Generic (Mod-7) Voice-Leading Spaces

2019 ◽  
Vol 63 (2) ◽  
pp. 167-207
Author(s):  
Leah Frederick
Keyword(s):  
Topological Space ◽  
Set Theory ◽  
Final Section ◽  
Voice Leading ◽  
Pitch Class ◽  
Mathematical Properties ◽  
Pitch Class Sets

This article constructs generic voice-leading spaces by combining geometric approaches to voice leading with diatonic set theory. Unlike the continuous mod-12 spaces developed by Callender, Quinn, and Tymoczko, these mod-7 spaces are fundamentally discrete. The mathematical properties of these spaces derive from the properties of diatonic pitch-class sets and generic pitch spaces developed by Clough and Hook. After presenting the construction of these voice-leading spaces and defining the OPTIC relations in mod-7 space, this article presents the mod-7 OPTIC-, OPTI-, OPT-, and OP-spaces of two- and three-note chords. The final section of the study shows that, although the discrete mod-7 versions of these lattices appear quite different from their continuous mod-12 counterparts, the topological space underlying each of these graphs depends solely on the number of notes in the chords and the particular OPTIC relations applied.

scholarly journals Interval Permutations

2013 ◽  
Vol 19 (3) ◽  
Author(s):  
Drew F. Nobile
Keyword(s):  
Set Theory ◽  
Theoretical Investigation ◽  
Song Cycle ◽  
Final Section ◽  
Pitch Class ◽  
Significant Relationships ◽  
Relationship Of ◽  
The Relationship ◽  
Pitch Class Sets

This paper presents a framework for analyzing the interval structure of pitch-class segments (ordered pitch-class sets). An “interval permutation” is a reordering of the intervals that arise between adjacent members of these pitch-class segments. Because pitch-class segments related by interval permutation are not necessarily members of the same set-class, this theory has the capability to demonstrate aurally significant relationships between sets that are not related by transposition or inversion. I begin with a theoretical investigation of interval permutations followed by a discussion of the relationship of interval permutations to traditional pitch-class set theory, specifically focusing on how various set-classes may be related by interval permutation. A final section applies these theories to analyses of several songs from Schoenberg’s op. 15 song cycle The Book of the Hanging Gardens.

Diatonic Connotations of Pitch-Class Sets

1997 ◽  
Vol 15 (1) ◽  
pp. 1-29 ◽  
Author(s):  
René Van Egmond ◽  
David Butler
Keyword(s):  
Set Theory ◽  
Theoretical Study ◽  
Class Relations ◽  
Tonal Music ◽  
Pitch Class ◽  
Tonal Pitch ◽  
Relationship Of ◽  
The Relationship ◽  
Pitch Class Sets

This is a music-theoretical study of the relationship of two-, three-, four-, five-, and six-member subsets of the major (pure minor), harmonic minor, and melodic (ascending) minor reference collections, using pitchclass set analytic techniques. These three collections will be referred to as the diatonic sets. Several new terms are introduced to facilitate the application of pitch-class set theory to descriptions of tonal pitch relations and to retain characteristic intervallic relationships in tonal music typically not found in discussions of atonal pitch-class relations. The description comprises three parts. First, pitch sets are converted to pitchclass sets. Second, the pitch- class sets are categorized by transpositional types. Third, the relations of these transpositional types are described in terms of their key center and modal references to the three diatonic sets. Further, it is suggested that the probability of a specific key interpretation by a listener may depend on the scale-degree functions of the tones.

scholarly journals Voice Leading Among Pitch-Class Sets: Revisiting Allen Forte’s Genera

2020 ◽  
Vol IV (2) ◽  
pp. 66-79
Author(s):  
Paulo Salles
Keyword(s):  
String Quartet ◽  
Voice Leading ◽  
Pitch Class ◽  
Similarity Relations ◽  
Pitch Class Sets

The theory of PC-set class genera by Allen Forte was an important contribution to the understanding of similarity relations among PC sets within the tempered system. The growing interaction between the universes of PC-sets and transformational theories has been explored the space between sets of the same or distinct cardinality, by means of voice-leading procedures. This paper intends to demonstrate Forte’s method along with proposals by other authors like Morris, Parks, Straus, Cohn, and Coelho de Souza. Some analysis demonstrates such operations in passages picked from Heitor Villa-Lobos’s works, like the Seventh String Quartet and the First Symphony.

scholarly journals Topological Games on Product Spaces and its Fuzzification

2013 ◽  
Vol 2 ◽  
pp. 11-15
Author(s):  
Bidyanand Prasad ◽  
BP Kumar
Keyword(s):  
Topological Space ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Perfect Information ◽  
Topological Spaces ◽  
Product Spaces ◽  
Pursuit And Evasion

This paper is concerned with the introduction of an infinite positional game of pursuit and evasion over an ideal of a topological space. A topological game has been played over some new D-product and C-product spaces of two Hausdorff topological spaces. Perfect information, decisions and goals in a game may not be feasible. Hence, fuzzy set theory has been applied in this paper to obtain better results. Academic Voices, Vol. 2, No. 1, 2012, Pages 11-15 DOI: http://dx.doi.org/10.3126/av.v2i1.8278

On Quasi Discrete Topological Spaces in Information Systems

2012 ◽  
Vol 3 (2) ◽  
pp. 38-52 ◽  
Author(s):  
Tutut Herawan
Keyword(s):  
Information System ◽  
Information Systems ◽  
Topological Space ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Topological Spaces ◽  
Discrete Topology

This paper presents an alternative way for constructing a topological space in an information system. Rough set theory for reasoning about data in information systems is used to construct the topology. Using the concept of an indiscernibility relation in rough set theory, it is shown that the topology constructed is a quasi-discrete topology. Furthermore, the dependency of attributes is applied for defining finer topology and further characterizing the roughness property of a set. Meanwhile, the notions of base and sub-base of the topology are applied to find attributes reduction and degree of rough membership, respectively.

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