Maximal evenness

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The major scale is maximally even. For example, for every generic interval of a second there are only two possible specific intervals: 1 semitone (a minor second) or 2 semitones (a major second).
The harmonic minor scale is not maximally even. For the generic interval of a second rather than only two specific intervals, the scale contains three: 1, 2, and 3 (augmented second) semitones.

In

specific intervals-in other words a scale whose notes (pcs) are "spread out as much as possible." This property was first described by John Clough and Jack Douthett.[1] Clough and Douthett also introduced the maximally even algorithm. For a chromatic cardinality c and pc-set
cardinality d a maximally even set is

where k ranges from 0 to d − 1 and m, 0 ≤ m ≤ c − 1 is fixed and the bracket pair is the

floor function. A discussion on these concepts can be found in Timothy Johnson's book on the mathematical foundations of diatonic scale theory.[2] Jack Douthett and Richard Krantz introduced maximally even sets to the mathematics literature.[3][4]

A scale is said to have

diatonic collection is both a well-formed scale and is maximally even. The whole-tone scale
is also maximally even, but it is not well-formed since each generic interval comes in only one size.

Second-order maximal evenness is maximal evenness of a subcollection of a larger collection that is maximally even. Diatonic triads and seventh chords possess second-order maximal evenness, being maximally even in regard to the maximally even diatonic scale—but are not maximally even with regard to the chromatic scale. (ibid, p.115) This nested quality resembles Fred Lerdahl's[6] "reductional format" for pitch space from the bottom up:

C E G C
C D E F G A B C
C D♭ D E♭ E F F♯ G A♭ A B♭ B C
(Lerdahl, 1992)

In a

chromatic theory.[7] Emmanuel Amiot has discovered yet another way to define maximally even sets by employing discrete Fourier transforms.[8][9]

Carey, Norman and Clampitt, David (1989). "Aspects of Well-Formed Scales", Music Theory Spectrum 11: 187–206.

References

  1. JSTOR 843811
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  2. ISBN 1-930190-80-8
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  3. S2CID 41964397
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  4. S2CID 117686406
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  5. JSTOR 745935
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  6. doi:10.1080/07494469200640161
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  7. ISBN 1-58046-266-9
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  8. S2CID 120481485
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  9. ISBN 9783319455808
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