“The
musician and the metronome” would have to be one of the most fascinating
relationships within the world of music. The musician, contemplative and
thoughtful as they work through a complex piece of music with great pride, then
meets the metronome, a stubborn little beast that is adamant they are correct
and that you must follow along with them. Yes, the complaints have been
registered by most, if not all, musicians over the years.
However,
there is a place or point in a musician’s practice when all of a sudden a
metronome is not available (or suspiciously lying in several broken pieces
conveniently next to a wall). A piece of music sitting on the stand is calling
upon the musician to play a certain amount of crotchet beats per minute, and
there is suddenly despair. “What if I play the music too fast?” “What if I play
it too slow?” “Will the metronome magically build itself together and tell me
I’m wrong and threaten to break me in several pieces for not getting it right?”
“Arrrrrrrrrgh!” Drama ensues.
But then,
the musician looks at their watch.
No, it’s not
a look of “How much longer do I need to remain cooped up in this room?”, or,
“If I look at this watch long enough the practice will mysteriously occur.”
Instead, they begin to work out the tempo of their piece of music.
For my last
entry, I criticised the proposed change of timetabling for students learning
music so their learning is standardised along with their counterparts that
don’t necessarily learn a musical instrument. This little series will exhibit
pro-active examples of how aspects of maths are used in music, and how students
can use this to (hopefully) overcome metronomic problems and establish a further
understanding of rhythm.
In music,
tempo is always a set ratio. When tempo is discussed in other activities,
whether it be fitness, sports, games, or even home crafts, there is still an
aspect of ratio involved. A tempo’s ratio in music is involved by calculating
how many beats per minute, or the ratio x:m where the value of “x” is
represented by the number of beats, and “m” is minutes (always being 1).
Unfortunately, this is often where the ratio remains as we lunge for the
metronome and look to it for answers.
It’s at this
point that we need to change the ratio. This is where we need to change from
the ratio of x:m to a new ratio of x:s. The value of “m”, where it was
remaining static at 1, will now to change to seconds (s). As is usually the
rule with ratios, the idea is to use whole values on both sides of the ratio
and not decimals/fractions.
Let’s start
with some basic examples. A tempo of 60 beats per minute (bpm), is quite
logical to follow. Because there are 60 seconds in a minute, we end up with a
ratio of 60:60. This reduces to 1:1 since 60, coincidentally, fits evenly into
both sides of the ratio. When a piece of music tells us that it is 60 bpm, we
look at the watch and count in time with the seconds ticking.
Another common
tempo is 120 bpm. This is where we set our initial ratio of 120:60, and we soon
discover that 60 is a multiple for both sides of the ratio. This reduces the
ratio to 2:1. When we look at the watch, two beats will be counted for every
second ticking. The tempo is asserted for 120 bpm, as well as the length of
quavers at 60 bpm.
Not all
tempi are going to reduce to a ratio of x:1, and this is where the rhythmic
complexity starts to establish itself. A tempo of 90 bpm is the first example
of this. With a ratio of 90:60 this will reduce to a ratio of 3:2 due to both
numbers being multiples of 30. The rhythm to commence will be crotchet triplets
across the two second ticks, three beats for every two seconds, and once the
watch is ignored as we continue to tap/click out the beat the tempo of 90 bpm
is established. What you may begin to establish after a while is how long the
notes are by taking the ratio and turning it into the fraction s/x. Each of the
notes, while the tempo is 60 bpm, will be 2/3 beat long.
How about
calculating the tempo for 80 bpm? Yes, it can be done! First, the ratio of
80:60 needs to be reduced. Since both numbers are multiples of 20, the ratio
will reduce to 4:3 – or counting four beats for every three seconds. Holy
dooly, how are we supposed to count four notes evenly in the space of three
seconds?! Let’s work out the fraction to determine the length of the notes. s/x
= 3/4 is our resulting length, or the value of dotted quavers. This could be
represented as the following...
These two bars appear different, but the length of note value for each of these bars is identical.
Once the
tempo is below 60 bpm, the ratio will start to be in favour of having a greater
value for the beats. For example, a tempo of 40 bpm is a ratio of 2:3, or 2
beats for every 3 seconds. The length of each beat calculated against the watch
will be 1.5 seconds each.
There will
be moments where a tempo marking is going to be ridiculous to work out. One of
my students has a piece with the tempo of 112 bpm, and they began to wonder how
they were going to calculate 28 beats in 15 seconds. Using some rounding, and
saving them from a brain melt, a tempo can be established that will be slightly
slower (27:15, or 9:5 = 108 bpm) or slightly faster (30:15, or 2:1 = 120 bpm).
If worse comes to worst, the metronome can be retrieved to give a definite
value of beat.
Give it a
try! Establish and work out that counting without having the metronome
automatically tell you. Give your brain a good rhythmical workout as you strive
for improved counting.
For the next
part of this series of metronomic proportions, I’ll be exploring how we can use
the metronome as part of an alternate form of working through scales using a
different form of calculation.
