Step ratio

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The step ratio of a scale is the ratio between the sizes of its steps. The step ratio of a binary scale is also known as hardness, alluding to the terms hard and soft used to name step ratios in TAMNAMS, or as Blackwood's R, after Easley Blackwood who described it for diatonic scales in The Structure of Recognizable Diatonic Tunings and referred to this ratio as R.

For binary scales, including moment-of-symmetry scales, the step ratio is usually written in the form x:y, where x and y represent the sizes of the large and small steps respectively (x > y). For example, in a scale with a step ratio of 2:1, each large step has twice the size of a small step. In general, for n-ary scales, the step ratio is usually written in the form x1:x2:...:xn, where x1, x2, ..., xn represent the size of each step in decreasing order of size.

The step ratio of a scale is mostly associated with its melodic shape. Scales whose steps are very similar in size may sound melodically smoother, softer, or more mellow. In contrast, scales with steps of very different sizes may sound jagged, dramatic, or sparkly.

A binary scale becomes "softer" as its step ratio approaches 1:1 and "harder" when approaching ∞:1 (or 1:0). At the extremes, either the large and small steps become equal, or the small step "collapses" to zero. In both cases, it degenerates into a unary scale (by omitting the zero-sized intervals if necessary), with a step ratio trivially equal to 1.

Relative interval sizes

Todo: clarify
This section relies on multiple assumptions which are not made clear to the reader (including the knowledge of advanced MOS concepts, which should minimally be linked to, or ideally avoided whenever possible).

Part of this perception stems from the fact that, as these x:y ratios change and pass certain critical rational values, the next MOS in the sequence changes structure entirely. For instance, when we have L:s > 2, the next MOS changes from "xL ys" to "yL xs". As an example, with the "5L 2s" diatonic MOS, if we have L/s < 2, the next MOS will be "7L 5s", and if we have L/s > 2, the next MOS will be "5L 7s". (At the point L/s = 2, we have that the next MOS is an equal temperament.)

Similar things happen with all of these rational points. As the L:s ratio decreases and passes 3/2, for instance, the MOS that is two steps after the current one changes. Again, as an example, with the familiar 5L 2s diatonic MOS sequence, if we have 3:2 < L:s < 2:1, the next two MOS's have 19 and 31 notes, whereas if we have L:s < 3:2, the next two MOS's have 19 and 26 notes.

Another way to look at this is using Rothenberg propriety: it so happens that, with one small exception, if a MOS has L:s < 2:1, it is "strictly proper", if it has L:s > 2:1, it is "improper", and if it has L:s = 2:1, it is "proper", all using Rothenberg's definition. The one exception is if the MOS has a single small step (e.g. it is of the form xL 1s), at which point it is always "strictly proper". Similarly we pass the L:s = 3:2 boundary, the next MOS changes from strictly proper to improper, and so on.

The special ratio L:s = φ is unique in that it is the only ratio in which the MOS is strictly proper, and all of the following MOS's are also strictly proper.

TAMNAMS naming system for step ratios

In TAMNAMS, a step ratio of 2:1 is named basic. Step ratios between 1:1 and 2:1 are named soft-of-basic, with 3:2 being soft, while step ratios between 2:1 and ∞:1 are named hard-of-basic, with 3:1 being hard. Other names are given to several other common ratios, as well as to ranges between these ratios.

It is also proposed to use pairwise naming for step ratios of ternary scales. For example, 3:2:1 would be named soft-basic, because 3:2 is soft and 2:1 is basic.