A rhythm is commonly written for many purposes as 2 measures in drum notation such as might be played repeatedly by a robot for the duration of a part lasting, let us say, 32 bars. A human drummer would of course never do this exactly. He usually does however play in such a way as to give the impression that the basic unit is being repeated constantly to some degree. The particular division of the basic rhythmic unit is a recognizable feature and repetition sets up the expectation that the same strokes will occur in the same places every 2 measures. A drummer would resent the suggestion that his playing provides a simplistic pattern through which other instruments can weave new and interesting melodies. We have an infinite number of ways to do rhythm, some of them whimsical. One cannot reduce one player to the rank of "guardian of tempo" simply because the entire band is always involved in making an actual rhythm. The regular and repetitious nature of rhythm is nonetheless acknowledged by all players : even if nobody is actually playing it, each musician has something like the "official" (intended) rhythm pattern going on in his head.

The repetition of 1 or 2 bars divided up in a certain way is merely a convenient means to express the idea that something of this nature constitutes a pleasant musical device. In the following development, I will consider a single bar as the repeating unit, "for the sake of clarity". This choice really doesn't affect the argument.

This page will say nothing about any given rhythm usually shown as notes on a staff, that is, where specific events occur at certain times. In this instance, I am interested in the path along which the rhythm flows. This path is represented as a framework of ruler divisions. I am most particularly interested in the case where a rhythm is played by several people using different "rulers". The case where the music suddenly shifts from one ruler to another is assumed to be no different : a "melodic" form of the same idea.

Any "everyday" ruler has a certain style of markings. American inch rulers tend to follow the principle of division by 2. Metric rulers follow divisions by 2, 5 and 10. For many scientists, this would be a matter of complete indifference. If you are a cabinet-maker and you are used to using inches, a metric ruler will seem impractical because many of the markings you expect (the markings in your head, if you will) don't match any you see on the ruler. If you are a musician, there is a world of difference.

It is a fact that all musical rules are set up and followed so that they can be broken. This could be called the principle of tension and resolution. If you don't break the rules, it ain't music. Mind you, if you don't follow any rules in the first place, it ain't music either.

When you change the rhythm, you generate pleasure and excitement. I would like to suggest that one often breaks some bits of rhythm while leaving others intact. Otherwise, the change would be totally unpredictable and place the band temporarily in a state of disarray and panic. This wouldn't be very musical. Of course, a completely arbitrary rhythm change works perfectly as long as it is done by one person, let us say, for 8 bars, after which, the rest of the band simply joins the new groove.

A common way to alter the rhythm in a structured manner so that the band can do it together is to keep the same tempo, double the tempo, or halve the tempo.

The following examples are often heard in jazz.

So-called "double time". The rhythm changes by going twice as fast.
	Note the complete agreement between the original tempo with triplet eighths and the "double" tempo with triplet sixteenths !

Playing "three on four". The rhythm changes by redividing the bar.
	Note the complete agreement between the 4/4 in triplet eighths and the 3/4 with "straight" eighths !

The "meters" on the left show the "count". Those on the right merely indicate formal division.

These diagrams are intended the show situations where new strokes are placed according to established "landmarks". This is good because the band can make the change together in lockstep. This also means that one section of the band can make the change while another section can not make the change, both being awake and assertive, with no alteration of the groove.

Am I really suggesting that a single ruler which contains all the divisions required for all kinds of "count" is a representation of "the groove" ? I think so. For instance, if one person played formal divisions of 8/8 to another person's 9/8, there could be much confusion and little appeal, beyond the almost unbearable tension this would produce. When you are improvising using divisions, you are attracted to such similar "ruler markings" as the band around you might have to offer and you don't go off and define your own without some good reason, such as the need for more (finer) divisions. The groove is a kind of common regularity into which all may "fall".

This is not to say that complete or partial mismatches should not occur. They do occur. Indeed, the most fundamental shift in jazz and musette involves one person playing 3 equally spaced strokes to another's 2. This creates a "polyrhythm" which is undeniably pleasant and appealing to the ear.

I would suggest however that complete or partial stroke matching in polyrhythms gives us a remarkable combination : one hears the expected and the unexpected together as one. In other words, the associations which I highlight in the above drawings are analogous to pictures of harmonics in consonant intervals in chords. Tonal harmony involves sending out waves of sound where consonance (resolution) and dissonance (tension) alternate and mingle in "interesting" ways. Rhythmic harmony similarly involves playing with simple shifts of meaning within a regular division of the bar (resolution) as well as "nutcase" shifts using "contradictory" methods to acheive a variety of irregular divisions (tension). [ Thank you, John, for pointing this out ! ]

A rhythm shift which requires a new ruler is not the easiest to play. Even one which displays a high degree of coincidence with the old ruler requires a lot of practice. This is equally true of different rhythms played together (polyrhythms) or one after another (rhythm changes).

There are many many phrases which produce rhythm changes without changing the meter (or the ruler) at all. You can for example play quarter notes for a while in groups of 2 and then throw in a burst of the same quarter notes in groups of three. The duration of one "count" doesn't change but the rhythm does. I don't go into this kind of "basic" change here because I believe it is trivial, obvious, and already known to all. I mustn't fail to point out however that it is just as "important" as the kind I describe in the pictures. By "important", I mean "fun" and "exciting" and "I do it all the time".

The purpose of this web page is to demonstrate in a fairly immediate and playable manner the basis of the widely accepted notion that rhythm and harmony are almost the same thing. A vague feeling is cool but I prefer a demonstration if I can get one. A lot of questions remain unanswered (to rot on the roadside).

	This is a "rhythmic ruler" for playing several levels of triplets. Amplitude versus time.
	This is the spectrum of a chord constructed on several levels of third harmonics. Amplitude versus frequency.

The striking similarities merely prove how easy it is to abuse mathematics. Any semi-conscious physical scientist would find the comparison flawed, to say the least. The idea is silly. But is it really? The musician is confronted with a tear in the tissue of time. The fault is with the way the body and brain work.

Time can be a frequency or the inverse, a period. To the left of the tear, we have the timbre and intonation of audible tones, that is pitch notation. To the right, we have rhythm and rhythmic notation. On the right, we usually think in terms of frequency and on the left, we think in terms of period, that is, ordinary time. Frequency is just the inverse of period. On the right, we have the realm of natural law. On the left, we have mere human convention. The "no man's land" between the two regions: we can't hear that low and we can't play that fast. No musical notation of any kind occurs here. Many non-tonal sub-frequencies produced by chords and many consonant "beat frequencies" do fall within the purple stripe however.

When close things don't match, the human mind always reaches for a common yardstick. The desire for coherence, even of the foggiest conceivable kind, is one of the most powerful driving forces in the human spirit. To bridge the gap, we have little choice, in choosing conventions, but to adopt regular rhythmic divisions, in mimicry of the natural laws which are unfolding on the other side. The pertinance of our terms in a physical sense is of no real importance. We can make do with screaming surrealism as long as we get some intuitive sense out of it. The intuitions of the most obtuse kind are usually no worse than the rest.