Generator sequence

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

Generator sequence (AGS) is a scale-building procedure first described by Scott Dakota. The notation AGS(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. An enumerated chord can be used instead for the argument: the chord's steps are the generators, and the chord's stacked end-to-end, e.g. AGS(4:5:6)[7] for Zarlino, which is syntactic sugar for AGS(5/4, 6/5)[7]. When x_r is stacked, we go back to x₁ and start stacking x₁ again, then x₂, ... Currently, the study of AGSs is dominated by constant structure AGS scales, which are obtained by stopping the stacking procedure at scale sizes that yield constant-structure scales. The term generator sequence^{[idiosyncratic term]} (GS) may be preferable, as alternating is usually used for sequences and series that repeat every two terms in mathematical terminology.

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

Terminology

• Consider a scale whose steps are all positive. Suppose that there exists a positive integer k such that for every generator x_i in the AGS recipe AGS(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 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.

When all of the above hold, this article calls the resulting scale well-formed GS (WFGS)^{[idiosyncratic term]}. 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 AGS 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.

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 AGS S = AGS(x₁, ..., x_r), a splitting^{[idiosyncratic term]} of S is a generator sequence AGS(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, AGS(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 AGS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino.

JI scales from WFGS series

Only CS sizes at least 5 are listed. Todo: check for larger CS sizes.

• The Zarlino series, AGS(5/4, 6/5) = AGS(4:5:6): 5, 7, 10, 17, 24, 41, 65
  • Other scales with the same AGS structure of two thirds adding up to 3/2 share the same CS sizes, including undecimal Zarlino (AGS(11/9, 27/22)), and Neogothic Zarlino (AGS(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, AGS(6:7:8): 5, 9, 14, 19, 24, 29
• AGS(3/2, 14/9): 5, 8, 13, 18
• The Zil series, AGS(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, AGS(9/5, 50/27): 5, 6, 7, 8, 15, 23, 38, 61, 99
• An unnamed 5-limit Mavila detemper, AGS(3/2, 3/2, 64/45): 5, 7, 9, 16, 25
• The Rhombi series, AGS(14/9, 11/7, 52/33, 81/52): 5, 8, 11, 14, 17, 31, 48, 65
• The Dwyn series: AGS(25/24 21/20 22/21 23/22 24/23 21/20 22/21 23/22 24/23): 15, 16, 31, 46
• AGS(13/11, 16/13, 77/64, 13/11, 16/13, 33/28): 7, 11, 15, 19
• A Magic detemper, AGS(13:16:20:25:31:39): 7, 10, 13, 16, 19, 22, 41
• AGS(30:42:57:80)
• AGS(19/14, 51/38, 23/17, 63/46, 19/14, 51/38, 23/17, 896/621)
• A Porcupine detemper, AGS(9:10:11:12)
• AGS(9:10:11:12, 9:10:11:12, 10/9, 11/10, 189/176)
• A Bleu detemper, AGS(22:24:26:28:31:33)