Organizing Just Intonation Pitches through Xenakis' Sieves and Prime Decomposition (original) (raw)

A Musical Scale Generated from the Ratio of Consecutive Primes

2015

This paper describes a 12-note musical scale generated from the ratio of consecutive prime numbers that is used as the basis for the author’s electronic composition The Music of the Primes. The scale is a type of extended just intonation, a tuning system whose intervals are derived from the harmonic (overtone) series and uses partials above the first six partials. The ratio of consecutive primes sequence makes a potentially infinite number of prime ratios available to the composer.

The Internal Representation of Pitch Sequences in Tonal Music

1981

A model for the internal representation of pitch sequences in tonal music is advanced. This model assumes that pitch sequences are retained as hierarchical networks. At each level of the hierarchy, elements are organized as structural units in accordance with laws of figural goodness, such as proximity and good continuation. Further, elements that are present at each hierarchical level are elaborated by further elements so as to form structural units at the next-lower level, until the lowest level is reached. Processing advantages of the system are discussed.

A general pitch interval representation: Theory and applications

Journal of New Music Research, 1996

Pitch and pitch-intervals are most often represented-in the western traditioneither by the traditional pitch naming system and the relating pitch-interval names, or as pitch-classes and pitch-class intervals. In this paper we discuss the properties, relationships and limits of these two representations and propose a General Pitch Interval Representation (GPIR) in which the above two constitute specific instances. GPIR can be effectively used in systems that attempt to represent pitch structures of a wide variety of musical styles (from traditional tonal to contemporary atonal) and can easily be extended to other microtonal environments. Special emphasis will be given to the categorisation of intervals according to their frequency of occurrence within a scale. Two applications of the GPIR will be presented: a) in a system that transcribes melodies from an absolute pitch number notation to the traditional staff notation, and b) in a pattern-matching process that attempts to discover repetitions within a melody.

JDubiel: A Java Applet for Pitch-Class Set Analysis

Proceedings of the International Computer Music Conference (ICMC 2009), Montreal, Canada, 2009

Computer programs that facilitate analysis of atonal music are indispensable tools for music theorists. However, most of the programs currently available on the Web are limited to the derivation of Tn/TnI types and interval vectors and basic twelve-tone operations and cannot handle rotational arrays of tone rows, transformational relations between pcsets, and so on. By contrast, “JDubiel,” written in Java, is platform-independent, easy to use, and ready for almost all analytical techniques employed in pc-set theory and transformational theory.

Towards a Harmonic Complexity of Musical Pieces

DATESO, 2014

Music information retrieval (MIR) is small, but fast growing discipline. It aims at extraction of relevant information from music in order to improve the work with multimedia information systems. A popular approach in MIR is to apply signal processing techniques to extract low-level features from a musical piece. However, in order to create helpful applications, high-level concepts, such as music harmony, need to be examined as well. In this paper, we present a new model for understanding music harmony in computer systems. The proposed model takes the challenge of filling the gap between music theory and mathematical structures. Inspired by the computational complexity, we then derive a new term, harmonic complexity, that can be evaluated for a musical piece, and we test it on a genre classification problem.

Characterization Of Intonation In Carnatic Music By Parametrizing Pitch Histograms

2012

Intonation is an important concept in Carnatic music that is characteristic of a raaga, and intrinsic to the musical expression of a performer. In this paper we approach the description of intonation from a computational perspective, obtaining a compact representation of the pitch track of a recording. First, we extract pitch contours from automatically selected voice segments. Then, we obtain a a pitch histogram of its full pitch-range, normalized by the tonic frequency, from which each prominent peak is automatically labelled and parametrized. We validate such parametrization by considering an explorative classification task: three raagas are disambiguated using the characterization of a single peak (a task that would seriously challenge a more naïve parametrization). Results show consistent improvements for this particular task. Furthermore, we perform a qualitative assessment on a larger collection of raagas, showing the discriminative power of the entire representation. The proposed generic parametrization of the intonation histogram should be useful for musically relevant tasks such as performer and instrument characterization.

Mathematics and Computation in Music

Springer eBooks, 2022

Drawing inspiration from both the classical Guerino Mazzola's symmetry-based model for first-species counterpoint (one note against one note) and Johann Joseph Fux's Gradus ad Parnassum, we propose an extension for second-species (two notes against one note).

Injecting Periodicities: Sieves as Timbres

2007

Although Xenakis’s article on sieves was published in 1990, the first extended reference to Sieve Theory is found in the final section of ‘Towards a Metamusic’ of 1967. There are certain differences among the two. The section of ‘Metamusic’ is titled ‘Sieve Theory’ whereas the 1990 article simply ‘Sieves’. The latter is more practical, as it provides a tool for treating sieves (with the two computer programmes). These two writings mark two periods: during the first, sieves were of the periodic asymmetric type, with significantly large periods; in the more recent, they were irregular with absent periodicity. Also, the option of a simplified formula only appears in the 1990 article: there is a progression from the decomposed formula to the simplified one. This progression reflects Xenakis’s aesthetic of sieves as timbres and is here explored under the light of the idea of injected periodicities.

A Rational Intonation Approach to Persian Music

Živá hudba, 2023

The correct intonation and size of intervals in Iranian classical music (Radif) has been debated since the beginning of the 20th century and has led to much discussion and produced differing opinions. In an attempt to arrive at a systematic approach, Iranian and Western theorists and musicologists have put forward several different theories. As a composer who has always been inspired by Iranian classical music, I have been paying more attention to tuning and intonation in recent years. I have been using a rational intonation approach to create my own sonic world. In this article, I present my approach to rational intonation for Iranian classical music. In order to achieve my goal, I first outline the methods used by other researchers working on this topic. By demonstrating the limitations of the methods used until now, I present my view on this subject in the second part. I also describe the method I have used to obtain intervals for a detailed interval palette in Iranian music. In this way, I can create and notate Iranian classical music without losing the exquisite nuances of the music. This is a personal approach to intervals in Iranian music that can be used both in the composing and notating of classical Persian music.