TERNARY RHYTHM NOTATION: A NOVEL GRAMMATICAL CONSTRUCTION OF MUSICAL RHYTHM OVER AN ALGEBRAIC LATTICE

Persistent Link:
http://hdl.handle.net/10150/612819
Title:
TERNARY RHYTHM NOTATION: A NOVEL GRAMMATICAL CONSTRUCTION OF MUSICAL RHYTHM OVER AN ALGEBRAIC LATTICE
Author:
CULLAN, MICHAEL JOSEPH
Issue Date:
2016
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
We present a novel musical notation, generated by a formal grammar, which symbolizes musical rhythms as strings over a ternary alphabet. This notative system is shown to be an improvement over the conventional binary rhythm notation, because it uniquely encodes any rhythm given in classical, sheet music notation. We leverage this grammatical system to compute the number of distinct rhythms up to a specified timeunit. We then define a binary operation on the strings generated by the rhythm grammar to coincide with the musical notion of ”additive rhythm.” We show that the set of musical rhythms with a given time-unit becomes a finite commutative monoid under this operation, and moreover that this operation imbues the set with an algebraic lattice structure. Finally, we use the algebraic properties of this representation of musical rhythms to define a number of similarity measures between rhythms, which we suggest to have applications in automated music transcription.
Type:
text; Electronic Thesis
Degree Name:
B.S.
Degree Level:
Bachelors
Degree Program:
Honors College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Gillette, Andrew

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleTERNARY RHYTHM NOTATION: A NOVEL GRAMMATICAL CONSTRUCTION OF MUSICAL RHYTHM OVER AN ALGEBRAIC LATTICEen_US
dc.creatorCULLAN, MICHAEL JOSEPHen
dc.contributor.authorCULLAN, MICHAEL JOSEPHen
dc.date.issued2016-
dc.publisherThe University of Arizona.en
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en
dc.description.abstractWe present a novel musical notation, generated by a formal grammar, which symbolizes musical rhythms as strings over a ternary alphabet. This notative system is shown to be an improvement over the conventional binary rhythm notation, because it uniquely encodes any rhythm given in classical, sheet music notation. We leverage this grammatical system to compute the number of distinct rhythms up to a specified timeunit. We then define a binary operation on the strings generated by the rhythm grammar to coincide with the musical notion of ”additive rhythm.” We show that the set of musical rhythms with a given time-unit becomes a finite commutative monoid under this operation, and moreover that this operation imbues the set with an algebraic lattice structure. Finally, we use the algebraic properties of this representation of musical rhythms to define a number of similarity measures between rhythms, which we suggest to have applications in automated music transcription.en
dc.typetexten
dc.typeElectronic Thesisen
thesis.degree.nameB.S.en
thesis.degree.levelBachelorsen
thesis.degree.disciplineHonors Collegeen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorUniversity of Arizonaen
dc.contributor.advisorGillette, Andrewen
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