as shown in figure 13. The time signature is 6/4, which is interpreted here as a sextuple meter due to the slowness of the tempo. Such 6/4 meter decomposes the bar into two subgroupings, each of three quarter notes. Then the simultaneous use of simple and compound meter divides the beat into two (in the left hand) and into three (in the right hand). Therefore, the full group of the metrical structure is this:

As a further illustration, consider figure 14 from the second movement of Bartok’s String Quartet No. 4. The second violin and viola are in 2/4; and the first violin and cello are in 6/8. Both 2/4 and 6/8 are duple meters. Thus the beats coincide in all four instruments. Nevertheless, the divisions do not. In the second violin and viola, the division is into two; whereas in the first violin and cello, the division is into three. Notice that there is no division between the level of the bar and the level of the beat. That is, the full group of the metrical structure is:

Notice that this means that the simultaneous divisions are each children of the beat - in accord with our algebraic theory of object-oriented inheritance.

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20 Complex Shape

Using the geometric concepts developed above, it becomes possible to give a powerful theory of musical composition. To do so, it is necessary to examine some of the main theory of complex shape developed in the book Leyton (2001). In