As we start to discuss symmetrical divisions of the octave, and gently move into later discussions of atonality, we’ll start using integer notation of pitch where appropriate.

Standard musical notation involves notes placed on a staff, with the noteheads designating a pitch to be played. While a notehead on a staff indicates a specific pitch to be played, enharmonic spellings are usually available that provide more choices (and sometimes ambiguity) about how to notate a pitch. C# is an enharmonic spelling of Db, etc. We’re beginning to see how these enharmonic spellings can be used to create ambiguity regarding harmonic progression and tonal areas. Dividing the octave symmetrically is a special case where using standard notation creates some cognitive dissonance regarding the appearance of the notation and the actuality of the sound.

Integer notation uses the numbers 0 – 11 to represent pitch, with C = 0, C#/Db = 1, E = 2, etc. to B = 11. In integer notation, there is no difference between C# and Db; both are represented as a 1 (one half-step away from C). A clock face can be used to show how the numbers continue to wrap around, with C always being 0. Up a whole step from 11 (B) is 1 (C#).

Instead of notating a perfectly symmetrical °7 chord as C# – E – G -Bb, which appears asymmetrical with the interval of the A2 from Bb – C#, you notate it as 1 – 4 – 7 – 10. The interval 10 up to 1 is clearly three half steps, just like the difference (interval) between the other numbers in the chord. And an enharmonic spelling of the °7, such as G – Bb – Db – Fb, which suggests a different function in tonal music, is clearly the same chord in integer notation (7 – 10 – 1 – 4; inverted to 1 – 4 – 7 – 10)

Integer notation will become central to how we analyze atonal works later on in the semester. For now, it will help us see symmetrical divisions of the octave.