vector

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].

[from Kees van Prooijen]

. . . . . . . . . . . . . . . . .

Simple vector addition of two ratios can be used to calculate intervals.

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.

The complement of this ratio, the "perfect 4th", has the ratio 4:3, which is equivalent to 2² * 3^-1, or [2 -1] in vector 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 vector 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

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.

For example:

The mistuning of certain "5ths" in the Werckmeister III temperament is 1/4 of a Pythagorean comma.

The Pythagorean comma is the difference between 12 "5ths" and 7 "8ves":

     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 vector notation is very simply [-19/4 12/4], which reduces to [-19/4 3].

So the "ratio" 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


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.


[from Joe Monzo, JustMusic: A New Harmony]

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updated:

2002.09.07
1999.12.16

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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

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