# Generalized Interval Systems (1): Preliminary Examples and Definition

# Generalized Interval Systems (1): Preliminary Examples and Definition

This chapter takes as its point of departure a figure showing two points *s* and *t* in a symbolic musical space. The arrow marked *i* symbolizes a characteristic directed measurement, distance or motion from s to t. It intuits such situations in many musical spaces, and *i* is called “the interval from *s* to *t*” when the symbolic points are pitches or pitch classes. The chapter begins by running through twelve examples of musical spaces: six involve pitches or pitch classes in melodic or harmonic relations; six involve aspects of measured rhythm. The general intuition at hand is then made formal by a mathematical model called a Generalized Interval System (GIS). A few basic formal properties of the model are explored. Then the twelve examples are reviewed to see how each instances the generalized structure.

*Keywords:*
Generalized Interval System, music theory, musical space, interval, pitch

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