Tone rows and tropes

Fripertinger, Harald; Lackner, Peter

doi:10.1080/17459737.2015.1070088

Cited by 11 publications

(3 citation statements)

References 17 publications

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“…Harald Fripertinger [14,15] discusses the enumeration of a variety of musical objects, such as intervals and chords, patterns of rhythms and motives, as well as tone rows and patterns of tropes. Fripertinger and Peter Lackner [16] examine in greater depth the enumeration of the latter two topics in an article that constitutes an entire special issue of Journal of Mathematics and Music. The enumeration of tropes is of particular relevance to this study, as it involves the action of a power group: a group that acts on hexachordal partitions of 12-tone rows by transposing or inverting the pitch-classes in the hexachords and/or by permuting the hexachords themselves.…”

Section: Enumeration Applications In Music Theorymentioning

confidence: 99%

Beat-Class Set Classes and the Power Group Enumeration Theorem

Peck¹

2021

MusMat

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The theory of beat-class sets originates in the work of Milton Babbitt, who demonstrates a correspondence between modular pitch-class spaces and metric spaces in the framework of total serialism. Later authors, particularly Richard Cohn, John Roeder, and Robert Morris, apply similar concepts to a variety of analytical situations, drawing on concepts and procedures from pitch-class set theory. In light of the correspondence between these theories, the universe of beat-class sets for a given modulus may be partitioned into equivalence classes similar to pitch-class set classes. This study investigates processes of enumerating these equivalence classes.We consider extensions to the theory of beat-class sets by including rhythms with more than one voice. Specifically, we examine equivalence classes of multiple-voiced beat-class sets using the Power Group Enumeration Theorem (PGET) of Frank Harary and Edgar M. Palmer. The PGET allows us to determine the numbers of equivalence classes of beat-class sets as determined by various groups of transformations: metric shift, retrogradation, and voice permutation, among others. Our results have implications for further applications in pitch-class set theory, serial theory, and transformational theory.

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Section: Enumeration Applications In Music Theorymentioning

confidence: 99%

Beat-Class Set Classes and the Power Group Enumeration Theorem

Peck¹

2021

MusMat

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“…For further explanation on this and comparisons between different systems, see [17,19,21,36,105,133,152,170]. 28 See [18,38,39,40,41,42,45,57,64,78,118,119,142,155,156]. Some of these works are discussed at length and in historical perspective in [150], see also [25].…”

Section: The Partitioning Of Pitch Spacesmentioning

confidence: 99%

Partitional Harmony: The Partitioning of Pitch Spaces

Feitosa¹

2020

MusMat

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In this preliminary work, we seek to present a brief historical review of the use of partitions in music, to provide a concise introduction to the theory of partitions, and lastly, through an extensive bibliographic revision and a thoughtful theoretical reflection, to lay the foundations of what we call partitional harmony - a comprehensive harmonic conception which relates the theory of partitions to several fields of post-tonal music theory. At the end, some basic operations (pitch, transposition, inversion, and multiplication) are defined and an illustrative musical application is provided, followed by our research prospects.

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“…The set of consonant triads, denoted Triads, as displayed on page 483 of [5]. been thoroughly explored by composers, music theorists, and mathematicians, see for example Babbitt [1], Forte [12], Fripertinger-Lackner [13], Hook [14], Hook-Peck [15], McCartin [21], Mead [22], Morris [23,24,25,26], and Rahn [29]. Indeed, the three recent papers [13,22,26] together contain over 100 references.…”

Section: Consonant Triadsmentioning

confidence: 99%

Hexatonic systems and dual groups in mathematical music theory

Berry

Fiore

2018

Involve

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Motivated by the music-theoretical work of Richard Cohn and David Clampitt on late-nineteenth century harmony, we mathematically prove that the P Lgroup of a hexatonic cycle is dual (in the sense of Lewin) to its T /I-stabilizer. Our point of departure is Cohn's notions of maximal smoothness and hexatonic cycle, and the symmetry group of the 12-gon; we do not make use of the duality between the T /I-group and P LR-group. We also discuss how some ideas in the present paper could be used in the proof of T /I-P LR duality by Crans-Fiore-Satyendra in [5].A slightly revised version of this manuscript will be published in Involve: A Journal of Mathematics, published by Mathematical Sciences Publishers.

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Tone rows and tropes

Cited by 11 publications

References 17 publications

Beat-Class Set Classes and the Power Group Enumeration Theorem

Beat-Class Set Classes and the Power Group Enumeration Theorem

Partitional Harmony: The Partitioning of Pitch Spaces

Hexatonic systems and dual groups in mathematical music theory

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