Periodic scale
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This is an expert page. It is written to allow experienced readers to learn more about the advanced elements of the topic. The corresponding beginner page for this topic is Scale. |
A periodic scale is a scale with a period. A periodic tuning system can be conceived analogously for tuning systems, if such a distinction is made.
Mathematical definition
A periodic scale may be defined in mathematical language as a type of quasiperiodic function from the integers to musical intervals; the integers in this case formalize the notion of "scale degrees". Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale s has a nonzero quasiperiod P and repetition interval O satisfying the following conditions
[math](1)\ s[0] = 0[/math]
[math](2)\ s[i + P] = s[i] + O[/math]
Scales written in the widely used Scala format are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, since arbitrarily high and low pitches go beyond the range of human hearing, this definition is a mathematical idealization, but it is much simpler to adopt the idealization than to worry about that. Neither Scala nor the above definition assumes that the scales are monotonically strictly increasing, but this condition, giving a monotone periodic scale, is often important to add:
[math](3)\ i \lt j\text{ implies }s[i] \lt s[j][/math]
Rotations
By a rotation or mode of a periodic scale s is meant a scale r such that r[i] = s[i + N] - s[N], where N is a fixed integer. Since s[i + P] - s[P] = s[i] there are only a finite number of rotations, equal to the number of notes of the scale reduced to the range of the interval of equivalence, 0 ≤ s[i] < O, which entails 0 ≤ i < P.
Classes
We may define an important function class(i) on the integers which gives the generic intervals of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties.
Step form and cumulative form
Given a periodic scale as defined above, we may define its step form as
[math]\Delta s[i] = s[i+1] - s[i],[/math]
where [math]\Delta s[i][/math] is the ith step of the scale. [math]\Delta s[/math] has the property [math]\Delta s [i + P] = \Delta s[i] \ \forall i \in \mathbb{Z}.[/math]
The step form [math]\Delta s[/math] and the cumulative form [math]s[/math] of a periodic scale are related by the fundamental theorem of finite-difference calculus:
[math]\displaystyle{\sum_{i=n_0}^{n_1} \Delta s[i] = s[n_1+1]-s[n_0] \ \text{for $n_1 \ge n_0$.}}[/math]
Thus, we may equivalently define a periodic scale as a periodic (in the usual mathematical sense) sequence of positive step sizes.
Scale properties
Constant structure
If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by Erv Wilson) means that i ≠ j implies class(i) ∩ class(j) = ∅. In academic music theory, this is called the partitioning property.
Propriety
If s is monotone, and if i ≤ j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i < j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called coherence. Note that strict propriety implies constant structure.
The set {s[i] | i ∈ ℤ} generates a group G, the group of the scale; this is a free, finitely generated subgroup of the reals ℝ. The rank of the scale is the rank of G.
Epimorphism
If there exists a homomorphism h: G → ℤ so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by Yves Hellegouarch. The name comes from the fact that h is an epimorphism onto ℤ.
Myhill's property
A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the trivalence property. Myhill's property is synonymous with strict MOS, though some authors prefer to identify MOS itself with Myhill's property.
Distributional evenness
A monotone scale in which every class comes in exactly n elements is n-distributionally even, or n-DE. If n = 2, then we can simply say that it is distributionally even. Distributional evenness is also synonymous with MOS, though some authors prefer a stricter definition of MOS identifying it with Myhill's property.
Convexity
The scale is convex if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod P is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval O is a ℤ-polytope in the lattice defined by a basis for G mod O.
Maximal evenness
Maximally even scales of n notes in m-edo are any mode of the sequence ME(n, m) = [floor(im/n) | i = 1, …, n], where the "floor" function rounds down to the nearest integer.