Andrew J. Milne

Spectral tools for dynamic tonality and audio morphing

Computer Music Journal Spectral Tools for Dynamic Tonality and Audio Morphing William Sethares sethares@ece.wisc.edu, Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706 USA. Andrew Milne andymilne@tonalcentre.org, Department of Music, P.O. Box 35, 40014, University of Jyvaskyla, Finland. Stefan Tiedje Stefan-Tiedje@addcom.de, CCMIX, Paris, France. Anthony Prechtl aprechtl@gmail.com, Department of Music, P.O. Box 35, 40014, University of Jyvaskyla, Finland. James Plamondon jim@thumtronics.com, CEO, Thumtronics Inc., 6911 Thistle Hill Way, Austin, TX 78754 USA

Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum

Andrew Milne, William Sethares, and James Plamondon *Department of Music University of Jyvaskyla Finland andymilne@tonalcentre.org **Department of Electrical and Computer Engineering University of Wisconsin-Madison Madison, WI 53706 USA sethares@ece.wisc.edu †Thumtronics Inc. 6911 Thistle Hill Way Austin, TX 78754 USA jim@thumtronics.com Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum

Modelling the similarity of pitch collections with expectation tensors

Models of the perceived distance between pairs of pitch collections are a core component of broader models of music cognition. Numerous distance measures have been proposed, including voice-leading, psychoacoustic, and pitch and interval class distances; but, so far, there has been no attempt to bind these different measures into a single mathematical or conceptual framework or to incorporate the uncertain or probabilistic nature of pitch perception. This paper embeds pitch collections in expectation tensors and shows how metrics between such tensors can model their perceived dissimilarity. Expectation tensors indicate the expected number of tones, ordered pairs of tones, ordered triples of tones, etc., that are heard as having any given pitch, dyad of pitches, triad of pitches, etc. The pitches can be either absolute or relative (in which case the tensors are invariant with respect to transposition). Examples are given to show how the metrics accord with musical intuition.

A midi sequencer that widens access to the compositional possibilities of novel tunings

We present a new Dynamic Tonality MIDI sequencer, Hex, that aims to make sequencing music in and across a large variety of novel tunings as straightforward as sequencing in twelve-tone equal temperament. It replaces the piano roll used in conventional MIDI sequencers with a two-dimensional lattice roll in order to enable the intuitive visualization and dynamic manipulation of tuning. In conventional piano roll sequencers, a piano keyboard is displayed on the left side of the window, and white and black note lanes extend horizontally to the right, into which a user can draw a sequence of notes. Similarly, in Hex, a button lattice is displayed in its own pane on the left side of the window, and horizontal lines are drawn from the center of each note to the right. These lines function as generalized note lanes, just like in piano roll sequencers, but with the added benefit that each note lanes height is always proportional to its pitch, even if the user changes the tuning. The presence of the button lattice on the left side of the window illustrates exactly which buttons a performer would play in order to replicate the sequence when playing a physical button lattice instrument.

Testing a spectral model of tonal affinity with microtonal melodies and inharmonic spectra

Tonal affinity is the perceived goodness of fit of successive tones. It is important because a preference for certain intervals over others would likely influence preferences for, and prevalences of, “higher-order” musical structures such as scales and chord progressions. We hypothesize that two psychoacoustic (spectral) factors—harmonicity and spectral pitch similarity—have an impact on affinity. The harmonicity of a single tone is the extent to which its partials (frequency components) correspond to those of a harmonic complex tone (whose partials are a multiple of a single fundamental frequency). The spectral pitch similarity of two tones is the extent to which they have partials with corresponding, or close, frequencies. To ascertain the unique effect sizes of harmonicity and spectral pitch similarity, we constructed a computational model to numerically quantify them. The model was tested against data obtained from 44 participants who ranked the overall affinity of tones in melodies played in a variety of tunings (some microtonal) with a variety of spectra (some inharmonic). The data indicate the two factors have similar, but independent, effect sizes: in combination, they explain a sizeable portion of the variance in the data (the model-data squared correlation is r2 = .64). Neither harmonicity nor spectral pitch similarity require prior knowledge of musical structure, so they provide a potentially universal bottom-up explanation for tonal affinity. We show how the model—as optimized to these data—can explain scale structures commonly found in music, both historical and contemporary, and we discuss its implications for experimental microtonal and spectral music.

Empirically testing Tonnetz , voice-leading, and spectral models of perceived triadic distance

We compare three contrasting models of the perceived distance between root-position major and minor chords and test them against new empirical data. The models include a recent psychoacoustic model called spectral pitch-class distance, and two well-established music theoretical models – Tonnetz distance and voice-leading distance. To allow a principled challenge, in the context of these data, of the assumptions behind each of the models, we compare them with a simple “benchmark” model that simply counts the number of common tones between chords. Spectral pitch-class distance and Tonnetz have the highest correlations with the experimental data and each other, and perform significantly better than the benchmark. The voice-leading model performs worse than the benchmark. We suggest that spectral pitch-class distance provides a psychoacoustic explanation for perceived triadic distance and its music theory representation, the Tonnetz. The experimental data and the computational models are available in the Online Supplement (http://dx.doi.org/10.1080/17459737.2016.1152517).

Computational creation and morphing of multilevel rhythms by control of evenness

We present an algorithm, instantiated in a freeware application called MeanTimes, that permits the parameterized production and transformation of a hierarchy of well-formed rhythms. Each “higher” rhythmic level fills in the gaps of all “lower” levels, and up to six such levels can be simultaneously sounded. MeanTimes has a slider enabling continuous variation of the ratios of the intervals between the beats (onsets) of the lowest level. This consequently changes—in a straightforward manner—the evenness of this level; it also changes—in a more complex, but still highly patterned manner—the evennesses of all higher levels. This specific parameter, and others used in MeanTimes, are novel: We describe their mathematical formulation, demonstrate their utility for generating rhythms, and show how they differ from those typically used for pitch-based scales. Some of the compositional possibilities continue the tradition of Cowell and Nancarrow, proceeding further into metahuman performance, and have perceptual and cognitive implications that deserve further attention.

Exploring the space of perfectly balanced rhythms and scales

Periodic scales and meters typically embody “organizational principles” – their pitches and onset times are not randomly distributed, but structured by rules or constraints. Identifying such principles is useful for understanding existing music and for generating novel music. In this article, we identify and discuss a novel organizational principle for scales and rhythms that we feel is of both theoretical interest and practical utility: perfect balance. When distributed around the circle, perfectly balanced rhythms and scales have their “centre of gravity” at the centre of the circle. The present article serves as a repository of the theorems and definitions crucial to perfect balance. It also further explores its mathematical ramifications by linking the existing theorems to algebraic number theory and computational optimizations. In the Online Supplement, http://www.dynamictonality.com/perfect_balance_files/, we provide audio samples of perfectly balanced rhythmic loops and microtonal scales, computational routines, and video demonstrations of some of the concepts.

Perfect Balance: A Novel Principle for the Construction of Musical Scales and Meters

We identify a class of periodic patterns in musical scales or meters that are perfectly balanced. Such patterns have elements that are distributed around the periodic circle such that their ‘centre of gravity’ is precisely at the circle’s centre. Perfect balance is implied by the well established concept of perfect evenness (e.g., equal step scales or isochronous meters). However, we identify a less trivial class of perfectly balanced patterns that have no repetitions within the period. Such patterns can be distinctly uneven. We explore some heuristics for generating and parameterizing these patterns. We also introduce a theorem that any perfectly balanced pattern in a discrete universe can be expressed as a combination of regular polygons. We hope this framework may be useful for understanding our perception and production of aesthetically interesting and novel (microtonal) scales and meters, and help to disambiguate between balance and evenness; two properties that are easily confused.