Journal of Mathematics and Music, Volume 12, Issue 1, March 2018

https://www.tandfonline.com/toc/tmam20/12/1

Journal of Mathematics and Music aims to advance the use of mathematical modelling and computation in music theory. The Journal focuses on mathematical approaches to musical structures and processes, including mathematical investigations into music-theoretic or compositional issues as well as mathematically motivated analyses of musical works or performances. In consideration of the deep unsolved ontological and epistemological questions concerning knowledge about music, the Journal is open to a broad array of methodologies and topics, particularly those outside of established research fields such as acoustics, sound engineering, auditory perception, linguistics etc.

Three issues of the Journal are published per annual volume, with one issue devoted exclusively to a single topic. The online edition of the Journal provides extended material in the form of sound files, applets, or similar electronic media. The Journal also welcomes the submission of book reviews.

Readership

Journal of Mathematics and Music is intended to serve the communities of music scholars, composers, mathematicians and computer scientists, particularly those with interdisciplinary interests. The Journal assumes a certain level of proficiency in these fields, appropriate to professionals and graduate students.

To join the SMCM, please visit http://www.smcm-net.info/.

Peer Review Policy

All submitted manuscripts are subject to initial appraisal by the Editor. If found suitable for further consideration, papers are subject to peer review by independent, anonymous expert referees. All peer review is double blind and submissions can be made online at http://mc.manuscriptcentral.com/tmam

Publishing Ethics

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Articles

**Predicting Babbitt’s orderings of time-point classes from array aggregate partitions in None but the Lonely Flute and Around the Horn **

`Brian Bemman & David Meredith `

Milton Babbitt (1916–2011) is credited with developing several techniques of 12-tone composition that extend beyond pitch. One such technique, the time-point system, figures prominently in his mature rhythmic practice. In many of his works based on the all-partition array, the possible orderings of pitch classes and time-point classes from an aggregate partition are the same. However, the number available is often large and his reasons for choosing one ordering over another are diverse and not clearly understood. In this article, we propose that, when constructing linear orderings of time-point classes from aggregate partitions in an all-partition array, Babbitt attempted to minimize both (1) their dissimilarity to the orderings of pitch classes from the same aggregate partitions and (2) the amount of counter-evidence they provide against a preselected beat. We first review two existing measures, Rothgeb’s dissimilarity measure using order inversions and Povel and Essens‘ clock induction model. We then present a way to determine the exact number of possible orderings of time-point classes (and pitch classes) without repetitions in a particular aggregate partition. Next, we introduce a novel heuristic, based on the aforementioned measures, for predicting from the available orderings of time-point classes those particular orderings chosen by Babbitt. We conclude by evaluating how well this heuristic predicts the orderings of time-point classes found in two of Babbitt’s works, None but the Lonely Flute (1991) and Around the Horn (1993).

This work was carried out as part of the project “Learning to Create” (Lrn2Cre8), which acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission [FET grant number 610859].

**Expanded interval cycles **

`Adam H. Berliner, David Castro, Justin Merritt & Christopher Southard `

Pitch space is commonly represented using the face of a clock, in which the 12 pitch-classes are mapped onto the elements of Z12. Combining this form of representation with modular arithmetic results in the emergence of significant rotationally symmetric patterns, which can be used to generate pitch content and to effect a modulation between closely related collections within a composition. We generalize the situation to rotationally symmetric patterns with any number of intervals, and give necessary and sufficient conditions for when an interval pattern yields an analogous symmetry.

**Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis **

`Alexandre Popoff, Moreno Andreatta & Andrée Ehresmann `

Abstract

In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-nets) constitute a useful framework with connections in both group theory and graph theory. Recent attempts at formalizing K-nets in their most general form have evidenced a deeper connection with category theory. These formalizations use diagrams in sets, i.e. functors C→Sets where C is often a small category, providing a general framework for the known group or monoid actions on musical objects. However, following the work of Douthett–Steinbach and Cohn, transformational music theory has also relied on the use of relations between sets of the musical elements. Thus, K-net formalizations should be extended further to take this aspect into account. The present article proposes a new framework called relational PK-nets, an extension of our previous work on poly-Klumpenhouwer networks (PK-nets), in which we consider diagrams in Rel rather than Sets. We illustrate the potential of relational PK-nets with selected examples, by analyzing pop music and revisiting the work of Douthett–Steinbach and Cohn.

**Conference Opening Remarks **

**Opening remarks of the Sixth Biennial International Conference on Mathematics and Computation in Music (MCM 2017) in Mexico City, June 26–29, 2017 **

`Emilio Lluis-Puebla `

This is an edited transcript of my opening remarks at MCM 2017. In the opening address, I presented a short summary of the last three decades of the field, mathematically speaking, and gave a brief description of how it began and what has been done. The website of the conference is http://www.mcm2017.org/.

**Art **

**Rational Melody XVI**

`Tom Johnson `

The Rational Melodies were written in 1982, but this drawing was made in 2017 to illustrate a performance of Rational Melody XVI at the Institut Henri Poincaré in Paris. It is constructed with a simple automaton of the type that I call “sandwiches” in the book Self-Similar Melodies.

As seen in the center of the drawing, the calculation of the sequence begins with 1-2-1 and then follows this sandwiching rule: If the difference between two numbers is one, form the next level by sandwiching in the note one degree higher, but if the difference is two, sandwich in the note between the two. Thus the progression begins:

1-2-1

1-3-2-3-1

1-2-3-4-2-4-3-2-1

1-3-2-4-3-5-4-3-2-3-4-5-3-4-2-3-1

etc.

The melody begins with the outside circle and continues with phrases closer to the center, each about half as long as the previous one, ending with 1-2-1. The musical interest has much to do with the fact that, unlike most “correct” mathematical sequences, the phrases alternate between those beginning 1-3-2-4 and those beginning 1-2-3-4.

The performance at the Institut Henri Poincaré was by flutist Amélie Berson, and all 28 Rational Melodies may be heard in ensemble versions on the New World CD of Ensemble Dedalus.

Tom Johnson, Composer, Paris, France