Step pattern product
The step pattern product is an operation on two abstract scale step patterns of the same length that produces a new abstract step pattern. The operation does not concern the information of the relative sizes of the scale steps (so that for example, diatonic and antidiatonic are represented by the same pattern AABAAAB). These step patterns are called factors, and their product is taken by doing the following:
- Take two step patterns. Conventionally, these are written with different sets of letters, for example AABAAAB and CCDCCDC.
- Pair up the entries in the step patterns: (A,C)(A,C)(B,D)(A,C)(A,C)(A,D)(B,C)
- Assign each ordered pair its own new symbol: EEFEEGH, again, conventionally written with a new set of letters.
This construction has an obvious generalization to the product of three or more scales.
Once the resulting step pattern has been acquired, one may arbitrarily assign the different types of steps to different relative sizes. Any permutation is possible, such as LLm1LLm2s, ssLssm1m2, etc; these scales are all sisters.
Utilization
The step pattern product can be used to construct complex scales such as the Zarlino scale from MOS scales. The Zarlino scale (and its sisters) may be represented by taking the product of ABABABA and AABAAAB, resulting in ABCBABC.
In general, every Fokker block can be expressed as the product of two or more MOS scales in a unique way. Fokker blocks are therefore equivalent to products of MOS scales of the same size. If one or both of the MOS scales are rotated into different modes relative to the original inputs, then the product Fokker block scale is not always a mode, but is often a dome of the original Fokker block instead.
Mathematical context
In general mathematical theory (and more advanced pages on the wiki), what this page refers to as "step patterns" are called "words", and the "step pattern product" is a "product word".
Examples
See also: Category:Product words