Generator sequence - Revision history

ArrowHead294: formatting + nowraps

2026-01-29T20:36:33Z

formatting + nowraps

← Older revision		Revision as of 20:36, 29 January 2026
Line 3:		Line 3:
	'''{{PAGENAME}}''' ('''GS''') is a scale-building procedure first described by [[Scott Dakota]].		'''{{PAGENAME}}''' ('''GS''') is a scale-building procedure first described by [[Scott Dakota]].

	The notation GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>) denotes a scale-building procedure where a ([[Periodic scale\|periodic]]) scale is built by stacking ''x''<sub>1</sub> first, ''x''<sub>2</sub> second, ..., reducing by the scale's [[equave]] when necessary. When ''x''<sub>''r''</sub> is stacked, we go back to ''x''<sub>1</sub> and start stacking ''x''<sub>1</sub> again, then ''x''<sub>2</sub>, ... This article adopts a convention where an enumerated chord can be used instead for part or whole of the argument, where the chord's steps are generators, for example writing [[Zarlino]] as GS(4:5:6)[7], which is syntactic sugar for GS(5/4, 6/5)[7].		The notation {{nowrap\|GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>)}} denotes a scale-building procedure where a ([[Periodic scale\|periodic]]) scale is built by stacking ''x''<sub>1</sub> first, ''x''<sub>2</sub> second, ..., reducing by the scale's [[equave]] when necessary. When ''x''<sub>''r''</sub> is stacked, we go back to ''x''<sub>1</sub> and start stacking ''x''<sub>1</sub> again, then ''x''<sub>2</sub>, ... This article adopts a convention where an enumerated chord can be used instead for part or whole of the argument, where the chord's steps are generators, for example writing [[Zarlino]] as GS(4:5:6)[7], which is syntactic sugar for GS(5/4, 6/5)[7].

	Currently, the study of GSs is dominated by certain [[constant structure]] GS scales, called ''guided generator sequence'' scales, which are obtained by taking a GS of detempered MOS generators and stopping the stacking procedure at the corresponding MOS scale sizes, which yields constant scales. In such a situation, we call the (logarithmic) average of the generators the ''guide generator''.		Currently, the study of GSs is dominated by certain [[constant structure]] GS scales, called ''guided generator sequence'' scales, which are obtained by taking a GS of detempered MOS generators and stopping the stacking procedure at the corresponding MOS scale sizes, which yields constant scales. In such a situation, we call the (logarithmic) average of the generators the ''guide generator''.

	Certain [[generator-offset property\|generator-offset]] scales are examples. For example, [[diasem]] is GS(8/7, 7/6) or GS(7/6, 8/7) depending on [[chirality]]. The trivial case GS(''x'') is stacking a single generator ''x'' to make a rank-2 scale, such as a [[MOS scale]].		Certain [[generator-offset property\|generator-offset]] scales are examples. For example, [[diasem]] is GS(8/7, 7/6) or GS(7/6, 8/7) depending on [[chirality]]. The trivial case GS(''x'') is stacking a single generator ''x'' to make a rank-2 scale, such as a [[MOS scale]].

	== Terminology ==		== Terminology ==
Line 14:		Line 14:

	This article describes scales with the following property as having a ''well-formed GS'' (WFGS){{idiosyncratic}}:		This article describes scales with the following property as having a ''well-formed GS'' (WFGS){{idiosyncratic}}:
	* There exists a positive integer ''k'' such that for every generator ''x''<sub>''i''</sub> in the GS recipe GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), every occurrence of ''x''<sub>''i''</sub> in the scale [[subtend]]s ''k'' steps. This automatically implies that the gap between the next higher equave and the result of stacking len(scale) ~~−~~ 1 of the generators in the recipe, called the ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps.		* There exists a positive integer ''k'' such that for every generator ''x''<sub>''i''</sub> in the GS recipe {{nowrap\|GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>)}}, every occurrence of ''x''<sub>''i''</sub> in the scale [[subtend]]s ''k'' steps. This automatically implies that the gap between the next higher equave and the result of stacking {{nowrap\|len(scale) − 1}} of the generators in the recipe, called the ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps.
	* The closing generator must be distinct from all of the generators used in the generator sequence and occur only once in the scale.		* The closing generator must be distinct from all of the generators used in the generator sequence and occur only once in the scale.

Line 21:		Line 21:
	''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub> stacked together is called the ''aggregate generator''.		''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub> stacked together is called the ''aggregate generator''.

	To exclude the case when the generator is a 1-step or a (len(scale) ~~−~~ 1)-step, the modifier ''non-step''{{idiosyncratic}} can be used.		To exclude the case when the generator is a 1-step or a ({{nowrap\|len(scale) − 1}})-step, the modifier ''non-step''{{idiosyncratic}} can be used.

	Given a choice of equave ''E'' and an GS ''S'' = GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence GS(w<sub>1</sub>, ..., w<sub>''r''</sub>) where each w<sub>''i''</sub> is a sequence of ''k'' = ''k''(''i'') intervals, ''y''<sub>''i''1</sub>, ..., ''y''<sub>''ik''</sub>, where ''y''<sub>''i''1</sub> + ... + ''y''<sub>''ik''</sub> ≡ ''x''<sub>''i''</sub> ~~modulo~~ ''E''. If ''k'' does not depend on ''i'', call the splitting ''uniform''{{idiosyncratic}}. For instance, the GS for Zil, GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160, 8/7, 7/6) is a uniform splitting of GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any 2/1-equivalent WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of GS(3/2), corresponding to a unique [[pergen]] with a 3/2 period.		Given a choice of equave ''E'' and an GS {{nowrap\|''S'' {{=}} GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>)}}, a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence {{nowrap\|GS(w<sub>1</sub>, ..., w<sub>''r''</sub>)}} where each w<sub>''i''</sub> is a sequence of {{nowrap\|''k'' {{=}} ''k''(''i'')}} intervals, ''y''<sub>''i''1</sub>, ..., ''y''<sub>''ik''</sub>, where {{nowrap\|''y''<sub>''i''1</sub> + ... + ''y''<sub>''ik''</sub> ≡ ''x''<sub>''i''</sub> (mod ''E'')}}. If ''k'' does not depend on ''i'', call the splitting ''uniform''{{idiosyncratic}}. For instance, the GS for Zil, {{nowrap\|GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160, 8/7, 7/6)}}, is a uniform splitting of {{nowrap\|GS(4/3, 4/3, 4/3, 27/20, 4/3)}}, which generates Zarlino. Any 2/1-equivalent WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of GS(3/2), corresponding to a unique [[pergen]] with a 3/2 period.
	<!-- todo: Non-WF GSes-->		<!-- todo: Non-WF GSes-->
	== Basic properties of generator sequences ==		== Basic properties of generator sequences ==
	A generator sequence can be analyzed in terms of its length and variety.		A generator sequence can be analyzed in terms of its length and variety.
	=== Length ===		=== Length ===
	The ''length'' of a generator sequence ''s'' is the length at which the GS repeats; it is the smallest ''n'' > 0 such that ''s''[''k'' + ''n''] = ''s''[''k'']. It is known that a length-2 WFGS gives rise to regular SV3 scales; see [[Ternary scale theorems]].		The ''length'' of a generator sequence ''s'' is the length at which the GS repeats; it is the smallest {{nowrap\|''n'' > 0}} such that {{nowrap\|''s''[''k'' + ''n''] {{=}} ''s''[''k'']}}. It is known that a length-2 WFGS gives rise to regular SV3 scales; see [[Ternary scale theorems]].

	=== Generator variety ===		=== Generator variety ===
	The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.		The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.

	There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1, if we assume that the ''p'' generators are linearly independent and that ''k'' stacked generators equave-reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.		There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety {{nowrap\|''p'' > 1}} and for any {{nowrap\|''k'' > 1}}, if we assume that the ''p'' generators are linearly independent and that ''k'' stacked generators equave-reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.

	One-period MOS scales have step variety 2 and generator variety 1, and certain [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2.		One-period MOS scales have step variety 2 and generator variety 1, and certain [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2.
Line 63:		Line 63:
	== MOS substitution ==		== MOS substitution ==
	[[MOS substitution]] is a procedure that yields ternary scales with binary generator sequences.		[[MOS substitution]] is a procedure that yields ternary scales with binary generator sequences.

	== Multi-GS ==		== Multi-GS ==
	To extend the GS construction to a multiple-period MOS that splits the 2/1 into p > 1 periods, we can take a MOS-sized CS generated with a WFGS, and take offset copies of this scale by a detempered version of p-edo. This is what Inthar calls a ''multi-(WF)GS''.		To extend the GS construction to a multiple-period MOS that splits the 2/1 into {{nowrap\|''p'' > 1}} periods, we can take a MOS-sized CS generated with a WFGS, and take offset copies of this scale by a detempered version of ''p''-edo. This is what Inthar calls a ''multi-(WF)GS''.

	An example: GS(9:10:11:12) × 5:7, intended as a detempering of [[Hedgehog]][14]:		An example: {{nowrap\|GS(9:10:11:12) × 5:7}}, intended as a detempering of [[Hedgehog]][14]:

			<pre>
	21/20		21/20
	10/9		10/9
Line 82:		Line 84:
	28/15		28/15
	2/1		2/1
			</pre>

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Aberrismic theory]]		[[Category:Aberrismic theory]]

Inthar: /* JI scales obtained from guided generator sequences */

2024-09-17T09:39:26Z

JI scales obtained from guided generator sequences

← Older revision		Revision as of 09:39, 17 September 2024
Line 38:		Line 38:

	== JI scales obtained from guided generator sequences ==		== JI scales obtained from guided generator sequences ==
	~~Only CS sizes between 5 and 84, inclusive, are listed.~~
	* The Zarlino series, GS(5/4, 6/5) = GS(4:5:6): 7, 10, 17, 34, 58, 82		* The Zarlino series, GS(5/4, 6/5) = GS(4:5:6): 7, 10, 17, 34, 58, 82
	* The Tas/[[diasem]] series, GS(6:7:8): 5, 9, 14, 19, 24, 29, 53		* The Tas/[[diasem]] series, GS(6:7:8): 5, 9, 14, 19, 24, 29, 53

Inthar: /* Ternary scales and WFGS */

2024-08-11T13:52:34Z

Ternary scales and WFGS

← Older revision		Revision as of 13:52, 11 August 2024
Line 59:		Line 59:
	== Ternary scales and WFGS ==		== Ternary scales and WFGS ==
	If a ternary [[billiard scale]] has a WFGS, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4 and the closing generator is excluded from the generator sequence.		If a ternary [[billiard scale]] has a WFGS, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4 and the closing generator is excluded from the generator sequence.
	~~=== Conjectures ===~~
			[[Aberrismic theory]] also makes use of generator sequences; see [[guide frame]]s and other articles in the [[:Category:Aberrismic theory\|aberrismic theory category]].

	== MOS substitution ==		== MOS substitution ==

Inthar: /* Multi-GS */

2024-08-09T18:16:39Z

Multi-GS

← Older revision		Revision as of 18:16, 9 August 2024
Line 85:		Line 85:
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Scale]]		[[Category:Scale]]
			[[Category:Aberrismic theory]]

Inthar at 11:14, 24 May 2024

2024-05-24T11:14:21Z

← Older revision		Revision as of 11:14, 24 May 2024
Line 35:		Line 35:
	There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1, if we assume that the ''p'' generators are linearly independent and that ''k'' stacked generators equave-reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.		There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1, if we assume that the ''p'' generators are linearly independent and that ''k'' stacked generators equave-reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.

	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.		One-period MOS scales have step variety 2 and generator variety 1, and certain [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2.

	== JI scales obtained from guided generator sequences ==		== JI scales obtained from guided generator sequences ==

Inthar: /* Generator variety */

2024-05-23T03:47:14Z

Generator variety

← Older revision		Revision as of 03:47, 23 May 2024
Line 33:		Line 33:
	The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.		The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.

	There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1, if we assume ''k'' generators reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.		There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1, if we assume that the ''p'' generators are linearly independent and that ''k'' stacked generators equave-reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.

	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.		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.

Inthar: /* Generator variety */

2024-05-23T03:40:47Z

Generator variety

← Older revision		Revision as of 03:40, 23 May 2024
Line 33:		Line 33:
	The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.		The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.

	There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1 if we assume ''k'' generators reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.		There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1, if we assume ''k'' generators reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.

	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.		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.

Inthar: /* Generator variety */

2024-05-23T03:38:22Z

Generator variety

← Older revision		Revision as of 03:38, 23 May 2024
Line 33:		Line 33:
	The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.		The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.

	There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1 it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.		There is in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1 if we assume ''k'' generators reduce to a step, it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.

	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.		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.

Inthar: /* Generator variety */

2024-05-23T03:35:38Z

Generator variety

← Older revision		Revision as of 03:35, 23 May 2024
Line 33:		Line 33:
	The ''generator variety''{{idiosyncratic}} is the number of generators in the generator sequence, not including the closing interval.		The ''generator variety''{{idiosyncratic}} is the number of generators in the generator 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 in general no simple relationship between a scale's [[step variety]] and its generator variety. For any generator variety ''p'' > 1 and for any ''k'' > 1 it is possible to construct a long WFGS so that all combinatorially possible sums of ''k'' generators (there are <math>{k + p - 1 \choose k}</math> of them) are obtained for scale steps.
	~~# 2'''x''' + 2'''y''',~~
	~~# '''x''' + 3'''y''',~~		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.
	~~# 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.

	== JI scales obtained from guided generator sequences ==		== JI scales obtained from guided generator sequences ==

Inthar: /* Generator variety */

2024-05-23T03:15:38Z

Generator variety

← Older revision		Revision as of 03:15, 23 May 2024
Line 31:		Line 31:

	=== Generator variety ===		=== Generator variety ===
	~~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.		The ''generator variety''{{idiosyncratic}} is the number of generators in the generator 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):