Tempo
Modulating tempos arises when we invert the dyadic tempo and by necessity change the time signature to the other integer, observe the example #7c. Here the bar length has increased as the time signature has changed to p7/5. The 7 changes to 50bpm while the 5 pattern becomes 35bpm with #7d being the inversion of #7c.
In regard to poly tempo music, the interpretation of traditional time signatures becomes a general one. A more precise method would be to replace the times signature with the actual tempo of each voice in bpm, but this idea also requires compromise as the interference of certain tempos involve a decimal point, 70.7bpm. There are numerous ways to express the technique of tempo polyphony, each with their own trade-offs. Therefore, one accepts the duplicitous nature of traditional time signatures to symbolise the intentions of poly tempo music.
In order to find the tempos of each voice in a polyrhythm the smallest digit divides its chosen tempo which is then multiplied by the larger digit.
Examples:
p5-3: 60bpm % 3 = 20 x 5 = 100bpm.
p7-4: 60bpm % 4 = 15 x 7 = 105bpm.
In examples #7a and #7b the dyadic tempo has been inverted with no change to the 5/4 time signature. The 7 remains 70bpm and the 5 pattern stays at 50bpm, resulting in no change to either tempo.
Modulating Tempos
Compound Tempos
The process of compound tempos is explored in the 3rd repertoire. Here we rhythmically modulate from one family to another and by doing so, one tempo will change and the other remains the same. Example N#6 is Unconditional, here a looped sequence of tempo modulations occur on page one of the guitar part. As you can see, Bar 1 contains a crotchet triplet from Family one followed by p5-3, also a member of Family 1, the 2nd bar manipulates p7-5 in the lower register, which is from Family 2, with p9-4 in the 3rd bar belonging to the same family as bar 1.
To alter our durations from crotchets to quavers, the code has to adjust. An illustration of this is example #5. Here the introduction is played by the Alto Sax in the tune Bi-two from the 2nd repertoire.
In the 2nd bar we have p7-5 whose code in crotchets is a2b3c45d6e7 / c4.5d, however in quavers the code becomes p14-5 a23b456c789d10.11.12.e1314. / b4.12e.
Two examples of the same passage have been given #5a is at 60 bpm and #5b is at 90 bpm. Only the primary code p7-5 is given in the music, although the notation indicates p14-5, again the intention in either case is clear.