# Scale properties simplified

A simplified explanation of the various properties of periodic scales.

## Definitions

Scale degree
The amount of steps subtended in an interval. (A perfect fifth falls on the 5th scale degree; so does a diminished fifth).
Interval
A specific musical interval (e.g. a major third or minor seventh).
Generic interval
A class of intervals which fall on the same scale degrees (e.g. thirds, fifths, sixths, etc). Generic intervals can also be likened to distances between note-heads on a traditional staff. A generic interval composed of k scale steps in any scale, diatonic or not, is called a "k-step" in TAMNAMS.

## Properties

Alternating generator (alt-gen, AG) property
A scale satisfies the alternating generator property if it satisfies the following equivalent properties:
• the scale can be built by stacking alternating generators, for example 7/6 and 8/7.
• the scale is generated by two chains of generators separated by a fixed interval; either both chains are of size m, or one chain has size m and the second has size m-1.
Constant structure
A scale has constant structure (CS) if all Intervals of the same size are also within the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the partitioning property in most academic literature.
Propriety
• Propriety: A scale is proper if there is no overlapping of generic interval classes. This means that no third is larger than a fourth, no fourth is larger than a fifth, etc.
• Strict propriety: A scale is strictly proper if the generic interval classes are disjoint. Replace the word "larger" with "larger-than-or-equal-to" in the definition above. The 12-tone diatonic scale is proper, but not strictly proper.
Epimorphism
• Weak epimorphism: A scale is weakly epimorphic if, under some val, all scale degrees are "filled," no matter which note you choose as the tonic.
• Epimorphism: A weakly epimorphic scale is epimorphic if it keeps rising in pitch as you go to higher scale degrees – the (n + 1)st degree is higher than the nth degree.
Symmetry
A scale is symmetrical if at least one mode of the scale is symmetrical. Therefore, every interval of that mode must have an inverse. These scales will always have an odd number of notes per period. They may not always have an odd number of notes per octave, however. The diatonic scale is symmetrical, but so is 12edo.
MOS/DE/Myhill's
• Distributional evenness: A scale is distributionally even (DE) if there are no more than two interval sizes for each generic interval class (e.g. major/minor thirds, perfect/augmented fourths, etc).
• Myhill's property: A scale has Myhill's property if there are exactly two interval sizes for each generic interval class (except octaves or other equivalence intervals such as tritaves).
• Trivalence property: Same as Myhill's property, but replace "two interval sizes" with "three interval sizes." The scale formed from the notes of a dominant 7th chord (e.g. C-E-G-Bb-C) is an example of a trivalent scale.
• Moment of symmetry: Both DE and Myhill's are essentially synonymous with MOS; Myhill's property is sometimes called "strict MOS".

The 12-tone diatonic scale has Myhill's property, and is also distributionally even.

The diminished scale is an MOS with a 1/4-octave period. Because there is only one interval size at the period, it does not have exactly two interval sizes per interval class. Therefore, it is MOS/DE, but doesn't have Myhill's property.

An EDO is a kind of degenerate MOS, in that it is distributionally even. It does not have Myhill's property. In other words, it has no more than two interval sizes for each generic interval class, but does not have exactly two interval sizes.

Convexity
Maximal evenness
Pepper ambiguity