Step pattern product: Difference between revisions

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The '''~~scale~~ product''' is an operation on two abstract [[~~Scale|scale patterns~~]] of the same length that produces a new abstract ~~scale~~ 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 ~~scale~~ patterns are called factors, and their product is taken by doing the following:

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 ~~scale~~ 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 ~~scale~~ 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 pair its own new symbol: ~~AABAABC~~.

* 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 ~~scale~~ 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|sisters~~.]]

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 ==

The ~~scale~~ 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.

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_blocks|~~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~~|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.

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 "~~scale~~ patterns" are called "words", and the "~~scale~~ product" is a "product word".

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, but this obscures the relation between the operation and scale patterns).

== Examples ==

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* [[2a5b * 3a4b]]

[[Category:~~math~~]]

[[Category:Math]]

[[Category:Scale]]

[[Category:~~Combinatorics_on_words~~]]

[[Category:Combinatorics on words]]

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-The '''scale product''' is an operation on two abstract [[Scale|scale patterns]] of the same length that produces a new abstract scale 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 scale patterns are called factors, and their product is taken by doing the following:
+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 scale 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 scale 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 pair its own new symbol: AABAABC.
+* 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 scale 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|sisters.]]
+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 ==
-The scale 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.
+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_blocks|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|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.
+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 "scale patterns" are called "words", and the "scale product" is a "product word".
+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, but this obscures the relation between the operation and scale patterns).
 == Examples ==
@@ Line 23: / Line 24: @@
 * [[2a5b * 3a4b]]
-[[Category:math]]
+[[Category:Math]]
 [[Category:Scale]]
-[[Category:Combinatorics_on_words]]
+[[Category:Combinatorics on words]]