There’s a very natural mapping between the equal-tempered musical octave and the face of a clock.  There are twelve notes in an octave and there are twelve hours on a clock. If we let twelve o’clock represent C, then the notes of a C major triad (C, E, G) would fall at twelve o’clock, four o’clock, and seven o’clock.  We can represent the triad by aligning the hour, minute, and second hands of the clock with the corresponding marks.  The spaces between any two hands of the clock show us the size of the interval between those two chord tones.

The idea of representing the octave as a clock is sometimes attributed to the composer Ernst Krenek and these octave-clock diagrams are sometimes called Krenek diagrams, or pitch constellations.  They’ve become very popular among people who study the so-called “geometry of music” and other connections between math and music.  I use them all the time in my own music practice when I’m trying to understand the structure of a chord or scale.  In fact, I’m so accustomed to drawing these diagrams that I sometimes assume most musicians are familiar with them, and I’m surprised when people say “Clock?  Huh?”  I’d like to use clock diagrams in some upcoming posts on music (particularly an upcoming investigation of the Nineteen Trichords), so I wanted to create some examples that show how these diagrams work.

Here you can see the notes of a C major root-position triad mapped out on the clock.  When looking at staff notation it’s common to read the notes from the bottom up: C, E, G.  On the clock, it’s a bit different: you take the lowest note of chord as the starting point and read clockwise till you’ve hit all the other notes.  You can see that the major third (M3) between C and E becomes a slice of 4 hours (corresponding to 4 semitones) on the clock face.  The minor third (m3) between E and G becomes a slice of 3 hours/semitones.  Finally, there’s a gray slice representing the non-sounding interval between the top note of the chord (G), and the bottom note transposed up an octave (click on the diagram for higher resolution):

C Major (Root Position) Clock Diagram

You can get an inversion of the chord by picking a different starting note on the clock face and reading clockwise from that note.  Here we start at E and read off the notes E, G, C, giving the triad in first inversion.  In this case we traverse a minor third (m3) and then a perfect fourth (P4):

C Major (First Inversion) Clock Diagram

Finally, you can transpose a chord by taking all the hands of the clock and rotating them as one unit  (so the spacing between the hands stays the same).  Here we transpose the C major triad up a semitone to C# major by moving all the hands clockwise by 1 hour.  Compare this diagram with the first diagram above:

C Sharp Major (Root Position) Clock DiagramIf you haven’t come across Krenek diagrams before it’s worth becoming familiar with them, because they make a lot of things about music a lot clearer. ■

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