Obscuring the Beat
Well, yes, but if all music were straight to the beat it would be boring. If you think about Bach's Jesu, Joy of Man's Desiring, the opening moving melodic line starts off the upbeat, even though the underlying bass line is firmly on the beat. The very idea of a hemiola is to shift from a complex (triplet) pattern to a simple (duplet) pattern and common in the music of Haydn and Mozart. As music progressed to the romantic composers, the beat became more and more sublime with irregular patterns creating a feeling of rubato even though the actual beat didn't change. Debussy was a master at creating music with no sense of pulse or one that seemed to float in hesitation as in his Prélude à l'après-midi d'un faune.
Mid way through the 20th Century composers moved so far in this direction that in Messian's Livre d'orgue (1951-2) - his only serial piece - the pulse is practically non-existent, even though the need to keep the pulse in the mind of the performer is critical in recreating the music. Ferneyhough and the School of New Complexity compose pieces where the metronome marking is often extremely slow, a quartet note equal to 50 to 60 beats a minute. However, the music itself is comprised of eighth notes to sixty-fourth notes (or even shorter) so the nuances of the subtle shifts in music can be accurately portrayed. The beat (or metre) exists, but not in any sense the audience can actually tap their foot to it.
Juxtapose this with popular music form. Latin rhythms spend as much if not more time off-beat than on. A mambo rhythm is comprised of one on-the-beat stress and then seven off-beat stresses. One of the biggest differences between rock and jazz music is the way the beats are stressed or laid back. Jazz tends to stress the two and four beats in a bar, with more of a triplet feel to the eighth notes, while rock stresses the one and three and the eighth notes are played straight. Obviously this is an over simplification, but a general rule of thumb for comparison purposes. In Hip Hop and other Urban forms, the stress of the beat is important as they are dance forms of music. However, it's common for the underlying drum pattern in the high hat to shift between stressing the beat to stressing the off beat - often this shift happening as the music moves from one eight bar section to another to create a different feel between the two sections.
Up to this point I've only spoken about relatively modern Western forms of music. Prior to Bach, the music of the Renaissance was fairly lose rhythmically. Madrigals have a metre to them, but the lines move independent of each other obscuring the pulse constantly. Looking back even further, plain chant may have rhythmic note values (or durations) but there is not an even metre to the progression. The time signature in music was developed in the late Renaissance. Indian Ragas are based on rhythmic and/or melodic movements, but how these ragas progress through a piece is wholly up to the performer. They may well migrate to be come unidentifiable from the original if held side by side, rather than seen as a progression from beginning to end (this is similar to Western motivic development). Jewish and Islamic chant is extremely free flowing avoiding any sense of beat and movement of the notes wholly in the mind and emotional state of the singer - even when accompanied by drums. There may be pulse points, places where the singer comes to a cadence in "time" with the drums, but in between these points the music is very free flowing. With the Gamelan, an Indonesians percussion ensemble, rhythm is very much a part of the music, but a strict pulse is rarely evident. Celtic folk music is very rhythmic, but listen to the stresses of a song like "The Rocky Road to Dublin" and you'll find the metre is in a 9/8 time signature (although I've seen it written in 4/4), with a hesitation beat in the phrase (a pain for bodhran players). Mongolian folk music has a very similar sound, with percussion playing a huge role in keeping the music propelled forward - but there is still a sense of hesitation in the music which keeps it interesting (and, even though culturally miles away from the Celts, very similar in sound).
So, what does this say about music and beat? Music is linear, it moves through time from a starting place and eventually comes to an end, but how it gets there isn't necessarily a straight line and certainly isn't one that followed a metered pace. Yes, there are pieces like Elgar's Pomp and Circumstance which plod along at a measured beat, which is why every high school and college in the US use this music for graduation - the students can keep in step (most of the time). These heavily metered pieces of music are the exception and not the rule. Humans like arrhythmic music. Perhaps we like some sort of a pulse to keep us in step as we dance, but the music itself needs to have something more, something different to the straight beat in order to hold our interest.