I'm sure you are dying for some examples about now! The first zig was from around 1900 up until the First World War and the two composers who were foremost in it were Arnold Schoenberg and Igor Stravinsky. The representative works are Erwartung by Schoenberg, a dramatic monologue:
And The Rite of Spring by Stravinsky:
After World War I, there was the urge to recoil from all futuristic excess and rehumanize art after the inhumanity of war. Stravinsky in particular changed his style completely and became a "neoclassicist" writing music that was imbued with the sounds of archaic styles. The first piece in this new style was Pulcinella based on Baroque themes, premiered in 1920:
Schoenberg's response was rather different. He consolidated by coming up with a new way to systematize composition using the 12 chromatic tones in a specific order. This also ensured that tonality would be eliminated. What audiences and critics reacted to was the level of dissonance that ensued, but Taruskin points out that one of the first things that Schoenberg did was use his new system to write little gavottes and gigues! In other words, this was another kind of neoclassicism. Here is his Suite for Piano, op. 25 composed 1921 to 1923:
The first zag lasted through the 40s, but after the Second World War a new zig began with a new generation of composers that included Pierre Boulez, Karlheinz Stockhausen and John Cage to name just the most prominent. All these composers shared the futuristic urge and thought of music in terms of experimental research. Again, the audience reception of this music was of no importance. What was, was technological expertise and mastery and the elimination of the personal expression of the composer. Boulez did it through intricate compositional strategies including serializing other parameters of sound besides pitch, such as rhythm and articulation. Stockhausen invented a number of complex ways of composing and with "moment form" gave up the need for a directional structure. Cage went even further and used chance procedures to eliminate any compositional plan and, in one famous instance, even notes themselves. Here are examples by each composer. First, Boulez, Structures for two pianos:
Next Momente by Stockhausen:
Finally the Concerto for Piano and Orchestra by John Cage:
This again is a repudiation of the previous zig. Instead of fragmented complex rhythms it has a simple and continuing pulse and instead of complex dissonances it has consonances. Two composers in particular took these two simple ideas and turned them into a new phase of music history. Philip Glass began with simple repetitive melodies and arpeggios. Here is Music with Changing Parts from 1970:
Steve Reich had a different approach, but the two basic features of recurring pulse and consonance were the same. Here is Six Pianos from 1973:
Both composers have traveled a considerable distance from these beginnings. Philip Glass is now writing symphonies and concertos that, while they continue his repetitive arpeggios, develop them more than before. Steve Reich has gone on to create masterpieces such as his piece Different Trains that manages to simultaneously be a personal memoire and a memorial to the victims of the Holocaust--and to do so in a completely non-melodramatic way. Here is a performance of Different Trains:
Though still over a continuous pulse and still consonant, there are rich layers of expression and meaning.
So that is a brief history of the last hundred years in music--classical music at least! Popular music has an entirely different history.
Both zigs were futuristic, utopian, experimental and ignored audiences. Both zags were concerned with audience response and turned away from the extremes of the zigs. The first zag referred directly to music of the past but the second made more of a new beginning, scrubbing down music to its essence and then rebuilding. The classical music world is still somewhat divided over this second zag. Even though its leading composers have won considerable popularity with audiences and are even influential on pop musicians, the more ossified institutions of classical music still do not, after forty years, regard them as quite respectable. Part of the reason might be that the musicians of the previous zig still wield considerable power in these institutions and regard the so-called "minimalism" with genuine horror and distain. Time is not on their side, however.
Let's end with one more piece by Steve Reich, the piece that really launched his popularity and influenced so many other musicians. This is Music for 18 Musicians, composed between 1974 and 1976 and released on record in 1978:
UPDATE: After thinking about it a bit, I decided to change part of the title of this post from "The New Tonality" to "The New Consonance" because that is closer to reality.