Linear algebra formalism: Difference between revisions

Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of just intervals (and as it turns out, the space of radical intervals) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear algebra:

** Every interval has a descending counterpart, which when stacked with that interval produces the unison (1/1). For example, 5/4 * 4/5 = 1/1.

Note that this is the fundamental definition of what it means for something to be "a vector"; vectors are defined as objects in spaces where these axioms apply.

 

Note that what we've described as multiplication is actually vector addition, and what we've described as exponentiation is actually multiplication of a vector (the interval) by a scalar (the exponent). Additionally, the unison is actually a zero vector. This makes sense if we think of intervals logarithmically, where multiplication of ratios becomes addition of cent values, the unison is 0 cents, and exponents become scale factors.  

Conventionally, vectors are notated as lists of numbers representing coordinates in space. In linear algebra, these coordinates are interpreted as scale factors on several arbitrarily chosen "basis vectors" representing the space, which are scaled and added together to produce the vector in question. The standard and most intuitive way of notating intervals as vectors is to take the entries of that interval's monzo and interpret them as vector coordinates, where each basis vector represents a different generator in the monzo's subgroup (usually a prime number). Because the entries of a monzo represent exponents on a prime, the intuition of stacking as addition carries over. For example, the interval 5/4, which has a monzo of 2.3.5 [-2 0 1], may be interpreted as the vector [-2 0 1⟩, where we use an [[Extended bra-ket notation|angle bracket ⟩]] on the right to indicate that it is a vector. This notation is called the ''generator-count vector'', and may in this case be called the ''prime-count vector'', since the generators are the primes. However, it is also common to call the vector itself a monzo (and hence, terms like "eigenmonzo"). {{Todo|complete intro|inline=1}}

{{Todo|complete section|inline=1}}A temperament mapping, in linear algebra, is represented by a matrix. A matrix is a grid of numbers, written like so:

This matrix can be thought of as a "function" that you apply to a vector (in this case, a generator-count vector) to get out another vector (here, another generator-count vector in the tempered space). This matrix has 3 columns, meaning the vector it takes as an "input" will have 3 elements (representing a rank-3 subgroup of JI), and it has 2 rows, meaning the vector you get out will have 2 elements (representing a rank-2 temperament). So, this is a "function" down from 3-dimensional space to 2-dimensional space. The columns tell us where to find the generators of our original subgroup in the space of tempered intervals. One of the key properties of linear algebra is that knowing that alone allows us to tell where every other interval ends up, and is why regular temperaments are covered so much in xenharmonic theory as opposed to irregular temperaments.

 

What follows is an explanation of several matrix operations in the mathematical language.  

== Vals and tuning maps ==

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