Step pattern product: Difference between revisions

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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.
The '''step pattern product''' is an operation on two abstract scale [[step pattern]]s 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, [[5L 2s|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:


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.
* 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.


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.
This construction has an obvious generalization to the product of three or more scales.


The importance of product words in music theory is that every [[Fokker_blocks|Fokker block]] can be expressed as the product word of two or more [[Distributional_Evenness|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 [[mode|modes]]), then the product Fokker block scale is not always a mode, but is often a [[Dome|dome]] instead.
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 LLm<sub>1</sub>LLm<sub>2</sub>s, ssLssm<sub>1</sub>m<sub>2</sub>, etc; these scales are all [[sister]]s.
[[Category:math]]
 
[[Category:Scale theory]]
== Related properties ==
[[Category:theory]]
* Every [[distributionally even]] ternary scale is the step pattern product of two MOS words. However, the step pattern product of two MOS scales (even if ternary) need not be distributionally even.
 
== 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 scale]]s 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 [[mode]]s 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]]''
* [[2a5b * 3a4b]]
 
[[Category:Math]]
[[Category:Scale]]
[[Category:Combinatorics on words]]

Latest revision as of 01:12, 5 January 2026

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.

Related properties

  • Every distributionally even ternary scale is the step pattern product of two MOS words. However, the step pattern product of two MOS scales (even if ternary) need not be distributionally even.

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

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