Generator sequence: Difference between revisions

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'''{{PAGENAME}}''' ('''GS''') is a scale-building procedure first described by [[Scott Dakota]].

The notation GS(''x''1, ..., ''x''''r'') denotes a scale-building procedure where a ([[Periodic scale|periodic]]) scale is built by stacking ''x''1 first, ''x''2 second, ..., reducing by the scale's [[equave]] when necessary. When ''x''''r'' is stacked, we go back to ''x''1 and start stacking ''x''1 again, then ''x''2, ... 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''1, ..., ''x''''r'')}} denotes a scale-building procedure where a ([[Periodic scale|periodic]]) scale is built by stacking ''x''1 first, ''x''2 second, ..., reducing by the scale's [[equave]] when necessary. When ''x''''r'' is stacked, we go back to ''x''1 and start stacking ''x''1 again, then ''x''2, ... 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''.

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 ==

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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''''i'' in the GS recipe GS(''x''1, ..., ''x''''r''), every occurrence of ''x''''i'' 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''''i'' in the GS recipe {{nowrap|GS(''x''1, ..., ''x''''r'')}}, every occurrence of ''x''''i'' 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.

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''x''1, ..., ''x''''r'' 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''1, ..., ''x''''r''), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence GS(w1, ..., w''r'') where each w''i'' is a sequence of ''k'' = ''k''(''i'') intervals, ''y''''i''1, ..., ''y''''ik'', where ''y''''i''1 + ... + ''y''''ik'' ≡ ''x''''i'' ~~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''1, ..., ''x''''r'')}}, a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence {{nowrap|GS(w1, ..., w''r'')}} where each w''i'' is a sequence of {{nowrap|''k'' {{=}} ''k''(''i'')}} intervals, ''y''''i''1, ..., ''y''''ik'', where {{nowrap|''y''''i''1 + ... + ''y''''ik'' ≡ ''x''''i'' (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.

== Basic properties of generator sequences ==

A generator sequence can be analyzed in terms of its length and variety.

=== 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 ===

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 {{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.

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 ==

~~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 Tas/[[diasem]] series, GS(6:7:8): 5, 9, 14, 19, 24, 29, 53

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== 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.

~~=== 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]] is a procedure that yields ternary scales with binary generator sequences.

== 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

10/9

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28/15

2/1

</pre>

[[Category:Terms]]

[[Category:Scale]]

[[Category:Aberrismic theory]]

@@ Line 3: / Line 3: @@
 '''{{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,&nbsp;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''.
-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,&nbsp;7/6) or GS(7/6,&nbsp;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 ==
@@ Line 14: / Line 14: @@
 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) &minus; 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.
@@ Line 21: / Line 21: @@
 ''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) &minus; 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-->
 == Basic properties of generator sequences ==
 A generator sequence can be analyzed in terms of its length and variety.
 === 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 ===
 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 {{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.
-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 ==
-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 Tas/[[diasem]] series, GS(6:7:8): 5, 9, 14, 19, 24, 29, 53
@@ Line 59: / Line 58: @@
 == 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.
-=== 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]] is a procedure that yields ternary scales with binary generator sequences.
 == 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>
 /20
 /9
@@ Line 82: / Line 84: @@
 /15
 /1
+</pre>
 [[Category:Terms]]
 [[Category:Scale]]
+[[Category:Aberrismic theory]]