Partitions, their classes, and multicolour evenness

Abstract

We extend the theory of maximally even sets to determine the evenness of partitions of the chromatic universe $U_c$. Interactions measure the average evenness of colour sets (partitioning sets) of $U_c$. For 2-colour partitions the Clough-Douthett maximal-evenness algorithm determines maximally even partitions. But to measure the evenness of non-maximally even partitions, it is necessary to use computational methods. Moreover, for more than two colour sets there is no simple algorithm that determines maximally even partitions. Again, we rely on computational methods. We also explore collections of partitions and partition-classes (orbits under a dihedral group) and construct tables that order partition-classes according to the evenness of their partitions. We use Bell numbers, Stirling numbers of the second kind, and integer partitions to enumerate relevant combinatorial objects related to our investigation.

Keywords:

• maximally even
• multicolour evenness
• squashed ordering
• partition-class
• partition tree
• integer partitions
• $\mathfrak{p}$-functions
• Bell numbers
• Stirling numbers of the second kind
• Z-relations

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Minimally even (me) sets and configurations are also discussed in Douthett (1999).

2 For other approaches related to the partitioning of $U_c$, there are a number of works that explore the partitioning of twelve-tone rows: see Alegant (2001), Alegant and Lofthouse (2002), Elliott (2022), Forte (1973), Hook and Douthett (2008), Mead (1988, 1989), Morris (1987, 1991, 2007), Peck (2013), Perle (1997), Rahn (1980), and Starr (1978) for example. In pcset theory there are number of related works that discuss voice leading metrics and measures and other mathematical aspects of evenness and smoothness. To name a few see Amiot (2007, 2009, 2016), Beauguitte (2011), Callender (2004), Callender, Quinn, and Tymoczko (2008), Carey (1998), Cartwright et al. (2019), Clampitt (2008), Cohn (1996, 1997, 2012), Demaine et al. (2009), Dominguez, Clampitt, and Noll (2009), Douthett (1999), Douthett, Clampitt, and Carey (2019), Douthett, Steinbach, and Hermann (2019), Douthett and Hook (2009), Douthett and Krantz (2007), Gamer and Wilson (1983, 2003), Gomez-Martin, Taslakian, and Toussaint (2009), Harasim, Schmidt, and Rohrmeier (2020), Hashimoto (2018, 2020), Hook (2008), Krantz and Douthett (2005, 2011), Krantz, Douthett, and Doty (1998), Lewin (1987, 1996), Milne, Bulger, and Herff (2017), Montiel and Gomez (2014), Noll (2014), Peck (2009), Plotkin (2010, 2013, 2019a, 2019b), Plotkin and Douthett (2013), Rahn (1991), Toussaint (2005), Tymoczko (2005, 2011), Yust (2013, 2016), and Žabka (2013).

3 An unordered partition is one in which no distinction is made between partitioning sets that have the same size. For example, the partitions 0,1,2,3 and 2,3,0,1 of $U_4$ are distinct as ordered partitions but are equivalent as unordered partitions.

4 Anderson (1987) defines the zero of a poset as a unique member a (if one exists) such that $a\le b$ for all b in the poset. While squashed ordering is not a partial ordering (no antisymmetry), in the context of our paper it seems reasonable to extend this definition of zero to squashed ordering to the smallest member of the chain of inequalities. What are called prime or normal form of a set-class by Rahn (1980) and Morris (1987, 1991) are the zeros of the set-class.

5 We pick $c=4$ for our examples because $c<4$ yields trivial information, and $c>4$ would generate pages of information. For those reasons $c=4$ is the Goldilocks zone for examples.

6 Technically we should write ${\mathfrak{p}}_4^{\mathcal{P}}\left(\right(4\left)\right)$, indicating (4) is the argument of the ${\mathfrak{p}}_4^{\mathcal{P}}$. But for simplicity we leave out one of the pairs of parentheses and write ${\mathfrak{p}}_4^{\mathcal{P}}\left(4\right)$. Similarly, we leave out the extra pair of parentheses for ${\mathfrak{p}}_4^{\mathcal{O}}$. We adopt this convention for all the variables. Also, the notation ${\mathcal{O}}_{\left(4\right)\text{ - }1}$ denotes (4) “dash” 1, not “minus” 1, throughout.

7 The term rotation index was introduced by Douthett (1999) since r determines the transposition of the ME set. Clough and Douthett used the term mode index.

8 While Douthett and Krantz (2007) gave the first general computational definition for maximally even sets, the seeds for this definition came 11 years earlier in their physics paper on energy extremes and the Ising model (see Douthett and Krantz 1996).

9 Douthett (1999) and Douthett and Krantz (2007) use the notation $\text{dis}{\text{t}}_c(a,b)$ for $\text{ar}{\text{c}}_c(a,b)$.

10 We have chosen the Dinner Table interaction because of its intuitive appeal. For the Ising model, interactions are generally decreasing convex functions such as $J\left(x\right)=e^{-\alpha x}$ or $J\left(x\right)=x^{-\alpha }$ where $\alpha >0$. Then ME partitions (called configurations in Physics) have minimum values (configurational energy) and me partitions have maximum configurational energy. In addition, for the Ising model the “average configurational energy” is taken over the total number of sites c (density). Averaging over the number of sites simplifies the calculation, and $A=\frac{1}{c}{\mathrm{\Phi }}_s(J,c)$. Again, for intuitive appeal we average over the number of chords.

11 The averages ${\text{A}}_1$, ${\text{A}}_2$, and ${\text{A}}_X$ are based on the interaction $J_{\text{D}}$ which is concave on $[1,c-1]$ while $A_Y$ is based on the interaction $-J_{\text{D}}$ which is convex on $[1,c-1]$.

12 This table is like Clough’s (1979) Diatonic Chord Table except that Clough defines equivalence only under transposition.

13 We are jumping a bit ahead here, but what we are asking is not a difficult task at this point. We will return to discuss n-color partitions for $n>2$ in more detail shortly.

14 Curiously in his original work on set-complexes in music, Forte (1964) remarks in his endnote 6 that in the rigorous mathematical sense the ICVs are not vectors. But that is exactly how we use them.

15 One might wonder why Forte (1973) did not list Z-related sets next to one another in his table. In his article that introduced his table of SCs, Forte (1964) lumped Z-related sets in the same SC. John Clough (1965) pointed out that sets not equivalent under transposition and inversion should be assigned to separate SCs. Forte (1965) initially took exception to Clough’s suggestion. But in his book, Forte (1973) must have reconsidered Clough’s suggestion, as he divided the Z-related sets into separate SCs. Rather than reorganize his SC table, he simply listed half the Z-related SCs to the end of their respective cardinal families.

16 In a sense, Y-partitions are self Y-related, but we prefer the term Y-partitions.

17 Rappaport (2005), for example, shows that if the vertex set of an inscribed k-gon is a ME set, then its area of the k-gon is maximum when compared with the area of all k-gons inscribed in a c-circle.

18 Section 6 will give a generating function that gives us an alternative way to determining this number.

19 This triangle is often referred to as the Stirling Triangle in the combinatorics literature.

20 For the sake of brevity, we have left out a considerable amount of supporting material for this theorem. We refer the inquiring reader to Theorem 4 in Harary and Palmer (1966), which is a slightly more general version of our theorem. The Orbit Theorem is tailored for unordered partitions.

21 Using de Bruijn’s Theorem, May and Wierman (2005, Appendix 1) derive our Table 8 in their work on percolation threshold bounds.

22 Note that ${\text{T}}_{2}P\text{ = }P$ since the partitions in ${\mathfrak{p}}_{12}^{\mathcal{P}}(3,3,2,2,2)$ are unordered.

23 The standard notation for the partition number of c is $p\left(c\right)$. However, in the context of this paper $|N\left(c\right)|$ is more appropriate.

24 This partition problem was the central problem discussed in the movie “The Man Who Knew Infinity” about the eminent Indian mathematician Srinivasa Ramanujan. Feitosa (2020) gives a rather detailed discussion on partition numbers and related combinatoric issues as they pertain to music theory. Those curious about these connections might refer to Feitosa’s paper, as it also includes an excellent bibliography.