Insiders’ Choice: Studying Pitch Class Sets Through Their Discrete Fourier Transformations

Noll, Thomas

doi:10.1007/978-3-030-21392-3_32

Cited by 5 publications

(4 citation statements)

References 12 publications

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“…The application enables its users to visualize the tonality of a piece of music viewed through the lens of the DFT without the requirement to understand its formal details. The app thus contributes to an exchange of knowledge and techniques for music analysis between In this spirit, we continue the work by Thomas Noll [8] and Jennifer Harding [5] to make DFT analyses of musical entities more easily accessible and reproducible for a broader readership, especially students.…”

Section: Introductionmentioning

confidence: 79%

midiVERTO: A Web Application to Visualize Tonality in Real Time

Harasim¹,

Affatato²,

Moss³

2022

Preprint

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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 harmonic function, 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.

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Section: Introductionmentioning

confidence: 79%

midiVERTO: A Web Application to Visualize Tonality in Real Time

Harasim¹,

Affatato²,

Moss³

2022

Preprint

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show abstract

“…The DFT has been used to analyze and compare the tonal languages of several composers, including Bach, Schubert and Scriabin (Noll, 2019;Yust, 2015), as well as existing approaches to key finding (Yust, 2017). It has also been applied to the study of changes in pitch-class distributions, both within pieces (Harding, 2020) and between compositions from different historical periods (Yust, 2019b).…”

Section: Analysing Music Using the Discrete Fourier Transform (Dft)mentioning

confidence: 99%

“…7. The natural hexachord has the largest unnormalized magnitude of all pitch-class sets, which is why some authors consider it to be the prototype for this coefficient (Noll, 2019). 8.…”

mentioning

confidence: 99%

Wavescapes: A visual hierarchical analysis of tonality using the discrete Fourier transform

et al. 2022

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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.

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“…In recent years it became increasingly popular among music theorists to study pitch-class sets and distributions via the discrete Fourier transform [1,8,15,17]. Since the DFT can be applied to music in symbolic formats (e.g., MIDI) without prior interpretation of the musical material by a music theorist, it is well suited for distantreading approaches in corpus studies, such as the comparison of different pieces' tonal organization at a high level of abstraction.…”

Section: Phantom Curvesmentioning

confidence: 99%

Phantom Curves

Moss

Affatato

Harasim

2022

9th International Conference on Digital Libraries for Musicology

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Scientific discovery can arise by many means. Results can be obtained from meticulous methodological rigor and carefully plannedthrough experiments, but oftentimes insights are also gained through spontaneous associations of previously unrelated facts or exploratory work. In particular for the latter mode of discovery, a stimulating environment is quintessential in order to allow creativity to flow [4,9]. Collaborative inter-and transdisciplinary research for which knowledge and expertise from different fields is brought into interaction, is especially likely to provide such environments. Interdisciplinary work can, however, be impeded by specialized expert knowledge that is difficult to communicate to researchers with different backgrounds. In such situations, appropriately designed tools may help to build bridges between different research traditions and provide interfaces to otherwise restricted knowledge.Here, we report an instance of scientific discovery made possible through interactive exploration using an online interface for music visualization. We document how the discovery of phantom curves,

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Insiders’ Choice: Studying Pitch Class Sets Through Their Discrete Fourier Transformations

Cited by 5 publications

References 12 publications

midiVERTO: A Web Application to Visualize Tonality in Real Time

midiVERTO: A Web Application to Visualize Tonality in Real Time

Wavescapes: A visual hierarchical analysis of tonality using the discrete Fourier transform

Phantom Curves

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