User:Inthar/Generator variety - Revision history

Inthar: Redirected page to Generator sequence#Generator variety

2024-05-23T03:36:30Z

Redirected page to Generator sequence#Generator variety

← Older revision		Revision as of 03:36, 23 May 2024
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	~~If a scale has a~~ [[~~generator~~ sequence~~]] or is a union of multiple offset copies of the same generator sequence, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.~~		#redirect [[Generator sequence#Generator variety]]

	~~There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator~~ variety ~~1, and [[MOS substitution~~]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because (4 generators) * (10 steps) = 40 steps ≡ 1 mod 13):
	~~# 2'''x''' + 2'''y''',~~
	~~# '''x''' + 3'''y''',~~
	~~# 2'''x''' + '''y''' + '''z''',~~
	~~# '''x''' + 2'''y''' + '''z'''.~~
	~~More trivially, a MOS generator when not stacked so that the resulting scale is a MOS size (equal to the denominator of a semiconvergent to the generator) will not have variety 2.~~

Inthar at 01:13, 23 May 2024

2024-05-23T01:13:20Z

← Older revision		Revision as of 01:13, 23 May 2024
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	If a scale has a [[generator sequence]] or is a union of multiple offset generator ~~sequences~~, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.		If a scale has a [[generator sequence]] or is a union of multiple offset copies of the same generator sequence, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.

	There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because (4 generators) * (10 steps) = 40 steps ≡ 1 mod 13):		There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because (4 generators) * (10 steps) = 40 steps ≡ 1 mod 13):

Inthar at 22:57, 22 May 2024

2024-05-22T22:57:25Z

← Older revision		Revision as of 22:57, 22 May 2024
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	If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.		If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.

	There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because 4*10 = 40 ≡ 1 mod 13):		There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because (4 generators) * (10 steps) = 40 steps ≡ 1 mod 13):
	# 2'''x''' + 2'''y''',		# 2'''x''' + 2'''y''',
	# '''x''' + 3'''y''',		# '''x''' + 3'''y''',

Inthar at 22:56, 22 May 2024

2024-05-22T22:56:43Z

← Older revision		Revision as of 22:56, 22 May 2024
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	If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.		If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.

	There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class:		There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because 4*10 = 40 ≡ 1 mod 13):
	# 2'''x''' + 2'''y''',		# 2'''x''' + 2'''y''',
	# '''x''' + 3'''y''',		# '''x''' + 3'''y''',

Inthar at 22:45, 22 May 2024

2024-05-22T22:45:21Z

← Older revision		Revision as of 22:45, 22 May 2024
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	If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.		If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.

	There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in ~~the~~ the generator interval class:		There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class:
	# 2'''x''' + 2'''y''',		# 2'''x''' + 2'''y''',
	# '''x''' + 3'''y''',		# '''x''' + 3'''y''',

Inthar at 22:45, 22 May 2024

2024-05-22T22:45:10Z

← Older revision		Revision as of 22:45, 22 May 2024
Line 1:		Line 1:
	If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.		If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.

	There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes:		There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the the generator interval class:
	# 2'''x''' + 2'''y''',		# 2'''x''' + 2'''y''',
	# '''x''' + 3'''y''',		# '''x''' + 3'''y''',

Inthar at 22:43, 22 May 2024

2024-05-22T22:43:27Z

← Older revision		Revision as of 22:43, 22 May 2024
Line 1:		Line 1:
	If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing ~~generator~~.		If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing interval.

	There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes:		There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes:

Inthar at 22:42, 22 May 2024

2024-05-22T22:42:04Z

← Older revision		Revision as of 22:42, 22 May 2024
Line 1:		Line 1:
	If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing generator.		If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing generator.

	There is no simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes:		There is no known simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes:
	# 2'''x''' + 2'''y''',		# 2'''x''' + 2'''y''',
	# '''x''' + 3'''y''',		# '''x''' + 3'''y''',

Inthar: /* Relationship between step variety and generator variety */

2024-05-22T22:41:04Z

Relationship between step variety and generator variety

← Older revision		Revision as of 22:41, 22 May 2024
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	If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing generator.		If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing generator.

	~~== Relationship~~ between step variety and generator variety ==		There is no simple relationship between a scale's [[step variety]] and its [[generator variety]]. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes:
	MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes:
	# 2'''x''' + 2'''y''',		# 2'''x''' + 2'''y''',
	# '''x''' + 3'''y''',		# '''x''' + 3'''y''',

Inthar at 22:38, 22 May 2024

2024-05-22T22:38:04Z

← Older revision		Revision as of 22:38, 22 May 2024
Line 1:		Line 1:
	If a scale has a [[generator sequence]], then the '''generator variety''' is the number of generators in that sequence, not including the closing generator.		If a scale has a [[generator sequence]] or is a union of multiple offset generator sequences, then the '''generator variety''' is the number of generators in that sequence, not including the closing generator.

	== Relationship between step variety and generator variety ==		== Relationship between step variety and generator variety ==