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In diatonic set theory
Diatonic set theory

Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and musical analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formed generated collection, the deep scale property, cardinality equals variety, and structure implies m...
 a generic interval is the number of scale step
Step

The term Step or Steps may refer to:* Walk* Dance step, the building block of many dances* Step dance, a dance style where the footwork is the most important part of the dance...
s between notes of a collection
Set (music)

In Set theory , a set is a collection of discrete entities, for example pitch sets, rhythm sets, and timbre sets . A set form is the arrangement of an ordered set: the prime form , inversion , retrograde , and retrograde inverse ....
 or scale. The largest generic interval
Interval (music)

In music theory, the term interval describes the relationship between the pitch of two notes.Intervals may be described as:*vertical if the two notes sound simultaneously...
 is one less than the number of scale members. (Johnson 2003, p.26)

In the diatonic collection the generic interval is one less than the corresponding diatonic interval: The largest generic interval in the diatonic scale being 7-1 = 6.

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 collection....
 is the quality of musical scales or collections with exactly two specific interval
Specific interval

In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle , in other words the number of half steps between note ....
s for every generic interval.






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In diatonic set theory
Diatonic set theory

Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and musical analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formed generated collection, the deep scale property, cardinality equals variety, and structure implies m...
 a generic interval is the number of scale step
Step

The term Step or Steps may refer to:* Walk* Dance step, the building block of many dances* Step dance, a dance style where the footwork is the most important part of the dance...
s between notes of a collection
Set (music)

In Set theory , a set is a collection of discrete entities, for example pitch sets, rhythm sets, and timbre sets . A set form is the arrangement of an ordered set: the prime form , inversion , retrograde , and retrograde inverse ....
 or scale. The largest generic interval
Interval (music)

In music theory, the term interval describes the relationship between the pitch of two notes.Intervals may be described as:*vertical if the two notes sound simultaneously...
 is one less than the number of scale members. (Johnson 2003, p.26)

In the diatonic collection the generic interval is one less than the corresponding diatonic interval:
  • Adjacent intervals, second
    Second

    The second , sometimes abbreviated sec., is the name of a units of measurement of time, and is the International System of Units SI base unit of time....
    s, are 1
  • Third
    Third

    Third may refer to:*3 , such as the 3rd of something*Fraction , such as 1/3*The Third *Third World, economically underdeveloped nations*Third-class degree, type of British undergraduate degree classification...
    s = 2
  • Fourth
    Fourth

    Fourth can refer to:* ?, one quarter* Fourth, the ordinal number following third...
    s = 3
  • Fifth
    Fifth

    Fifth may refer to:* Fifth Amendment to the United States Constitution, as in the expression "Taking the Fifth".* One fifth, a quintile, or 20% of a certain amount...
    s = 4
  • Sixth
    Sixth

    Sixth can refer to:* Sixth_Amendment_to_the_United_States_Constitution to the U.S. Constitution...
    s = 5
  • Seventh
    Seventh

    Seventh may refer to:*Seventh_Amendment_to_the_United_States_Constitution to the U.S. Constitution...
    s = 6
The largest generic interval in the diatonic scale being 7-1 = 6.

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 collection....
 is the quality of musical scales or collections with exactly two specific interval
Specific interval

In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle , in other words the number of half steps between note ....
s for every generic interval. In other words, each generic interval can be made from one of two possible different specific intervals.

Source

  • Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1-930190-80-8.