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.

After encountering Julian Hook's work on uniform triadic transformations, Jack Douthett wrote letters to John Clough suggesting further group-theoretic generalizations of Hook's idea. These letters have been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.

Jack Douthett wrote a number of letters to John Clough and Richard Cohn concerning Cohn's “P-Relations,” single-semitone voice-leading relationships. The ideas in these letters led to graph-theoretic and geometric models. The following selection has been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.

Mathematical Music Theory, pp. 3-19 (2018) Free AccessChapter 1: From Musical Chords to Twin PrimesJack Douthett, David Clampitt, and Norman CareyJack DouthettUniversity of New Mexico, Albuquerque, NM, USA, David ClampittThe Ohio State University, Columbus, OH, USA, and Norman CareyCUNY Graduate Center, New York, NY, USAhttps://doi.org/10.1142/9789813235311_0001Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: The following sections are included: Introduction Myhill's Property and Well-Formed Scales Complementary WF Scales Cardinality Equals Variety Dual CCV Bibliography FiguresReferencesRelatedDetails Mathematical Music TheoryMetrics History PDF download

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 using FiPS, and map familiar transformational graphs to maximally even coordinate spaces, thereby separating transformational groups from the scalar contexts in which they are most often explained. We also look at differences in the definition and usage of maximal evenness in continuous space, and examine the impact of scalar context on voice-leading distance.

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 Point-Symmetry, which were left out of ‘Scalar Context in Musical Models’.

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 fraction analysis. We show that generated scales can be optimized to preserve the pure-tone character of the scale by compromising the modulation/transposition properties by a least amount. The other four measures are generalizations that apply to scales generated by multiple intervals. These measures are directly applied to individual scales and numerically compared. We apply this formalism to pure intervals of import in antiquity and show that this approach yields the historically important just 12-tone system as an optimum scale. Finally, we show how this formalism can be applied to the analysis of non-standard scales.

Submitted for the 4CF11 Meeting of The American Physical Society The Role of Higher Harmonics In Musical Interval Perception RICHARD KRANTZ, Metropolitan State College of Denver, JACK DOUTHETT, University of New Mexico (retired) — 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 even with a relatively small addition of higher harmonics the perfect fifth emerges as a consonant interval with more, musically important, just intervals emerging as consonant as more and more energy is shifted into higher frequencies. Richard Krantz Metropolitan State College of Denver Date submitted: 10 Sep 2011 Electronic form version 1.4

At the 2007 Helmholtz Workshop in Berlin, two seemingly disparate papers were presented. One of these, by Julyan Cartwright, Diego Gonzalez, 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, maximally even (ME) sets, and their relationship to the one-dimensional antiferromagnetic Ising model. Both these approaches lead to a fractal structure involving Farey series known as a devil's staircase. Why is this? What is the connection between them? The ME sets approach is related to the Ising model of statistical physics; on the other hand, the forced oscillator model relates to the circle map of dynamical systems. Thus we find ourselves facing a deeper question: what are the links between these two paradigms, the Ising model and the circle map, that are fundamental to statistical physics on the one hand and to dynamical systems on the other? Here we present the two halves of the work side by side, so that the construction of the two arguments that arrive at a devil's staircase in music can be seen together.

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 are needed to identify well-formed scales, and a number of equivalent definitions for well-formed scales with irrational generators are explored. Continued fraction convergents are also important in determining equal-tempered scales (ETSs) with good fits for just intervals. Best ETSs of the first, second, and third kind are all dependent on continued fractions. Best ETSs of the second kind are determined by principal convergents, while principal, last-half, and sometimes middle convergents are needed to determine best ETSs of the first and of the third kinds. Hall's Remarkability Function and Krantz and Douthett's Desirability Functions for multiple target intervals are compared. For multiple target intervals, the Remarkability Function is problematic in that a single interval can dominate the fitting measure of a collection of target intervals. This is a problem not shared by the Desirability Function. The formalism throughout this paper is designed to accommodate analysis of non-standard scales (scales with closure intervals other than the octave), as demonstrated in the analysis of the Bohlen-Pierce scale, which closes at the tritave (3:1).

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 is constructed that allows these ordered pairs to be added consistent with our expectations. Similarly, accidental modifiers numerically identify the number of sharps or flats attached to a given note: 0 indicates no attached accidentals, negative integers indicate the number of flats attached, and positive integers indicate the number of sharps attached. These modifiers are linked with diatonic classes as ordered pairs that mimic common note names. For example, the note G ♭ is represented by (–1,4) and Gx by (2,4). Intervals and notes represented by these ordered pairs are said to be in MD-notation (MD for modifier-diatonic). A group action and generalized interval system are defined for intervals and notes in MD-notation. An implied quartertone system is also discussed.

(douthett@unm.edu) holds two Master ofMusic degrees, one in performance and the other in theoryand composition, and received his Doctorate ofMathematics from the University of New Mexico. He haspublished in the disciplines of mathematics, physics,acoustics, and music theory, and in 1993, he and JohnClough received the Society of Music Theory’s OutstandingPublication Award for their work on maximally even setsand scale theory. He is currently Visiting Professor at theUniversity of New Mexico. In his free time, he enjoys hikingand visiting art galleries.

Until now, diatonic systems and neo-Riemannian transformations have generally been considered separately, and group and graph theoretic approaches have dominated the neo-Riemannian and transformational theory literature. In this paper, an alternative approach will be explored; techniques similar to those used in the study of dynamical systems in science will be adopted to study neo- Riemannian theory and its connection to diatonic theory.

Uniform Triadic Transformations and the Twelve-Tone Music of Webern I? 9 Julian Hook and JackDouthett FREQUENTLY A THEORETICAL PRINCIPLE finds application in situa tions very different from those forwhich itwas originally devised. The annals of science, medicine, and technology are teeming with serendipitous discoveries and unsuspected applications, and the music theory literature offers a few examples of itsown. Allen Forte's theory of pitch-class sets has been applied in the rhythmic domain, in the form of beat-class sets, and in a mod-7 (rather than mod-12) universe, in the development of diatonic set theory.Many of the fundamental concepts in transformation theory grew out of considerations arising in the study of post-tonal music, but soon found gainful employment in the neo Riemannian literature in the service of tonal and triadic structures.1 92 PerspectivesofNew Music This paper turns the last idea on its head: we borrow a technique originally formulated for the study of triadic transformations and apply it,with minimal modification, to investigate the twelve-tone music of Anton Webern. The technique in question involves uniform triadic transformations, or UTTs. UTTs were introduced as an extension of neo-Riemannian theory, a means of simultaneously generalizing, simp lifying,and uniting a broad swath of diverse work in triadic transforma tion theory. The central observation of this paper is that the objects on which UTTs act need not be triads at all. Closely analogous relation ships obtain among other types of musical structures, notably twelve tone rows, therebymaking such structures amenable to UTT analysis. Moreover, our examples (drawn fromWebern's Variations for Piano, Op. 27; String Quartet, Op. 28; and Concerto for Nine Instruments, Op. 24) will demonstrate that such an analysis can yield musical insights that might escape notice in a conventional twelve-tone analysis.2 Introduction to UTTs As originally formulated (and as their name implies), UTTs act on triads, represented here by ordered pairs of the form A = (r, a). Here r is an integermod 12 signifying the root of the triad (following the usual convention C = 0), and a is a sign indicating the triad's mode: + for major, ? for minor. Thus (0, +) represents the C-major triad, while (10, ?) represents B-flat minor. The set of 24 triads is denoted T. The components r and a of a triad A may be written as ta and a a at times when confusion with other r and a is possible. We will often abbreviate the ordered-pair notation, writing (for example) 0+ and 10~ as short hand for (0, +) and (10, -). A uniform triadic transformation (UTT) is a transformation defined on the set T, represented by an ordered triple U= (a, t+, f) (always written in angle brackets). Here a is a 4- or ? sign (called the sign ofU and denoted au when necessary), and t+ and t~ are integers mod 12, called the transposition levels of U for major and minor triads, respectively. If U= (av, t+, t~) is a UTT and A = (r, a a) is a triad, then U acts on A by transposing its root upward by either t+ or t~ semitones, depending on whether F) The action of U on any other triad can be deduced from these. Ifwe know that aUTT transformsC major to E minor, then itmust be of the form (?, 4, x), and so automatically itmust transform T>\> major to F minor, D major to Ftfminor, and so on: the definition of the UTT action ensures that itmust transform all major triads "the same way," according to the rules specified by t+ and av. Likewise, any UTT that transforms Bb minor to F major must be of the form (?, x, 7) and must transform B minor to Ftfmajor, C minor to G major, and so on. This property is the uniformity condition to which the name "UTT" refers; because of uniformity, the behavior of any UTT is completely known once its action is specified on one major triad and one minor triad. Many familiar triadic relationships can be represented as UTTs. For instance, for any n = 0, 1, . . . , 11, theUTT Tn = (+,?,?) represents the operation of pc-set transposition on triads, and P = (?, 0, 0...

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 equivalent under rotation and reflection, and a simple algorithm is constructed that generates these configurations. A number of equivalent conditions that determine a maximally even configuration are established. These equivalent conditions permit maximally even configurations to apply to a number of seemingly disparate problems including the dinner table and concentric circles problems, the one-dimensional antiferromagnetic Ising model, and musical scales.

Based on previous work by Krantz and Douthett [1, 2, 3] and Sethares [7, 8], briefly reviewed in Section 2, a universal measure of the equal-tempered musical scale consonance is developed in Section 3. Preliminary results applying this, so-called, Integrated Scale Desirability Function for scales made up of complex tones show that higher frequency components of complex tones are necessary to explain the emergence of our usual 12-tone equal-tempered musical scale. These results are discussed in Section 4.

This conclusion presents some closing thoughts on the concepts covered in the preceding chapters of this book. The book hypothesizes that the cognition of musical structure occurs through particular types of relationship that the mind constructs – typically subconsciously – between musical events that are the same or similar, through which one is felt to imitate another or others. It sets out a model that develops this thinking in more detail, distinguishing identities that occur by chance from those that are likely to figure in the perceived structural equation. The book draws a distinction too between relationships assumed to function as percepts during the normal course of listening and those realized as concepts in the act of analysis. Zygonic meta-analysis facilitates an exploration of the underlying similarities in their conceptual architecture, and points up the necessity in all such enterprises of assessing the probable …