Perfect balance
A non-empty finite set of real numbers S in the range [math]\displaystyle{ [0, 1) }[/math] is called perfectly balanced if a wheel with an equal weight placed at angle [math]\displaystyle{ 2\pi x }[/math] for each [math]\displaystyle{ x \in S }[/math] has its center of gravity exactly at the hub. Mathematically, this is given by the equation [math]\displaystyle{ \sum_{x \in S} e^{2\pi i x} = 0 }[/math].
In the context of musical tunings, a perfectly balanced set can be converted to a periodic scale by taking the frequency ratio [math]\displaystyle{ 2^x }[/math] for each [math]\displaystyle{ x }[/math], producing a scale that repeats at the octave. Any other interval of equivalence may be chosen, but for the sake of this article octave-equivalence is assumed. Perfectly balanced sets have been investigated in the context of generating repeating rhythms as well, such as in the freeware app XronoMorph.
In general, the balance of a set is given by [math]\displaystyle{ B = 1 - \frac{1}{K}\left|\sum_{x \in S} e^{2\pi i x}\right| }[/math] where [math]\displaystyle{ K }[/math] is the size of the set. A set is perfectly balanced iff its balance is the maximum value of 1.
Within EDOs
The connection to music theory was originally noted by David Lewin in 1959, who noted that certain scales in 12edo had this property, which he called the "exceptional property." For example, the hexatonic scale {0, 3, 4, 7, 8, 11} (one of Messiaen's modes of limited transposition) is perfectly balanced. This is because the 3edo triads {0, 4, 8} and {3, 7, 11} are each perfectly balanced, and if two sets are perfectly balanced and disjoint then so their union.
The only perfectly balanced scale within a prime EDO N is the equally spaced scale containing every tone in N-EDO. For a composite EDO N, one can construct new perfectly balanced scales by superimposing equally spaced scales of size k where k is a divisor of N, such that the scales are transposed so that no pitches coincide. For example, in 18edo the 3edo {0, 6, 12} and the 2edo {1, 10} may be combined to form the perfectly balanced scale {0, 1, 6, 10, 12}.
This leads to the question of whether every perfectly balanced scale within an EDO is decomposable into a union of one or more equally spaced disjoint scales. This turns out to be false, and counterexamples can occur when N has three or more distinct prime factors. The smallest EDO with three distinct prime factors is 30edo, and indeed it has six such scales up to transposition:
{0, 1, 7, 13, 19, 20}
{0, 1, 2, 12, 13, 19, 20}
{0, 1, 7, 11, 17, 18, 24}
{0, 1, 7, 8, 14, 18, 20, 24}
{0, 1, 2, 8, 12, 18, 19, 20}
{0, 1, 2, 8, 12, 14, 18, 20, 24}
Searching for these scales is nontrivial. Milne et al. computed all such patterns for products of three distinct primes up to N = 102.
Outside EDOs
It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition. The space of perfectly balanced scales of size K > 1 forms a K-dimensional manifold, which is in general complex and poorly understood.
Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. This is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (u, v), project the points back onto the original circle by dividing by the norm, then compute the cost function [math]\displaystyle{ \left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2 }[/math] where [math]\displaystyle{ \mathbf{x} }[/math] and [math]\displaystyle{ \mathbf{y} }[/math] are vectors of the x- and y-coordinates. Use any standard unconstrained optimization procedure to find u and v so that the cost function is minimized. It can be seen that the cost is 0 iff perfect balance is achieved.
Due to the convexity of the problem the minimum is guaranteed global, but it may not always exist if the original scale is too unbalanced. It is unclear from sources whether the minimal cost is always 0 if it exists, but this seems to be the case in practice.
For example, a perfectly balanced approximation to Ptolemy's intense diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following Scala file:
Ptolemy's diatonic scale approximated with perfect balance 7 212.52434916042787 406.31587841801996 521.6495827358298 721.9824694633705 894.3281503142967 1088.9892227955181 2/1
The effect here is rather subtle, as the diatonic scale is already close to balanced. As a more dramatic example, the perfectly balanced version of the 12edo harmonic minor scale displaces it by the cent values [0.0, +22.96, +40.23, +57.08, +44.32, +32.06, +0.59].
Search procedures for perfectly balanced scales under other optimization criteria are conceivable. Minimizing harmonic entropy is one such approach.