Coordination of Interval Sizes in Seven-Tone Collections

Rahn, Jay

doi:10.2307/843809

Cited by 25 publications

(13 citation statements)

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“…As can be seen in Table 4 associated with what Rahn [4] (p. 37) calls the "neapolitan ascending melodic minor scale" or "whole tone scale with filler tone". Table 4 also illustrates the following lemma.…”

Section: If S ⊆ [P ]mentioning

confidence: 94%

“…Let R µ,n denote the set of non-contradictory sets of cardinality n when the chromatic cardinality is μ. Rahn [4] (p. 37) observes that R 12,7 = D ∪ M ∪ H ∪ W. From Eq. 17 above, it follows that C(D) = R 12,7 .…”

Section: Definition 1 If P Is a Pitch Class Set And There Exist Intementioning

confidence: 99%

“…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%

“…In other words, the set of minimal simple cycle sets associated with the diatonic spectra is equal to the union of the set of tonal scale sets, T , and the set of sets associated with what Rahn [4] (p. 37) calls the "whole-tone scale with filler tone", W.…”

Section: If S ⊆ [P ] T For Some Specified Pitch Class Set P Then Lmentioning

confidence: 99%

“…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. This scale type has been recognized within tonal theory since at least the mid-19th century.…”

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: If S ⊆ [P ]mentioning

confidence: 94%

Section: Definition 1 If P Is a Pitch Class Set And There Exist Intementioning

confidence: 99%

“…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: If S ⊆ [P ] T For Some Specified Pitch Class Set P Then Lmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%