Ternary parallelogram scales are MOS substitution - Revision history

Inthar: /* Pitch-class group */

2026-05-04T17:38:08Z

Pitch-class group

← Older revision		Revision as of 17:38, 4 May 2026
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	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.		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 if the gcd of the coordinates ''v''<sub>''i''</sub> of '''v''' ∈ ℤ<sup>''r''</sup> is 1~~, then~~ the quotient group ℤ<sup>''r''</sup>/{{angbr\|'''v'''}} is torsion-free; this can be proven using Bézout's identity.		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 ===		=== Parallelogram scale ===

Inthar: /* Pitch-class group */

2026-05-04T17:23:33Z

Pitch-class group

← Older revision		Revision as of 17:23, 4 May 2026
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	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.		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 if the gcd of the coordinates ''v''<sub>''i''</sub> of '''v''' ∈ ℤ<sup>''r''</sup> is 1, then the quotient group ℤ<sup>''r''</sup>/{{angbr\|'''v'''}} is torsion-free; this ~~is a theorem of group theory~~.		Below we take it as known that if the gcd of the coordinates ''v''<sub>''i''</sub> of '''v''' ∈ ℤ<sup>''r''</sup> is 1, then the quotient group ℤ<sup>''r''</sup>/{{angbr\|'''v'''}} is torsion-free; this can be proven using Bézout's identity.

	=== Parallelogram scale ===		=== Parallelogram scale ===

Inthar: /* Pitch-class group */

2026-05-04T17:18:16Z

Pitch-class group

← Older revision		Revision as of 17:18, 4 May 2026
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	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.		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 if the gcd of the coordinates ''v''<sub>''i''</sub> of '''v''' is 1, then the quotient group ℤ<sup>''r''</sup>/{{angbr\|'''v'''}} is torsion-free; this is a theorem of group theory.		Below we take it as known that if the gcd of the coordinates ''v''<sub>''i''</sub> of '''v''' ∈ ℤ<sup>''r''</sup> is 1, then the quotient group ℤ<sup>''r''</sup>/{{angbr\|'''v'''}} is torsion-free; this is a theorem of group theory.

	=== Parallelogram scale ===		=== Parallelogram scale ===

Inthar: /* Pitch-class group */

2026-05-04T17:17:12Z

Pitch-class group

← Older revision		Revision as of 17:17, 4 May 2026
Line 7:		Line 7:
	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.		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 if the gcd of the ~~step signature~~ is 1, then the ~~corresponding pitch class~~ group ~~of the scale word~~ is torsion-free; this is a theorem of group theory.		Below we take it as known that if the gcd of the coordinates ''v''<sub>''i''</sub> of '''v''' is 1, then the quotient group ℤ<sup>''r''</sup>/{{angbr\|'''v'''}} is torsion-free; this is a theorem of group theory.

	=== Parallelogram scale ===		=== Parallelogram scale ===

Inthar: /* Pitch-class group */

2026-05-04T17:05:29Z

Pitch-class group

← Older revision		Revision as of 17:05, 4 May 2026
Line 7:		Line 7:
	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.		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 if the gcd of the step signature is 1, then the corresponding pitch class group of the scale word is torsion-free.		Below we take it as known that if the gcd of the step signature is 1, then the corresponding pitch class group of the scale word is torsion-free; this is a theorem of group theory.

	=== Parallelogram scale ===		=== Parallelogram scale ===

Inthar: /* Pitch-class group */

2026-05-04T16:55:53Z

Pitch-class group

← Older revision		Revision as of 16:55, 4 May 2026
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	=== Pitch-class group ===		=== 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.		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 if the gcd of the step signature is 1, then the corresponding pitch class group of the scale word is torsion-free.

	=== Parallelogram scale ===		=== Parallelogram scale ===

Inthar: /* Open problems */

2026-04-10T01:36:56Z

Open problems

← Older revision		Revision as of 01:36, 10 April 2026
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	# Prove a converse to this theorem.		# 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.		#* <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: LLmLLsLLmLLsLLs is of type 10L(2m3s))		#* 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:Math]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]
	[[Category:Combinatorics on words]]		[[Category:Combinatorics on words]]
	[[Category:Pages with open problems]]		[[Category:Pages with open problems]]

Inthar: /* Open problems */

2026-04-09T01:37:53Z

Open problems

← Older revision		Revision as of 01:37, 9 April 2026
Line 85:		Line 85:
	# Prove a converse to this theorem.		# 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.		#* <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.		#* 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: LLmLLsLLmLLsLLs is of type 10L(2m3s))
	[[Category:Math]]		[[Category:Math]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]
	[[Category:Combinatorics on words]]		[[Category:Combinatorics on words]]
	[[Category:Pages with open problems]]		[[Category:Pages with open problems]]

Inthar: /* Open problems */

2026-04-09T01:31:35Z

Open problems

← Older revision		Revision as of 01:31, 9 April 2026
Line 85:		Line 85:
	# Prove a converse to this theorem.		# 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.		#* <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.
	[[Category:Math]]		[[Category:Math]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]
	[[Category:Combinatorics on words]]		[[Category:Combinatorics on words]]
	[[Category:Pages with open problems]]		[[Category:Pages with open problems]]

Inthar: /* Open problems */

2026-04-09T01:30:21Z

Open problems

← Older revision		Revision as of 01:30, 9 April 2026
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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'' - 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.		# 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 and the step signature's gcd = 1 => 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.
	[[Category:Math]]		[[Category:Math]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]
	[[Category:Combinatorics on words]]		[[Category:Combinatorics on words]]
	[[Category:Pages with open problems]]		[[Category:Pages with open problems]]