Jack Douthett

Central New Mexico · Math, Science, and Engineering

We extend the theory of maximally even sets to determine the evenness of partitions of the chromatic universe Uc. Interactions measure the average evenness of colour sets (partitioning sets) of Uc. For 2-colour partitions the Clough-Douthett maximal-evenness algorithm determines maximally even partitions. But to measure the evenness of non-maximall...

This chapter explores the connection between Cohn cycle and other hypercubes neo-Riemannian topics. The generalized middle layer conjecture and its musical applications is also discussed.

This Chapter gives a music theoretic equivalent statement statement to the twins prime conjecture. This musical statement adapts well-formed scales, maximally even sets, and other modern mathematical-music theory concepts to trace musical chords to twin primes.

In this paper, we examine relationships between signature transformations, Filtered Point-Symmetry (FiPS), and voice-leading spaces, with a strong emphasis on the role of scalar context in each model. While these models overlap, the differences between them are of substantial analytical importance. We determine how to expand the signature group usi...

It is exciting to have this opportunity to read and react to three other substantial papers. Marek Žabka's idea to advocate this format deserves substantial praise, as does Rick Cohn's organization of the most recent Clough Conference. In order to offer our reactions, we need to briefly expand our discussion of two relevant concepts in Filtered Poi...

We develop formalism for generating pure-tone systems that best approximate the modulation/transposition properties of equal-tempered scales. We define six measurements to determine the closeness of scales generated by pure intervals to equal-tempered scales. Two measures apply to scales generated by single intervals and rely heavily on continued f...

Using an alternative parameterization of the roughness curve we make direct use of critical band results to investigate the role of higher harmonics on the perception of tonal consonance. We scale the spectral amplitudes in the complex home tone and complex interval tone to simulate acoustic signals of constant energy. Our analysis reveals that eve...

At the 2007 Helmholtz Workshop in Berlin, two seemingly disparate papers were presented. One of these, by Julyan Cartwright, Diego González, and Oreste Piro, dealt with a nonlinear dynamical model for pitch perception based on frequency ratios and forced oscillators, while the other, by Jack Douthett and Richard Krantz, focused on musical scales, m...

Some time ago two apparently dissimilar presentations were given at the 2007 Helmholtz Workshop in Berlin. One by J. Douthett and R. Krantz focused on the commonality between the mathematical descriptions of musical scales and the long-ranged, one-dimensional, anti-ferromagnetic Ising model of statistical physics. The other by J. Cartwright, D. Gon...

Numbers called quality modifiers are used to identify interval qualities: 0 numerically represents perfect, 1/2 represents major, –1/2 represents minor, and so on. These modifiers are linked with diatonic class intervals as ordered pairs that mimic common interval notation. For example, a minor third is represented by (–1/2, 2). A binary operator i...

Although it is common practice to borrow tools from mathematics to apply to physics or music, it is unusual to use tools developed in music theory to mathematically describe physical phenomena. So called ``Maximally Even Set'' theory fits this unusual case. In this poster, we summarize, by example, the theory of Maximally Even (ME) sets and show ho...

How should men and women be seated around a dinner table to maximize conversation between members of the opposite sex? What can be said about the distribution of points around two concentric circles? How are the white and black keys on the piano keyboard organized? What spin configuration in the Ising model minimizes energy? These four problems hav...

Convex (concave) interaction weighting functions are combined with circular configurations of black and white sites to determine configurations that have minimum (maximum) weight. These configurations are called maximally even configurations. It is shown that for a given number of black and white sites, all maximally even configurations are equival...

Classical algorithms for principal and intermediate continued fraction convergents provide convenient ways of obtaining information about musical scales. It is shown that the principal convergents of the generators of generated scales provide a way of identifying scales with best Pythagorean type commas. Both principal and intermediate convergents...

Using recent developments in music theory, which are generalizations of the well-known properties of the familiar 12-tone, equal-tempered musical scale, an approach is described for constructing equal-tempered musical scales (with "diatonic" scales and the associated chord structure) based on good-fitting intervals and a generalization of the modul...

Cyclic configurations of white and black sites, together with convex (concave) functions used to weight path length, are investigated. The weights of the white set and black set are the sums of the weights of the paths connecting the white sites and black sites, respectively, and the weight between sets is the sum of the weights of the paths that c...

By extending the formalism of maximally even sets to the one-dimensional antiferromagnetic Ising model with arbitrary-range interaction in an applied magnetic field it is shown that the system exhibits a devil’s-staircase magnetic phase diagram. A “generalized” similarity dimension (a generalization for nonuniform fractals) of the phase diagram is...

Connections between parsimonious structures and modes of limited transportation from three set classes are explored. A graph-theoretic approach proves useful in illustrating the symmetries inherent in parsimonious structures and modes of limited transposition. Four parsimonious graphs called mode graphs are constructed. Each mode graph consists of...

The one-dimensional antiferromagnetic spin-1/2 Ising model is investigated using the formalism of Maximally/Minimally Even sets. The salient features of Maximally/Minimally Even set theory are introduced. Energy and spin content vectors are defined to facilitate the use of interval spectra used in Maximally/Minimally Even set theory. It is shown th...

Using three criteria a desirability function is developed which allows comparison of the reasonableness of equal-tempered musical scales to approximate just musical intervals. Unlike most other measurements, this scale is not dominated by very fine octave divisions. This feature allows for comparison of systems with few divisions to the octave with...

Two questions are addressed: (1) By what path did ancient Indian musicians make their way from a “chromatic” universe of 22 microtonal divisions of the octave (the śrutis) to a “diatonic” set of seven degrees? and (2) What features do the resulting scale structures have in common with later versions of the diatonic scale in the West? The authors ex...

It is shown that every graph in any of the following three classes contains a cycle of length 0mod3: (i) cubic graphs, (ii) subdivisions of 3-connected cubic graphs with order n≥10 and (iii) subdivisions of graphs with order n and 3n-5 or more edges. The bound in (iii) is sharp. The proof of (ii) is by induction on n, the base case being establishe...