There is much talk about the relationship between mathematics and music, which
mostly consists of speculation by those on the outside, concerning some of the
obvious things they have in common. Yet there isn't much said by those on the
inside, and as a former student of mathematics and a lifetime musician, I will
attempt to shed some light on the subject. I think that the degree that you
can understand the relationship between music and mathematics is proportional
to your understanding of both music and mathematics. The more you know of both,
the more you will know of the relationship, and attempts to peer into the shadows
from the outside will not yield much more than some wonderment. The important
thing to realize is that numbers and math are not cold and lifeless, and that
music, which is a tangible incarnation of numbers, reflects in its beauty and
emotion some of the beauty and emotion in the world of mathematics.
I graduated from the University of Maryland with a Magna Cum Laude Phi Beta
Kappa degree in mathematics in 1974. I had an aptitude for mathematics,
and I must confess that something made me choose the life of the musician instead.
The two have many similarities, in that they have strong intellectual, spiritual
and creative foundations. I think I chose music because I can participate in
the world. When you are really doing mathematics, the people, places and events
in the world are distractions from your work. When you are really doing music,
you can be just as deeply involved in the mathematical beauty of the music and
the theory, yet you can be at the party. You can only really share mathematical
beauty with other mathematics people, yet you can share the equivalent music
beauty with anyone, and they can enjoy it on some level. There is an essential
element of communication in music that I think is what made me choose that life.
Most of us I think have more of an idea of what music is than we know what math
is. If we were to poll people on the street, they would probably associate math
with numbers and calculators-- things that really are arithmetic. My guess is
that as many mathematicians don't balance their checkbooks as non-mathematicians,
though they can prove that it can be balanced. Math is about thinking. Math
is about problem solving. Math is about working with what you do know to give
you a framework and a method of exploring and understanding what you don't know,
about seeing relationships and patterns. Mathematics is a mind-set, and an attitude
when you face something you do not understand. But there is also a beauty and
a wonder about mathematics that only insiders know about. Words like elegant
and beautiful are used constantly by mathematicians to describe paths
of reasoning and proofs.
Certainly many tasks in the life of a musician fall into this category. Arranging
a melody on an instrument and finding fingerings that correspond to certain
sequences of notes is definitely a type of math problem. Playing the same melody
on different instruments is math, as is playing a stringed instrument and changing
the tuning. And when you find the best key to play a certain melody on a guitar,
for example, there is a sensation that is known to math insiders as elegance.
Mathematicians praise each other for the elegance of a proof, referring to the
esthetic beauty of it. When you write a new piece of music, when you find the
best fingering on a stringed instrument for a sequence of notes, or when you
arrange a piece of music for an ensemble, you can experience nearly identical
sensations of elegance. As you learn about music and about chord theory, you
learn to recognize chord changes, and you experience a mathematics of musical
structure also. Playing harmonies, playing the same song in different keys,
taking solos on unfamiliar songs-- these things all involve recognizing the
structure of a piece of music. Good musicians can often listen to a song, observe
the musical structure, and play along with it, without really knowing it or
rehearsing it, because they recongnize patterns and familiar shapes. This type
of thinking is very much like the way you think when you study mathematics.
Both music and math have concepts, and special symbols. What is a musical key?
What is a number? The definitions of things in both disciplines are somewhat
circular and vague, unless you understand what they are. You cannot define a
number, but you know what they are much of the time and you can use them. It's
no different with a musical notion like a minor key. Once you know what it means
you can spot one, though you cannot really define it rigorously.
There are many things in music that are obviously math-related, and many musical
notions can be explained in numbers. But it is important to note that numbers
are not some way to describe music-- instead think of music as a way to listen
to numbers, to bring them into the real world of our senses.
The ancient Greeks figured out that the integers correspond to musical notes.
Any vibrating object makes overtones or harmonics, which are a series of notes
that emerge from a single vibrating object. These notes form the harmonic series:
1/2, 1/3, 1/4, 1/5 etc. The fundamental musical concept is probably that of
the octave. A musical note is a vibration of something, and if you double
the number of vibrations, you get a note an octave higher; likewise if you halve
the number of vibrations, it is an octave lower. Two notes are called an interval;
three or more notes is a chord. The octave is an interval common to all
music in the world. Many people cannot even distinguish between notes an octave
apart, and hear them as the same. In western music, they are given the same
letter names. If you blow across a coke bottle and it produces the note F, and
you drink enough so that the air remaining in the bottle is twice as much, the
note will be also an F, but an octave lower. If you shorten a string exactly
in half, it makes a note an octave higher; if you double its length, it makes
a note an octave lower. You can think of the concept of octave and the number
2 as being very closely associated; in essence, the octave is a way to listen
to the number 2.
If you shorten a string to 1/3 its length, a new note is produced, and the second
most fundamental musical concept, that of a musical 5th emerges. We call it
a 5th, because it is the 5th scale note of the Western do-re-mi scale, but it
represents the integer 3. (Incidentally, the 5th is the only interval other
than the octave that is common to all musics in the world.) Strings of a violin
are tuned a 5th apart. Men and women often sing a 5th apart, and most primitive
harmony singing involves octaves and fifths. In fact, they say that when you
are learning to tune a stringed instrument, you can only trust your ear to hear
octaves and fifths, and you should not rely on your ability to compare other
musical intervals properly. The next note in the harmonic series corresponding
to the number 4 is 2 times 2 and thus a second octave. The number 5 produces
a new note, called the musical 3rd. The 3rd is the other note in the fundamental
chord, called the major triad, which is made up of 1st, 3rd and 5th notes of
the Western scale. The number 6 produces a note an octave higher than the 5th,
and it is also a very harmonious note. The number 7 produces the first dissonant
note in the harmonic series, which has some numerological and religious significance.
Also of spiritual and numerological interest-- the next dissonant overtones
are the 11th and the 13th.
If you build a musical system out of these integer notes, it is what is now
called the Pythagorean scale, as used by the ancient Greeks. If you bore
holes in a flute according to integer divisions, you will produce a musical
scale. Oddly enough, if you try to build complex music from these notes, and
play in other keys and using chords, dissonances show up, and some intervals
and especially chords sound very out of tune. Our Western musical scale paralleled
the evolution of the keyboard, and finally reached its modern form at the time
of J.S. Bach, who was one of its champions. After a few intermediate compromise
temperings, as systems of tuning are called, the so called even-tempered or well-tempered system was developed. Even-tempering makes all the notes
of the scale equally and slightly out of tune, and divides the error equally
among the scale notes to allow complex chords and key changes and things typical
of western music. Our ears actually prefer the Pythagorean intervals, and part
of learning to be a musician is learning to accept the slightly sour tuning
of well-tempered music. Tests that have been done on singers and players of
instruments that can vary the pitch (such as violin and flute) show that the
players and singers tend to sing the Pythagorean or "sweeter" notes whenever they
can. More primitive ethnic musics from around the world generally do not use
the well-tempered scale, and musicians run into intonation problems trying to
play even Blues and Celtic music on modern instruments. Old bagpipes had the holes drilled in places that sound "sour" to modern ears. The modern musical scale
divides the octave into 12 equal steps, called half-tones. 12 is an important
number on Western music, and it is oddly also an important number in our time-keeping
and measurement systems. The frets of a guitar are actually placed according
to the 12th root of 2, and 12 frets go halfway up the neck, to the octave, which
is halfway between the ends of the strings. On fretted instruments we are actually playing
irrational numbers! And any of you who have trouble tuning your guitars might
get a clue as to why they are so hard to tune. Our ears don't like the irrational
numbers, but we need them to make complex chordal music. The student of music
must learn to accept the slight dissonances of the Western scale in order to
tune the instrument and to play the music.
Studying mathematics can also assist you in daily life as a musician. I cannot
tell you how many times I have actually needed to solve an equation or refer
to one of my math textbooks, but the answer is a very small integer. I think
the only time I ever needed to do that was to compute how many combinations
of a guitar capo that allows you to selectively capo any combination of strings
at a given fret, rather than just clamp across all the strings as capos have
traditionally done. I am not talking so much about solving the little algebra
problems of life like changing money when you tour in foreign countries. Though
it is a good exercise to go to England, pay British pounds to buy liters of
gasoline, and try to figure out 1) what miles per gallon you are getting 2)
what you are actually paying in US $ for a gallon of gasoline. That's kind of
tricky, though it is junior-high school math involved.
Being a former math student makes it easier I think for me to use and understand
my computer, which is an essential tool for a working musician today. We have
to have mailing lists, and print out mailing labels to advertise our concerts
using various Boolean and/or statements. Print out a list of everybody who has
signed up in the last 2 years who either lives in northern Mass, coastal NH
or Southern Maine, but only if they are media, and sort them by zip code. It's
a math problem.
When you are setting up a sound system for a band, you might have a 16 channel
mixer, with a monitor send and an effects send. How can you plug in your wires
to send a mix to the main amps to send to the audience, send another mix to
the monitors for the band to hear, and maybe a 3rd mix to a radio feed or a
tape recorder. The wiring of sound systems and the routing of signals is a type
of mathematics. The noise in a signal is determined by a theory called gain
structure, where it passes through from 5 to 15 different devices and wires
of different lengths through pre-amps, delays, choruses, reverbs, mixers, tuners;
learning to understand and optimize your use of these things is definitely a
math problem. Troubleshooting a sound system 30 minutes before the gig is a
math problem. One of the speakers is not working. Why? Is it the speaker? Is
it a bad wire? Is it the channel of the amp? A fuse? Is it the connector jack,
or the mixer? Solving these kinds of problems is a form of mathematics, where
you systematically eliminate possible problems and de-bug the system. Do you
switch wires, speakers, or amp channels to find the broken one, and in what
order?
The phone calls between band members to book your gigs and arrange rehearsals
are a math problem. How do you notify everybody with the fewest calls? How do
you all get together to rehearse? Do you take one or 2 or 3 cars to the gig?
This is the hardest part of all. When the gig is close to home, and everybody
lives near each other, it does not matter if everybody drives their own car,
because the number of miles is small. If you are driving 800 miles on a tour,
then the answer is simple, since you all travel together, But what if you live
an hour apart and the gig is 2 or 3 hours away? Should you car-pool? And sometimes
only some vehicles are big enough, and as a musician, you are forced to become
an expert at routing theory.
Because I studied math, I know about the mailman problem, and Euler's bridge
problem, and the famous brain teasers about the fox, the chicken and the cabbage,
and the cannibals and the missionaries. These all involve traveling to several
destinations, or transporting things in boats across a river, and problems of
their type have been around in various forms for centuries. Some of these are
very tricky problems, and you face versions of them every day as a musician.
These old brain-teasers are exactly the kinds of things musicians do all the time. How do you get all the people and their gear to the gig most efficiently, if the string bass ony fits in one of the vehicles, and one of the drivers has another gig earlier in the day and they need the power amp and speakers? I used to study math puzzle books when I was a child, and the solutions to some
of the problems of how to get the people or foxes and chickens across the river
in the boat are not simple at all.
How do you go to n places in the shortest
possible route? I remember from my schooling that the equations are not solvable
about pretty small n- it is either 5 or 7 -- I forget They use computers to
do trial and error on these problems, and the airlines and the postal service
and UPS and FedEx spend tremendous amounts of time and money trying to solve
very complex "routing theory" problems of this nature. My guess is that computer networks deal
with the same issues, and the algorithms they use to get e-mail around are not
unlike what you do when you are a musician on tour. You have to get n vehicles
and p people to various destinations, by various routes and at various times.
How do you all get to the gig, unload the gear, check into the hotel, do sound
check, eat dinner, change clothes, do the performance, then pack up? It can
take several band members an entire lunch meeting to figure out logistics of
one gig! You want to minimize the time anybody has to spend waiting, and you
want to minimize travel time and expenses.
The gig is in Albany NY, 5 hrs away. We need to sound check at 5 pm. What time
do you leave home and in what in order do you go to the post office, bank, the
printer and the cleaners, then pick up the bass player in Cambridge, to meet
the keyboard player who will leave his car at the Holiday Inn in Worcester,
where you will pick him up, allowing for rush hour or holiday traffic? There
is an incentive to finding the smartest solution to the problem, since you save
time and gas if you do it right. If you have a regular gig somewhere, you can
compare different methods and routes and see what is simplest and fastest.
Sometimes you deal with an ever-changing math problem. If your band is all in
one bus, then your bus is the hub, and the problem is actually conceptually
simpler. When you only have 3 or 4 people involved, it can be amazingly hard
to come up with a travel plan that suits all the needs of all the people. And
you end up doing many of the things that FedEx does. They use the Hub method--
they fly everything to Memphis first. Sometimes I use a hotel room as a hub.
This stuff is maddeningly difficult, and a big part of the life of a musician.
Do you put all your stage wires and gear into one big trunk, or do you have
several smaller suitcases? Sometimes you need all of it, and sometimes you just
need some of it, so how do you minimize carrying around all your gear, and not
make any one container too heavy or too crowded? Which things do you need to
buy 2 of, so you have one at home and one for the road? Most of us in the business
have a pile of smaller suitcases, which is probably not unlike the way computer
networks send little packets of data around rather than huge chunks.
They are now developing theories of traffic, and when I travel and I see a traffic
jam, I am reminded that there are equations that describe this, and there are
real reasons why one guy with his hood up can cause a 2 mile backup. Queuing
theory I think they call it.
Having some background in mathematics gives you a reference point for approaching
a problem to be solved. I am sure that in the life of a salesman or a housewife
or many other careers there are math problems, and even if you do not actually
end up in a science career, you will benefit greatly from studying math. Developing
a patient attitude toward problem solving is a big part of what mathematcis
is all about, and there is no end to the uses in the world of a working musician
for those kinds of skills. And if nothing else, when you play in a band with
n members, you are really good at dividing by n at the end of the gig when you
get paid! I used to be able to divide the money up in my head. Which is why
I don't play in a band for a living anymore.
© by HARVEY REID
This web site
concerns the music and life of acoustic musician, writer & music educator Harvey Reid.
If
you don't find what you want, or if you have comments or questions, please email
to
This web site
concerns the music and life of acoustic musician, writer & music educator Harvey Reid.
If
you don't find what you want, or if you have comments or questions, please email
to
