Sets of pitch classes can be changed or “transformed” by transposing or inverting them.  Pitch classes are all the notes one or more octaves apart from one another and their enharmonic equivalents. Since all C’s belong to the same pitch class, once you ascend an octave you are right back where you started.  Pitch-class space is circular.  We can therefore display pitch classes around a circle like this:

Pitch-Class Circle, or “Clockface Diagram”

I refer to this as a “clockface diagram” because the twelve pitch-classes match the hours on a clock.  We often refer to pitch-classes by their positions in this diagram.  Pitch-class F-sharp is 6 because it is at the six o’clock position.  Pitch-class E is 4 because it is at the 4 o’clock position.  Notice that C is at the zero, or 12 o’clock position.  The use of numbers for pitch classes helps avoid the confusion that can arise from enharmonically equivalent note names:  Pitch-class C# or Db is simply pitch-class 1.

Pitch classes form sets when they cohere aurally as a melodic entity, such as a motive, or as a harmonic entity, such as a chord.  Larger sets, often called collections, may consist of both melodic and harmonic content combined.

When we represent a set of pitch classes by circling note-names on a clockface diagram, we can easily see intervals between pitch classes.  We can also picture transformations.  Transposition is spinning motion.  Transposition by 4 semitones (T4) is four clockwise clicks around the clockface.  Inversion is flipping motion across an axis that cuts the clockface down the middle.

The following videos will present the ideas of transposition and inversion with frequent references to linear pitch space, as represented by a conventional musical staff, and to circular pitch-class space, as pictured on a clockface diagram.


The following is a transformational analysis of the whole-tone passage that opens scene 1 from Leos Janacek’s The Cunning Little Vixen.  Here is the score for the passage: Janacek. Cunning Little Vixen, opening scene

   Transformational Graph for Janacek’s Cunning Little Vixen

In the following two videos, I will refer to the following worksheet: Worksheet on Inversion. Inverting Sets

   Inverting Sets

   Labeling Inversions Between Sets

The next couple of videos step through the process of identifying transformations between sets found in the opening of Schoenberg’s Op. 11.  Here is the score with select sets labeled: Schoenberg. op.11 no.1.opening

   How to Find Transformations, Part I

   How to Find Transformations, Part II

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