The aim of this paper is to argue that complementation is an operation similarly fundamental to music theory as transposition and inversion. We focus on studying the chromatic complement mapping that translates diatonic seventh chords into 8-note scales which can also be interpreted as rhythmic beat patterns. Such complements of diatonic seventh chords are of particular importance since they correspond to the scales popularized by the Jazz theorist Barry Harris, as well as to rhythms used in African drum music and Steve Reich's Clapping Music. Our approach enables a systematic study of these scales and rhythms using established theories of efficient voice leading and generalized diatonic scales and chords, in particular the theory of second-order maximally even sets. The main contributions of this research are (1) to explicate the correspondence between voice leadings and rhythmic transformations, (2) to systematize the family of Barry Harris scales, and (3) to describe classes of voice leadings between chords of different cardinality that are invariant under complementation.

Many structural aspects of music, such as tonality, can be expressed using hierarchical representations. In music analysis, so-called keyscapes can be used to map a key estimate (e.g., C major, F minor) to each subsection of a piece of music, thus providing an intuitive visual representation of its tonality, in particular of the hierarchical organization of local and global keys. However, that approach is limited in that the mapping relies on assumptions that are specific to common-practice tonality, such as the existence of 24 major and minor keys. This limitation can be circumvented by applying the discrete Fourier transform (DFT) to the tonal space. The DFT does not rely on style-specific theoretical assumptions but only presupposes an encoding of the music as pitch classes in 12-tone equal temperament. We introduce wavescapes, a novel visualization method for tonal hierarchies that combines the visual representation of keyscapes with music analysis based on the DFT. Since wavescapes produce visual analyses deterministically, a number of potential subjective biases are removed. By concentrating on one or more Fourier coefficients, the role of the analyst is thus focused on the interpretation and contextualization of the results. We illustrate the usefulness of this method for computational music theory by analyzing eight compositions from different historical epochs and composers (Josquin, Bach, Liszt, Chopin, Scriabin, Webern, Coltrane, Ligeti) in terms of the phase and magnitude of several Fourier coefficients. We also provide a Python library that allows such visualizations to be easily generated for any piece of music for which a symbolic score or audio recording is available.

This paper describes a data-driven framework to parse musical sequences into dependency trees, which are hierarchical structures used in music cognition research and music analysis. The parsing involves two steps. First, the input sequence is passed through a transformer encoder to enrich it with contextual information. Then, a classifier filters the graph of all possible dependency arcs to produce the dependency tree. One major benefit of this system is that it can be easily integrated into modern deep-learning pipelines. Moreover, since it does not rely on any particular symbolic grammar, it can consider multiple musical features simultaneously, make use of sequential context information, and produce partial results for noisy inputs. We test our approach on two datasets of musical trees -- time-span trees of monophonic note sequences and harmonic trees of jazz chord sequences -- and show that our approach outperforms previous methods.

Functional harmony is an integral part of many repertoires in the Western musical practices, including both diatonic and extended tonality. In the latter context, music-theoretical accounts suggest that the three octatonic equivalence classes (OECs) consisting of pitch-classes related by stacked minor-third intervals may be associated with tonic (T), dominant (D), and subdominant (S) functions. Whether this theoretical description of music is also relevant to the perception of music has not yet been tested empirically. In this study, 100 participants familiar with Western repertoires were presented with jazz chord progressions containing chord substitutions. When each stimulus had been played, participants predicted how many more chords they would have expected to hear before the progression could reach a plausible conclusion. We computed the similarity of responses for pairs of stimuli containing different harmonic substitutions and modeled such similarity values based on different measures of harmonic relatedness between substitutions. Data show that the OEC membership of substitutions strongly predicts the similarity of participants' completion ratings. Bayesian mixed-effects modeling of similarity values further showed a categorical distinction between D and S as functional categories, on one hand, and T, on the other hand. The data also appear to reflect the prevalent influence of rock and pop repertoires on the participants, encouraging further research into the influence of stylistic diversity and musical expertise. Overall, results contribute to the characterization of listeners' implicit knowledge of the principles of harmonic structure in extended tonality and support the relevance of OECs not only as descriptors of extended-tonal compositional practices but also parsimonious predictors of perceived functionality.

We introduce phantom curves, a novel music-theoretical concept based on the discrete Fourier transform (DFT), and document the creative process that led to their discovery. In particular, we emphasize the importance of interactive web applications for music visualization and analysis. This is demonstrated using the example of the application midiVERTO which affords interactions with the pitch-class content of musical pieces encoded in MIDI format without requiring in-depth understanding of the underlying mathematics. We illustrate the analytical value of studying families of phantom curves by applying the concept to music from a Broadway musical, a video game, and a Hollywood movie. This process of discovery thus testifies to the fact that digital tools can bridge disciplinary boundaries between music theory and mathematics, and this interaction can generate new scientific knowledge.

Chord prediction in jazz is one of the ways in which machine learning can be used in the field of music. Using the jazz songs included in the Weimar Jazz Database , this study investigates whether an LSTM model including melody information in the input data can achieve higher chord prediction accuracy than a similar model using only chord progressions as input. A statistical trend is found suggesting that the inclusion of melody information indeed improves prediction performance, but this improvement is not statistically significant. The way in which prediction accuracy changes for specific tunes is analyzed using two example tunes. Our investigation also shows that for the model with melody information there is substantial improvement on root note prediction.

This paper presents a web application for visualizing the tonality of a piece of music—the organization of its chords and scales—at a high level of abstraction and with coordinated playback. The application applies the discrete Fourier transform to the pitch-class domain of a user-specified segmentation of a MIDI file and visualizes the Fourier coefficients' trajectories. Since the coefficients indicate different musical properties, such as triadicity and diatonicity, the application isolates aspects of a piece's tonality and shows their development in time. The aim of the application is to bridge a gap between mathematical music theory, musicology, and the general public by making the discrete Fourier transform as applied to the pitch-class domain accessible without requiring advanced mathematical knowledge or programming skills up front.

We introduce phantom curves, a novel music-theoretical concept based on the discrete Fourier transform (DFT), and document the creative process that led to their discovery. In particular, we emphasize the importance of interactive web applications for music visualization and analysis. This is demonstrated using the example of the application midiVERTO which affords interactions with the pitch-class content of musical pieces encoded in MIDI format without requiring in-depth understanding of the underlying mathematics. We illustrate the analytical value of studying families of phantom curves by applying the concept to music from a Broadway musical, a video game, and a Hollywood movie. This process of discovery thus testifies to the fact that digital tools can bridge disciplinary boundaries between music theory and mathematics, and this interaction can generate new scientific knowledge.

Studies in psycho-linguistics have provided compelling evidence that theoretical syntactic structures have cognitive correlates that inform and influence language perception. Generative grammar models also present a principled way to represent a plethora of hierarchical structures outside the domain of language. Hierarchical aspects of musical structure, in particular, are often described through grammar models. Whether such models carry perceptual relevance in music, however, requires further study. To address the descriptive adequacy of a grammar model in music, unfamiliar musical phrases consisting of chord progressions within the Jazz idiom were used, and zero to three chords were cut from the end of each phrase. A total of 150 participants were then presented with these stimuli and asked to provide a Closure Response, that is to predict how many more chords (0, 1, 2, or 3) were expected before the chord progression was complete. Simultaneously, a grammar model of hierarchical structure as well as a bigram model were trained over a corpus of 150 expert-annotated Jazz tunes. The models were then used to estimate probability distributions of Closure Responses in the stimuli presented to the participants. Bayesian mixed-effects models reveal that the models carry predictive value for the participants’ response distributions and that the hierarchical model contains incremental predictive information over the bigram model. The present results suggest that – akin to language – hierarchical relationships between musical events have a cognitive correlate, which influences the perception and interpretation of music.

Author(s): Harasim, Daniel; O’Donnell, Timothy; Rohrmeier, Martin Alois | Abstract: Musicians and listeners perceive dependency structures between musical events such as chords and keys. Music theory postulates the goal-directedness of such dependencies, which manifests in formal grammar models as right-headed (head-final, left-branching) phrase structure. Goal-directedness has a direct cognitive interpretation; dependencies that point forward in time can be understood as creating expectation, and the empirical correlates of this relationship are a topic of current psychological research. This study presents a computational grammar model that represents the abstract concept of headedness but does not encode properties specific to music. Bayesian grammar learning is applied to infer a grammar for Jazz and its headedness proportions from a corpus of Jazz-chord sequences. The results show that the inferred grammar is right-headed. A second simulation using artificial data was conducted to verify the correct functionality of the headedness induction. The goal-directedness of Jazz harmony is thus demonstrated to be learnable without music-specific prior knowledge.

Author(s): Harasim, Daniel; Finkensiep, Christoph; Bigo, Louis; Giraud, Mathieu; Leve, Florence; Sears, David R. W.; Shanahan, Daniel; Rohrmeier, Martin Alois | Abstract: Music is highly complex and provides a rich variety of insights into the human mind, its mental structures, and processes. Experienced musicians are able to create complex structures in real time effortlessly, yet there is at present no successful model of full musical structure. The integration of different musical aspects such as melody, rhythm, voice leading, and form as well as the representation of long-term structure are particularly challenging. To open new possibilities for the study of higher-order structure in music and its perceptual correlates, cognitive music research would benefit from further mutual integration of theoretical, mathematical, computational, and psychological research, similar to advancements in linguistics. This symposium therefore focuses on the formal understanding and empirical investigation of music-theoretically motivated research questions in music cognition. It connects perspectives from music theory, behavioral research, corpus research, and computational modeling, and aims to initiate interdisciplinary discussions about the currently most challenging topics related to the cognition of higher-order structures in music.

Tonality is one of the most central theoretical concepts for the analysis of Western classical music. This study presents a novel approach for the study of its historical development, exploring in particular the concept of mode. Based on a large dataset of approximately 13,000 musical pieces in MIDI format, we present two models to infer both the number and characteristics of modes of different historical periods from first principles: a geometric model of modes as clusters of musical pieces in a non-Euclidean space, and a cognitively plausible Bayesian model of modes as Dirichlet distributions. We use the geometric model to determine the optimal number of modes for five historical epochs via unsupervised learning and apply the probabilistic model to infer the characteristics of the modes. Our results show that the inference of four modes is most plausible in the Renaissance, that two modes–corresponding to major and minor–are most appropriate in the Baroque and Classical eras, whereas no clear separation into distinct modes is found for the 19th century.

Scales are a fundamental concept of musical practice around the world. They commonly exhibit symmetry properties that are formally studied using cyclic groups in the field of mathematical scale theory. This paper proposes an axiomatic framework for mathematical scale theory, embeds previous research, and presents the theory of maximally even scales and well-formed scales in a uniform and compact manner. All theorems and lemmata are completely proven in a modern and consistent notation. In particular, new simplified proofs of existing theorems such as the equivalence of non-degenerate well-formedness and Myhill's property are presented. This model of musical scales explicitly formalizes and utilizes the cyclic order relation of pitch classes.

Grammatical models which represent the hierarchical structure of chord sequences have proven very useful in recent analyses of Jazz harmony. A critical resource for building and evaluating such models is a ground-truth database of syntax trees that encode hierarchical analyses of chord sequences. In this paper, we introduce the Jazz Harmony Treebank (JHT), a dataset of hierarchical analyses of complete Jazz standards. The analyses were created and checked by experts, based on lead sheets from the open iRealPro collection. The JHT is publicly available in JavaScript Object Notation (JSON), a human-understandable and machine-readable format for structured data. We additionally discuss statistical properties of the corpus and present a simple open-source web application for the graphical creation and editing of trees which was developed during the creation of the dataset.

Tonal harmony is one of the central organization systems of Western music. This article characterizes the statistical foundations of tonal harmony based on the computational analysis of expert annotations in a large corpus. Using resampling methods, this study shows that 1) the rank-frequency distribution of chords resembles a power law, i.e. few chords govern a large proportion of the data; 2) chord transitions are referential and chord predictability is significantly affected by distinguished chord features; 3) tonal harmony conveys directedness in time; and 4) tonal harmony operates differently at the hierarchical levels of chords and keys. These results serve to characterize tonal harmony on empirical grounds and advance the methodological state-of-the-art in digital musicology.

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 on one hand, and the global properties of voice-leading spaces on the other. In particular, we aim to identify those neo-Riemannian chord connections which can be integrated into the diatonic system and those which cannot. Starting from Jack Douthett’s approach of filtered point symmetries, we generalize diatonic triads as second-order Clough-Myerson scales and compare the resulting Douthett graph to the respective Betweenness graph. This paper generally strengthens the integrationist position, for example by presenting a construction of the hexatonic and octatonic cycles that uses the principle of minimal voice leading in the diatonic system. At the same time it provides a method to detect chromatic wormholes, i.e. parsimonious connections between diatonic chords, which are not contiguous in the system of second order Clough-Myerson scales.

Music is hierarchically structured, both in how it is perceived by listeners and how it is composed. Such structure can be elegantly captured using probabilistic grammatical models similar to those used to study natural language. They address the complexity of the structure using abstract categories in a recursive formalism. Most existing grammatical models of musical structure focus on one single dimension of music–such as melody, harmony, or rhythm. While these grammar models often work well on short musical excerpts, accurate analysis of longer pieces requires taking into account the constraints from multiple domains of structure. The present paper proposes abstract product grammars–a formalism which integrates multiple dimensions of musical structure into a single grammatical model–along with efficient parsing and inference algorithms for this formalism. We use this model to study the combination of hierarchically-structured harmonic syntax and hierarchically-structured rhythmic information. The latter is modeled by a novel grammar of rhythm that is capable of expressing temporal regularities in musical phrases. It integrates grouping structure and meter. The combined model of harmony and rhythm outperforms both single-dimension models in computational experiments. All models are trained and evaluated on a treebank of hand-annotated Jazz standards.

4.5 years after its first publication (see below), this is the first revised version of the ABC. In the meantime, the [DCML corpus initiative](https://www.epfl.ch/labs/dcml/projects/corpus-project/) has advanced and this update has as a main goal to harmonize the ABC with all other annotated corpora that have been and will be published. This includes the following changes: ### Upgrade to MuseScore 3 * All scores have been converted to [MuseScore](https://musescore.org/download) 3.6.2 format and can be found in the folder `MS3`. * The harmony labels have been moved to MuseScore's "Roman Numeral Analysis" layer of the left-hand staff. ### New folder and file structure * The `code` folder was removed since the old Julia code has been replaced by the Python library [ms3](https://pypi.org/project/ms3/). * The MuseScore files are contained in `MS3` and for each movement there are a couple of other files available, identified by their file names: * The folder `notes` contains one TSV file per movement with all note heads (not every note head represents an onset). * The folder `measures` contains one TSV file per movement with all measure-like units * The folder `harmonies` contains one TSV file per movement with all harmony annotation labels * The folder `reviewed` contains two files per movement: * A copy of the score where all out-of-label notes have been colored in red; additionally, modified labels ( w.r.t. v1.0) are shown in these files in a diff-like manner (removed in red, added in green). * A copy of the harmonies TSV with six added columns that reflect the coloring of out-of-label notes ("coloring reports") * The file `warnings.log` lists those labels where over 60 % of notes within the label's segment are not expressed by the label. Potentially, most of them are semantically incorrect. The folders are automatically kept up to date by the [dcml_corpus_workflow](https://github.com/DCMLab/dcml_corpus_workflow) which calls the command `ms3 review -M -N -X -D` on every change. Information on what the columns in the TSV files contain can be found in the [documentation for ms3](https://johentsch.github.io/ms3/columns). ### Changes to the data **A full diff of all changes applied with version 2.0 can be seen [here](https://github.com/DCMLab/ABC/commit/8bd699a9b5b00dba3214c6626575f8368279b965).** * The scores have been aligned by [tunescribers.com](https://tunescribers.com/) with the Henle and Breitkopf editions provided in the `pdf` folder and indicated in its README. * Systematic changes to the harmony labels: * With the harmony labels moved to the Roman Numeral Analysis layer, no initial `.` are needed anymore. * `V9` is not part of the DCML harmony annotation standard and has been replaced by `V7(9)` or `V7(+9)`. * Corrected `vii` chords in major keys that had often been wrongly labeled as `#vii`. * Obvious errors have been corrected in many places. Thanks to @craigsapp, @lancioni, @malcolmsailor, @MarkGotham, @napulen and @tymoczko for reporting quite a few of them!

The ABC dataset consists of expert harmonic analyses of all Beethoven string quartets (opp. 18, 59, 74, 95, 127, 130, 131, 132, 135, composed between 1800 and 1826), encoded in a human- and machine-readable format (MuseScore format). Using a modified Roman Numeral notation, the dataset includes the common music-theoretical set of harmonic features such as key, chordal root, chord inversion, chord extensions, suspensions, and others. The accompanying Data Report has been published by Frontiers in Digital Humanities.