Generator sequence

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. 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 (GS) may be preferable, as alternating is usually used for sequences that repeat every two terms.

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

When all generators x_i in the AGS recipe AGS(x₁, ..., x_r) subtend the same number of steps (not depending on i), we call the resulting scale well-formed GS (WFGS). This automatically implies that the leftover interval after stacking len(scale) − 1 of the generators in the recipe (analogous to the imperfect generator in MOS scales) also subtends this number of steps. 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.

To exclude the case when the generator is a 1-step or a (len(scale) − 1)-step, the modifier non-step can be used.

Given a choice of equave E and a GS S = GS(x₁, ..., x_r), a refinement 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 refinement uniform. 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 refinement of GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino.

Series arising from well-formed generator sequences

Only CS sizes at least 5 are listed.

• The Zarlino series, AGS(5/4, 6/5): 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(7/6, 8/7): 5, 9, 14, 19, 24, 29
• The Tri-Stone series, 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
• AGS(7/5, 19/14, 80/57)
• AGS(19/14, 51/38, 23/17, 63/46, 19/14, 51/38, 23/17, 896/621)
• AGS(10/9, 11/10, 12/11)
• AGS(10/9, 11/10, 12/11, 10/9, 11/10, 12/11, 10/9, 11/10, 189/176)

Conjectures about AGS scales

• Let n be the length of a constant structure well-formed AGS scale S with equave E. Then n is the cardinality of an E-equivalent primitive MOS generated by the guide generator of S.