Maximally Even Sets

Clough, John; Douthett, Jack

doi:10.2307/843811

Cited by 160 publications

(114 citation statements)

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“…If S is a discrete set of pitches (e.g., the ones that can be produced on a normal piano) and we define a specific interval to be one between pitches in this discrete S and a generic interval to be the code onto which each specific interval is mapped, then a contradiction in Rothenberg's sense occurs when the ordering of two specific intervals is the opposite of the ordering of their corresponding generic intervals. The concept of contradiction defined in the more recent papers of Rahn [4] (p. 36) and Clough and Douthett [8] (p. 125) is therefore identical to Rothenberg's concept in the special case where S is a discrete set of pitch points. Rothenberg [10] (pp.…”

Section: C(d) C(p) and Rothenberg's Concept Of Proprietymentioning

confidence: 76%

“…The concept of a contradiction discussed by Rahn [4] (p. 36) and Clough and Douthett [8] (p. 125) can be defined as follows.…”

Section: C(d) and Contradictionsmentioning

confidence: 99%

“…For example, S({3, 2, 7, 10, 8, 0, 5}) = 0, 2, 3, 5, 7, 8, 10 . Following Clough and Myerson [9] (p. 262) and Clough and Douthett [8] (p. 95), let the k-spectrum, e k (P ), of a pitch class set, P , be defined as follows:…”

Section: Minimal Simple Cycle Sets and The Diatonic Spectramentioning

confidence: 99%

“…The sets D and M are each closed under inversion. However, the inversion of a harmonic minor scale set is in the transpositional equivalence class [{0, 1,3,5,6,8, 9}] T , denoted here by H maj . This is the set class associated with what Rahn [4] (p. 41) calls the "major harmonic scales", an example of which is C-D-E-F-G-A -B-C.…”

Section: Introductionmentioning

confidence: 99%

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Tonal Scales and Minimal Simple Pitch Class Cycles

Meredith

2011

Mathematics and Computation in Music

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Abstract. Numerous studies have explored the special mathematical properties of the diatonic set. However, much less attention has been paid to the sets associated with the other scales that play an important rôle in Western tonal music, such as the harmonic minor scale and ascending melodic minor scale. This paper focuses on the special properties of the class, T , of sets associated with the major and minor scales (including the harmonic major scale). It is observed that T is the set of pitch class sets associated with the shortest simple pitch class cycles in which every interval between consecutive pitch classes is either a major or a minor third, and at least one of each type of third appears in the cycle. Employing Rothenberg's definition of stability and propriety, T is also the union of the three most stable inversional equivalence classes of proper 7-note sets. Following Clough and Douthett's concept of maximal evenness, a method of measuring the evenness of a set is proposed and it is shown that T is also the union of the three most even 7-note inversional equivalence classes.

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Section: C(d) C(p) and Rothenberg's Concept Of Proprietymentioning

confidence: 76%

“…The concept of a contradiction discussed by Rahn [4] (p. 36) and Clough and Douthett [8] (p. 125) can be defined as follows.…”

Section: C(d) and Contradictionsmentioning

confidence: 99%

Section: Minimal Simple Cycle Sets and The Diatonic Spectramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tonal Scales and Minimal Simple Pitch Class Cycles

Meredith

2011

Mathematics and Computation in Music

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“…Another concept used to explain the properties of pitch and rhythm that has received much attention in music theory is 'maximal evenness' (Clough & Douthett, 1991;London, 2004;Rahn, 1996;Taylor, 2012;Toussaint, 2013). In simple terms, a rhythm is maximally even if the strokes are distributed as evenly as possible within the cycle.…”

Section: Maximal Evennessmentioning

confidence: 99%

Comparing Timeline Rhythms in Pygmy and Bushmen Music

Poole

2018

EMR

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Combining theories of African rhythm from ethno/musicology and findings from anthropological research and population genetics with musical analyses based on transcriptions and computational phylogenetic techniques, this article compares rhythms used in Pygmy and Bushmen music in an attempt to provide new perspectives on an old debate that these musical cultures may share a common heritage. To do this, the comparative analyses focus on timelines: foundational rhythmic features that provide the structural basis of the music. The findings suggest that Pygmy and Bushmen timelines are interrelated and that most are organised according to the principles of 'rhythmic oddity' and maximal evenness. Generative theory suggests that commonly used rhythmic cells, in particular the 3:2 pattern, form the structural basis of many Pygmy/Bushmen timelines as well as many other timelines featured in African and African-derived musics. Timelines are also multi-purpose musical devices used in various different social contexts and their structure appears to be resilient to radical change. Phylogenetic analysis of timelines provides no clear Pygmy/Bushmen ancestral timeline, although it is possible that foundational rhythms such as the 3:2 pattern may have featured in the music of a common ancestral group.

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The Geometry of Musical Rhythm

Toussaint

2005

Discrete and Computational Geometry

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Maximally Even Sets

Cited by 160 publications

References 0 publications

Tonal Scales and Minimal Simple Pitch Class Cycles

Tonal Scales and Minimal Simple Pitch Class Cycles

Comparing Timeline Rhythms in Pygmy and Bushmen Music

The Geometry of Musical Rhythm

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