# Is math a music theory

## Information on mathematical music theory

### Thomas Noll

Selected mathematical approaches to music theory and analysis are briefly portrayed below. The selection made here focuses primarily on the work of Guerino Mazzola and his colleagues, as well as on work that was suggested by them. A discussion of the results is not the aim of this text.

A special feature of the Mazzola School - compared to the schools of mathematical music theory shaped by John Clough and David Lewin in the United States, for example - is the particularly pronounced role that mathematical theory formation plays. The music-theoretical examination of these investigations brings with it two problems of understanding. The first concerns the presence of mathematical content itself, the often strictly formal representation of which cannot be inferred without a certain amount of practice. The real difficulty, however, lies in the epistemological exploration of the musical or music-theoretical meanings of that mathematical content. The present text tries to avoid formal representations of mathematical content and to focus on the interpretative work of the mathematical music theorist. This happens at the price of omissions or trivializations. This text is therefore not suitable as a basis for argumentation for an active discussion of the research discussed here, but it is intended to help the interested reader to work out one.

Mazzola's first monograph, Groups and Categories in Music. Draft of a mathematical music theory from 1985 combines four lines of investigation: (a) the development of a theoretical framework, (b) investigations into specific music-theoretical questions, (c) analyzes of music examples with regard to the questions examined, (d) experimental exploration of mathematical ideas in a compositional study .

The first line of investigation - the development of a mathematical metalanguage for music theory - plays a central role in Mazzola's work. In structural mathematics, the meaning of a mathematical object is revealed not least through its relation to other mathematical objects. If one wants to load a concrete mathematical structure with music-theoretical content, then, strictly speaking, the mathematical meaning of that structure can only be transferred into music theory if one also ›interprets‹ all those other structures with music theory to which the selected structure has informative mathematical relationships . At first glance, this thought alone seems to be a case for Ockham's knife, but a sidelong glance at the principle of least effect successfully applied in physics shows that the problems lie deeper. The formulation of this principle is based on theoretically expanding the world of the physically factual in order to better understand it. Of course, nothing stands in the way of the use of Ockham's knife in the case of a clever surgical conceptual instinct, even in mathematical music theory. The argument is intended to exemplify that there are good reasons for not just using isolated mathematical structures as the starting point for music-theoretical interpretation.

With the increasing breadth of music theoretical, music analytical, music technological and other topics, Mazzola has since worked in parallel on expanding the mathematical theoretical language and his extensive monograph The Topos of Music - Geometric Logic of Concepts, Theory, and Performance, published in 2002, is dedicated to the mathematical, semiological and related issues and epistemological issues special attention.

### Music-theoretical interpretations of affine mappings

This section summarizes some results of mathematical-music-theoretical investigations on questions of harmony and counterpoint. What connects these investigations from a mathematical point of view is the special role of affine mappings on tones, intervals, chords or chord structures. The following preliminary consideration should motivate people to focus on such images:

The concatenation of musical intervals is a music theory meaningful interpretation of adding vectors. There is a direct and an indirect way of doing justice to such an additive structure. The act of transposing a chord or a set of chords can be interpreted as a translation; H. as a mapping that moves every point of a space by a certain vector. So translations are direct manifestations of adding. The act of mirroring a chord or a structure of chords at a fixed mirroring center, on the other hand, has the property that all internal interval relationships are preserved (even if each individual interval reverses its direction). In particular, each mirrored sum agrees with the sum of the mirrored summands. Mathematical mappings with the property that the images of sums agree with the sums of the images (or that the images of multiples agree with the multiples of the images) are called linear. They only embody addition indirectly by respecting it.

Reflections with different mirror centers are interconnected by translations. The combination of linear mappings and translations is called affine mappings. 'Affine' means that the directions that are available from every point are also viewed in context. The abstract term ›fourth‹, for example, not only includes the individual isolated quart instances from each reference note, but also their connection as a ›quartz field‹, which extends over the entire pitch space via translation from reference note to reference note.

Among the investigations with the aim of reformulating music-theoretical questions with the help of affine images and possibly better understanding them, a special modulation model by Mazzola deserves first of all^{[1]} Mention to which other authors later contributed. It draws inspiration from Arnold Schönberg's theory of harmony and tries to identify traces of an interaction between the target key and the original key by examining the levels of the target key used in a modulation. Keys are described as abstract diatonic levels in the twelve-tone system, i.e. H. without distinction of a tonic and without distinction of the tone sex. The interaction of these step structures is described by translations or reflections. One starts with a minimum set of steps that clearly characterizes the target key, such as {II, V}, {IV, V}, {II, III}, {III, IV} or {VII} and notes the tones used. Then one looks for the tones of the starting key that interact with these and adds the common tones of both tones below to the initially noted set of tones. In addition to the initially selected (unambiguously identifying) levels, those levels of the target key are added as modulating levels, which are also formed from the (possibly) expanded set of tones. For example, let C be the starting diatonic and G the target diatonic with the inversion induced by the key change from C-H as an interaction, and only the VII level {F #, A, C} is chosen as the minimum characteristic for the G diatonic. The interaction then mediates tone-by-tone between F # and F, A and D, as well as C and B and thus relates the VIIth level of G to the VIIth level of C. The tones B and D are common tones of both diatonic scales and become the already F #, A, C added to memorized tones. III and V are added as modulating levels.

The mathematical model imposes restrictions on this process in the sense of an economic principle. The result is a list of 26 interaction modulations. A further counterfactual experiment by Daniel Muzzulini ^{[2]}is based on the idea that ladder-specific triads can generally be understood as a sequence of two double steps on a 7-note scale. His transfer of the above procedure to the corresponding systems of ›triad structures‹ for all 7-note scales in the twelve-tone system results in a special position for the diatonic, because the number of 26 interaction modulations is the smallest of the 34 scales in which all 12 ›keys‹ can be used via interaction modulation can reach. Possible connections with other special properties of the diatonic system in the twelve-tone system (maximal evenness, myhill property, cardinality = variety) have not been investigated.

Another investigation by Mazzola^{[3]} is dedicated to the role of consonances and dissonances in two-part counterpoint and is motivated by the pronounced dichotomy character of this pair of terms, which is not covered by the psychoacoustic levels of description. The complementarity between the consonances and dissonances is described by an affine mapping between the two halves of the dichotomy, which Mazzola calls the autocomplementarity function. In the following description we number the intervals in the circle of fifths, i.e. 0 = prime / octave, 1 = fifth, 2 = major second, 3 = major sixth, 4 = major third, 5 = major seventh, 6 = tritone, 7 = minor second , 8 = minor sixth, 9 = minor third, 10 = minor seventh, 11 = fourth. The dichotomy halves are then {0, 1, 3, 4, 8, 9} and {2, 5, 6, 7,10, 11}. The autocomplementarity function exchanges every interval x with the interval 5x + 2 mod 12, i.e. the major second 2 with the prime 0, the major seventh with the major sixth 3, the tritone 6 with the minor sixth 8, the fourth 11 with the minor third 9 as well as the minor seventh 10 with the major third 4 and the minor second 7 with the fifth 1.

This mapping has interesting properties from the point of view of the minor and major third relationship of intervals. On the one hand, it is easy to see that the consonances have many thirds related to one another (thirds to the prime and fifth, the sixths to the octave, etc.). H. the consonances are close together as a family on the third torus. The same goes for the dissonances among each other.

In addition, the autocomplementarity function is a central reflection of the third torus, i. H. it exchanges the Kleinterz kinship cycles {0, 3, 6, 9} and {2, 5, 8, 11} with each other, while it mirrors the cycle {1, 4, 7, 10} in itself. It also exchanges the diametrically opposite major third relationship cycles {1, 5, 9} and {7, 11, 3} or {2, 6, 10} and {0, 8, 4} with each other. For the complementarity of the consonances and dissonances, this means that the interrelated intervals are far apart.

The dichotomy half {0, 1, 3, 4, 8, 9} of consonances has another remarkable property, namely that it is a monoid; H. that all products of these 6 numbers (mod 12) can always be found again under these numbers. In many cases, this does not apply to the dissonances among each other, especially to the multiplication of each dissonance by itself. I have considered this property as a possible mathematical explanation for the markedness of the dissonances, which e.g. B. expressed in their need for resolution. In particular, the conclusive consonances 0, 1, 4, 9 are idempotent, i.e. H. they agree with their own potencies: 0^{2}=0, 1^{2}=1, 4^{2}=4, 9^{2}= 9 mod 12. A theoretical meaning of the interval multiplication arises in connection with the morphology of the chords (see below).

Mazzola used the auto-complementarity function in the context of a model for purely consonant progressions, which contains a derivation for the prohibition of parallel fifths and Fux's prohibitions on tritones.

The consonance / dissonance dichotomy can also be discussed in the form of a counterfactual argument. In purely combinatorial terms, there are 924 possibilities to select six consonant intervals from the twelve intervals {0, 1, 2, ...., 11}. But only one of the dichotomies obtained in this way has the three properties of autocomplement, the distance between the dichotomy halves and the markedness: the ›factual‹ consonance / dissonance dichotomy {0, 1, 3, 4, 8, 9} / {2, 5, 6, 7, 10, 11}.

In our own research on the morphology of the chords ^{[4]}Tone perspectives - meaning the 144 affine images of the twelve-tone system - are used to model chord representation in functional harmony. Those tone perspectives that depict a chord in themselves are seen as stabilizing properties of the same. In this approach, the self-perspectives of the major and minor triads are associated with tonal functions. The approach combines Hugo Riemann's concept of a relational tone thinking with the mathematical consideration that tones and intervals are related to one another via affine mappings. The relation of the fifth C-G to the major sixth G-E means, strictly speaking, the relation of this fifth to a fifth shifted ›triple fifth‹ (modulo octave). The tone perspective determined with this then also transfers every other tone x to the tone 3x + 1 mod 12. So the major sixth GE is in turn related (due to the affine connection) to the minor third EG, since each of the three partial fifths is based on the factor 3 is mapped into a sixth, and thus the sixth G - E (as a 'triple fifth') on a 'ninefold', in other words: on a minor third. The decisive factor for the designation of this tone perspective as the self-perspective of the C major triad is the fact that the note E itself is related to a tone of this triad - in this case to G.

An important part of these investigations is the study and interpretation of the connection between several sound perspectives. The self-perspectives of a chord form a family that is closed in pairs: a monoid. This leads to points of contact with Hugo Riemann's suggestion to compare musical thinking with logical thinking. In the context of the toposlogic of monoid actions on chords, generalized truth values acquire genuine musical meaning^{[5]}.

### Classification and quasi-analysis of musical structures

Another branch of research, which Harald Fripertinger in particular has systematically developed in addition to Mazzola, is the classification of mathematical structures, including those of particular importance in music theory. Behind this is the need to formulate and understand mathematical-music-theoretical statements in a suitable context. In particular, the paradigm of affine mapping plays a preferred role. Complete lists of representatives or at least their numbers are available for chords or motifs in different tone systems and rhythmic periods.

The systematic search for mathematical structures with musically interesting properties is also interesting from a compositional perspective. So Dan Tudor has Vuza^{[6]} Constructed a special kind of rhythmic canons whose voices on the one hand never meet, but on the other hand cover every beat. The peculiarity of these canons is that neither the inner rhythm, which the voices ever play with a time delay, nor the outer rhythm of the voice inserts is completely regular. The simplest known example has 6 voices with the period 72. The inner rhythm is (0, 1, 5, 6, 12, 25, 29, 36, 42, 48, 49, 53) and the outer rhythm is (0, 8 , 16, 18, 26, 34). Emmanuel Amiot, Moreno Andreatta and Harald Fripertinger have since worked on the investigation of such structures.

Canons, for their part, are a very special case of a complex musical structure being put together from components. Mazzola has devoted extensive research to the general theme of global composition. These are mathematical objects whose definitions are inspired by Bernhard Riemann's concept of manifold. This is because the definitions of these global objects are already composed of several locally compatible definitions. Against this background, several authors have developed specific methods for the mathematical investigation of pieces of music.

A fundamental difficulty of interpretation in musical analysis is that while isolating parts and examining their paradigmatic and syntagmatic relationships one can gain valuable insights, it is not automatically justified to regard these parts as components of music in the constitutive sense. This objection is general, although it is exacerbated by some approaches developed in mathematical music theory. It is therefore helpful if in the following we understand 'analysis' in the sense of Carnap's concept of quasi-analysis.

The approaches to the metrical and melodic analysis have been worked out for experimental purposes within the framework of the software project RUBATO, which experimentally links musical analysis and musical design.^{[7]}

In the internal metric analysis, the incidence relationships of local meters, i.e. H.examined by arithmetic sequences of notes. ›Inner metric analysis‹ means that these meters are exemplified in the score itself through notes or other events. The incidence relationships are quantified by internal metric weights. Anja Volk has shown in many experiments that metricity in the musical sense can be investigated on the basis of the periodicities of that inner metric weight.

In the case of melodic analysis, two forms of paradigmatic relationships come into play: symmetry and similarity. The interest in a (quasi) component of the melody depends on the presence of other components which, apart from inversion, augmentation, diminution, etc., match or are similar to it. To determine the similarity, shape paradigms are modeled with the help of topologies. Chantal Buteau has carried out extensive research on this. A consequence of this purely paradigmatic interest is that the concrete syntagmatic combinatorics of the interesting components results as an empirical result, which can either be quantified as a weight or in turn can serve as an occasion for further investigations. Andreas Nestke has carried out numerous combinatorial studies on this.

### literature

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_____ (2004), "Paradigmatic Motivic Analysis", in: Perspectives of Mathematical and Computational Music Theory, ed. by Guerino Mazzola, Thomas Noll and Emilio Lluis-Puebla, Osnabrück: epOs Music, 343–365.

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