Pitch-Class Set Theory Basics

Pitch-Class Set Theory Basics

• Pitch class is defined by notes sharing the same name but differing in octave.
• Each pitch class corresponds to an integer from 0 to 11.

Individual Pitch Classes

• 0 corresponds to C.
• 1 corresponds to C# (or Db).
• 2 corresponds to D.
• 3 corresponds to D# (or Eb).
• 4 corresponds to E.
• 5 corresponds to F.
• 6 corresponds to F# (or Gb).
• 7 corresponds to G.
• 8 corresponds to G# (or Ab).
• 9 corresponds to A.
• 10 corresponds to A# (or Bb).
• 11 corresponds to B.

Modulus Operations

• Mod-12 for positive numbers requires subtracting 12.
• Mod-12 for negative numbers requires adding 12 to return to a positive pitch class.

Pitch Class Intervals

• Pitch class interval indicates the exact number of half steps between two notes.
• Ordered pitch class interval is assessed by moving in an ascending (clockwise) direction.
• Unordered pitch class interval (or interval class) measures the shortest distance between two pitch classes; only six possible intervals exist.
• Intervals larger than six must be simplified to their complement.

Collections of Pitch Classes

• A pitch class set is an unordered collection of distinct pitch classes.
• Specific terms define the number of notes in a set:
  • Dyad for a 2-note set.
  • Trichord for a 3-note set.
  • Tetrachord for a 4-note set.
  • Pentachord for a 5-note set.
  • Hexachord for a 6-note set.

Interval Class Vector and Normal Order

• The interval class vector is a tally of the interval classes present in a set.
• Normal order arranges pitch classes in ascending order (mod-12) to achieve the most compact representation.
• To determine the normal order:
  • Exclude the largest interval.
  • List remaining pitch classes in ascending order.
  • For ties, list potential normal orders and select the one with the smallest ordered intervals.

Additional Pitch Classes

• Db corresponds to 1.
• Eb corresponds to 3.
• Gb corresponds to 6.
• Ab corresponds to 8.
• Bb corresponds to 10.