Modulo 12 

Using a Dodecaphonic Scale

(In Search Of My Muse #1.04 ) 

Since the twenties, Classical contemporary music composers have experimented with different musical scales in order to create new timbre colors and by the same token convey new emotions in their music. Indeed, in addition to the standard seven-note diatonic scale, composers have tried whole-tone scales, modal scales and twelve-tones-per-octave dodecaphonic scales.

Historically, Austrian composer Arnold Schoenberg was the first to propose a new radical system, with the hope to get rid of the tyranny of the tonal diatonic system: atonality was born!

Obviously, an in-depth analysis of Dodecaphonic scale composition is beyond the scope of this article : I will thus only give you a quick overview on how the system is used.

Pitch Class Set

As in a chromatic scale, there are 12 tones per octave in a dodecaphonic scale. Contrary to conventional notations, this scale uses integers values instead of staff notations: i.e 0,1,2,3,4,5,6,7,8,9,10,11 and 0 (12=0) corresponding to the chromatic scale C, C#, D, D#, E, F, F#, G, G#, A, A#, B and C.

Note: A table is shown hereunder, including two sets of slide rulers, to help you compute and visualize, at a glance, the relationships existing between the Pivot tone (PT) and all elements of your Modulo 12 serie (top part of the illustration).

1) Select a serie of integers values, randomly, within the set. These integers values represent the twelve-tones of your atonal melody. From now on, this will be your original serie Pitch Class Set or PCS. 

For example, if your PCS melody is 4,10,2,6,9,5,11,3,7,8,0,1 then, tone 4 is the main Pivot Tone (PT) in the serie (see lower-right illustration).

2) As a convention, all tones within the PCS must always be played in the same order.

3) Some tones can be repeated consecutively. 

For example: 4,10,2,2,2,6,9...

4) Any tones of the row can be transposed by n, in any octave. 

For example:

4+10=14-12= 2

10+10=20-12=8

2+10=12-12=0



5) PCS series can be played in four different ways by a single voice: 

Original serie. 
Retrograde serie : i.e the serie is flipped over downwards, 
Inverted serie : i.e the serie is flipped over backwards) 
Inverted retrograde serie : i.e the serie is flipped over backwards, then downwards. 


6) If needed, you can split your PCS into 3 or 4 parts: this will be your Pitch Class Subsets (PCSS).

For example: [4,10,2], [6,9,5] , [11,3,7] and [8,0,1] 

or [4,10,2,6], [9,5,11,3], [7,8,0,1]

7) All tones in your PCSS can undergo circular permutations.

For example:

4,10,2

10,2,4

2,4,10....

8) All tones in your PCS or PCSS can be multiplied by n (factorials)

For example: 

4*5=20 and 20-12= 8 

10*5=50 and 12*4=48; so, 50-48=2 

2*5=10

9) The serie can be played vertically, in order to build chords or to morph between harmonics).

10) Ideally, the range of the Modulo 12 scale should remain within 2 1/2 octaves.

11) Notation should be freely enharmonic: use flats, sharps and natural values.

12) Go back to 1....



Creative adaptations

As you have seen by now, a Modulo 12 serie gives birth to a succession of intervals and transpositions which are permutated constantly in a functional manner. Indeed, the serie organizes, in a coherent way, all melodic, harmonic and chordal variations vs. time. All other parameters, like events duration, are only different aspects of the same musical reality (see lower-left illustration showing angular relationships between several tones events vs. time ( notice that corresponding durations are plotted on the Z plane). Experiment and discover new ways to organize your sounds more efficiently.

Now, some of you might wonder why they should be concerned with this atonal serial system, considering that their music is a mix of tonal and/or hard-core electronic noises!

Remember, that in order to be fully creative you need an open mind and have the willingness to bend the rules to your own advantage, whenever it is necessary... Try to find ways to adapt the above Modulo 12 system of rules to your own musical compositions.

For example, If you have an analog synth and an hardware sequencer having four independent rows outputs (such as the 4x16 steps Serge TKB), you could program a tonal or atonal sequence and use the above system of rules creatively: row A= original sequence, row B= inverted sequence, row C= retrograde sequence and row D =inverted-retrograde sequence.

You could also use output rows A,B,C,D in parallel to build up chords or use the 16x4 steps A+B+C+D output - in conjunction with a vertical clock - to morph between harmonics, for each step. 

Evidently, you could also play, the original sequence in row A with the inverted sequence in row B, simultaneously.

Finally, you could use the Hold, UP/ Down and Reset functions of your sequencer to make selective circular permutations within n steps.

Be creative....

André 

 7/3/2000

(Extracts from 'Ye Olde Timer's Analogue Cookbook' by André C. Stordeur) 

André Stordeur has taught analog modular synthesis since 1973. He studied with David Wessel at the I.R.C.A.M, in Paris, and with American composer Morton Subotnick.