Step pattern product: Difference between revisions

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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:

* Take two step patterns, for example AABAAAB and ~~BBABBAB~~.

* 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,B)(A,B)(B,A)(A,B)(A,B)(A,A)(B,B)

* 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: ~~AABAACD~~.

* 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 ~~LLmLLms~~, ~~ssLssLm~~, etc; these scales are all [[sister]]s.

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 [[sister]]s.

== 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 ==

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== 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".

It does not matter what symbols are used for the two inputs (which may use different sets of symbols) or the output (in fact, the output is usually written with a different set of symbols to the two inputs).

== Examples ==

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 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:
-* Take two step patterns, for example AABAAAB and BBABBAB.
+* 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,B)(A,B)(B,A)(A,B)(A,B)(A,A)(B,B)
+* 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: AABAACD.
+* 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 LLmLLms, ssLssLm, etc; these scales are all [[sister]]s.
+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.
+== 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 ==
@@ Line 16: / Line 19: @@
 == 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".
-It does not matter what symbols are used for the two inputs (which may use different sets of symbols) or the output (in fact, the output is usually written with a different set of symbols to the two inputs).
 == Examples ==