# Product word

A word is a sequence of letters from some finite alphabet. Words can be used to represent scales, or families of scales; for example "aabaaab" represents the major scale if 'a' represents a whole step and 'b' a half step.

Given two words of the same length ("factors"), their **product word** is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose *n*th letter is the ordered pair of the *n*th letters of its factors.

For example, the product word of "aabaaab" and "xxyxyxy" is "(a,x)(a,x)(b,y)(a,x)(a,y)(a,x)(b,y)". For brevity, we can substitute each ordered pair of letters by a new single letter and say this is equivalent to the word "rrsrtrs". This construction has an obvious generalization to the product of three or more words.

The importance of product words in music theory is that every Fokker block can be expressed as the product word of two or more distributionally even scales in a unique way. Fokker blocks are therefore equivalent to product words of DE scales of the same size. If one or both of the DE scales are rotated (into different modes), then the product Fokker block scale is not always a mode, but is often a dome instead.