Twelve-tone technique

Twelve-tone technique (also dodecaphony) is a system of musical composition devised by Arnold Schoenberg. Music using the technique is called twelve-tone music. Josef Matthias Hauer also developed a similar system using unordered hexachords, or tropes, at the exact same time and country but with no connection to Schoenberg.

The technique

The basis of twelve-tone technique is the tone row or set, an ordered arrangement of the twelve notes of the chromatic scale (the twelve equal tempered pitch classes), or, rather, an ordered arrangement of intervals which produce those notes. When the technique is strictly applied, an entire piece must be built up from statements of any transposition of this tone row in strict order or transformations of this row. Both melody and harmony may be created in this way. The set may be used in succession or simultaneously, the latter of which may be ordered up or down, or not. Given twelve pitch classes, there are 12! (12 factorial) possible tone rows, though invariance often reduces the number of distinct rows.

Related Topics:
Tone row - Set - Chromatic scale - Pitch class - Interval - Transformation - Melody - Harmony - Factorial

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The initial tone row, or set form, used is called the prime series (P), untransposed it is P0. P can be used starting on any one of the twelve notes of the chromatic scale (Pχ) - so long as the intervals are the same, the rows are equivalent. Pχ = P0 + χ.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Additionally, P can be transformed in three basic ways: it can be turned backwards to get the retrograde (R) or turned upsidedown to give the inversion (I) or the reverse contour direction. I(χ) = 12 - Pχ. These three transformative techniques can be combined to give the retrograde inversion (RI). As with the prime series, R, I and RI can be transposed to any note of the chromatic scale.

Related Topics:
Retrograde - Inversion - Retrograde inversion

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

thus:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

More recently, composers such as Charles Wuorinen have also used multiplication of the row. However, there are only a few numbers which one may multiply a row by and still end up with twelve tones. Multiplication is indicated by MX, X being the multiplier. As with the other compound operations multiplication is carried out and then transposition. P0 = M10, I0 = M110, M70=I(M50). Thus, for the untransposed form of all:

Related Topics:
Charles Wuorinen - Multiplication

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Even numbers remain unchanged under M7 and all odd numbers become transposed by a tritone. The chromatic scale may be mapped onto the circle of fourths with M5, and the circle of fifths with M7.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Suppose the prime series is as follows:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Then the retrograde is the prime series in reverse order:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The inversion is the prime series with the intervals inverted (so that a rising minor third becomes a falling minor third):

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

And the retrograde inversion is the inverted series in retrograde:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

P, R, I and RI can each be started on any of the twelve notes of the chromatic scale, meaning that 47 permutations of the initial tone row can be used, giving a maximum of 48 possible tone rows. However, not all prime series will yield so many variations because tranposed transformations may be identical to each other. This is known as invariance. A simple case is the ascending chromatic scale, the retrograde inversion of which is identical to the prime form, and the retrograde of which is identical to the inversion (thus, only 24 forms of this tone row are available).

Related Topics:
Permutation - Invariance

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

When rigorously applied, the technique demands that one statement of the tone row must be heard in full (otherwise known as aggregate completion) before another can begin. Adjacent notes in the row can be sounded at the same time, and the notes can appear in any octave, but the order of the notes in the tone row must be maintained. Durations, dynamics and other aspects of music other than the pitch can be freely chosen by the composer, and there are also no rules about which tone rows should be used at which time (beyond them all being derived from the prime series, as already explained).

Related Topics:
Octave - Dynamics

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Schoenberg's idea in developing the technique was for it to act as a replacement for tonal harmony as a basic grounding force for music. As such, twelve-tone music is usually atonal, and treats each of the 12 semitones of the chromatic scale with equal importance, as opposed to earlier classical music which had treated some notes as more important than others (particularly the tonic and the dominant note).

Related Topics:
Tonal - Harmony - Atonal - Semitone - Chromatic scale - Tonic - Dominant note

~ ~ ~ ~ ~ ~ ~ ~ ~ ~