Step pattern product: Difference between revisions
I don't see how relabeling would obscure the pattern, I'd say it's less confusing |
Specify "ordered" pair, links, categories, misc. edits |
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The '''scale pattern product''' is an operation on two abstract [[ | The '''scale 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 scale patterns are called factors, and their product is taken by doing the following: | ||
* Take two scale patterns, for example AABAAAB and BBABBAB. | * Take two scale patterns, for example AABAAAB and BBABBAB. | ||
* 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 scale patterns: (A,B)(A,B)(B,A)(A,B)(A,B)(A,A)(B,B) | ||
* Assign each pair its own new symbol: AABAACD. | * Assign each ordered pair its own new symbol: AABAACD. | ||
This construction has an obvious generalization to the product of three or more scales. | 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 [[ | 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]]s. | ||
== Utilization == | == Utilization == | ||
The scale 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. | The scale 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 [[ | 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 == | == 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 pattern product" is a "product word". | 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 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). | 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). | ||
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* [[2a5b * 3a4b]] | * [[2a5b * 3a4b]] | ||
[[Category: | [[Category:Math]] | ||
[[Category:Scale]] | [[Category:Scale]] | ||
[[Category: | [[Category:Combinatorics on words]] | ||
Revision as of 00:19, 28 June 2025
The scale 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 scale patterns are called factors, and their product is taken by doing the following:
- Take two scale patterns, for example AABAAAB and BBABBAB.
- Pair up the entries in the scale patterns: (A,B)(A,B)(B,A)(A,B)(A,B)(A,A)(B,B)
- Assign each ordered pair its own new symbol: AABAACD.
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 sisters.
Utilization
The scale 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 "scale patterns" are called "words", and the "scale 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
See also: Category:Product words