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Jeremy Arden
POSTER: Old Tricks New Media: Schillinger Techniques are Relevant to All Kinds of
Contemporary Music Irrespective of Style
Music associated with the Concre`te tradition, dependent on synthesis and sampling, is
frequently rhythmically complex and highly unpredictable. The timbres associated with this genre
may be dense and, in terms of information theory, feature a very low rate of redundancy. This
article describes how Schillinger techniques can be used to structure the medium- and large-scale
form of electroacoustic music, determining the onset and relative weighting of complex
spectromophologically conceived sounds. The same procedures are also highly useful in live
electroacoustic performances and music that involves improvisation. Against this background
the author describes how Schillinger techniques play a part in his own composition practice and
feed into his electroacoustic music, helping him to maintain a degree of control when composing
with complex and unusual sounds, noise-based timbres and intricate textures, the very things that
test the threshold of our mental capacities.
Keywords: Schillinger System; Composition; Electroacoustic; Temporal Ratio; Rhythmic
Structure; Tabla
Brian Bemman
Anatomy of the Six-part, All-Partition Array as Used by Milton Babbitt: Preliminary
Efforts Towards a Computational Method of Automatic Generation
For Milton Babbitt (1916–2011), twelve-tone techniques were indispensable forms of musical
composition because they are fundamentally derived from mathematical constructs. Perhaps his
most complex form of composition is the all-partition array. Its large-scale structure is formed by
concatenated permutations of 12-tone rows under the common class of musical equivalence
relations, transposition, inversion, retrograde and retrograde inversion, and then organized into
pairs of hexachordally combinatorial rows. The resultant pitch-class space is then partitioned
into segment lengths of 12 integers or fewer distributed into a specified number of parts. Both
its organization and number of parts are determined in part by the unique mathematical
properties of all-combinatorial hexachords. Through analyses of two works representative of two
types of six-part arrays, initial ground work is established for the automatic generation of the allpartition array structure. Preliminary concerns in the computational process are also addressed.
Olaf Beyersdorff and James Hall
POSTER: Visualisation of MANTRA by Stockhausen
Stockhausen's piece MANTRA is one of the most rigorously structured compositions in the
whole musical literature. The piece evolves from a single musical formula (the Mantra) which
governs the whole composition in a dominant mathematical way. The piece is split into thirteen
cycles, each inspired by a master note from the mantra. Each of these cycles contains a further
twelve mantras, again, each inspired by a master note from the same basic mantra. This iterative
behaviour is analogous to iterative and recursive methods for solving problems within the area of
computer science.
While listening to the piece, it is difficult for even a strong musician to follow the
complex structure. We evaluated existing products for musical visualisation, which tend to only
exhibit exact repetitions in musical structure. They were therefore found insufficient of
demonstrating the complex structure of this piece in an effective way.
This project analyses the complex composition of MANTRA and visualises its structure
through a computer program that displays the structural context of the music during the
performance. This enables the listener to follow and indeed interactively explore the complex
flow of the composition. This way the user can experience the rich structure that otherwise
inevitable largely remains hidden.
We view this project as a first case study for visualising complex musical structures that
are inspired by mathematics. Our technique could potentially be transferred to many
compositions from the literature.
Marc Brooks
Four-dimensional Music? – The Hidden Mathematical Influences on Schoenberg’s
String Quartet No. 1
Schoenberg believed that he was able to access nature through his unconscious and then channel
it into his music. Carl Dahlhaus saw this as a secular version of an earlier protestant theology in
which strong subjective emotion was the guarantee of truth. He did not doubt the sincerity of
Schoenberg’s faith in his own anthropocentric ‘aesthetic theology’, but saw the ‘roots’ of such
religion – and any truth thereby uncovered – as lying beyond the purview of the historian. The
purpose of this paper is to ask whether this apparently sensible view might be challenged by
taking seriously the thesis of Alain Badiou’s Being and Event (1988) that ‘mathematics is ontology’.
Of particular relevance is the realist corollary that there are truths that transcend language and
historical situation which then place an irresistible demand on those who follow.
Without the need to resurrect any spurious grand narrative of musical progress or any
metaphysical notion of music as a window onto the absolute, I claim that certain aspects of
Schoenberg’s style derive their aesthetic originality from developments in contemporaneous
mathematics. I shall demonstrate this via two routes. Historically, by showing the influence of
Helmholtz’s writing about non-Euclidean space on Blaue Reiter expressionism, particularly on the
techniques of non-logicism and non-parallelism. And analytically, by showing that Berg’s wellknown analytic rewriting of the String Quartet No.1 – so that it is ‘easier to understand’, in his
words – amounts to transformation of a four-dimensional conception back into (ordinary) three
dimensions.
It turns out that the music does capture something of the truth about nature after all, but
only if nature is the liberating matrix of ontological possibilities unlocked by mathematics and
not the mythical-mystical origin to which Schoenberg himself wished to return.
K. Nicholas Carlson and Nick Collins
POSTER: Musical Applications of the Fractal Flame Algorithm
Iterated function systems have been used by a number of composers to create novel electronic
music (Gogins 1991, di Scipio 1999, Qiu et. al. 2012). One variation that has not previously been
applied in musical mappings is the fractal flame algorithm (Draves and Reckase 2003, as available
in software such as Electric Sheep and Apophysis). The 'information content' of a chaotic
attractor is plotted in a colourful and complex image of some beauty; the hope in application in
music is to exploit some of the same radical mathematically gifted aesthetic.
Musical mappings explored here include the direct synthesis of sound from a fractal
flame, and symbolic mappings of event streams (e.g. of MIDI, score and pattern data). For
sound mappings, spectral synthesis directly from 2-D colour images is perfectly plausible (or
indeed, other maps from image to sound once the fractal flame algorithm generates images). But
a more direct mapping approach allows the creation of a signal generator with its outputs from
the fractal flame's final form or from successive iterations in building a flame. Event generation
simply runs similar algorithms, but at the slower rate of rhythmic events; of course, synthesis and
event level can be re-combined, allowing a Xenakisian unification of multiple time levels of a
composition.
Stace Constantinou
POSTER: Is it Possible to Use Prime Numbers as an Analogue for Musical
Composition?
In my compositions I have applied the prime number sequence in various ways; to
electroacoustic music, solo instrumental music, and ensemble writing.
From Euclid to Riemann, no one has been able to fully unlock the secrets of the prime
number sequence. How this enigmatic mathematical sequence is applied to music will be the
focal point of this poster.
Composers need to ask themselves several questions, when devising a method by which
to apply mathematical material to a musical work. Is it important that the mathematical nature
be present in the musical material? Is it possible to transfer the mathematical nature to the
musical aesthetic? If so, to what extent can the transferal be considered a success? If the
purpose of using a mathematical pattern or device is to not realise an analogue in the music, then
what is point of using it all?
Solutions include: when used in combination with the harmonic series, the prime number
sequence can be used to form any number of new harmonic fields, which can then be translated
into music notation and applied during the compositional process; these form intervallic ratios
that are then translated into microtonal staff notation. Prime numbers can also be used to create
rhythmic materials, as well as shape the musical structure. Finally, combining materials created
using prime and non-prime numbers it is possible to form a variegated musical fabric.
This poster raises the questions above, and provides potential answers with a view to
stimulate thought, and discussion.
Mark Gotham
Mathematical Models for Metrical Theory
Many standard accounts of musical metre (such as Lerdahl and Jackendoff 1983) assign equal
value to each structural level, however empirically-tested cognitive preferences (see London 2012
for a summary) suggest a more nuanced quantification by weighting them on the basis of pulse
salience (Parncutt 1994). These principles are modelled well by a Gaussian distribution relating
the inter onset interval of a pulse to its salience and, with this improved account of the relative
importance of each pulse level, an additive system combining them can at last stand as a worthy
model of metre. This paper advances two applications of this model.
First, the criteria for selecting tempo to optimise the net salience of a metre are
identified. It is shown that maximal salience interacts in simple, categorical ways with the tempo,
metrical structure, and the number of metrical levels represented. This model suggests a set of
‘default’ tempi specific to various categories of metres. The division of metres into these
categories is intriguingly counter-intuitive. The model also provides a more rigorous basis for the
definition of what it means for music to be ‘fast’ or ‘slow’ in those contexts.
Second is a model of categorical metrical listening after Large and Palmer's 2002 waves
of expectation. This paper adapts Large and Palmer’s model to include the more subtle
quantification of metrical weight described above. The implications include a categorical disparity
between metrical positions formerly considered equal, and the prospective development of a
quantification of metrical instability.
Andrew Gustar
Statistics in Historical Musicology
Statistics is used extensively in music analysis, perception, and performance studies. However,
statisticians have largely ignored the many music catalogues, databases, dictionaries,
encyclopaedias, lists and other datasets compiled by institutions and individuals over the last few
centuries. Such datasets present fascinating historical snapshots of the musical world, and
statistical analysis of them can reveal much about the changing characteristics of the population
of musical works and their composers, and about the datasets and their compilers. The author
has applied this methodology to several case studies covering, among other things, music
publishing and recording, composers’ migration patterns, nineteenth-century biographical
dictionaries, and trends in key signatures. This presentation uses these case studies to illustrate
the insights to be gained from quantitative techniques; the statistical characteristics of the
populations of works and composers; the limitations of the predominantly qualitative approach
to historical musicology that has formed our received narrative of music history; and some
practical and theoretical issues associated with applying statistical techniques to musical datasets.
Alan Marsden
KEYNOTE: Music, Mathematics, Morality and Motion
What do we want mathematics to do for music? What do we want music to do for mathematics?
What should we as scholars of music and mathematics be doing? Certainly, where possible, we
should use mathematics to verify the claims we make about music, and we should cease to
tolerate unverified claims. Mathematics has made some definite contributions to the
understanding of music in the past, and will do in the future. I attempt to classify these
contributions and suggest some potentially profitable avenues of enquiry. In general, musical
mathematicians should move away from the static fields of group theory, topology, tuning and
scales, and examine more the potential for mathematics to explicate the dynamic aspects of
music through statistical analysis and information theory. Among other examples, I propose an
information-theoretic interpretation of Schenkerian analysis. Contributions of music to
mathematics, on the other hand, are less evident. Suspecting that the mental processes of
composing music and composing mathematical solutions are not so different, I speculate that
presentations of mathematical information in musical form might lead to insights. Even if they
do not, they can sound interesting!
René Mogensen
‘Timbre Pitch Space’ with audio signal paths: a transformation analysis of Kaija
Saariaho’s NoaNoa (1992)
Towards an analytical typology of musical interactions between instrumentalist and
electronics
The composition NoaNoa (1992) for flute and electronics by Kaija Saariaho contains an
interesting economy of pitch structure contents, in spite of its very dramatic gestural character. It
seems that much of the drama in the piece comes from the integration of changing timbral
aspects, including electronic sound, with pitch structure developments in the flute. My own
hearing of the piece distinguishes four pitch structures that are transformed, while also moving
through a complex timbral space. I am especially interested in understanding how this timbral
space is dependent on the combinations and interactions of flute with electronics.
Perhaps a contextual transformation network analysis, which takes account of audio signal
paths, could illuminate the structural listening perspective for some of the ‘highly interactive’ness which is claimed by the composer (Chabot 1993) for the piece? To examine this possibility I
employ an adaptation of the abstract algebra-based analytical approach from Lewin (2011
[1993]). Coinciding pitch structures and timbres are examined as dot product timbre-pitch
classes. This allows me to formulate a succinct view of Lewinian ‘passes’ through the theoretical
pitch-timbre space. The passes are transformation network maps that indicate ‘paths’ which go
from ‘input’ timbre-pitch classes, through transformation functions, to other classes in the pitchtimbre space. I propose that these paths can be interpreted as a typology of Saariaho's dynamic
use of electronics in relation to her variations in pitch structures: In other words, the paths
indicate types of changes in the use of instrument-electronics interactions.
Thomas Noll and Tom Fiore
Voice Leading and Transformation
This paper investigates linear transformations on voicings and paves the way for a
transformational approach to the study of voice-leading. It implements ideas on linear
representations of contextual transformations that we briefly introduced in the outlook of [1].
From a music-theoretical point of view one may distinguish between two main motivations for
voicing transformations: (1) transformations with harmonic motivation are inherited from a
certain root form transformations and are then extended to all chord inversions through the
conjugation with voice permutations. (2) Transformations with intervallic motivation are
genuinely sensitive towards the intervallic structure of a given voicing. The investigation of the
paper is addressed to the latter type. The mathematical core of the paper consists in the study of
a suitable subgroup ̃ of
, where denotes the number of voices. As a first concrete
music-theoretical result we represent Julian Hook's 288-element group of uniform triadic
transformations in terms of voicing transformations for n = 3. Another result is the interpretation
of the hexagonal common-tone regions as orbits under suitable subgroups of order 6 (recall that
this is not possible in the context the triadic action of the classical Neo-Riemanian PLR-group).
Furthermore we construct cross-type transformations on the basis of voice-doublings.
References
[1] Thomas M. Fiore, Thomas Noll, and Ramon Satyendra. Incorporating voice permutations
into the theory of neo-riemannian groups and lewinian duality.
http://arxiv.org/abs/1301.4136.
Michelle Phillips
Why Large Scale Mathematical Patterns in Music are Unlikely to be Heard, Using the
Golden Section as an Example
One of the reasons that music is explored in terms of mathematics is that patterns are deemed by
composers, musicologists or listeners to add to the aesthetic value of the piece. Such a claim
assumes that these relationships may be heard. However, rarely do studies account for processes
of aural perception. Of the 110 publically available studies which suggest that the golden section
features in one or more musical works, only a quarter mention perception, and none seek to
explore whether this ratio may be heard.
Previous studies of the experience of duration during music listening suggest that this
may be warped by music’s modality, tempo, and extent of harmonic modulation. Also, studies of
how large-scale form in music is processed cognitively suggest that local factors may be more
than global structures.
Empirical studies which I have undertaken will be presented in support of this paper’s
claim. Results suggest that estimation of duration whilst listening to music may be altered by
sense of enjoyment, sense of completeness of the musical extract, concurrent occupation of
working memory, and variation in listening level (volume) within the musical extract
These studies therefore show no support for the claim that the golden section in the
large-scale structure in musical form can be perceived by the listener (although it is not denied
that such forms may lend insight into the compositional process). Hence claims of aesthetic
worth of a piece due to such an underlying framework may be flawed.
Matthew Sallis
Godel, Plato, Xenakis, Symbolic Music and Numbers
I have developed Xenakis’ Symbolic Music outlined in his book Formalized Music using the
method of Gödel Numbering. This assigns an integer to pitch vectors and rhythm vectors which
are then raised as a power to successive prime numbers. If each discreet pitch and rhythm vector
in a piece of music is given an integer value the music may be encoded as a single Gödel
Number: the product of successive prime numbers raised to a power. The symbolic encoding
contained within the Gödel Number can be decoded using prime factorisation.
Since a Gödel Number behaves as an integer it can be manipulated mathematically to
produce a symbolic representation of the music it encodes. It is argued that this symbolic
representation is a kind of musical DNA encoding the choices made by the composer. A lute
dance, Dowland’s Midnight, by John Dowland has been Godelised in the manner described. The
musical DNA produced has been mutated into a new realization of the same lute dance.
Since the original lute dance and its mutated form equal the same Gödel Number they
could be regarded as two manifestations of the same music. This fact gives a new slant to the
ideas of Platonic music in which a composition pre-exists its discovery by a composer.
Kelvin Thomson
POSTER: PRESENT TiME for Piano Quartet and TiME before AND TiME after for
Septet: Musical Affinity with Finite Projective Geometry, T.S. Eliot, Bach and others
Numbers and mathematical systems frequently provide inspiration for my compositions and
compositional strategies. They act as objective counterparts to my other various subjective
creative stimuli and often render musically surprising results.
PRESENT TiME (2010) for piano quartet evolved from investigating the latent
musical translation possibilities of the visually compelling diagrams representing the 2x2 case of
Steven H. Cullinane’s Diamond Theorem. He used finite projective geometry to explain ‘the
surprising symmetry properties of some simple graphic designs—found, for instance, in quilts.’
These diagrams were superimposed onto his new arrangement of the 64 Hexagrams of the I
Ching (a number-grid of coordinates geometrically describing the natural group of
transformations of the hexagrams) to form a sequentially shifting and filtering mechanism, used
to generate pitch and rhythm material.
This number-based system served as the nexus of a range of stratified extra- and intramusical concepts and techniques. Meta-structural and narrative elements were derived from T.S.
Eliot’s ‘Burnt Norton’; micro-structural elements and rationale from Bach’s Goldberg Variations;
key zones from a ‘processed’ Hebridean air; rhythms from contemporary club music; canon,
hocket, cantus firmus and temporal dislocation. TiME before AND TiME after (2011) is a
reworking of sections III–V of PRESENT TiME, for clarinet, bassoon, trumpet, percussion,
piano, violin and cello.
My poster will demonstrate the methodology employed to formulate and deploy the
compositional palette for these compositions, with some contextual notes and examples from
the scores. Audio-visual recordings and full scores will be available for perusal.
Scores and recordings are available at:
<http://www.kelvinthomson.com/#/ensemble/4564768137>
Valerio Velardo
POSTER: Music, Memes and Mathematics
This poster proposes a formal model of the memetics of music based on recursion, which describes
the fundamental processes of musical evolution. From the musical point of view, the model
draws upon Dawkins’ theory of memes and Jan’s memetics of music. From the mathematical point of
view, the model relies on infinite dimensional analysis and formal systems.
The poster addresses the following questions: what is a musical meme? how can we
represent musical memes in abstract spaces? How do atomic musical particles generate musical
memes (musemes)? How/why do musemes evolve over time?
The poster has a threefold aim. First, it describes music using abstract spaces. Secondly,
it provides a recursive grammar showing how complex musical structures are generated from
atomic musical particles. Finally, it accounts for the musical evolutionary process based on the
cultural selection of musemes.
The poster provides a single theoretical framework that attempts to describe different
phenomena such as music, musical memes and musical evolution. Likewise, it proposes a new
way of representing music and provides an evolutionary model of music suitable to serve as the
basis of computer programs which analyse musical evolution within multi-agent societies.
Susannah Wixey
A Graph Theoretic Approach to Musical Harmony
In his book, ‘A Geometry of Music: Harmony and Counterpoint in the Extended Common
Practice’, Dmitri Tymoczko sets out what he considers to be the five key components of
tonality. He then uses geometrical interpretations of music theoretic relationships to explore
these features and how they constrain composers who wish to create music with a sense of
tonality. Here we interpret the same voice-leading relations as a mathematical graph, and employ
results from graph theory and complex network theory to visualise, characterise and quantify
different harmonic frameworks.
In particular, classical graph theory defines notions of connectivity and traversability,
ideas which stem from the famous ‘Bridges of Konigsberg problem’ which asks whether all
bridges in a certain town can be crossed exactly once on a single walk. This now simple question
is closely related to the Travelling Salesperson problem, one of the most computationally
intensive algorithmic problems around today. The recent field of complex network theory aims
to find solutions to dynamical problems defined on graphs and networks by extracting the most
useful information from them in a concise manner. Although the graphs obtained from voiceleading in any reasonable harmonic structure are small compared to typical complex networks,
the notions of centrality, algebraic connectivity and communicability are still appropriate
diagnostics for musical analysis.