In
diatonic set theory
Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the ...
, a generated collection is a
collection
Collection or Collections may refer to:
* Cash collection, the function of an accounts receivable department
* Collection (church), money donated by the congregation during a church service
* Collection agency, agency to collect cash
* Collectio ...
or
scale formed by repeatedly adding a constant
interval in
integer notation
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
, the generator, also known as an
interval cycle, around the
chromatic circle until a complete collection or scale is formed. All scales with the
deep scale property can be generated by any interval
coprime with (in twelve-tone equal temperament) twelve. (Johnson, 2003, p. 83)
The C major diatonic collection can be generated by adding a cycle of
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
s (C7) starting at F: F-C-G-D-A-E-B = C-D-E-F-G-A-B. Using integer notation and modulo 12: 5 + 7 = 0, 0 + 7 = 7, 7 + 7 = 2, 2 + 7 = 9, 9 + 7 = 4, 4 + 7 = 11.
The C major scale could also be generated using cycle of
perfect fourths (C5), as 12 minus any coprime of twelve is also coprime with twelve: 12 − 7 = 5. B-E-A-D-G-C-F.
A generated collection for which a single
generic interval
In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members. (Johnson 2003, p. 26)
A specific interval is the clo ...
corresponds to the single generator or interval cycle used is a MOS (for "moment of symmetr
or well formed generated collection. For example, the diatonic collection is well formed, for the perfect fifth (the generic interval 4) corresponds to the generator 7. Though not all fifths in the diatonic collection are perfect (B-F is a diminished fifth, tritone, or 6), a well formed generated collection has only one
specific interval
In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members. (Johnson 2003, p. 26)
A specific interval is the clo ...
between scale members (in this case 6)—which corresponds to the generic interval (4, a fifth) but to not the generator (7). The major and minor
pentatonic scales are also well formed. (Johnson, 2003, p. 83)
The properties of generated and well-formedness were described by
Norman Carey
Norman or Normans may refer to:
Ethnic and cultural identity
* The Normans, a people partly descended from Norse Vikings who settled in the territory of Normandy in France in the 10th and 11th centuries
** People or things connected with the Norm ...
and
David Clampitt
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...
in "Aspects of Well-Formed Scales" (1989), (Johnson, 2003, p. 151.) In earlier (1975) work, theoretician
Erv Wilson
Ervin Wilson (June 11, 1928 – December 8, 2016) was a Mexico, Mexican/United States, American (dual citizen) music theory, music theorist.
Early life
Ervin Wilson was born in a remote area of northwest Chihuahua (state), Chihuahua, Mexico, wher ...
defined the properties of the idea, and called such a scale a ''MOS'', an acronym for "Moment of Symmetry".
While unpublished until it appeared online in 1999, this paper was widely distributed and well known throughout the
microtonal music
Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, also called "microintervals". It may also be extended to include any music using intervals not found in the customary Western tuning of tw ...
which adopted the term. the paper also remains more inclusive of further developments of the concept.
For instance, the
three-gap theorem implies that every generated collection has at most three different steps, the intervals between adjacent tones in the collection (Carey 2007).
A degenerate well-formed collection is a scale in which the generator and the interval required to complete the circle or return to the initial note are equivalent and include all scales with equal notes, such as the
whole-tone scale. (Johnson, 2003, p. 158, n. 14)
A
bisector is a more general concept used to create collections that cannot be generated but includes all collections which can be generated.
See also
*
833 cents scale
The 833 cents scale is a musical tuning and scale proposed by Heinz Bohlen based on combination tones, an interval of 833.09 cents, and, coincidentally, the Fibonacci sequence.Bohlen, Heinz (last updated 2012).An 833 Cents Scale: An experimen ...
*
Cyclic group
*
Distance model
*
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
References
Sources
*
*Carey, Norman and Clampitt, David (1989). "Aspects of Well-Formed Scales", ''Music Theory Spectrum'' 11: 187–206.
*Clough, Engebretsen, and Kochavi. "Scales, Sets, and Interval Cycles", 79.
*Johnson, Timothy (2003). ''Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals''. Key College Publishing. .
External links
Original concept of MOS as presented in a 1975 letter by Erv Wilson
{{Set theory (music)
Diatonic set theory