value of p (including 1afdo and 1ifdo).
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.