Scale properties simplified

(Redirected from Myhill's property)

A simplified explanation of the various properties of periodic scales.

Definitions

Interval
A specific musical interval (e.g. a major third or minor seventh).
Generic interval (ordinal category)
A class of intervals which fall on the same scale degrees. In the diatonic scale, these classes are the set of seconds, the set of thirds, the set of fifths, 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, can be called a "k-step" (terminology taken from TAMNAMS).
Scale degree
The number of steps subtended by a generic interval from the tonic. In conventional music theory, degrees of the diatonic scale are usually given 1-indexed ordinal names corresponding to the diatonic intervals themselves, for example the fifth degree for the degree occupied by the fifth. However, in xenharmonic music, 0-indexed names are preferred by some people for degrees of non-diatonic scales: the k-degree is the degree represented by the k-step generic interval from the tonic.

Properties

Alternating generator (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
• Epimorphism: A JI scale is epimorphic if, under some val, all scale degrees are "filled," no matter which note you choose as the tonic, and successive degrees are always increasing. Without the second condition, the scale is only weakly epimorphic.
• Epimorph val/temperament: A val that witnesses that a JI scale is epimorphic is called the epimorph val of the scale, and a temperament supported by an epimorph val is an epimorph temperament. Many low-accuracy edos and temperaments are useful as epimorph vals and temperaments, and these temperaments imply structure rather than tuning; a CS scale may be constructed as a detempering of the low-accuracy tuning implied by such a temperament.
• Example: 5-limit Zarlino is a 2.3.5 JI scale that is epimorphic under the val 7 11 16], and the 2.3.5 temperaments dicot and meantone are both epimorph temperaments for Zarlino.
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.
Myhill's property and MOS
• Myhill's/MOS property: A scale has Myhill's property if there are exactly two interval sizes for each (reduced) interval class not including the equave. A scale is a MOS scale if there are no more than two interval sizes for each generic interval class not including the equave. This is equivalent to a scale being Myhill with a smaller equave. Myhill's property is sometimes called "strict MOS".
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.

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, 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.

Arity, binary, ternary, n-ary
The number of distinct step sizes occurring in a given scale. Arity disregards other properties, such as rank or maximum variety. For example, 12edo melodic minor is a binary scale which is not rank-2 or MV2 (MOS).
Convexity
Maximal evenness
Pepper ambiguity