Rhythmic Symbols, Duration, Tempo, Metric Modulation
Robert Morris
1) Rhythmic symbols.
Unlike the standard notation for pitch,1 the standard rhythmic symbols are variables. This
is because the duration of rhythmic symbols such as eighth notes changes depending on
the tempo.
2) Durations.
If we know the duration of one rhythmic symbol, we know the duration of all the rest.
For instance, if a quarter note lasts 2 seconds, an eighth notes last one second and a
sixteenth-sextuple note lasts a third of a second.
3a) Calculation.
For the purpose of calculation we shall use the following symbols:
W
=
w
H
=
h
Q
=
q
E
=
e
S
=
x
T
=
(thirty-second note)
1
When we write F# on the first space of the treble clef, the symbol stands for one pitch
only (given A, a minor third higher, is set to 440 Hz). The use of transposition for
instruments like the horn or string bass is an exception in one sense, but within the parts
written for these instruments the pitches written on the staff are invariable. Thus, to
within a given instrumental part, a pitch symbol on the staff is a constant, not a variable.
Morris on Rhythmic symbols and so forth page 1
We understand from the standard relations in rhythmic notation that
2T = S;
2S = E;
2E = Q;
2Q = H;
2H = W
H
= Q;
2
Q
= E;
2
E
= S;
2
S
= T.
2
and
W
= H;
2
3b) Tuplets.
Dividing rhythmic symbols by n, for an n-tuplet note, represents tuplets.
Therefore,
£
Q
=E
3
and
∞
E
=X
5
More complex tuplets can be defined just as easily; for instance,
septuplet-note, seven of which equal a
E +S
is the value of a
7
ek
We can define tuplet notes values in terms of W.
Q W
Q
Q
Q W
=
(since 3 = Q and 4Q = W so 12
= W and
=
)
3 12
3
3
3 12
or
Q W
=
.
5 20
Morris on Rhythmic symbols and so forth page 2
3c) Special time signatures.
Some composers have used special time signatures based on these identities; for instance,
5
the time signature
indicates there are five sixteenth note quintuplets in a measure
20
1
12
(which is the same duration of a measure of time).
time indicates that each
4
14
measure has 12 eighth note septuplets; like this:
12 ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶
EEEEEEEEEEEE
14
|
The composer might even be more explicit, specifying the 12 “beats” to be grouped into
four equal parts:
3× 4 ¶ ¶ ¶
EEE
14
¶ ¶ ¶
¶ ¶ ¶
¶ ¶ ¶
EEE EEE EEE
|
These types of time signature can be used instead of metric modulations.
3d) Adding durations.
As we have already implied, to add together n identical rhythmic values, we simply
multiply by n. So, E + E + E = 3E
When we have to add together different rhythmic values, we have to obey the rules of
adding fractions taught in high school.
As an example. let us add together the three rhythmic symbols: E
We have
£
h
Q
+H+E
3
We transform each value so it is in relation to Q.
Q
Q
+ 2Q +
3
2
Morris on Rhythmic symbols and so forth page 3
q
.
We perform the addition of fractions.
 2 Q  6
 3 Q
  +  2Q +  
 2 3  6
 3 2
2
12
3
Q+ Q+ Q
6
6
6
2Q +12Q + 3Q 17
5
= Q = 2 Q
6
6
6
Thus, the result is equal to a half note followed by 5 sextuplet notes. If q = 1 second, the
result is equal to 2.83333... seconds.
3e) Dotted values.
To prevent ambiguities (confusing mensuration dots, periods and decimal points), we will
not use dotted values in our calculations. So qk = 3E.
However, we can formalize dotted values so that a rhythmic symbol S followed by n dots
S
is equivalent to ∑k = 0,n −1 k
2
Therefore,
hkkk = ∑k= 0,2 Hk
2
=
H H H H H H
+ + = + + = H + Q + E = 7E .
2 0 21 2 2 1 2 4
Morris on Rhythmic symbols and so forth page 4
4a) Metronome markings.
“B MM”, where B is a number means that a metronome produces B (even) clicks in a
minute. So 67 MM means the metronome produces 67 clicks in a minute. 120 MM means
there are 120 clicks in a minute or 2 clicks a second.
Now the duration between clicks is equal to
1
60
minutes, or
seconds.2 If we use d for
B
B
the duration between clicks, we have
d=
60
seconds. This gives us the number of seconds per duration.
B
We can also write
1 B
= seconds. This gives us the number of durations per second.
d 60
1
(#durations per second) are reciprocals, so
d
1
that we need only invert the fraction equaling d to get the fraction equaling
d
Note that d (#seconds per duration) and
Example:
Let us set a metronome to 80. So B = 80.
The duration of each click is
60 3
3
seconds since: d =
=
80 4
4
1
1 80 4
1
There are 1 clicks per second because: =
= =1
d 60 3
3
3
(As indicated above,
3
4
and are reciprocals.)
4
3
Note that B and d are inversely proportional. d × B = 1 minute or 60 seconds. We can
say B is the frequency of the clicks and d is the wavelength.
2
Morris on Rhythmic symbols and so forth page 5
4b) Tempos.
Tempos are given by the marking “N = B MM”, where N is a rhythmic value and B is a
number. For instance,
q = 69 MM
Since d =
Q=
or
hk = 132 MM.
60
seconds, the duration of the rhythmic value is d. So in the first case above,
B
60 20
60
5
=
seconds; and in the second case, 3Q =
= seconds.
69 23
132 11
4c) Duration of pieces.
We can calculate the duration of a piece of music from the counts of measures and the
tempo marking.
60 2
= seconds per qk ;
90 3
2
there are 4 in a measure and 32 measures. Thus the duration of the piece is × 4 × 32 or
3
1
85 seconds.
3
If we have 32 measure piece in 12/8 time at qk = 90 MM, 3E =
5
time with a measure equaling 108 MM. The
8
5
3
11
piece is 7 measures long: 4 measures of , 1 measure of , 1 measure of ,
8
8
8
5
and 2 measures of . The duration of the piece is therefore equal to
8
108 9
4 × 5E + 3E +11E + 2 × 5E = 44E. Now 5E =
= seconds and E =
60 5
9
9
9
seconds.
÷5 =
=
5
5 × 5 25
9
The duration of the piece is 44E = 44 ×
= 15.84 seconds.
25
Another example. The piece starts out in
Morris on Rhythmic symbols and so forth page 6
5a) Metric modulations.
Metric modulations may be written in two ways. At some juncture in a piece, A) the
metronome indication changes, or B) some rhythmic value before the juncture is set to
equal a different rhythmic value after the juncture. Here is an example of each type of
indication.
A) q = 80 MM |
B) E = E
q = 100 MM
£
5b) Calculating metric modulations.
In A) above , the performer may want to know the relation between the beat of the first
tempo to that of the second. In B), the performer may want to know what the metronome
indication is after (or before) the change of tempo. (In this case, we have to know the
metronome indication of the music either before or after the tempo change.)
Some times the composer gives both A) and B) at once; if so, he or she has to know how
A) and B) are related. If we understand this, then we can calculate B) from A) and vice
versa. We start by setting up two metronome tempo indications.
N1 = B1 MM
and
N2 = B2 MM
and N1 (in seconds) = N2 (in seconds).
Morris on Rhythmic symbols and so forth page 7
5c) Relating tempos (part 1).
We define a variable P that relates the first tempo the second tempo.
This gives: PN1 = P
60 60
=
= N2
B1 B2
We can determine P from knowing both N1 and N2, or B1 and B2.
PN1 = N 2 and PB2 = B1 (derived from P
60 60
= )
B1 B2
5d) Relating tempos (part 2).
If P relates the first tempo to the second,
1
relates the second tempo to the first
P
This is why:
We have PN1 = P
60 60
=
= N2
B1 B2
Multiplying everything times
1
we get
P
1
1 60 1 60 1
= N2
PN1 = P =
P
P B1 P B2 P
Simplifying,
N1 =
60 1 60 1
=
= N 2 and
B1 P B2 P
N1 =
B
1
N 2 and B2 = 1
P
P
Morris on Rhythmic symbols and so forth page 8
5e) Converting metric modulation A) into B).
For this example, we go back to section 5a) and work with the indication
q = 80 MM
|
q = 100 MM
4
seconds.
3
60 60
We set B2 to 100. We can solve for P, using the formula, P =
B1 B2
Let us set N1 to Q, and B1 to 80. Thus in the first tempo N1 = Q =
P
60 60
80 4
=
=
=
80 100 100 5
So P =
4
.
5
Now PN1 =
4 5Q 4 Q
4
5Q
and PN1 = ×
= N2
N1 = N 2 . Since N1 = Q we let N1 =
=
5 5
5
5
5
Thus, the duration of N2 in the first tempo equals 4 quintuplet notes or Q
∞
This means we can write the relation between the two tempi in the manner of B). We set
N2 to Q so we write:
∞
Q =Q
Putting A) and B) together we have
∞
q = 80 (Q = Q) q = 100.
Morris on Rhythmic symbols and so forth page 9
There is a practical problem here if the music in the first tempo of q = 80 does not have
quintuple divisions. We can fix this by setting a new duration, M1 in the first tempo to a
new duration M2 in the second tempo. The relation from the first tempo to the second is
P, so PM1 = M2. (Remember that P =
Now let M1 = Q, M 2 = PQ =
so we write: Q =
5
.)
4
5
Q =Q+S
4
Q X and
q = 80 MM (Q = Q X) q = 100 MM
5f) Converting metric modulation B) into A).
Here we will work with the equivalence supposedly found in a score:
he = h
The first tempo is defined as q = 80 MM. We want to determine the second tempo,
that is, q = ? MM.
60
60
seconds, we have Q =
seconds.
B
80
3
15
3
So Q = seconds. We have to find the value of h e or 5E. From Q, E = , and 5E =
4
8
8
Starting with the first tempo and the formula d =
7
7
seconds. Now we know that in the second tempo, h = 1 seconds. Since the
8
8
H 15 1 15
tempo is to be set for quarter notes not half notes, Q = = × = . Using the
2 8 2 16
formula again, we have
or 1
Q=
15 60
60
16 × 60
and
= . B=
= 16 × 4 = 64 . So the second tempo is q = 64 MM.
16 B
B
15
Morris on Rhythmic symbols and so forth page 10
So we can write:
q = 80 MM (h e = h) q = 64 MM.
By the way, the value of P in PN1 = N 2 is
4E 4
4
since P(5E ) = 4 E and P =
= .
5E 5
5
6) Summary.
Given “B MM”...
d=
60
seconds: # of seconds per duration between clicks.
B
1 B
= seconds: # of durations per second.
d 60
Tempos are given by the marking “N = B MM”, where N is a rhythmic value and B is a
number.
In metric modulations with two tempi, respectively N1 = B1 MM and N2 = B2 MM...
PN1 = P
60 60
=
= N2 .
B1 B2
P relates the first tempo to the second, such that
PN1 = N 2 and PB2 = B1
and
N1 =
1
B
N 2 and B2 = 1 .
P
P
Morris on Rhythmic symbols and so forth page 11