Symmetrical Scales Explained
Progressive rock and avant-garde metal draw quite a lot of theory ideas from classical music and jazz, and symmetrical scales are among them. I have written up this explanation of symmetrical scales in order to help people understand my analysis of ‘Larks Tongues in Aspic’ by King Crimson, but hopefully it will be useful to anyone interested in the subject.
Traditional major and minor scales are diatonic scales - seven note scales which alternate between whole and half steps in an uneven (asymmetrical) manner, allowing our ear to easily tell where we are in the scale. For example, a major scale alternates whole and half steps in the following way, starting from the root note (aka the first note in the scale): whole, whole, half, whole, whole, whole, half. If a note is preceded by a half step and followed by three whole steps, a trained ear can easily recognize that it is the fourth note in the scale.
Whole steps (W) and half steps (H) in a C Major Scale:
Symmetrical scales lack these points of reference, as they consist of regular intervallic patterns. This makes it very difficult for a listener to tell where a given note is located in the scale. The other main property that differentiates symmetrical scales from standard diatonic scales is transposition. A major scale can be transposed any number of times due to its asymmetry - you can start a major scale on any of the 12 chromatic tones, and it will form a distinct set of notes each time. Symmetrical scales can only be transposed a limited number of times before you run into a scale with the same notes.
The most basic symmetrical scale is the whole tone scale, a six-note scale consisting exclusively of whole steps. The whole tone scale can only be transposed once, meaning there are only two whole tone scales. The first is C, D, E, F#, G#, A# and the second is C#, D#, F, G, A, B:
If you play a whole tone scale starting on D, the pitches will be identical to the first scale. If you play a whole tone scale starting on D#, the pitches will be identical to the second scale. These two scales complement each other - if you put all of the notes of the two scales together, you get a chromatic scale. Whole tone scales were used extensively by 20th century French composer Claude Debussy. ‘Voiles’, the second piece in Book 1 of Debussy’s piano preludes, is a great example of his use of whole tone scales. King Crimson’s ‘Fracture’, the closing song on ‘Starless and Bible Black’, is built entirely out of whole tone scales.
The second most basic symmetrical scale is an eight-note scale which alternates whole steps and half steps. It is called the whole-half scale or diminished scale in jazz, and it is called the octatonic scale (the term I will use) in classical music. There are only three transpositions of the octatonic scale:
1. C, D, E♭, F, F#, G#, A, B
2. C#, D, E, F, G, A♭, B♭, B
3. C, D♭, E♭, Fb, Gb, G, A, B♭
Note that the note spellings here are somewhat arbitrary because our system of naming notes is designed around diatonic scales - there are seven letters for seven notes. Any of the sharps here could just as easily be spelled as flats, and vice versa. All non-diatonic music has this notation problem.
There are many different ways to use octatonic scales. The Messiaen motive that I discussed towards the end of Rock Meets Classical Part 3 is a great example of pure octatonicism. On the title track from King Crimson’s ‘Red’ (1974), Robert Fripp used the octatonic scale to create a melody and chordal accompaniment, which gives the song a sense of harmonic movement. By contrast, the Swedish metal band Meshuggah prefers to use octatonicism as a static mode to create a sense of mystery - check out the quiet break and guitar solo in ‘Bleed’, or the entire song ‘Closed Eye Visuals’. Stravinsky’s ballet ‘The Firebird’ also provides a good example, as it uses the octatonic scale to create a mystical feeling around the ballet’s fantastical elements, while using extended Romantic-era harmonies for the rest.
The octatonic scale is also called the diminished scale because the notes easily form diminished triads and sevenths. The scale can be divided into two complimentary diminished seventh chords. For instance, the first scale can be viewed as a C-E♭-F#-A diminished seventh plus the diminished seventh one step above - D-F-G#-B. These chords are also symmetrical - you can invert them, but the intervals between each note will not change - they will still all be minor thirds (C-E♭-F#-A inverts to E♭-F#-A-C which inverts to F#-A-C-E♭, etc.). In contrast, if you invert a major triad, which consists of a major third followed by a minor third (for example, C-E-G), the intervals change - a first inversion major triad consists of a minor third stacked below a perfect fourth (E-G-C).
Similarly, augmented triads (all major thirds, like C-E-G#) are formed from whole tone scales. Due to the symmetry of these triads, you cannot form augmented sevenths by stacking another major third on top of the chord - for example, the note a major third above G# is C, so you’ve just returned to the starting position. Whole tone scales can also be divided into two complimentary augmented triads - for example, C-E-G# and D-F#-A#.
20th century French composer Olivier Messiaen formed a list of symmetrical scales, which he called ‘modes of limited transposition’, and used them extensively in his music. He excluded the whole tone scale, as he felt its possibilities had already been thoroughly explored by Debussy, but widely used the octatonic scale. If you’re interested in learning about more symmetrical scales, check out Messiaen’s complete list.
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