Microtonal Music

In Praise of Lou Harrison

by Joel Taylor

published in The Open Space Magazine, fall 2003

Some thoughts about Lou Harrison's music with emphasis on the experimental microtonal works, and the works for Gamelan. Some thoughts about Lou Harrison's music with emphasis on the experimental microtonal works, and the works for Gamelan.

A Computational Model for Rule-Based Microtonal Music Theories and Composition

by Torsten Anders

Anders, T., E. Miranda (2011). A Computational Model for Rule-Based Microtonal Music Theories and Composition. Perspectives of New Music. 48(2).

This paper presents a computational model for microtonal music theories and composition based on the constraint... more This paper presents a computational model for microtonal music theories and composition based on the constraint programming paradigm. The fundamental layer of this model is its pitch representation, which introduces variables for pitches, pitch classes, and (chord or scale) degrees, as well as their their dependencies. Constrainable representations for higher level pitch-related concepts such as chord and scale objects are defined. The model has been implemented in Strasheela, so that this model can be used together with other Strasheela feature (e.g., temporal score object hierarchies).

This paper demonstrates the proposed model in a number of case studies that implement microtonal theories of harmony, melody and counterpoint. These case studies also showed how the model supports various equal temperaments. We modeled a diatonic cadence in 12-tone equal temperament (12-TET); a 7-limit harmony progression in 31-TET and adaptive just intonation; a chord figuration of a chord from La Monte Young's The Well-Tuned Piano in 41-TET; and finally first a melody and and then harmonic counterpoint with Paul Erlich's static symmetrical major scale in 22-TET.

The presented model supports equal temperaments only; we are currently working on an extension for arbitrary regular temperaments including just intonation.

The Anti-systematic System. Hans Zender's Third Way between Lachenmann and Ferneyhough

by Håvard Enge

The German composer and conductor Hans Zender (born 1936) has addressed the question of pluralism in increasingly... more The German composer and conductor Hans Zender (born 1936) has addressed the question of pluralism in increasingly pointed ways in both his works and his theoretical essays. In this paper, I will discuss his theory of microtonal harmony, which he developed in the 1990s and refers to as an “anti-systematic system”. I will emphasize how Zender situates his theory as a “third way” between the school-building ideas of Lachenmann and Ferneyhough. As well as offering a middle way between the approaches to harmony in the works of these dominant contemporary modernists, Zender’s “anti-systematic system” suggests ways in which the experience of pluralism might be integrated in a flexible and open-ended, but still essentially modernist way of composition.

A great microtonal survey

by Mark Lindley

A review, published in the OUP journal, “Early Music” v.37/3 (August 2009), of Patrizio Barbieri, “Enharmonic Instruments and Music, 1470-1900.”

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

by Andrew Milne

Prechtl, A., Milne, A. J., Holland, S., Laney, R. Sharp, D. B. (2012). Computer Music Journal, 36(1), 42–54.

This is a preprint version of an article that has been accepted for publication in Computer Music Journal, 36(1).

We present a new Dynamic Tonality MIDI sequencer, Hex, that aims to make sequencing music in and across a large... more 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 lane's 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.

The Tone Diamond

by Andrew Milne

Milne (2007). Unpublished report.

The Tone Diamond controls the trade-offs between tuning regularity, harmonicity of timbre, and sensory dissonance in... more The Tone Diamond controls the trade-offs between tuning regularity, harmonicity of timbre, and sensory dissonance in Dynamic Tonality.

Hex Player—a virtual musical controller

by Andrew Milne

Milne, A. J., Xambó, A., Laney, R., Sharp, D. B., Prechtl, A., & Holland, S. (2011). Hex Player—a virtual musical controller. In A. R. Jensenius, A. Tveit, R. Godøy, & D. Overholt (Eds.), Proceedings of the 2011 International Conference on New Interfaces for Musical Expression (NIME11) (pp. 244–247). Oslo, Norway.

In this paper, we describe a playable musical interface for tablets and multi-touch tables. The interface is a... more In this paper, we describe a playable musical interface for tablets and multi-touch tables. The interface is a generalized keyboard, inspired by the Thummer, and consists of an array of virtual buttons. On a generalized keyboard, any given interval always has the same shape (and therefore fingering); furthermore, the fingering is consistent over a broad range of tunings. Compared to a physical generalized keyboard, a virtual version has some advantages—notably, that the spatial location of the buttons can be transformed by shears and rotations, and their colouring can be changed to reflect their musical function in different scales.

We exploit these flexibilities to facilitate the playing not just of conventional Western scales but also a wide variety of microtonal generalized diatonic scales known as moment of symmetry, or well-formed, scales. A user can choose such a scale, and the buttons are automatically arranged so their spatial height corresponds to their pitch, and buttons an octave apart are always vertically above each other. Furthermore, the most numerous scale steps run along rows, while buttons within the scale are light-coloured, and those outside are dark or removed.

These features can aid beginners; for example, the chosen scale might be the diatonic, in which case the piano’s familiar white and black colouring of the seven diatonic and five chromatic notes is used, but only one scale fingering need ever be learned (unlike a piano where every key needs a different fingering). Alternatively, it can assist advanced composers and musicians seeking to explore the universe of unfamiliar microtonal scales.

Scratching the scale labyrinth

by Andrew Milne

Milne, A. J., Carlé, M., Sethares, W. A., Noll, T., & Holland, S. (2011). In C. Agon, E. Amiot, M. Andreatta, G. Assayag, J. Bresson, & J. Mandereau (Eds.), Mathematics and Computation in Music: Third International Conference, MCM 2011, Paris, France, June 2011 (Lecture Notes in Computer Science, Vol. 6726, pp. 180–195). Berlin Heidelberg: Springer-Verlag.

The original publication is available at http://www.springerlink.com/content/n257556xw8390457/?CFID=70130633&CFTOKEN=29786250.

In this paper, we introduce a new approach to computer-aided microtonal improvisation by combining methods for (1)... more In this paper, we introduce a new approach to computer-aided microtonal improvisation by combining methods for (1) interactive scale navigation, (2) real-time manipulation of musical patterns and (3) dynamical timbre adaption in solidarity with the respective scales. On the basis of the theory of well-formed scales we offer a visualization of the underlying combinatorial ramifications in terms of a scale labyrinth. This involves the selection of generic well-formed scales on a binary tree (based on the Stern-Brocot tree) as well as the choice of specific tunings through the specification of the sizes of a period (pseudo-octave) and a generator (pseudo-fifth), whose limits are constrained by the actual position on the tree. We also introduce a method to enable transformations among the modes of a chosen scale (generalized and refined “diatonic” and “chromatic” transpositions). To actually explore the scales and modes through the shaping and transformation of rhythmically and melodically interesting tone patterns, we propose a playing technique called Fourier Scratching. It is based on the manipulation of the “spectra” (DFT) of playing gestures on a sphere. The coordinates of these gestures affect score and performance parameters such as scale degree, loudness, and timbre. Finally, we discuss a technique to dynamically match the timbre to the selected scale tuning.

Modelling the similarity of pitch collections with expectation tensors

by Andrew Milne

Milne, A. J., Sethares, W. A., Laney, R., Sharp, D. B. (2011). Journal of Mathematics and Music, 5(1), 1–20.

This is a preprint of the article as submitted for consideration in the Journal of Mathematics and Music; Journal of Mathematics and Music is available online at: http://www.tandf.co.uk/journals/titles/17459737.asp. This preprint is called "Expectation Arrays: Modelling the Similarity of Pitch Collections"; following peer review, the name was changed to "Modelling the Similarity of Pitch Collections with Expectation Tensors".

The final (postprint) version is available on request (email me). MATLAB routines used to calculate the examples in the paper can be downloaded from http://sethares.engr.wisc.edu/pitchmetrics.html.

Models of the perceived distance between pairs of pitch collections are a core component of broader models of music... more 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, nor 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.

X_System

by Andrew Milne

Milne, A. J., Sethares, W. A., & Plamondon, J. (2006). Report commissioned by Thumtronics, Inc.

Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum

by Andrew Milne

Milne, A. J., Sethares, W. A., & Plamondon, J. (2007). Computer Music Journal, 31(4), 15–32

This article introduces the idea of tuning invariance, by which relationships among the intervals of a given scale... more This article introduces the idea of tuning invariance, by which relationships among the intervals of a given scale remain the “same” over a range of tunings. This requires that the frequency differences between intervals that are considered the “same” are “glossed over” to expose underlying similarities. This article shows how tuning invariance can be a musically useful property by enabling (among other things) dynamic tuning, that is, real-time changes to the tuning of all sounded notes as a tuning variable changes along a smooth continuum.

On a keyboard that is (1) tuning invariant and (2) equipped with a device capable of controlling one or more continuous parameters (such as a slider or joystick), one can perform novel real-time polyphonic musical effects such as tuning bends and temperament modulations—and even new chord progressions—all within the time-honored framework of tonality. Such novel musical effects are discussed briefly in the section on dynamic tuning, but the bulk of this article deals with the mathematical and perceptual abstractions that are their prerequisite.

Tuning continua and keyboard layouts

by Andrew Milne

Milne, A. J., Sethares, W. A., & Plamondon, J. (2008). Journal of Mathematics and Music, 2(1), 1–19

This is an electronic version of an article published in Journal of Mathematics and Music, 2(1). Journal of Mathematics and Music is available online at:
http://www.informaworld.com/smpp/content~content=a793326484~db=all~order=page

Previous work has demonstrated the existence of keyboard layouts capable of maintaining consistent fingerings across a... more Previous work has demonstrated the existence of keyboard layouts capable of maintaining consistent fingerings across a parametrized family of tunings. This paper describes the general principles underlying layouts that are invariant in both transposition and tuning. Straightforward computational methods for determining appropriate bases for a regular temperament are given in terms of a row-reduced matrix for the temperament-mapping. A concrete description of the range over which consistent fingering can be maintained is described by the valid tuning range. Measures of the resulting keyboard layouts allow direct comparison of the ease with which various chordal and scalic patterns can be fingered as a function of the keyboard geometry. A number of concrete examples illustrate the generality of the methods and their applicability to a wide variety of commas and temperaments, tuning continua and keyboard layouts.

New tonalities with the Thummer and The Viking

by Andrew Milne

Milne, A. J., & Prechtl, A. (2008). In A. Crossan & T. Kaaresoja (Eds.), Proceedings of the 3rd International Haptic and Auditory Interaction Design Workshop (Vol. 2, pp. 20–22). Jyväskylä, Finland.

In this paper we explain the theoretical background of Dynamic Tonality using the Thummer, a new musical interface,... more In this paper we explain the theoretical background of Dynamic Tonality using the Thummer, a new musical interface, and The Viking, a software synthesizer written especially for it. Dynamic Tonality is a musical audio routine that allows for novel tunings and enables the user to relate – to an arbitrary degree – these tunings with the partials of their notes. The Viking features Dynamic Tonality and works with any MIDI instrument, but when paired with the Thummer (or another two-dimensional interface) it creates a system of fingering invariance across chords and tunings. Thus, the Thummer and The Viking render non-standard tunings more physically, pedagogically, and aesthetically accessible.

Spectral tools for Dynamic Tonality and audio morphing

by Andrew Milne

Sethares, W. A., Milne, A. J., Tiedje, S., Prechtl, A., & Plamondon, J. (2009). Computer Music Journal, 33(2), 71–84

The Spectral Toolbox is a suite of analysis–resynthesis programs that locate relevant partials of a sound and allow... more The Spectral Toolbox is a suite of analysis–resynthesis programs that locate relevant partials of a sound and allow them to be resynthesized at specified frequencies. This enables a variety of routines including spectral mappings (changing all partials of a source sound to fixed destination frequencies), spectral morphing (continuously interpolating between the partials of a source sound and those of a destination sound), and what we call Dynamic Tonality (a novel way of organizing the relationship between a family of tunings and a set of related spectra). A complete application called the TransFormSynth provides a concrete implementation of Dynamic Tonality.

Metrics for pitch collections

by Andrew Milne

Milne, A. J., Sethares, W. A., Laney, R., Sharp, D. B. (2010). Proceedings of the 11th International Conference on Music Perception and Cognition (pp. 77–80). University of Washinton, Seattle, USA.

Models of the perceived distance between pairs of pitch collections are a core component of broader models of the... more Models of the perceived distance between pairs of pitch collections are a core component of broader models of the perception of tonality as a whole. Numerous different 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 framework, nor to incorporate the uncertain or probabilistic nature of pitch perception (whereby tones with similar frequencies may, or may not, be heard as having the same pitch).

To achieve these aims, we embed pitch collections in novel multi-way expectation arrays, and show how metrics between such arrays can model the perceived dissimilarity of the pitch collections they embed. By modeling the uncertainties of human pitch perception, expectation arrays indicate the expected number of tones, ordered pairs of tones, ordered triples of tones and so forth, that are heard as having any given pitch, dyad of pitches, triad of pitches, and so forth. The pitches can be either absolute or relative (in which case the arrays are invariant with respect to transposition).

We provide a number of examples that show how the metrics accord well with musical intuition, and suggest some ways in which this work may be developed.