A non-zero vector is a line-segment with both magnitude and direction in some space, and also the set of coordinate points which specify that magnitude and that direction. A zero vector is the number zero and has no direction.
Regarding the coordinate definition, for an integer n, a vector is an ordered n-tuple of numbers in an n-dimensional space, of the form [x, y, z ... n] .
Representations of an interval as a row (or column) of numbers is a vector. Vector addition is a common operation.
In general an n-dimensional vector space is a collection of elements that can be represented by n numbers. Operations are addition and scalar multiplication (multiplication with a single value). In the lattice context the dimensions are of course the relevant primes and the vector elements are the power indices [exponents].
Simple vector addition of two ratios can be used to calculate intervals.
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For example: The pythagorean "perfect 5th" has the ratio 3:2. This is equivalent to 2-1 * 31, or simply [-1 1] in vector notation. There is a standardized type of vector called a "monzo", used to represent the prime-factor exponents of musical pitch ratios; this monzo is written |-1 1, > . The complement of this ratio, the "perfect 4th", has the ratio 4:3, which is equivalent to 22 * 3-1, or [2 -1, > in monzo notation. The difference between them is the pythagorean "whole tone" with ratio 9:8, which is equivalent to 2-3 * 32, or [-3 2, > in monzo notation. This can be calculated by regular fractional math: 3 4 3 3 9 - ÷ - = - * - = - 2 3 2 4 8 or by vector addition:
2 3
[-1 1, > 3/2
- [ 2 -1, > ÷ 4/3
----------- = -----
[-3 2, > 9/8
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Vector addition is especially useful in tuning calculations when dealing with ratios containing very large numbers in their terms (such as the pythagorean comma), and when utilizing fractional portions of ratios, as in temperaments such as meantone or well-temperaments. In the former case, the exponents are much smaller to deal with and the numbers are added and subtracted instead of multiplied and divided. In the latter case, simple fractional math can be used instead of having to deal with roots and powers.
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For example: The mistuning of certain "5ths" in the Werckmeister III well-temperament is 1/4 of a pythagorean comma. The Pythagorean comma is the difference between 12 "5ths" and 7 octaves:
2 3
[-12 12, > (3/2)12 = 312/212 = 531441/4096 531441/4096
- [ 7 0, > ÷ (2/1)7 = 128 * 1/128
------------- = --------- = ---------------
[-19 12, > 531441/524288
1/4 of the Pythagorean comma is (531441/524288)(1/4), which in rational-monzo vector notation is very simply [-19/4 12/4, >, which reduces to [-19/4 3, > . So the rational-monzo vector of the "Werckmeister 5ths" is thus:
2 3 ~cents
[-1 1, > 3:2 ratio = "perfect 5th" 701.9550009
- [-19/4 3, > 1/4 Pythagorean-comma - 5.865002596
--------------- ---------------
[ 15/4 -2, > Werckmeister 5th 696.0899983
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Besides its obvious mathematical advantages, another example which i think shows the usefulness of vector notation is my Lattice diagrams comparing rational implications of various meantone chains.
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