Download Mathematical Properties of the Octatonic “Diminished

Mathematical Properties of the Octatonic “Diminished” Scale
History
The octatonic or diminished scale is thought to have been first discovered by the early
Persian culture in the 7th century AD. It was not, however, used in Western music until at
least the 1730s (Scarlatti) and was not recognised or written about until 1797. Composers
only started to deliberately use this scale in their compositions in the mid-19th century (Liszt,
Stravinsky, Messaien etc). The scale is otherwise known as Messaien Mode 2 of this Modes
of Limited Transposition (meaning there are only a small number of such scales). (Source:
Wikipedia, 18/10/2014)
Description
The “Diminished” scale (to use its jazz terminology, as there are other octatonic or 8-note
scales) consists of a series of notes separated alternately by a tone and a semitone. For
example,
C D Eb F Gb Ab A B = C diminished scale.
There are only 3 different diminished scales, as if one transposes it up three semitones one
repeats the initial scale, only starting a minor third higher. There are two modes, one
starting with a tone and the second starting with a semitone (see Uses, 2 nd para).
Uses
In classical music this is used to provide a sense of exoticism and mystery, or to produce
musical tension, such as in The Rite of Spring (Stravinsky). Interestingly although Bach often
used diminished chords, I have failed to find the diminished scale in any of his works, which I
find quite surprising since he includes most other advanced harmonies (such as flattened
6ths, major 7ths and unusual chromatic sequences). It also, of course, serves as a scale
upon which to write melodies over a diminished chord.
In jazz the scale is very widely used, especially (and almost exclusively) by modern jazz
musicians, as a scale to improvise on not only over diminished chords, but also over certain
dominant 7th chords (namely 13b9 chords, e.g. C13b9 which consists of C, E, G, Bb, Db and A
natural). One uses the second mode, described above under “Description”, for this
purpose. It can also be used as a rich source of “voicings” or alternative ways of playing a
dominant 7th chord. See Wikipedia for the number of triads (three-note chords separated
by a third) that can be formed from this scale.
Mathematical properties.
I have not found reference to this observation in any other material available on the
internet. Whether it is something I have discovered I am not sure, but it would seem to me
that a lot of mathematicians are musical and vice versa, so I would have thought someone
would have discovered it before me!
If one takes the notes of the second mode of the diminished scale in the order they appear
in the harmonic series, i.e. (for the second mode of Bb diminished, which begins on C):
C C’ G E Bb F# A Db Eb
These correspond to the following harmonics:
1 2 3 5 7 11 13 17 19
Which are of course the first 9 prime numbers!
This probably reflects the fact that the notes are not very harmonically related to the root of
the scale (for the second mode, this is the starting note), giving it the most dissonant sound
of any commonly used scale. However, even the untrained ear can tell the difference
between a chord or phrase played using this scale and one using random unrelated notes,
which sound completely dissonant.
Unfortunately, as each octave is obtained simply by multiplying the harmonic value by 2, the
prime number pattern is not continued beyond the first octave. Bang goes a method of
calculating prime numbers! However, as above the first octave many notes can be
described by more than one harmonic (they are so close as to be indistinguishable by the
unaided ear) it should be possible to construct a scale consisting entirely of prime numbers.
What this scale would sound like and whether it would have any uses would need a
synthesizer capable of producing a continuous increase in frequency and a program to
extract the required frequencies that correspond to prime number harmonics. Something I
am sure Messaien would have loved!
Further reading
More on triads to be derived from the diminished scale, voicings and progressions is to be
found in The Jazz Piano Book by Mark Levine (Sher Music Co.) and many other advanced
modern jazz educational and academic publications.
Forthcoming publications by the same author:
Mathematical and musical properties of the melodic minor (ascending) scale.
Jazz Piano Voicings – Covering all commonly used scales (Shawnuff Music).
Dorothy Shaw October 2014