atonal, dodecaphonic, serial, twelve-tone
are all used as descriptions for certain types of modern music, notably that of Schoenberg and his followers. But there is an essential difference, not always respected,
between atonal and the others. Atonal, the only one of the four old enough to appear in the OED Supp., and there defined as ‘a style of composition in which there is no conscious reference to any scale or tonic’, is a purely negative term, vague and intrinsically subjective, applicable to any music that does not give a feeling of a key. The others are positive terms for rigid methods of composition which, treating all twelve notes of the octave as of equal and independant value (hence dodecaphonic and twelve-tone), prescribe certain rules governing the sequence in which they should
be used (hence serial). That music so composed will naturally be atonal is incidental; Schoenberg himself disapproved of the use of the word as a descriptive label
for his music.
A Dictionary of Modern English Usage