Music
Determining factors of numbers or frequencies determines their degree of harmonicity, affinity or attraction. [Dale Pond]
In Mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3, 5, and the polynomial x2ˆ4 factors as (xˆ2)(x + 2). In all cases, a product of simpler objects is obtained.
The aim of factoring is usually to reduce something to "basic building blocks" (least common denominator), such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Vieta's formulas relate the coefficients of a polynomial to its roots.
The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an "expanded" polynomial, written as just a sum of terms.
Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.
A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.
Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a subjective function with an injective function. This situation is generalized by factorization systems. Wikipedia, Factoring
You can factor numbers/frequencies here: Wolfram Mathematica, Factoring
See Also
Affinity
Aliquot
Fast Fourier Transform
Figure 8.14 - Some Basic Waveforms and their constituent Aliquot Parts
Finite Element Analysis
Fundamental
Graduation
Harmonic
Harmonicity
Harmony
Interval
Law of Cycles
Modal Analysis
Music
Note
Octave prime
Overtone Series
Overtone
Partial
prima materia
PRIMARY IMPULSES
prime mover
Prime Neutral Center
Prime Numbers
Prime Third
prime thirds
prime
Ramsay - Rationale of the Limit of Musical Mathematical Primes
Ramsay - Tendencies of the Notes form Proximity and Affinity
three primary centers
three primary ratios
two prime conductors