Cardinality equals variety
Encyclopedia
The musical operation of scalar transposition
shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition
shifts every note in a melody by the same distance in pitch class
space. In general, for a given scale S, the scalar transpositions of a line L can be grouped into categories, or transpositional set classes
, whose members are related by chromatic transposition. In diatonic set theory
cardinality equals variety when, for any melodic line L in a particular scale S, the number of these classes is equal to the number of distinct pitch classes in the line L.
For example, the melodic line C-D-E has three distinct pitch classes. When transposed diatonically to all scale degrees in the C major scale, we obtain three interval patterns: M2-M2, M2-m2, m2-M2.
Melodic lines in the C major scale with n distinct pitch classes always generate n distinct patterns.
The property was first described by John Clough
and Gerald Myerson in "Variety and Multiplicity in Diatonic Systems" (1985) (Johnson 2003, p.68, 151). Cardinality equals variety in the diatonic collection and the pentatonic scale
, and, more generally, what Carey and Clampitt (1989) call "nondegenerate well-formed scales." "Nondegenerate well-formed scales" are those that possess Myhill's property
.
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
shifts every note in a melody by the same distance in pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
space. In general, for a given scale S, the scalar transpositions of a line L can be grouped into categories, or transpositional set classes
Set theory (music)
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...
, whose members are related by chromatic transposition. In diatonic set theory
Diatonic set theory
Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and...
cardinality equals variety when, for any melodic line L in a particular scale S, the number of these classes is equal to the number of distinct pitch classes in the line L.
For example, the melodic line C-D-E has three distinct pitch classes. When transposed diatonically to all scale degrees in the C major scale, we obtain three interval patterns: M2-M2, M2-m2, m2-M2.
The property was first described by John Clough
John Clough
John Clough born September 13, 1984 is a rugby league player who currently plays for the Blackpool Panthers rugby league team. Clough is the club Captain. A former Lancashire Academy representative, John plays at hooker. John was born in St Helens and is brother of St Helens player Paul...
and Gerald Myerson in "Variety and Multiplicity in Diatonic Systems" (1985) (Johnson 2003, p.68, 151). Cardinality equals variety in the diatonic collection and the pentatonic scale
Pentatonic scale
A pentatonic scale is a musical scale with five notes per octave in contrast to a heptatonic scale such as the major scale and minor scale...
, and, more generally, what Carey and Clampitt (1989) call "nondegenerate well-formed scales." "Nondegenerate well-formed scales" are those that possess Myhill's property
Myhill's property
In diatonic set theory Myhill's property is the quality of musical scales or collections with exactly two specific intervals for every generic interval, and thus also have the properties of maximal evenness, cardinality equals variety, structure implies multiplicity, and be a well formed generated...
.
Further reading
- Clough, John and Myerson, Gerald (1985). "Variety and Multiplicity in Diatonic Systems", Journal of Music Theory 29: 249-70.
- Carey, Norman and Clampitt, David (1989). "Aspects of Well-Formed Scales", Music Theory Spectrum 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.
Source
- Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1-930190-80-8.