Riemann’s musical grammar has set a precedent as a concept. A modern variation can be found, for example, in the harmony taught at jazz schools. It is somewhat different because it has to provide the improvising musician with appropriate tools. The polyphonic way of thinking with its emphasis on contrapuntal structures is no longer applicable, as it is impossible to create complex contrapuntal sequences spontaneously in real time. On the one hand, the musicians would have to know what their fellow musicians have in mind, and on the other hand, they would have to interpret a complex system of rules fluently. That would be too much of a good thing.

Although counterpoint certainly plays a role in jazz – especially in written arrangements for larger ensembles such as big bands – and it has also developed its own contrapuntal rules, the emphasis is clearly on harmony. The lack of polyphony is compensated for in the improvised solo with harmonic timbre. The harmonic rules of jazz therefore determine how harmony schemes can be varied. They state, for example, which other chords can be used to replace a chord without changing the basic structure of a piece. Complications are particularly popular.

The jazz harmony theory of David Liebman, for example, shows how simple harmony schemes can be replaced by more complicated ones. Liebman’s rule of tritone substitution states, for example, that a chord can be replaced by a chord one tritone away, a technique that is particularly popular in bebop. This means

four beats Dm7 – four beats G7 – eight beats C

can be replaced by

four beats Abm7 – four beats Db7 – eight beats C

There are variants of this rule, such as a mixed form in which both variants come into play:

two beats Abm7 – two beats Dm7 – two beats Db7 – two beats G7 – eight beats C

This can be supplemented by tricks such as a chromatic transposition or aiming:

two beats Abm7 – two beats A7 – two beats Db7 – two beats B7 – eight beats C

The substitution rules can also be used to characterize jazz styles in a meaningful way. The saxophonist John Coltrane, for example, preferred to substitute the above harmony sequence as follows:

two beats Dm7 – two beats Eb7 – two beats Ab – two beats B7 – two beats E – two beats G7 – four beats C