From: dijk~orldaccess.nl Date: Mon, 26 Aug 96 20:55:25 GMT Subject: RE: The Importance Of The Circle Of Fifths.
Wim Dijkgraaf's reply to 'The Circle Of Fifths'.
For me, the circle of fifths is the most helpfull tool to recognise and understand pitch structures and harmonic relationships. I shall explain why. I'll try to explain my point of view about the importance of the circle of fifths in relation to completely understand the systems in our musical language.
Our tone system is based on 12 different pitches.
The term 'pitch-class' refers to all the pitches that are octave duplications of one another. There are only twelve pitch-classes. Because of our tonal heritage, we are used to naming these twelve pitch-classes with seven letter-names (A to G) plus sharp or flats. To clarify the status of pitches in both tonal and non-tonal music, many theorists now use a new system of note names when analyzing music. Pitch-classes are usualy named by numbers instead of letters.
Just as pitch-class is the grouping of all pitches of the same type, intervall-class is the grouping of all intervals of the same type. Each interval-class includes an interval, its complement, and all compounds of the interval and its complement. So, there are only 6 different interval classes:
[1,11],[2,10],[3,9],[4,8],[5,7],[6]
For some people, this text will be a little bit abra-ka-dabra. It's very simple to understand when you think of a 'clock'. A clock has only 12 hours. Each hour is a intervall-class. All the lines with different lengths you can draw between two hours in the clock are the six interval classes. You can only draw lines between hours 12&1, 12&2, 12&3, 12&4, 12&5, 12&6. Six lines which can be turn clockwise and anti-clockwise (=transposing).
Every pitch-construction can be represented as a figure in the clock (=the circle of (perfect)fifths = the circle of(perfect) fourths).
By drawing all possible figures in the clock, one has all possible pitch-combinations possible in our twelve tone system.
Combination possibilities in our twelve-tone system: Duotonic: (12.11/2!-6)/12 + 1= 6 (= the six intervall-classes) Tritonic: (12.11.10/3! -6)/12 +1 = 19 (=19 triads) Tetratonic: (12.11.10.9 / 4! -15)/12 +3 = 43 (=43 tetrachords) Pentatonic: (12.11.10.9.8 /5!)/12 = 66 (=66 pentatonic combinations) Hexatonic: (12.11.10.9.8.7 /6! - 24)/12 +5 = 80 Heptatonic: (12.11.10.9.8.7.6 /7!) /12 = 66 (inversion of pentatonisism) Octotonic: ......................=43 (inversion of tetratonisism) Enneatonic: .....................=19 (inversion of tritonisism) etc.
When you draw a augmented triad, you see a triangle with three sides with the same length (I don't know the english term for a 'gelijkbenige driehoek').
When you draw a diminished seven chord, you see a rectangle.
The major scale is half the circle.
A pentatonic scale are five hours beside each other.
Draw a minor triad and a major triad and see they are exactly the same figures, but inverted.
These are the symmetrical figures & pitch structures possible in our 12 tone system: * OM= Olivier Messiaen (French Composer who loved symetrical pitch structures). Hour: Name:
12&6 Tritone 12&4&8 Augmented triad 12&3&6&9 Diminished seventh chord 12&1&6&7 A seventh chord (non-tonal) 12&2&6&8 A seventh chord (difficult tonal chord, some possibilities) 12&2&4&6&8&10 The whole-tone scale! (OM Modus 1) 1&2&4&5&7&8&10&11 Half step / Whole step (diminished scale) Inversion of dimminished seventh chord (OM Modus 2). 1&2&3& 5&6&7& 9&10&11 Inversion of augmented triad (OM Modus 3). 3&4&5&6& 8&9&10&11 OM Modus 4 (Inversion of 12&1&6&7) 12&1&2& 6&7&8 OM Modus 5 1&3&4&5&7&9&10&11 OM Modus 6 (Inversion of 12&2&6&8) 1&2&3&4&5&7&8&9&10&11 OM Modus 7 (Inversion of tritonus) 12&3&4&7&8&11 OM Modus 8 or The Augmented Scale (see book: The AUGMENTED Scale In Jazz, by Walt Weiskopf & Ramon Ricker) 12&1&3&6&7&9 OM Modus 9 12&1&4&6&7&10 OM Modus 10
(Modus 9&10 are never mention by Olivier Messiaen but are also symetrical tone combinations!)
- --------------------------------------------------------------------------------- By representing pitch-structures as figures, one can instantly recognise patterns . Even 12 tone rows in dodecafonic music can be drawn as line constructions!
Because of the fact, that functonal harmony has the perfect fifth (=perfect fourth) as the basis, all of the most common harmonic formuleas are hours in the clock that lay beside of each other.
Just draw the following harmonic patterns as line construction in the clock:
II, V, I (for instance: D,G,C) VI, II, V, I (for instance: A,D,G,C) I,IV,VII,III,VI,II,V,I (for instance: C,F,B,E,A,D,G,C) I,bIII, bVI,bII (for instance: C,Eb,Ab,Db and look at the analogy with (I,)VI,II,V,I A lot of jazz pieces don't modulate outside the bVII and #IV borders: #IV,VII,III,VI,II,V,I,IV,bVI (the enneatonic perfect fifth tone system)
- ---------------------------------------------------------------------------------- What about drawing rhythmic patterns from the 12/8 time into the clock? Every puls being a pitch-class!! Very nice to look at the clave patterns and the analogy with a major scale and the pentatonic scale!
This know-how is not new at all. Pythagoras already made the calculations and constructed different tone-systems based on prime-numbers with far more interesting tone-combination possibilities. In compositions for electronic and digital circuits (computer music), other tone-systems are used very frequently. In the Netherlands, Fokker builded a 31 tone organ based on Pythagoras' theory, for which the dutch composer Peter Schat (who does use the tone-clock intensively), composed at least one piece.
