taxicab metric

The measurement of the number of steps along each axis (i.e., each dimension) of the lattice-diagram, which in the case of the rectangular prime-factor lattice counts the value of the exponents in the prime-factorization of the interval under consideration, and in the case of the triangular basic-interval lattice, the numer of steps 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.

The precise value of a taxicab metric is given by a prime-factor exponent vector.

The taxicab metric often plays a role in determining the finity of a tuning, in that the general procedure used by theorists to find the "kernel" periodicity-block of a tuning is to first limit the dimensionality of the lattice to certain prime-factors (and perhaps certain exponent-values for them), then to measure the instance of each pitch in the tuning which is the shortest distance by the taxicab metric along the combined prime or basic-interval axes, from the origin which is prime-factor exponent vector n⁰ and ratio 1/1, thus ensuring that the boundaries of the kernal periodicity-block are as close as possible to the origin-point.

(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.)


[from Joe Monzo, JustMusic: A New Harmony ]

---

Updated:

2002.11.29 -- page created

---

(to download a zip file of the entire Dictionary, click here)

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here

I welcome feedback about this webpage: corrections, improvements, good links. Let me know if you don't understand something. return to the Microtonal Dictionary index return to my home page return to the Sonic Arts home page