Parallelogram substring scale: Difference between revisions

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A '''~~quasi-~~parallelogram scale''' is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. Such scales are currently being investigated in ternary generator-offset theory.

A '''parallelogram substring scale''' is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. Such scales are currently being investigated in ternary generator-offset theory.

== Mathematical definition ==

An '''e'''-equivalent scale is a '''~~quasi-~~parallelogram''' if there exist integers ''m'' > 0, ''n'' > 0, 0 ≤ ''a'' < ''n'', 0 ≤ ''b'' < ''n'', a vector '''a''', and two linearly independent vectors '''v''' and '''w''' such that the set of notes in the scale as a subset of the lattice of '''e'''-equivalent pitches is

An '''e'''-equivalent scale is a '''parallelogram substring''' if there exist integers ''m'' > 0, ''n'' > 0, 0 ≤ ''a'' < ''n'', 0 ≤ ''b'' < ''n'', a vector '''a''', and two linearly independent vectors '''v''' and '''w''' such that the set of notes in the scale as a subset of the lattice of '''e'''-equivalent pitches is

<math>

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== Open problems ==

# Classify all MOS-substitution ~~quasi-~~parallelogram scales.

# Classify all MOS-substitution parallelogram substring scales.

# Classify all ternary ~~quasi-~~parallelogram scales. <s>Conjecture: All ternary ~~quasi-~~parallelogram scales are MOS substitution scales.</s> (counterexample: 5sL/5sR diachrome is 2-row QP)

# Classify all ternary parallelogram substring scales. <s>Conjecture: All ternary parallelogram substring scales are MOS substitution scales.</s> (counterexample: 5sL/5sR diachrome is 2-row QP)

[[Category:Pages with open problems]]

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-A '''quasi-parallelogram scale''' is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. Such scales are currently being investigated in ternary generator-offset theory.
+A '''parallelogram substring scale''' is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. Such scales are currently being investigated in ternary generator-offset theory.
 == Mathematical definition ==
-An '''e'''-equivalent scale is a '''quasi-parallelogram''' if there exist integers ''m'' > 0, ''n'' > 0, 0 &le; ''a'' < ''n'', 0 &le; ''b'' < ''n'', a vector '''a''', and two linearly independent vectors '''v''' and '''w''' such that the set of notes in the scale as a subset of the lattice of '''e'''-equivalent pitches is
+An '''e'''-equivalent scale is a '''parallelogram substring''' if there exist integers ''m'' > 0, ''n'' > 0, 0 &le; ''a'' < ''n'', 0 &le; ''b'' < ''n'', a vector '''a''', and two linearly independent vectors '''v''' and '''w''' such that the set of notes in the scale as a subset of the lattice of '''e'''-equivalent pitches is
 <math>
@@ Line 31: / Line 31: @@
 == Open problems ==
-# Classify all MOS-substitution quasi-parallelogram scales.
+# Classify all MOS-substitution parallelogram substring scales.
-# Classify all ternary quasi-parallelogram scales. <s>Conjecture: All ternary quasi-parallelogram scales are MOS substitution scales.</s> (counterexample: 5sL/5sR diachrome is 2-row QP)
+# Classify all ternary parallelogram substring scales. <s>Conjecture: All ternary parallelogram substring scales are MOS substitution scales.</s> (counterexample: 5sL/5sR diachrome is 2-row QP)
 [[Category:Pages with open problems]]