[Assam] FW: On Mathematics and Music

[Barua, Rajen]

Thanks Mukulda for posting the article.
Music and Mathematics always mesmerize me and many others like me. When
I wrote the article recently  "Music, Math and Mozart" (Assam Tribune,
January 2007), I wrote it after years of intoxication as a guy who knows
Math (to the limit that Engineering allows) and loves it and as one who
is rather an outsider to the Music world except as a lover of music. But
when Harvey writes this article, he writes as one who in inside both in
Math and Music. I did not know that Harvey is a famous Musician.  That
makes the difference. Looking at his biodata, I am equally impressed at
his talents.
(Harvey's brief biodata is posted below for general information). Not
only that, this guy 'has been called a "giant of the steel strings" and
"one of the true treasures of American acoustic music."'
 
BTW, Indian Music has the same octave (Sa-Re-Ga-Ma) as the Western
(Do-Re-Me), but I came to know that Indian music has much more
subtelities of divisions in between.   
Rajen
 
For a wealth of information on this musical genius, one may look into
this site.
 
http://www.woodpecker.com/harveyreid.html 

 

"THE MASTER MINSTREL"


 HR PHOTOS <http://www.woodpecker.com/images/harvey_reid.jpg> 


Songwriter and multi-instrumentalist Harvey Reid has honed his craft
over the last 30 years in countless clubs, festivals, street corners,
cafes, schools and concert halls across the nation. He has been called a
"giant of the steel strings" and "one of the true treasures of American
acoustic music." He has absorbed a vast repertoire of American
contemporary and roots music and woven it into his own colorful,
personal and distinctive style. His 19 recordings on Woodpecker Records
showcase his mastery of many instruments and styles of acoustic music,
from hip folk to slashing slide guitar blues to bluegrass, old-time,
Celtic, ragtime, and even classical.


Reid's skills and versatility on the guitar alone mark him as an
important new voice in acoustic music. He won the 1981 National
Fingerpicking Guitar Competition and the 1982 International Autoharp
competition. Yet he's also a veteran musician with a long list of studio
and band credits, a strong flatpicker who has won the Beanblossom
bluegrass guitar contest, a versatile and engaging singer, a powerful
lyricist, prolific composer, arranger and songwriter, a solid mandolin
and bouzouki player, and a seasoned performer and captivating
entertainer. And he plays the 6-string banjo and the autoharp like
you've never heard.

Reid started playing guitar in his early teens in Maryland, and quickly
fell in with the now-legendary DC-area bluegrass scene. After
street-fiddling and playing old-time and bluegrass music for close to 8
years, he moved into his van in the late 1970's and began pursuing solo
acoustic songs & instrumental music, primarily fingerstyle acoustic
guitar and autoharp. After stints in a bluegrass band in Colorado,
playing Telecaster in a country band in Virginia, and a winter in
Nashville, Reid migrated to Northern New England, playing 5 nights a
week on the "blue-collar" folk circuit in Maine and New Hampshire, while
developing his own personal blend of American acoustic music.


  _____  

From: assam-bounces at assamnet.org [mailto:assam-bounces at assamnet.org] On
Behalf Of mc mahant
Sent: Monday, March 26, 2007 4:57 AM
To: assam at assamnet.org
Subject: [Assam] FW: On Mathematics and Music



Assamnetters discussed on this topic less than a month back.
How about this one -- seen by my niece/forwarded.

 


  _____  

Date: Mon, 26 Mar 2007 00:10:33 -0700
From: mmita_2001 at yahoo.com
Subject: On Mathematics and Music
To: mikemahant at hotmail.com


On Mathematics and Music
 
 
HARVEY REID
 
(c) November 1995
 
This was written for a talk I gave at the Choate-Rosemary Hall School as
part of the mathematics lecture series. I have never given any other
such talks, and it has been 22 years since I have studied anything
mathematical...
 
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 22 years ago. 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 meaans 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. 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 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-p; 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. 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 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.
 



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