802:
32:
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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,
63:-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. Clough and Douthett also introduced the maximally even algorithm. For a chromatic cardinality
157:. A discussion on these concepts can be found in Timothy Johnson's book on the mathematical foundations of diatonic scale theory. Jack Douthett and Richard Krantz introduced maximally even sets to the mathematics literature.
129:
27:
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).
689:
39:
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 (
80:
655:
598:
587:
428:
Douthett, Jack; Krantz, Richard (2007). "Maximally Even Sets and
Configurations: Common Threads in Mathematics, Physics and Music".
412:
682:
463:
Douthett, Jack; Krantz, Richard (2007). "Dinner Tables and
Concentric Circles: A harmony of Mathematics, Music, and Physics".
862:
846:
883:
781:
675:
825:
345:
574:. Eastman Studies in Music: 72-106. Ed. J. Douthett, M. Hyde, and C. Smith. University of Rochester Press, NY.
351:
Carey, Norman and
Clampitt, David (1989). "Aspects of Well-Formed Scales", Music Theory Spectrum 11: 187–206.
709:
801:
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is also maximally even, but it is not well-formed since each generic interval comes in only one size.
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theory. Emmanuel Amiot has discovered yet another way to define maximally even sets by employing
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and iterated maximally even sets have been constructed. This approach has implications in
19:
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405:
Foundations of
Diatonic Theory: A Mathematical Based Approach to Musical Fundamentals
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724:
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192:
769:
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196:
68:
24:
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441:
570:
Douthett, Jack (2008). "Filter Point-Symmetry and
Dynamical Voice-Leading".
341:
325:
337:
533:
Lerdahl, Fred (1992). "Cognitive
Constraints on Compositional Systems".
498:
Carey, Norman; Clampitt, David (1989). "Aspects of Well-Formed Scales".
648:
Music
Through Fourier Space: Discrete Fourier Transform in Music Theory
519:
389:
667:
511:
381:
30:
18:
611:
Armiot, Emmanuel (2007). "David Lewin and
Maximally Even Sets".
124:{\displaystyle D={\left\lfloor {\frac {ck+m}{d}}\right\rfloor }}
671:
572:
Music and
Mathematics:Chords, Collections, and Transformations
368:
Clough, John; Douthett, Jack (1991). "Maximally Even Sets".
172:
sizes, and a scale with Myhill's property is said to be a
180:
is both a well-formed scale and is maximally even. The
83:
336:, and leads to some interesting connections between
809:
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8:
59:has either one or two consecutive integers
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676:
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153:− 1 is fixed and the bracket pair is the
95:
90:
82:
360:
430:Journal of Combinatorial Optimization
191:p.115) This nested quality resembles
7:
14:
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613:Journal of Mathematics and Music
863:Structure implies multiplicity
847:Generic and specific intervals
477:10.1080/07468342.2008.11922294
55:(scale) is one in which every
1:
188:Second-order maximal evenness
195:'s "reductional format" for
465:College Mathematics Journal
346:discrete Fourier transforms
900:
826:Cardinality equals variety
407:. Key College Publishing.
798:
705:
646:Armiot, Emmanuel (2016).
625:10.1080/17459730701654990
580:10.1017/9781580467476.006
557:10.1080/07494469200640161
535:Contemporary Music Review
442:10.1007/s10878-006-9041-5
403:Johnson, Timothy (2003).
160:A scale is said to have
75:a maximally even set is
710:All-interval tetrachord
370:Journal of Music Theory
16:Concept in music theory
715:All-trichord hexachord
125:
44:
28:
833:(Deep scale property)
500:Music Theory Spectrum
334:Neo-Riemannian theory
126:
34:
22:
858:Rothenberg propriety
842:Generated collection
765:Pitch-interval class
199:from the bottom up:
81:
37:harmonic minor scale
884:Musical terminology
849:(Myhill's property)
782:Similarity relation
328:approach, spinning
178:diatonic collection
699:Musical set theory
330:concentric circles
121:
61:specific intervals
53:maximally even set
45:
29:
871:
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174:well-formed scale
170:specific interval
162:Myhill's property
137:ranges from 0 to
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57:generic interval
41:augmented second
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318:(Lerdahl, 1992)
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619:(3): 157-172.
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506:(2): 187–206.
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376:(35): 93–173.
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71:cardinality
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46:
43:) semitones.
831:Common tone
755:Pitch class
750:Permutation
197:pitch space
25:major scale
813:set theory
792:Z-relation
720:Complement
355:References
51:theory, a
633:120481485
543:CiteSeerX
485:117686406
342:chromatic
326:dynamical
164:if every
878:Category
821:Bisector
811:Diatonic
730:Identity
450:41964397
338:diatonic
141:− 1 and
117:⌋
93:⌊
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631:
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520:745935
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390:843811
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176:. The
145:, 0 ≤
133:where
69:pc-set
629:S2CID
516:JSTOR
481:S2CID
446:S2CID
386:JSTOR
324:In a
775:List
652:ISBN
595:ISBN
584:ISBN
409:ISBN
340:and
67:and
35:The
23:The
770:Set
621:doi
576:doi
553:doi
508:doi
473:doi
438:doi
378:doi
303:B♭
297:A♭
291:F♯
282:E♭
276:D♭
47:In
880::
627:.
615:.
582:.
551:.
537:.
514:.
504:11
502:.
479:.
469:39
467:.
444:.
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432:.
384:.
374:35
372:.
348:.
309:C
306:B
300:A
294:G
288:F
285:E
279:D
273:C
268:C
265:B
260:A
255:G
250:F
247:E
242:D
237:C
232:C
221:G
214:E
205:C
149:≤
691:e
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380::
151:c
147:m
143:m
139:d
135:k
112:d
108:m
105:+
102:k
99:c
88:=
85:D
73:d
65:c
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