Generator sequence: Difference between revisions
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:''Note: This page is chiefly maintained by Inthar. Terms indicated as idiosyncratic are his coinages, not necessarily Scott Dakota's.'' | :''Note: This page is chiefly maintained by Inthar. Terms indicated as idiosyncratic are his coinages, not necessarily Scott Dakota's.'' | ||
'''{{PAGENAME}}''' (''' | '''{{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 of 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 | Currently, the study of GSs is dominated by [[constant structure]] GS scales, called guide generator sequence scales, which are obtained by using detemperings of MOS generators andstopping the stacking procedure at the correspondibg MOS scale sizes, which yields constant scales scales. | ||
Certain [[generator-offset property|generator-offset]] scales are examples. For example, [[diasem]] is | 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 == | ||
* The term '''generator sequence'''{{idiosyncratic}} (GS) may be preferable, as ''alternating'' is usually used for sequences and series that repeat every two terms in mathematical terminology. | * The term '''generator sequence'''{{idiosyncratic}} (GS) may be preferable, as ''alternating'' is usually used for sequences and series that repeat every two terms in mathematical terminology. | ||
* Consider a scale whose steps are all positive. Suppose that there exists a positive integer ''k'' such that for every generator ''x''<sub>''i''</sub> in the | * Consider a scale whose steps are all positive. Suppose that 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 ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. Suppose also that the imperfect generator is distinct from all of the generators used in the generator sequence and occurs only once in the scale. | * 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 ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. Suppose also that the imperfect generator is distinct from all of the generators used in the generator sequence and occurs only once in the scale. | ||
When all of the above hold, this article calls the resulting scale ''well-formed GS'' (WFGS){{idiosyncratic}}. In such a situation, we call the (logarithmic) average of the generators the ''guide generator''. The choice of "well-formed" is informed by the well-formed property of single-period MOS scales: the property that each occurrence of the generator subtends the same number of steps. The reason that WFGSes are studied is that the sequence yields CS scales at sizes of MOS scales generated by the guide generator and with the same period used by the WFGS, as long as the MOS scale in question is not too large. In summary, WFGS scales are made by detempering a MOS's generator chain into a stacked generator sequence, and the MOS sizes of the guide generator can help predict the sizes at which the | When all of the above hold, this article calls the resulting scale ''well-formed GS'' (WFGS){{idiosyncratic}}. In such a situation, we call the (logarithmic) average of the generators the ''guide generator''. The choice of "well-formed" is informed by the well-formed property of single-period MOS scales: the property that each occurrence of the generator subtends the same number of steps. The reason that WFGSes are studied is that the sequence yields CS scales at sizes of MOS scales generated by the guide generator and with the same period used by the WFGS, as long as the MOS scale in question is not too large. In summary, WFGS scales are made by detempering a MOS's generator chain into a stacked generator sequence, and the MOS sizes of the guide generator can help predict the sizes at which the GS scale will be CS. This is because WFGS is designed to be exactly the right condition such that when one equates all of the generators of a WFGS chain, one gets a MOS scale which will be CS (this MOS has an abstract generator so there are no concerns about linear independence). As, by assumption, there are no steps in the original scale that are negative relative to the MOS, the original scale will thus be CS as well. | ||
''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''. | ||
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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 (len(scale) − 1)-step, the modifier ''non-step''{{idiosyncratic}} can be used. | ||
Given a choice of equave ''E'' and an | 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. | ||
<!-- todo: Non-WF GSes--> | <!-- todo: Non-WF GSes--> | ||
=== Proof that a WFGS scale is constant structure === | === Proof that a WFGS scale is constant structure === | ||
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== JI scales from WFGS series == | == JI scales from WFGS series == | ||
Only CS sizes at least 5 are listed. Todo: check for larger CS sizes. | Only CS sizes at least 5 are listed. Todo: check for larger CS sizes. | ||
* The Zarlino series, | * The Zarlino series, GS(5/4, 6/5) = GS(4:5:6): 5, 7, 10, 17, 24, 41, 65 | ||
** Other scales with the same | ** Other scales with the same GS structure of two thirds adding up to 3/2 share the same CS sizes, including undecimal Zarlino (GS(11/9, 27/22)), and Neogothic Zarlino (GS(14/11, 13/11) with [[364/363]] tempered), although the latter may break at higher sizes depending on how the intervals are tuned. | ||
* The Tas/[[diasem]] series, | * The Tas/[[diasem]] series, GS(6:7:8): 5, 9, 14, 19, 24, 29 | ||
* | * GS(3/2, 14/9): 5, 8, 13, 18 | ||
* The Zil series, | * The Zil series, GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160, 8/7, 7/6): 5, 9, 14, 19, 24 | ||
* The Porcusmine series, | * The Porcusmine series, GS(9/5, 50/27): 5, 6, 7, 8, 15, 23, 38, 61, 99 | ||
* An unnamed 5-limit Mavila detemper, | * An unnamed 5-limit Mavila detemper, GS(3/2, 3/2, 64/45): 5, 7, 9, 16, 25 | ||
* The Rhombi series, | * The Rhombi series, GS(14/9, 11/7, 52/33, 81/52): 5, 8, 11, 14, 17, 31, 48, 65 | ||
* The Dwyn series: | * The Dwyn series: GS(25/24 21/20 22/21 23/22 24/23 21/20 22/21 23/22 24/23): 15, 16, 31, 46 | ||
* | * GS(13/11, 16/13, 77/64, 13/11, 16/13, 33/28): 7, 11, 15, 19 | ||
* A "Magic" detemper, | * A "Magic" detemper, GS(13:16:20:25:31:39): 7, 10, 13, 16, 19, 22, 41 | ||
* | * GS(30:42:57:80) | ||
* | * GS(19/14, 51/38, 23/17, 63/46, 19/14, 51/38, 23/17, 896/621) | ||
* A Porcupine detemper, | * A Porcupine detemper, GS(9:10:11:12) | ||
* | * GS(9:10:11:12, 9:10:11:12, 9:10:11, 189/176) | ||
* A "Bleu" detemper, | * A "Bleu" detemper, GS(22:24:26:28:31:33) | ||
* A Machine detemper, | * A Machine detemper, GS(8/7, 9/8, 112/99, 9/8) | ||
== Ternary scales and WFGS == | == Ternary scales and WFGS == | ||
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== Extensions == | == Extensions == | ||
These are extensions proposed to the | These are extensions proposed to the GS concept by others. | ||
=== Multi-(A)GS === | === Multi-(A)GS === | ||
To extend the | 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''. | ||
An example: | An example: GS(9:10:11:12) × 5:7, intended as a detempering of [[Hedgehog]][14]: | ||
21/20 | 21/20 | ||