Periodic scale
|
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, or in layman's terms, a "table" that maps integers (which formalize the notion of "scale degrees") to intervals given in cents (hence, an additive notation will be used, with the stacking of intervals notated by addition). In this case, a periodic scale s has a nonzero quasiperiod P (the period in scale steps) and repetition interval O, also notated s[P] (the period in cents) where by adding P to the scale degree, O is always added to the resulting interval.
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 strictly increasing, but this condition, giving a monotone periodic scale, is often important to add.
Here is the above in terms of mathematical statements:
[math]\displaystyle{ (1)\ s[0] = 0 }[/math]
[math]\displaystyle{ (2)\ s[i + P] = s[i] + O }[/math]
[math]\displaystyle{ (3)\ i \lt j\text{ implies }s[i] \lt s[j] }[/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 in scale steps 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.
Modes
A mode, or "rotation", of a periodic scale is a scale r such that r[i] = s[i + N] - s[N], where N is a fixed integer; in other words, it is the same scale pattern, but starting on a different scale degree. 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] < s[P], which entails 0 ≤ i < P.
Classes
A class is a category of all intervals spanning a specified number of scale degrees, such as seconds, thirds, fourths etc in diatonic, or the generalization to any kind of scale.
In mathematical terms, we can define a function class(k) on the integers which gives sets representing the generic intervals of a periodic scale. For some integer k, the set class(k) consists of all intervals [math]\displaystyle{ s[k+i] - s[i] }[/math]. Equivalently, it is all the intervals found on the same degree of the different modes of the scale, or all the intervals between notes a given number of scale steps apart. Since s is quasiperiodic, class(P) only contains the period O, but the rest may contain multiple intervals.
Step form and cumulative form
Given a periodic scale, we may call the function defined above the "cumulative form", and we may define its step form as
[math]\displaystyle{ \Delta s[i] = s[i+1] - s[i], }[/math]
where [math]\displaystyle{ \Delta s[i] }[/math] is the ith step of the scale. [math]\displaystyle{ \Delta s }[/math] has the property [math]\displaystyle{ \Delta s [i + P] = \Delta s[i] \ \forall i \in \mathbb{Z}. }[/math]; in other words, the "step form" of the scale repeats across periods.
The step form [math]\displaystyle{ \Delta s }[/math] and the cumulative form [math]\displaystyle{ s }[/math] of a periodic scale are related by the fundamental theorem of finite-difference calculus:
[math]\displaystyle{ \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]; in other words, the size of a scale degree can be obtained by summing up all the step sizes that build up to it.
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) = ∅ - that is, class(i) and class(j) have no elements in common, or in other words, there are no true enharmonic equivalents. 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 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 linear map h: G → ℤ so that h(s[i]) = i, then s is weakly epimorphic with the map 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 first considered by Yves Hellegouarch.[1] The name comes from the fact that h is an epimorphism onto the integers (i.e. the map h is surjective).
Myhill's property
A monotone scale in which every class but classes nP have exactly two elements is a MOS with period P (as opposed to a fraction of P; that is, a strict MOS), and thus has Myhill's property. If every such class has exactly three elements, it has the trivalence 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 and is thus a MOS (of a more general form). 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.
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.
Numerical properties
See also
References
- ↑ Yves Hellegouarch, A Mathematical Interpretation of Expressive Intonation, in Mathematics and Art, p. 141-148, Springer-Verlag, 2002