Generator sequence

Note: This page is chiefly maintained by Inthar. Terms indicated as idiosyncratic are his coinages, not necessarily Scott Dakota's.

Generator sequence (GS) is a scale-building procedure first described by Scott Dakota.

The notation GS(x₁, ..., x_r) denotes a scale-building procedure where a (periodic) scale is built by stacking x₁ first, x₂ second, ..., reducing by the scale's equave when necessary. When x_r is stacked, we go back to x₁ and start stacking x₁ again, then x₂, ... 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 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

This concept was formerly known as AGS or alternating generator sequence and was renamed to its current name since alternating indicates a sequence of 2 generators with period 2.
One can use arity terminology for the number of distinct generators in a GS. When we have 2 (resp. 3) distinct generators, the generator sequence is binary (resp. ternary).

This article describes scales with the following property as having a well-formed GS (WFGS)⁠ ⁠^{[idiosyncratic term ]}:

There exists a positive integer k such that for every generator x_i in the GS recipe GS(x₁, ..., x_r), every occurrence of x_i in the scale subtends 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.
The closing generator must be distinct from all of the generators used in the generator sequence and occur only once in the scale.

Whereas guided GS is a procedure, WFGS provides a stopping condition for the procedure of guided GS described above. 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.

x₁, ..., 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 term ]} can be used.

Given a choice of equave E and an GS S = GS(x₁, ..., x_r), a splitting⁠ ⁠^{[idiosyncratic term ]} of S is a generator sequence GS(w₁, ..., w_r) where each w_i is a sequence of k = k(i) intervals, y_i1, ..., y_ik, where y_i1 + ... + y_ik ≡ x_i modulo E. If k does not depend on i, call the splitting uniform⁠ ⁠^{[idiosyncratic term ]}. 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.

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.

Generator variety

The generator variety⁠ ⁠^{[idiosyncratic term ]} 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.

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

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
GS(3/2, 14/9): 5, 8, 13, 18, 31, 49, 67, 85
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, 43
The Porcusmine series, GS(9/5, 50/27): 5, 6, 7, 8, 15, 23, 38, 61, 99
An unnamed 5-limit Mavila detemper, GS(3/2, 3/2, 64/45): 5, 7, 9, 16, 25, 84
The Rhombi series, GS(14/9, 11/7, 52/33, 81/52): 5, 6, 8, 11, 14, 17, 31, 48, 65
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, 77
GS(13/11, 16/13, 77/64, 13/11, 16/13, 33/28): 7, 11, 15, 19
A "Magic" detemper, GS(13:16:20:25:31:39): 7, 10, 13, 16, 19, 22, 41
GS(30:42:57:80): 5, 7, 9, 11, 13, 15, 17, 19, 36, 53
GS(19/14, 51/38, 23/17, 63/46, 19/14, 51/38, 23/17, 896/621): 5, 6, 7, 8, 9, 10, 11, 20
A Porcupine detemper, GS(9:10:11:12): 5, 6, 7, 8, 15, 22, 29
GS(9:10:11:12, 9:10:11:12, 9:10:11, 189/176): 5, 6, 7, 8, 15, 22
A "Bleu" detemper, GS(22:24:26:28:31:33): 5, 6, 7, 8, 9, 17
A Machine detemper, GS(8/7, 9/8, 112/99, 9/8): 5, 6, 11, 17, 28, 45
A Slendric detemper, GS(8/7, 147/128, 8/7): 5, 6, 11, 16, 21, 26, 31, 36, 41, 77

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.

Aberrismic theory also makes use of generator sequences; see guide frames and other articles in the 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.

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

21/20 10/9 7/6 11/9 77/60 12/9 7/5 3/2 14/9 5/3 77/45 11/6 28/15 2/1