(musTh 212) Pitch-Class Sets and Interval Vectors, Part 1

Pitch-Class Sets

Pitch-Class Sets are unordered collections of pitch-classes.

Normal Form. The normal form of a pitch-class set is the most compressed way of writing the set in ascending pitch-class order. Compressed means traversing the shortest interval from first pitch-class to last, and placing smaller intervals first.

Transposition. A pitch-class set can be transposed by n (arbitrary number of) semitones within one octave.

Inversion. Inversion of pitch-classes within a set mirrors around 0. The pitch-class set [3,5,8] inverts:

• 0 – 3 = 9
• 0 – 5 = 7
• 0 – 8 = 4

Arrange the new pitch-classes into ascending order [4,7,9] to get the inversion around 0 (T0I). To invert around any other pitch-class, first invert around 0 then add the transposition number. T2I of [3,5,8] = the inversion [4,7,9] with 2 added to each pitch-class = [6,9,11].

Interval-Class Content – Interval Vector

We can take inventory of the interval classes within a melody or pitch-class set. Interval classes must be between 1 and 6 (inclusive). The process of extracting interval classes from pitch-class sets is methodical. For example, for the pitch classes 5, 0, 7, 6:

• start by finding the normal form of the pitch-class set: [5,6,7,0]
• starting with the first note of the set, find the interval classes that it forms with the remaining notes: 5 -> 6, 5 ->7, 5 -> 0 gives 110010 (or 1 instance of interval class 1, 5-6; 1 instance of interval class 2, 5-7; zero instances of interval classes 3 and 4; 1 instance of interval class 5, 5->0 = 7, inverted to 5; and zero instances of interval class 6).
• proceed to the next pitch class in the set and do the same with all intervals above that note: 6 -> 7 and 6->0. We don’t need to count 6 -> 5, as that was already counted in the first pass. This gives 1 instance of interval class 1 and 1 instance of interval class 6 (100001).
• repeat the previous step with each pitch class that is not the last in the set. In this case, 7 -> 0 is the last step, with 1 instance of interval class 5 (000010).
• Add up the interval content for each pass. Note, these are not binary numbers, but rather independent columns.
  1  1  0 0 1 0
  1  0 0 0 0 1 +
  0 0 0 0 1 0 +
  2 1  0 0 2 1 is the resulting interval vector for the set.