new Tuning Dictionary entry: taxicab metric

hello all,

a new Tuning Dictionary entry:

taxicab metric
http://sonic-arts.org/dict/taxicab.htm

feedback appreciated.

-monz
"all roads lead to n^0"

--- In tuning@y..., "monz" <monz@a...> wrote:
> hello all,
>
>
> a new Tuning Dictionary entry:
>
> taxicab metric
> http://sonic-arts.org/dict/taxicab.htm
>
>
>
> feedback appreciated.

this paragraph is correct:

"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. "

except that in the triangular lattice, each plane has three basic
intervals, not two.

the other stuff doesn't look correct to me. the taxicab metric
provides, like the euclidean metric, a *single number* for the
distance between any two pitches. not a list of exponents or anything
like that. the "kernel" simply refers to any complete basis for the
temperament or finity. you may be thinking of the TM-reduced basis,
which simply means that choice of kernel in which the basis vectors
are the shortest possible taxicab length in the Tenney rectangular
lattice. it's called "TM" after Tenney and Minkowski -- the latter
came up with the procedure for basis reduction.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:

> the "kernel" simply refers to any complete basis for the
> temperament or finity.

sorry -- that would simply be a "basis" -- the "kernel" actually
refers to the entire, infinite set of unison vectors for the
temperament

> you may be thinking of the TM-reduced basis,
> which simply means that choice of kernel in which the basis
vectors
> are the shortest possible taxicab length in the Tenney
rectangular
> lattice.

that should be the choice of basis, not the choice of kernel!