Musical Structures: Summation and Subdivision

Have you ever heard about ‘summation’ and ‘subdivision’ in music? Meanwhile, these mathematical concepts are quite applicable to music analysis. You can see examples of ‘summation’ and ‘subdivision’ almost everywhere in music. But what does it mean?

Summation in Music

All melodies consist of segments – phrases. Every phrase has its own construction. The relationships between phrases create a shape of a melody. When several equal in length phrases are followed by another one that is twice as long as each of them, the structure is called ‘Summation’.

Sometimes the longer segment is followed by one more phrase which is twice as long as a previous one. The new formed structure is called ‘Progressive Summation’. We perceive this final part as a kind of ‘conclusion’ which sets up completeness of a musical shape.

This is a typical example of summation in music – the popular ‘Jingle Bells’ song.

Summation in Music

Jingle Bells,1 bar
Jingle Bells,1 bar
Jingle all the way.2 bars

1+1+2 (Summation)

What fun it is to ride in a one horse open sleigh.4 bars

1+1+2+4 (Progressive Summation)

Subdivision in Music

Subdivision is the opposite structure in which several rather short phrases follow a long one. The Christmas song “Ding Dong Merrily on High’ gives a good example of subdivision in music.

Subdivision in Music

As shown above, there are six short 1-bar phrases coming after two long 4-bar musical ‘sentences’. At the end of the song we can see a two-bar phrase that completes the whole structure. It forms a new compound structure: ‘Subdivision with Ending’.

4 bars+4 bars + 1bar + 1bar + 1bar + 1bar + 1bar+ 1bar (Subdivision)

1bar + 1bar + 1bar + 1bar + 1bar+ 1bar+ 2 bars (Summation)

4 bars+4 bars + 1bar + 1bar + 1bar + 1bar + 1bar+ 1bar + 2 bars (Subdivision with Ending)

Semantically, in summation the longest phrase ‘sums up’ all previous musical phrases. On the contrary, in subdivision the first long phrase states the main idea while subsequent short ones ‘explain’ the details. These structures are very important in music; that’s why they are used so often. Try to find out yourself!