Editing Maximal evenness (section)

{{Short description|Concept in music theory}}
[[Image:Maximal evenness seconds.png|thumb|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).]]
[[Image:Not maximal evenness seconds.png|thumb|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 [[scale (music)]] theory, a [[maximally even set]] (scale) is one in which every [[generic interval]] has either one or two consecutive integers [[specific interval]]s-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.<ref>{{cite journal |last1=Clough |first1=John |last2=Douthett |first2=Jack |title=Maximally Even Sets |journal=Journal of Music Theory |date=1991 |volume=35 |issue=35 |pages=93–173 |doi=10.2307/843811 |jstor=843811 }}</ref> Clough and Douthett also introduced the maximally even algorithm. For a chromatic cardinality ''c'' and [[Pitch class|pc-set]] cardinality ''d'' a maximally even set is

<math>D = {\left \lfloor \frac{ck + m}{d} \right \rfloor }</math>

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.<ref>{{cite book |last1=Johnson |first1=Timothy |title=Foundations of Diatonic Theory: A Mathematical Based Approach to Musical Fundamentals |date=2003 |publisher=Key College Publishing |isbn=1-930190-80-8}}</ref> Jack Douthett and Richard Krantz introduced maximally even sets to the mathematics literature.<ref>{{cite journal |last1=Douthett |first1=Jack |last2=Krantz |first2=Richard |title=Maximally Even Sets and Configurations: Common Threads in Mathematics, Physics and Music |journal=Journal of Combinatorial Optimization |date=2007 |volume=14 |issue=4 |page=385-410|doi=10.1007/s10878-006-9041-5 |s2cid=41964397 }}</ref><ref>{{cite journal |last1=Douthett |first1=Jack |last2=Krantz |first2=Richard |title=Dinner Tables and Concentric Circles: A harmony of Mathematics, Music, and Physics |journal=College Mathematics Journal |date=2007 |volume=39 |issue=3 |page=203-211|doi=10.1080/07468342.2008.11922294 |s2cid=117686406 }}</ref>

A scale is said to have [[Myhill's property]] if every [[generic interval]] comes in two [[specific interval]] sizes, and a scale with Myhill's property is said to be a [[well-formed scale]].<ref>{{cite journal |last1=Carey |first1=Norman |last2=Clampitt |first2=David |title=Aspects of Well-Formed Scales |journal=Music Theory Spectrum |date=1989 |volume=11 |issue=2 |pages=187–206 |doi=10.2307/745935 |jstor=745935 }}</ref> The [[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<ref>{{cite journal |last1=Lerdahl |first1=Fred |title=Cognitive Constraints on Compositional Systems |journal=Contemporary Music Review |date=1992 |volume=6 |issue=2 |page=97-121|doi=10.1080/07494469200640161 |citeseerx=10.1.1.168.1343 }}</ref> "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 [[dynamical]] approach, spinning [[concentric circles]] and iterated maximally even sets have been constructed. This approach has implications in [[Neo-Riemannian theory]], and leads to some interesting connections between [[diatonic]] and [[chromatic]] theory.<ref>{{cite journal |last1=Douthett |first1=Jack |title=Filter Point-Symmetry and Dynamical Voice-Leading |journal=Music and Mathematics:Chords, Collections, and Transformations |date=2008 |volume=Eastman Studies in Music |page=72-106. Ed. J. Douthett, M. Hyde, and C. Smith. University of Rochester Press, NY|doi=10.1017/9781580467476.006 |isbn=9781580467476 }} {{ISBN|1-58046-266-9}}.</ref> Emmanuel Amiot has discovered yet another way to define maximally even sets by employing [[discrete Fourier transform]]s.<ref>{{cite journal |last1=Armiot |first1=Emmanuel |title=David Lewin and Maximally Even Sets |journal=Journal of Mathematics and Music |date=2007 |volume=1 |issue=3 |page=157-172|doi=10.1080/17459730701654990 |s2cid=120481485 }}</ref><ref>{{cite book |last1=Armiot |first1=Emmanuel |title=Music Through Fourier Space: Discrete Fourier Transform in Music Theory |date=2016 |publisher=Springer |isbn=9783319455808}}</ref>

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