John Cage said time is the only valid organizing principle of music,
because it is the only parameter common to sound and silence, the basic elements of music.
In most improvised music, time is either periodic or aperiodic.
So-called classic jazz is largely periodic, and so-called free music is largely aperiodic.
In addition to periodic and aperiodic time there is polyperiodic time,
just as in addition to tonal and atonal harmony there is polytonal harmony.
Harmonically, improvised music has thoroughly explored tonality, polytonality and atonality.
Rhythmically, it has thoroughly explored periodicity and aperiodicity but not polyperiodicity.
Here is grounds for a new organizing principle of improvised music.
A pulse field is a polyperiodic texture comprising two or more pulses.
As an organizing principle of composed music,
it has been used extensively by Stravinsky, Ives, Messiaen, Nancarrow and Ligeti among others.
A machine (such as Nancarrow’s player piano) can readily be programmed to play any pulse field whatsoever.
To ask musicians to improvise in pulse fields is an interesting alternative proposition.
A pulse field is polytemporal rather than polyrhythmic, for unlike polyrhythms it is relative not hierarchical.
Any pulse in a field can be expressed or perceived in terms of the other(s).
Consider for example a field of three pulses in the temporal ratio of 3:4:5,
i.e. measured in units of 3, 4 and 5 beats.
The 4 and 5 beat pulses can be heard in terms of the 3:
or the 3 and 5 in terms of the 4:
or the 3 and 4 in terms of the 5:
In each case the composite rhythm is the same.
A pulse field resolves to a simultaneity at the lowest common denominator of the values of its respective beats.
For example, the 3:4:5 field resolves after 60 beats = 3(20) = 4(15) = 5(12).
The problem with aperiodic time in improvised music is that
it lacks markers where improvisers can choose to interact in a particular way
(such as by changing one or more parameters of what they’re playing).
Polyperiodic time provides markers at points of simultaneity.
In the 3:4:5 field, for example, there are 10 such markers.
In a musical manifestation of the distributive law,
the 3 and 4 beat pulses coincide every 12 beats or 5 times per cycle;
3 and 5, every 15 or 4 times;
4 and 5, every 20 or 3 times;
all three, every 60 or once.
In other words:
(3 x 4) x 5 = (3 x 5) x 4 = (4 x 5) x 3 = (3 x 4 x 5) x 1 = 60
A temporal ratio is a pitch interval (frequency ratio) in macrocosm.
That is, pitch is pulse speeded up to where the ear perceives tone instead.
For example, 3:4 is the ratio of frequencies in a perfect fourth.
This aspect can be used to organize pulse fields harmonically.
For example, the pulse field 3:4:5 is a macrocosm of a first inversion minor triad (in just intonation).
The sequence 1:2:3:4:5:6:7:8… represents the ratios of intervals in the undertone series, the inverse of the overtone series.
Pulse fields or rhythms based upon them can be expressed musically in a variety of ways:
registrally, by playing different pulses in different registers;
articulately, by playing different pulses in different articulations;
dynamically, by playing different pulses in different dynamics;
harmonically, by playing different pulses in different keys;
and so on, in any combination.
In an ensemble situation, each player may play all or part of a field.
In a trio improvising a 3-pulse field, for example,
each player may play a different pulse,
or one player may play 2 pulses out of 3,
or any player may switch pulses at any 2-pulse simultaneity.
All pulse fields are palindromes.
Pulses in a field can be subdivided in common or not.
For example, the pulses in the 3:4:5 field may all be subdivided in sixteenth notes.
In this case, the 3 pulse is 3 sixteenths or a dotted ei
the 4 pulse is 4 sixteenths or a quarter note,
and the 5 pulse is 5 sixteenths or a quarter plus a sixteenth note.
Alternatively, each pulse can be subdivided in triplets.
Expressing three pulses all subdivided in triplets is more difficult
but allows for truly rhythmically canonic playing.
Following (in PDF format) are pulse-field exercises designed to develop facility with temporal harmony.
Arbitrarily, every field (except some in triplets) is built on the undertone series of B6:
1 B6 2 B5 3 E5 4 B4 5 G4 6 E4 7 C#4 8 B3 9 A3
3:4:5 / 3
4:5:6 / 3
4:5:7 / 3
4:6:7 / 3
5:6:7 / 3
5:6:8 / 3
5:7:8 / 3
6:7:8 / 3
6:7:9 / 3
6:8:9 / 3
7:8:9 / 3