Begin with three sections, in the proportion of 4 to 3 to 7:
Note that the sum of first two sections is the same length as the third (4+3=7), forming symmetrical halves.
Repeat this pattern faster, then slower:
Not just arbitrarily faster and slower, but in the same proportion, 4 to 3 to 7:
Then take this larger unit and repeat it similarly on yet a higher level of structure, 4 and 3 and 7:
This is the overarching structure of the Kyrie: three proportional sections, occurring at three levels of form. The three largest-level sections are separated by transitions, and there follows an extended coda, also utilizing these proportions:
The many levels of structure are reminiscent of a great cathedral. It's a very grand scheme, yet subtly presented, with plenty of surface ornament at every point that makes it easy to miss the larger shapes.
In this illustration, blue, red, & green hues indicate sections of lengths 4, 3, & 7, respectively, on all levels of structure. Grey indicates added elements, which still utilize some of the same proportions, but more freely. The surface structure is shown in the darkest shades, and lighter shades are used for deeper-level (less immediately apparent) proportions. The lighter shades are also used (below) for smaller surface events, some of them overlapping.
The 4-3-7 proportions also occur on smaller levels of structure: metric, rhythmic, and even melodic and harmonic. Many of these additional details are represented in the drawing. Note the relationships revealed when each section is divided in halves, quarters, and eighths. For example, the 7-section divided in half often contains a smaller 4-3-7; conversely, the midpoint of the 4+3 sections combined occurs 7/8 of the way through the 4-section. The 7-sections in the central (Christe) section of the piece, in compound-meter, are divided into thirds; the first two into equal thirds, and the last in the proportion 2-2-3 (4+3), with a 3+4 proportion underlaid in other parts.
The intricate harmonic and melodic materials in the piece are also governed by the same 4-3-7 relationships. The harmonies are overtonal throughout, so the pattern of root motion through the piece can be shown with a single line. It is overlaid in yellow, and it consists mostly of descending fourths (4:3) and ascending septimal minor thirds (7:6). Note how the 8-6-7 root progression corresponds with the surface structure, and how it is extended to become a dovetailing progression of 64-48-56-42-49.
Here is the entire drawing thusfar (a work in progress), without barlines but including the tonality map, at a scale of one horizontal pixel = one eighth note:
Mark Johnson 21 November 01