Step ratio

The melodic sound of a MOS is not just affected by the tuning of its intervals, but by the sizes of its steps. MOSes with L more similar to s can sound smoother/softer/more mellow. MOSes with L much larger than s can sound jagged/dramatic/sparkly. For extreme tunings, the step pattern of the MOS will become increasingly ambiguous; this is as much a feature as a bug - it depends on your intent. The step ratio or hardness, the ratio between the sizes of L and s, is thus important to the sound of the scale. The step ratio has also been called Blackwood's R, after Easley Blackwood who described it for diatonic mosses and referred to this ratio as R.

Relative interval sizes

Part of this perception stems from the fact that, as these L:s 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

Main article: TAMNAMS #Step ratio spectrum