Polyrhythmic
sequences and polymetric expressions
Do
not skip this section! It introduces one of the most useful features of BP2,
and examples are easy to perform. (Lazy readers will find them in file
"-da.PeriodNotatio
n".)
Display
the "Data" window and type cmd-j or select "Type from MIDI" in the "Misc" menu
(see §1.4). Now, play a few notes on the MIDI keyboard. These are
immediately transcribed to the "Data" window, for instance:
C5
D5 G5 F5 G5 C5 D5 D#5 D5 C5 A#4 C5
Now
select this sentence and type cmd-p. A dull performance, isn't it? Keep in
mind that BP2 allows all kinds of manipulations of note durations (see
§1.8) and time patterns (see §9). For the moment we will use a plain
metronome to demonstrate rhythmic possibilities at the level of the
score.
We call this dealing with
symbolic
durations.
The
period notation
Insert
periods (indicating beats), for instance:
C5.D5.G5.F5
G5.C5 D5.D#5.D5.C5 A#4.C5
Play
this example. Symbolic durations have been resized automatically so that the
sum of durations remains constant over each 'beat'. In western terminology,
the item would be said to contain crotchets and quavers. In Indian
terminology, the same would be described as tempo changes from
hargun
(speed 1) to
dogun
(speed 2).
This
period
notation
is a recent feature of BP2. Previously, the simple (although less flexible)
way of notating speed changes was:
/1
C5 D5 G5/2 F5 G5 C5 D5/1 D#5 D5/2 C5 A#4/1 C5
A
slash followed by an integer number specifies the
tempo,
more precisely the
density
of sound-objects
(in units per beat). We call it an
explicit
tempo marker.
Tempo markers are still used in the present version, and may be combined with
the period notation. Read on...
You
may try all kinds of arrangements of periods on a sequence. You will find that
changes of tempo may introduce fractional ratios. Try for instance:
C5
D5.G5 F5.G5 C5 D5.D#5 D5 C5.A#4 C5
Here,
the duration of each note in "G5 C5 D5" and "D#5 D5 C5" is 2/3d that of the
first note "C5". Is it possible to represent this example with explicit tempo
markers? The answer is given by BP2: select the item and click "Expand
selection" on the Control panel (accessed with cmd-=). The explicit notation
is the following:
/3
C5_ _.D5_ _.G5_ _.F5_ _.G5_ C5._ D5_.D#5_ D5._ C5_.A#4_ _.C5_ _
in
which '_' indicates a prolongation of the preceding note. The initial part "/3
C5_ _" means that the tempo is "speed 3" (
tigun
in Indian notation), but since "C5" is followed by two prolongation symbols its
resulting duration (3/3 = 1) is the same as in the preceding examples.
There
is an important rule regarding tempo: in a sequence, the tempo of sound-objects
or notes in the first beat sets the initial tempo, whatever the beat division.
The initial tempo may be the default one "/1" unless otherwise specified.
Examples will make it clear. First we introduce an explicit initial tempo:
/2
C5 D5.G5 F5.G5 C5 D5.D#5 D5 C5.A#4 C5
Now,
something more tricky: although the third beat "G5 C5 D5" contains three notes
we want it to be performed at "speed 2". This will force BP2 to resize other
beats accordingly. The notation is:
C5
D5.G5 F5./2 G5 C5 D5.D#5 D5 C5.A#4 C5
What
is the duration of the initial "C5" in this case? It is 3/2 that of each note
in the third (and fourth) beats. Since the latter are forced to speed 2, the
resulting duration for "C5" is 3/2 x 1/2 = 3/4. This is shown by the expanded
notation of the same example:
/4
C5_ _ D5._ _ G5_._ F5_ _.G5_ C5_.D5_ D#5_.D5_ C5_.A#4_ _ C5._ _--
Note
that for the sake of consistency BP2 completed the last beat with two silences
'-'.
At
this stage you might start figuring out that BP2 is smart in dealing with
rhythmic problems. Imagine doing the previous examples on a MIDI sequencer
using event lists or common musical notation... But there is more to come!
What
happens if several explicit tempo markers are inserted? For instance,
C5
D5.G5 F5./2 G5 C5 D5./2 D#5 D5 C5.A#4 C5
is
equivalent to the preceding example. One of the two markers is redundant.
Putting several markers in the same sequence is useless and may even lead to
inconsistency. For instance,
C5
D5.G5 F5./2 G5 C5 D5.D#5 D5 C5./3 A#4 C5
results
in conflicting durations and will be rejected by the interpreter. A good
strategy is to use no more than one explicit tempo marker in an entire musical
item, and let the interpreter adjust durations accordingly. This explicit
tempo marker is often in the beginning of the item, but it may be inserted
elsewhere, notably in case the beginning tempo cannot be specified by an
integer number.
Silences
(notated '-') may of course be inserted in sequences and will be resized like
simple notes or sound-objects, for instance:
C5.D5.-
- F5 G5.C5
A
sequence of several silences '-' may be replaced with an integer number. Thus,
the preceding notation is equivalent to:
C5.D5.2
F5 G5.C5
More
generally, silences may be defined as integer ratios. For instance, the item
C5.D5.3/2
F5 G5.C5
is
interpreted as:
/7
C5_ _ _ _ _ _ D5_ _ _ _ _ _ - _ _ F5_ G5_ C5_ _ _ _ _ _
Sometimes
it is difficult to calculate the duration of a silence, or it is more
convenient to leave it unspecified so that the interpreter determines its value
it in the current context. Silences with unspecified durations are called
undetermined
rests,
notated '_res
t'
or '...'. Try the following
C5.D5.G5..../5
F5 G5 C5.D5 D#5.D5 C5.A#4.C5
or
equivalently:
C5.D5.G5.
_rest/5 F5 G5 C5.D5 D#5.D5 C5.A#4.C5
Given
the context, the interpreter estimates that the simplest value for the
undetermined rest will be 2/5 of a beat. The resulting interpretation is:
/1
C5 D5 G5/5 - - F5 G5 C5/2 D5 D#5 D5 C5/1 A#4 C5
Data
file "-da.ShowPeriod
s"
contains a few examples demonstrating the period notation.
Let's
finish with a quizz: can you write a piece of music which keeps accelerating,
using a unique symbol for the duration? The answer is 'yes', and the solution
is given §4.11 of the Reference Manual.
Polymetric
expressions
All
features presented in §1.12.1 apply to
polymetric
expressions,
which contain simultaneous musical fragments. The notation{A, B,...} means
that expressions A, B,..., are performed simultaneously with identical
(symbolic) durations. Expressions separated by commas are called
fields
of the polymetric expression.
The
condition on equal durations is similar to the one in sequence operations. In
fact, the same
polymetric
expansion algorithm
is used. For example, the sequence
/1
C4 D4 E4 F4 G4.C5 D5 E5
is
interpreted:
/3
C4_ _ D4_ _ E4_ _ F4_ _ G4_ _ C5_ _ _ _ D5_ _ _ _ E5_ _ _ _
Similarly,
the
polymetric
expression
{C4
D4 E4 F4 G4, C5 D5 E5}
leads
to a two-line score represented in a table, the
phase
diagram:
Note
that a polymetric expressio
n
may contain several levels of curled brackets {} and any its fields may in turn
be notated in period notatio
n.
Other features such as explicit tempo markers and undetermined rests apply in
the same way shown with period notation.
Typical
examples of rhythmic items combining period notation with polymetric expressio
ns
are given in the data file "-da.checkPol
y".
Simple examples will also be introduced further in the present document.
Read
more about polymetric expressio
ns
in the reference manual §4.10.