Since TIME is more fundamental than PITCH and
     since Rhythm is the structure of the world of TIME,
Rhythm really seems to be the best place to start.
     We will procede methodically from the 2 Dimensions of TIME.

Beats and Off-beats

The Quantitative Dimension of Duration (time-length)



The natural law which dominates this dimension is the law of Regularity which produces a series of equal durations or, if one prefers, a given duration divided into equal parts. Although it might not be artistically or aesthetically preferable to limit oneself to equal durations, it is certainly the simplest, most fundamental possibility, the ideal way to undertake the study of the structure of Rhythm.

The Qualitative Dimension of Alternation (of Beats and Off-beats)



A series of equal durations creates what we call a "rhythmic level". Each time-length (duration) of the smaller (upper) level could be represented by any musical note-value. As long as this level of note-values is perceived alone and isolated, no particular "quality" is felt in any of these values. However, if 2 levels are perceived simultaneously, the larger level will impose a feeling of "quality" on the members of the smaller level. The simplest manner to add a larger level to the existing one is to have its members twice the length of those of the existing level, thus following the law of Simplicity.

This new larger level,
where each member would be identified
by a musical note-value twice that of the upper level,
is, of course, also following the law of Regularity.
Moreover, it imposes a sensation of "quality"
on the note values of the smaller level.

Two of the shorter notes, grouped within the same longer note,
     are placed differently and project different sensations:
1. the first shorter note, whose attack coincides with that of the longer note,
     is "on the beat" (called simply: "Beat"), while
2. the second shorter note, whose attack does not coincide with that of the larger note,
     is "off the beat" (called simply: "Off-beat").
Inversely, we could say that any note value is divided into 2 (equal) parts,
     the first on the beat (a Beat) and the second off the beat (an Off-beat).
An Off-beat has a quality, a feeling of "rise" to it,
     whereas a Beat has rather a quality of "fall".
For this reason, Off-beats have been inscribed higher on the illustration than Beats,
     following the natural gesture of beating time which places them higher.
For all intents and purposers, the words "beat", "fall", and "thesis" (in Gregorian chant) are synonymous,
     and the terms "Off-beat", "rise", and "arsis" are also synonymous.
It does not seem necessary, for the present, to apply these relationships and the concept of quality to any greater number of levels. All levels will be composed of alternating Beats and Off-beats, following the Law of Hierarchy. The quality of the members of any level will always correspond to the position of the members of the next larger level.

The Association of Off-beats

At any given level, each Off-beat "associates" with a neighboring Beat of the same level, either
     the Beat immediately preceding it or the Beat immediately following it.
These associations are determined by harmonic, melodic, or linguistic factors
     which either impose links at certain spots, thereby creating associations,
     or prevent links where there are breaths, thereby imposing them
          at the only possible alternative position.

The Rebound Association
A rebound association is shown at the spot indicated "X". The Off-beat is associated with the preceding Beat and acts as a "rebound" from this Beat (as in the word "mor-ning"). Since "movement" always precedes "rest", the movement is here on the Beat, the rest on the Off-beat. Any breath will evidently be after the Off-beat. This rebound association is part of a procedure of subdivision; both Beat and Off-beat, in this association, belonging to the same value of the larger level. We are merely subdividing the larger value into its 2 equal parts. The example given here consists of only 2 note-values in Rebound Association. We will later see a larger number of values in Rebound Association which we will call Rebound Grouping.

The Pick-up Association
A pick-up association is shown at the spot indicated "Y". The Off-beat is associated with the following Beat and acts as a pickup to this Beat (as in the word "to-day"). In this case, the movement is on the Off-beat, the rest on the Beat, a far more "natural" situation. Any breath will evidently be before the Off-beat. The pick-up association is part of a procedure of grouping. The Off-beat and the Beat, in this association, belong to different values of the larger level. These pick-up "cells" are placed upon the larger values of the next level like two rows of bricks with the joints of the 2 levels at alternate positions and never coinciding. The example given here consists of only 2 note-values in Pick-up Association. We will later see a larger number of values in Pick-up Association which we will call Pick-up Grouping.

What about Bar-lines ?
In Rebound Grouping, a group starts BEFORE the first note of a bar and includes the WHOLE bar.
In Pick-up Grouping, a group starts AFTER the first note of a bar and includes the first note of the NEXT bar.

Pick-up Grouping will dominate the larger levels of rhythmic structure.
Rebound Grouping, merely subdividing the larger time-values,
will be reserved almost exclusively for the smaller levels of rhythmic structure.

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