  # taxicab metric

[Joe Monzo]

The measurement of a musical interval's distance along the "streets" "avenues", vertical "shafts", and whatever other higher-dimensional pathways are needed for a particular prime-space, where each pathway represents an axis (i.e., a dimension) of the harmonic lattice diagram.

In the case of the rectangular prime-factor lattice the route travels the number of steps along a prime-axis (or another path parallel to that prime-axis) as indicated by that prime's exponent in the interval's monzo.

In the case of the triangular basic-interval lattice, the route travels the shortest path of basic-intervals which compose that interval.

This is in contradistinction to the Euclidean distance, which is a straight line between the two points representing the two pitches composing the interval. The taxicab metric can be likened to a different "route" in which planes of the lattice representing the combinations of (in the rectangular lattice) any two prime-factors or (in the triangular lattice) any two basic-intervals, are divided up into rectangular or triangular "blocks" like those of a typical city street pattern. The taxicab can't travel along the straight Euclidean line, but instead must travel along the individual "streets" of the prime or basic-interval axes.

In the Tenney lattice, where the length of one step along each prime axis is log(prime), the precise value of a taxicab metric is given by log(n*d), where n is the numerator and d is the denominator of the ratio of the interval.

(For 2-dimensional examples of this, see any of the 5-prime-limit "tiling" bingo-card lattices for various EDOs; for a 3-dimensional example, see the 7-prime-limit graphics at the bottom of the meride entry.)

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