Parallelogram substring scale: Difference between revisions
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This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''. | This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''. | ||
A parallelogram substring scale with full first and last rows is a '''parallelogram scale'''. | |||
== Ternary scales with this property == | == Ternary scales with this property == | ||
=== Examples === | === Examples === | ||
* All non-Fraenkel balanced primitive MV3 scales | * All non-Fraenkel balanced primitive MV3 scales | ||
* All ''a'''''x'''(''b'''''y'''(''a'' - ''b'')'''z''') MOS substitution scales if gcd(''a'', ''b'') = 1 | * All ''a'''''x'''(''b'''''y'''(''a'' - ''b'')'''z''') [[MOS substitution]] scales if gcd(''a'', ''b'') = 1 | ||
* All MOS substitution scales where: | * All MOS substitution scales where: | ||
** The template MOS is primitive | ** The template MOS is primitive | ||
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* All multiperiod MOS substitution scales (e.g. 4L(10m10s)) | * All multiperiod MOS substitution scales (e.g. 4L(10m10s)) | ||
== Mathematical facts == | |||
=== Ternary parallelogram scales are MOS substitution === | |||
:''Main article: [[Ternary parallelogram scales are MOS substitution]]'' | |||
== Open problems == | == Open problems == | ||
# Classify all MOS-substitution parallelogram substring scales. | # Classify all MOS-substitution parallelogram substring scales. | ||
# Classify all ternary parallelogram substring scales. Conjecture: All ternary parallelogram substring scales are MOS substitution scales. | # Classify all ternary parallelogram substring scales. | ||
#* <s>Conjecture: All ternary parallelogram substring scales are MOS substitution scales.</s> (Numerous counterexamples, e.g. LLmLmLmLmLLs) | |||
# Classify all ternary full parallelogram scales (PS with full first and last rows). | # Classify all ternary full parallelogram scales (PS with full first and last rows). | ||
[[Category:Pages with open problems]] | [[Category:Pages with open problems]] | ||