Structure implies multiplicity

2024-07-17T01:27:24Z

162.83.141.55: ME does not imply C=V or SM

In [[diatonic set theory]] '''structure implies multiplicity''' is a quality of a collection or [[scale (music)|scale]]. For collections or scales which have this property, the interval series formed by the shortest distance around a diatonic [[circle of fifths]] between members of a series indicates the number of unique [[interval (music)|interval]] patterns (adjacently, rather than around the circle of fifths) formed by [[diatonic transposition]]s of that series. Structure refers to the intervals in relation to the circle of fifths; multiplicity refers to the number of times each different (adjacent) interval pattern occurs. The property was first described by [[John Clough]] and [[Gerald Myerson]] in "Variety and Multiplicity in Diatonic Systems" (1985). ({{harvnb|Johnson|2003|pp=68, 151}})

Structure implies multiplicity is true of the [[diatonic collection]] and the [[pentatonic scale]], and any subset.

For example, [[cardinality equals variety]] dictates that a three member diatonic subset of the C major scale, C-D-E transposed to all [[scale degree]]s gives three interval patterns: M2-M2, M2-m2, m2-M2.

[[Image:Cardinality equals variety CDE.PNG|400px|three member diatonic subset of the C major scale, C-D-E transposed to all scale degrees]]
[[File:Structure implies multiplicity circle of fifths.png|thumb|C-D-E on the circle of fifths]]

On the circle of fifths:
C G D A E B F (C)
1 '''2''' 1 '''2''' 1 2 '''3'''
E and C are three notes apart, C and D are two notes apart, D and E two notes apart. Just as the distance around the circle of fifths between forms the interval pattern 3-2-2, M2-M2 occurs three times, M2-m2 occurs twice, and m2-M2 occurs twice.

[[Cardinality equals variety]] and structure implies multiplicity are true of all collections with [[Myhill's property]].

==References==
*{{cite book |last=Johnson |first=Timothy |date=2003 |title=Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals |publisher=Key College Publishing |isbn=1-930190-80-8}}

==Further reading==
*Clough, John and Myerson, Gerald (1985). "Variety and Multiplicity in Diatonic Systems", ''Journal of Music Theory'' 29: 249-70.
*Agmon, Eytan (1989). "A Mathematical Model of the Diatonic System", ''Journal of Music Theory'' 33: 1-25.
*Agmon, Eytan (1996). "Coherent Tone-Systems: A Study in the Theory of Diatonicism", ''Journal of Music Theory'' 40: 39-59.

{{Set theory (music)}}

[[Category:Diatonic set theory]]

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Structure implies multiplicity