Ternary parallelogram scales are MOS substitution: Difference between revisions

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=== Pitch-class group ===

The ''pitch-class group'' of a scale word ''w'' in letters {{nowrap|'''x'''1, ..., '''x'''''r''}} with [[step signature]] {{nowrap|'''e''' ∈ ℤ''r''{{angbr|'''x'''1, ..., '''x'''''r''}}}} is the abelian group {{nowrap|C(''w'') :{{=}} ℤ''r''{{angbr|'''x'''1, ..., '''x'''''r''}}/{{angbr|'''e'''}}.}} The pitch-class group is associated with a canonical map π that sends every step vector to its pitch class.

Below we take it as known that the gcd of the coordinates ''v''''i'' of '''v''' ∈ ℤ''r'' is 1 iff the quotient group ℤ''r''/{{angbr|'''v'''}} is torsion-free; this can be proven using Bézout's identity.

=== Parallelogram scale ===

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# Conjecture: Ternary scales that are parallelogram substrings with full row length ''m'', full column length ''n'', and cardinality {{nowrap|''mn'' - 1}} are MOS substitution.

# Conjecture: Ternary scales that are parallelogram substrings with full row length ''m'', full column length ''n'', and cardinality {{nowrap|''mn'' - 2}} are MOS substitution.

# Prove a converse to this theorem. ~~(something like "~~If the template MOS of a MOS substitution scale has m > 1 periods ~~+ ...~~ => m × n parallelogram scale")

# Prove a converse to this theorem.

#* <s>If the template MOS of a MOS substitution scale has m > 1 periods and the step signature's gcd = 1 => m × n parallelogram scale.</s> This is false as stated, since no 5L(3m7s) scale is a parallelogram scale.

#* Conjecture: If the template MOS of a MOS substitution scale has m > 1 periods, the step signature's gcd = 1, and the filling MOS is a multiMOS, then the scale is a parallelogram scale. (But the filling MOS need not be a multiMOS for a parallelogram scale: LLmLLsLLmLLsLLs is of type 10L(2m3s))

#* Conjecture: If a MOS substitution scale's template MOS is a multiMOS, and its step signature's gcd = 1, then its generator structure is a (possibly) *shifted* full parallelogram, i.e. either a full parallelogram or the first and last rows' lengths add up to the full row length.

[[Category:Math]]

[[Category:Pages with proofs]]

[[Category:Combinatorics on words]]

[[Category:Pages with open problems]]

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 === Pitch-class group ===
 The ''pitch-class group'' of a scale word ''w'' in letters {{nowrap|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}} with [[step signature]] {{nowrap|'''e''' ∈ ℤ<sup>''r''</sup>{{angbr|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}}}} is the abelian group {{nowrap|C(''w'') :{{=}} ℤ<sup>''r''</sup>{{angbr|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}}/{{angbr|'''e'''}}.}} The pitch-class group is associated with a canonical map π that sends every step vector to its pitch class.
+Below we take it as known that the gcd of the coordinates ''v''<sub>''i''</sub> of '''v''' ∈ ℤ<sup>''r''</sup> is 1 iff the quotient group ℤ<sup>''r''</sup>/{{angbr|'''v'''}} is torsion-free; this can be proven using Bézout's identity.
 === Parallelogram scale ===
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 # Conjecture: Ternary scales that are parallelogram substrings with full row length ''m'', full column length ''n'', and cardinality {{nowrap|''mn'' - 1}} are MOS substitution.
 # Conjecture: Ternary scales that are parallelogram substrings with full row length ''m'', full column length ''n'', and cardinality {{nowrap|''mn'' - 2}} are MOS substitution.
-# Prove a converse to this theorem. (something like "If the template MOS of a MOS substitution scale has m > 1 periods + ... => m × n parallelogram scale")
+# Prove a converse to this theorem.
+#* <s>If the template MOS of a MOS substitution scale has m > 1 periods and the step signature's gcd = 1 => m × n parallelogram scale.</s> This is false as stated, since no 5L(3m7s) scale is a parallelogram scale.
+#* Conjecture: If the template MOS of a MOS substitution scale has m > 1 periods, the step signature's gcd = 1, and the filling MOS is a multiMOS, then the scale is a parallelogram scale. (But the filling MOS need not be a multiMOS for a parallelogram scale: LLmLLsLLmLLsLLs is of type 10L(2m3s))
+#* Conjecture: If a MOS substitution scale's template MOS is a multiMOS, and its step signature's gcd = 1, then its generator structure is a (possibly) *shifted* full parallelogram, i.e. either a full parallelogram or the first and last rows' lengths add up to the full row length.
 [[Category:Math]]
 [[Category:Pages with proofs]]
 [[Category:Combinatorics on words]]
 [[Category:Pages with open problems]]