A Geometric Property of the Octatonic Scale

An elementary geometric proof of Cohn’s theorem on dyadic parti-tioning of musical scales is presented. 1 Praeludium Most Western European music, past and present, is built upon the dia-tonic scale (witness Julie Andrews in The Sound of Music leading the von Trapp children: “Doe, a deer, a female deer...”). However, many alternative scales have found use throughout music history. For example [1], the Arabs in the 7th century A.D. utilized a scale known as “zer ef kend ” (“string of pearls”) composed of alternating half and whole steps (concepts which will be reviewed below). There was a resurgence of interest in this scale (now called the octatonic scale) during the late 19th and early 20th centuries led by Rimsky-Korsakov, Stravinsky, Debussy and Bartók. Interest in the octatonic scale persists to this day. The influential 1974 al-bum Red by King Crimson opens with composer/guitarist Robert Fripp burn-ing through a sequence of ascending octatonic scales. In the 1997 movie adap-tation of Robert A. Heinlein’s science fiction classic Starship Troopers, Basil Poledouris ’ soundtrack employs pulsating brass and string octatonic scales to presage appearances of the “bugs”. This octatonic scale has the property that it may be partitioned into pairs of pitches of the same “type ” in the maximum number of ways. This property (to be further elaborated upon below) is called dyadic partitioning and it allows the composer wide latitude in the construction of coherent musical structures. Cohn’s Theorem [2] states that there is no other scale possessing this particular combinatorial property [3]. 2418 B. J. McCartin In this paper, the geometrical model that was presented in [4] will be em-ployed in order to provide an elementary proof of this important musical theo-rem. However, in order to make the present account reasonably self-contained, we first review some requisite music theory fundamentals together with the basics of our geometric musical model