Jazz improvisers are used to looking at music in terms of chords and modes – by looking at the common notes between each chord they gain an understanding of how to navigate melodically through the changes in harmony. In classical music such examples can be seen in composers’ treatment of the idea of key modulations, where the tonal center of the mode gradually shifts from one key to the next.
The simplest (and usually the most commonly found) modulation within western tonality is a key modulation from the tonic to its fifth – C Major to G Major, for example. When playing within a key it is important to understand what notes exist within it – for C Major, CDEFGAB, while G Major, GABCDEF#. The popularity of the dominant modulation lies in the abundance of common tones between them, allowing for relatively smooth transitions from one key to the next.
At first it is helpful to play strictly within one mode to get the feel of how it sounds in its own context. (In this case, in strict C Major or strict G Major.) The task then becomes to modulate from C Major to G Major in the smoothest possible fashion. When attempting to bridge them together, it is important to pay attention to notes that exist out of the scale – the transition from C Major to G Major will produce a new note, the F#, which will initially sound “wrong” to the ear.
A significant reason why many of the composers of the past are remembered today lies in their ability to modulate from key to key with as little perceived dissonance as possible. Often it is the ability to take the audience to another place without them necessarily being aware of it that gives music its influential power – this is an effect that can also be achieved through improvisation if done carefully. Whether playing individually or with a group, the improviser should have a fairly good idea of what key they are playing in, when the key center is shifting, and when it has shifted, even if they may not know exactly when. Initial attempts may be awkward and may end up sounding forced, but with enough practice these bumps can eventually be smoothed out. (As reference, one might study Bach inventions as examples of these modulations happening within notational practices.)
A shift from the tonic to a key a tritone anyway is usually difficult because of the lack of common-tones between the two keys (C D E F G A B -> F#G#A#BC#D#E#), but can bridged together by making pit-stops through other key areas (CDEFGAB -> ABC#DEF#G# -> F#G#A#BC#D#E#). In tonal music, the idea of “dissonance” can often easily be replaced by the idea of “distant” – it’s not that the relationship between C major and F# major is necessarily “dissonant”, but only “distant” in the sense that it takes a longer route to get there.
Using the same principal musicians may be able to learn how to modulate from any key to any key, or from any mode to any mode. These exercises can also be utilized in non-western and non-equal-tempered structures, as long as the common-tones are clearly understood. (Personally I've become fairly attached to the idea of modulating between the "white" and "black" notes on the keyboard, using keys like G and C minor as a bridge between them, but that's just me.) The basic idea of the exercise is to acknowledge the similarities and differences between modal constructions, then attempt to move in between them. These techniques are especially helpful in cross-cultural or polystylistic musics where an inter-leaving of modality becomes an important function – one might notice common intervals between the modal constructions of non-western and western scales, for example, which would allow a musician to invoke more than one style in a simultaneous fashion.
Here is a fugue improvisation done by the A-Tribute Ensemble done in the key of G minor. Chris (cello) starts us off with a theme that the rest of us imitate. After some minutes of doing that, we eventually modulate to D major -- it's not clear exactly where we "landed" into the next key, although by 1:30 it's pretty clear that we've modulated into somewhere new and by 1:45 we've made a collective decision to go back to G minor. It seems like in this particular case, we didn't feel too comfortable in the new key!