# Thomas NollEscola Superior de Música de Catalunya (ESMUC) · Music Theory and Composition

Thomas Noll

Dr. phil. Dipl. math

## About

80

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Introduction

Mathematical Music Theory

Additional affiliations

September 2004 - March 2016

December 1998 - June 2005

## Publications

Publications (80)

The theory of well-formed modes is a modal refinement of the theory of well-formed scales. The mathematical approach is based on various results from the subdiscipline of algebraic combinatorics on words. Section 1 provides anchors and motivations for this investigation both in music theory and in mathematics and traces some earlier cross-connectio...

Through the application of algebraic combinatorics on words to the study of diatonic modes, the paper characterizes the ascending authentic Ionian mode among the others in terms of divider incidence. This property characterizes positive standard words among their conjugates with respect to plain adjointness. The plain adjoint of the Ionian step-int...

Pairwise well-formed modes are investigated as substitutions on words over a three-letter alphabet. Motivating examples are the major and minor modes, here described in terms of the substitutions a↦ac,b↦ba,c↦cab and a↦ab,b↦ca,c↦bac. Pairwise well-formedness is investigated through the inspection of the two-letter-mergings of a substiution, namely c...

The article investigates an extension of the theory of well-formed modes and proposes a model of the major and minor modes in harmonic tonality. The established theory of well-formed modes is well adapted to the description of the medieval diatonic modes. Its core is the conversion of the circle-of-fifths encoding into the circle-of-steps encoding...

One of the most telling features of the application of combinatorics on words in mathematical music theory is the extent to which aspects of the history of theory are captured by the word theory model. Well-formed modes, including the usual diatonic modes, are modeled by Christoffel words and their conjugates. A class of Sturmian morphisms produces...

The description of the Major and Minor modes as fillings of a triadic division of the octave offers the possibility to study them as Pairwise Well-Formed Modes. As a consequence one obtains two projections: the diatonic projection yields the well-known Ionian and Aeolian modes and provides a link between the triadic modes and the pseudo-classical m...

The paper offers an integration of the theory of structural modes, functional theory and diatonic scale degrees. In analogy to the parsimonious voice leading between generic diatonic triads we study parsimonious function leading between embedded structural modes. A combinatorics of diatonic embeddings of structural modes is given. In four analytica...

This contribution responds to a growing interest in the application of Discrete Fourier Transform (DFT) to the study of pitch class sets and pitch class profiles. Theoretical fundaments, references to previous work and explorations of various directions of study have been eloquently assembled by Emmanuel Amiot. Recent pioneering work in the applica...

This paper studies the “integration” problem of nineteenth-century harmony—the question whether the novel chromatic chord transitions in this time are a radical break from or a natural extension of the conventional diatonic system. We examine the connections between the local behavior of voice leading among diatonic triads and their generalizations...

This article focuses on the interpretation of the fundamental bass in terms of structural modes. The aim is to develop an analytical interpretation that relates tonality and tonal form to the fundamental bass. The approach is anchored in the Ramellian tradition by considering the fundamental bass as an autonomous level of analysis. A contiguity pri...

One of the most significant attitudinal shifts in the history of music occurred in the Renaissance, when an emerging triadic consciousness moved musicians towards a new scalar formation that placed major thirds on a par with perfect fifths. In this paper we revisit the confrontation between the two idealized scalar and modal conceptions, that of th...

Abstract Rameau [15] redundantly defines the subdominant (1) as the fifth under the tonic and (2) as the scale degree immediately below the dominant. In the context of scale theory this motivates the interpretation of this defini- tion as an equation. It states that the diazeuxis (the difference between the generator and its octave complement) is a...

Rameau [15] redundantly defines the subdominant (1) as the fifth under the tonic and (2) as the scale degree immediately below the dominant. In the context of scale theory this motivates the interpretation of this definition as an equation. It states that the diazeuxis (the difference between the generator and its octave complement) is a step inter...

The conjugation class of a special Sturmian morphism carries a natural linear order by virtue of the two elementary conjugations \(conj_a\) and \(conj_b\) with the single letters a and b, with the standard morphism of the class as the smallest element in this order. We show that a lexicographic order on the morphisms of the given conjugation class...

One of the most significant attitudinal shifts in the history of music occurred in the Renaissance, when an emerging triadic consciousness moved musicians towards a new scalar formation that placed major thirds on a par with perfect fifths. In this paper we revisit the confrontation between the two idealized scalar and modal conceptions, that of th...

The engagement with the music-theoretical reasoning of Carl Dahlhaus encourages the elaboration of links between older and younger traditions of systematic thought in music theory. The core of the question is the incorporation of musictheoretical content from ‚unscientific‘ or ‚dogmatic‘ historical sources into a body of accepted knowledge a system...

Motivated by analytical methods in mathematical music theory, we determine the structure of the subgroup J of GL(3,Z12) generated by the three voicing reflections. As applications of our Structure Theorem, we determine the structure of the stabilizer H in Sigma3 semi-direct product J of root position triads, and show that H is a representation of H...

The paper revisits results from scale theory through the study of modes. Point of departure is the nested hierarchy of tri- ads embedded into diatonic modes embedded into the chromatic scale. Generic diatonic triads can be described as stacks of thirds or as stacks of triple-fifths. These two possibilities lead to different generalizations and it i...

This essay advocates the integration of mathematical reasoning into the teaching of music theory. It offers a didactical pathway through a series of attractive results from well-known algebraic approaches to the study of diatonicity. In particular, strategies are discussed that let the students re-enact processes of investigation and discovery. The...

Diatonic Modes can be modeled through automorphisms of the free group F
2 stemming from special Sturmian morphisms. Following [1] and [2] we associate special Sturmian morphisms f with linear maps E(f) on a vector space of lattice paths. According to [2] the adjoint linear map E(f) ∗ is closely related to the linear map E(f
∗ ), where f and f
∗ are...

A familiar problem in neo-Riemannian theory is that the P, L, and R
operations defined as contextual inversions on pitch-class segments do not
produce parsimonious voice leading. We incorporate permutations into
T/I-PLR-duality to resolve this issue and simultaneously broaden the
applicability of this duality. More precisely, we construct the dual...

We begin the development of a categorical perspective on the theory of
generalized interval systems (GIS's). Morphisms of GIS's allow the analyst to
move between multiple interval systems and connect transformational networks.
We expand the analytical reach of the Sub Dual Group Theorem of Fiore--Noll
(2011) and the generalized contextual group of...

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 u...

In this paper we take the three tonal functions tonic, subdominant and dominant out of their usual theoretical domicile—the combinatorics of fifth-related triads enriched by a dialectical interpretation—and
redeploy them within an alternative theoretical framework: the combinatorics of the modes of the musical tetractys, enriched by musical-theoret...

The paper investigates an extension of Christoffel duality to a certain family of Sturmian words. Given an Christoffel prefix w of length N of an Sturmian word of slope g we associate a N-companion slopegN∗ such that the upper Sturmian word of slope gN∗ has a prefix w∗ of length N which is the upper Christoffel dual of w. Although this condition is...

The goal of this article is to clarify the relationship between the topos of
triads and the neo-Riemannian PLR-group. To do this, we first develop some
theory of generalized interval systems: 1) we prove the well known fact that
every pair of dual groups is isomorphic to the left and right regular
representations of some group (Cayley's Theorem), 2...

Der Tonkreisel ist ein kombinatorisches Musikinstrument und wurde im Jahre 2011 als ständiges Exponat für die Ausstellung Erlebnisland Mathematik in den Technischen Sammlungen in Dresden geschaffen.1 In zweierlei Sinne handelt es sich dabei um ein “Kind unserer Zeit”: Das Exponant benutzt elektronische Bodensensoren, einen versteckten Computer und...

The attribute "well-formed" is used homonymously in two different approaches to the study of musical tone relations. The present article explores commonalities and differences of the underlying theoretical ideas. Fred Lerdahl (2001) uses the attribute "well- formed" (in extension of the GTTM tradition) in order to formulate constraints for the cons...

This paper studies the mathematical basis for a new study of modes of well-formed (WF) scales, and presents a new characterization of special standard Sturmian morphisms.
We introduce WF words, which coincide with the step-interval patterns of modes of well-formed scales. WF words can be represented as conjugates of some Christoffel word (generaliz...

Norman Carey and David Clampitt observed in [4] that each region has two well-formed scales as its prefixes. If one looks
at this finding from the viewpoint of word theory, one observes that regions are central words and the two prefixes are their
independent periods. More precisely, each region, understood as a word in a two-letter alphabet, conta...

With a few exceptions (Chemillier and Truchet 2003), (Chemillier 2004), musical scale theory and combinatorial word theory have remained unaware of each other, despite having an intersection
in methods and results that by now is considerable. The theory of words has a long history, with many developments coming
in the last few decades; see Lothaire...

Many properties of the mechanical word s = (b(n+1)gcb ngc)n=0,1,2,... play a strikingly illuminating role in music theory when g = log2(3/2). This number represents the pitch height ratio between the musical intervals fifth and octave. The word s exemplifies the discrete rendering of the kernel of the linear pitch height form p : Z2 ! R with p(m,n)...

This paper presents a transformational approach to musical intervals with particular focus on their constitutive role for well-formed scales. These scales have the property that their binary step-interval pattern is maximally even. Transposition classes of well-formed scales are therefore characterized by two step intervals and their characteristic...

The mathematical study of the diatonic and chromatic uni-verses in the tradition of David Lewin (9) and John Clough (6) is a point of departure for several recent investigations. Surprisingly, Lewin's original idea to apply finite Fourier trans-form to musical structures has not been further investigated for four decades. It turns out that several...

We present an automated, mathematical approach to the paradigmatic analysis of the melodic content of a piece of music. We consider all melodic segments of consecutive notes, however, segments of different sizes are processed separately. We compare and group these segments using a similarity measure, which accounts for standard symmetry transformat...

The paper aims at clarifying the pedagogical relevance of an algebraic-oriented perspective in the foundation of a structural and formalized approach in contemporary computational musicology. After briefly discussing the historical emergence of the concept of algebraic structure in systematic musicology, we present some pedagogical aspects of our M...

cote interne IRCAM: Andreatta06a

Xenakis' tone sieves belong to the first examples of the-oretical tools whose implementational character has con-tributed to the development of computation in music and musicology. According to Xenakis' original intuition, we distinguish between elementary sieves and compound ones and trace the definition of sieve transformations along the sieve co...

The present article argues in favor of a discipline of Mathematical Music Theory (see [16], [18] for earlier attempts). By reviewing and re-interpretating known results, we draw further conclusions and formulate working hypotheses. Especially, we recapitulate a known fact about diatonic triads and seventh chords in connection with an analogous fact...

The article studies the topos Sets T of actions of an 8-element monoid T on sets. It is called the triadic topos as T is isomorphic to the monoid of affine transformations of the twelve tone system Z 12 , leaving a given major or minor triad invariant. The subobject classifier Ω of this topos and its Lawvere-Tierney-Topologies j are calculated. We...

http://www.gmth.de/zeitschrift/artikel/199.aspx

http://www.gmth.de/zeitschrift/ausgabe-2-2-2005/inhalt.aspx

The article discusses collective performance of score-based music at two levels of description: collective behavior and musical score. Two specifications of the concept of coherence are presented. Intentional coherence applies to the intentionality of a performing ensemble and characterizes the way how distributed collective knowledge about a piece...

Opuscope is an initiative targeted at sharing musical corpora and their analyses between researchers. The Opuscope repository will contain musical corpora of high quality which can be annotated with hand-made or algorithmic musical analyses. So, analytical results obtained by others can be used as a starting point for one's own investigations. Expe...

Das Vorstellen der Tonbewegungen ist ein wirkliches Mitmachen der-selben mit dem Willen; die Seele, der lebendige Menschengeist, führt selbst diese Bewegungen aus und erfreut sich in ihnen seines Da-seins, seiner Wirkungskräfte. (Hugo Riemann [23], p.15). Abstract In extrapolation of Wilhelm Wundt's suggestion to apply Weber-Fechner's law to apperc...

This article provides an introduction to basic geometric investigations of the 12-tone sytem and its subsets, the chords. The various denitions and results are intended to lay a theoretical basis for 12-tone-based explo- rative and empirical research on occidental harmony. The entire approach is motivated by the assumption that the 12 tones do not...

The article proposes a conceptual framework for a special type of experiments in harmonic analysis and discusses aspects of its implementation in a software tool – called HarmoRubette. The framework comprises three basic components, namely (1) a harmonic configuration space, HARM, equipped with a harmonic tensor quantifying the transitions between...

The article investigates aspects of globality with respect to music the-ory and especially mathematical and computer-aided music theory. The local/global dichotomy is applied (a) to the discipline such from a cul-tural semiotic point of view, (b) to the strategies of scientific knowledge management dogmatics, modeling and hermeneutics, and (c) to t...

Thesis--TU Berlin, 1995. Includes bibliographical references (p. 145-148).

During the second half of the twentieth century, algebraic methods have been increasingly recognised as powerful approaches to the formalisation of musical structures. This is evident in the American music-theoretical tradition as well as in the European formalised approach to music and musicology. We mention the mathematician and composer Milton B...

Harmonische Ausweichungen bei durchgehender Grundtonart äußern sich typischerweise in der solidarischen Verknüpfung zweier Phänomene: der Alteration von Stufen der Grundtonart und der virtuellen Verrückung des tonalen Bezugs. Im Falle einer Ausweichung in die Tonart der V. Stufe wird z.B. die vierte Stufe der Grundtonart hochalteriert und die V. St...

In extrapolation of Wilhelm Wundt's suggestion to apply Weber-Fechner's law to apperception and in slight modification of David Lewin's definition of generalized interval systems we propose a four-dimensional space A as a model for an active tone system. This space A is also a group acting on itself. Its elements are interpreted in two ways, (1) as...

To deal with musical ambiguity is an emprically di cult but promising task. The paper draws a connection between very general issues of mental activity on the one hand and concrete phenomena of tonal ambiguity on the other. We think that a convincing understanding of ambiguity in the surface tonality of musical pieces (like V/I-I/V substitutions or...