More than you'd think initially, much 20th-century music keeps adhering to 18th- and 19th-century principles, structures, and institutions. Despite the century's reputation for experimenting, in terms of the instruments, the orchestral make-up, operatic form, pedagogical matters, and more, 20th-century music does not drastically break from the past.
The real and first important 20th-century musical innovations regarded the struggle against "functional tonality," a search for new forms of expression and underlying structural organizing principles. At the turn of the century, Western music had been distinguished traditionally by functional tonality, what we might loosely call a system whereby music is constructed on a basic scale formation, major or minor, within which hierarchies of sound (intervals, chords) prevail, expressed as points of stability and instability. Certain combinations of tones represent goals; other combinations suggest movement away from these. The tonic chord is a home base; most chords are triads; cadences are more or less predictable and involve harmonic resolution. So saying a piece is in Ab major is more than saying the first and last triads are Ab major (Ab-C-Eb). Both horizontal and vertical orders are established, and every musical event has a functional role in the sequence: direction and motion, expectation, resolution, dissonance or consonance, intensity, relaxation, cadence, phrase, punctuation, tempo, rhythm. Primary goals (e.g., the dominant-to-tonic or V-I cadence) bring secondary goals in modulations away from primary centers of gravity. Organization of sound determines aspects of our experience. That's functional tonality. 20th-century composers sought to establish their own premises, to find new modes of expression and organization.
At the turn of the century, the artistic and intellectual centers were Paris -- France was undergoing La belle époque (1871-1914) -- and Vienna. The musical world was still dominated by the late romantic aesthetic, and in music one heard increasingly dramatic secondary dominants, modulations, altered chords -- all the expansions of tonality and chromaticism that ultimately would undermine that very tonality.
The musical scene was also dominated by the influence of the German composer Richard Wagner (1813-1883), who had expanded the palette of orchestral and harmonic technique with increased chromaticism and incipient "atonality." Wagner's opera, Tristan und Isolde (1856-1859), is a clear example of chromaticism and seeming atonality, without absolute chaos: the music is "still based on the listener's expectation that one musical event implies another -- much of Tristan is built on the very idea of the defeat of expectation" (Salzman 5). In other words, patterns do urge us in directions, but amid the woven leitmotifs, there is no stable key area, and real resolution is postponed. We experience "expectation defeated by false and evasive resolution, harmonic delay, and long-range suspension" (Salzman 10). Listen to the opening Prelude / Liebestod (Love-Death) for a good example of tonal disorientation. Although Tristan emerged much earlier, its influence at the end of the century was significant.
Arnold Schoenberg (1874-1951) begins as a post-Wagnerian; this can be heard in Verklärte Nacht (1899) -- the pseudo-program music "Transfigured Night" (based on a poem vs. "pure music") -- and other works in early years of 1900s. Strauss and Mahler also figure in here among the post-Wagnerians. Eventually Schoenberg would sum up 20th-century music as "the emancipation of the dissonance," composing music with no tonal center at all in which all notes are equally related.
Somewhat outside this Germanic tradition was France. In the late 1800s Paris once again became an intellectual and artistic center in the Western world. A conscious search for new forms combined with fin-de-siècle abstracted sensuality. French music tended to be more fluid and poetical than the metrical "verse" construction of classical Italian and German music: less directional, more coloristic, with its "metrical, rhythmic, and phrase flexibility closely related to the free, non-accentual character of the French language" (Salzman 14). In his small-scale pieces, Satie, for example, used "disconnected, static, objectified sound -- in an abstract way" divorced from structural tonal principle (Salzman 19). More influential though was Claude Debussy (1862-1918), who brings us fully into 20th-century music.
In short, chromaticism; the extended and freer use of dissonance; the establishment of harmonic and melodic freedom; the use of harmonic, melodic, and structural ideas derived from folk music and early Western music; the concept of the structural interrelationships between all the parts of a musical composition; the discovery of the distant past and of non-Western music; the vast expansion of instrumental technique and color; the new freedom, complexity, and independence of rhythm, dynamics, and tone color -- all these modern ideas have roots deep in the last century. (Salzman 3)
Bonds, Mark Evan. A History of Music in Western Culture. Upper Saddle River, NJ: Prentice Hall, 2003.
Morgan, Robert P., ed. Music and Society: Modern Times. Englewood Cliffs, NJ: Prentice Hall Inc., 1993.
Ross, Alex. The Rest Is Noise: Listening to the Twentieth Century. NY: Farrar, Straus and Giroux, 2007.
Salzman, Eric. Twentieth-Century Music: An Introduction. 2nd ed. Englewood Cliffs, NJ: Prentice-Hall Inc., 1974.