The Physics
of 
Music
by
Tim Burns

Physics is used to describe many things, including things in our everyday lives that we have no clue how they work.  Physics can explain sound and how we hear things.  It can also describe music and what intervals of notes should be used.  Physics can not, however, explain why we like certain sounds, and why certain sounds sound appealing to us.  It can explain what intervals produce notes, and what notes should sound good at a given time, but it can not explain why we perceive these notes to sound good.
Sound occurs when a source emits a vibration at a certain frequency.  That sound then travels through a medium, and into a receiver.  In the case of the sound of a tree crashing and a human hearing it; the tree crashes causing air to vibrate or oscillate at a certain frequency, a human’s eardrums then detect these vibrations.  In this case the tree crashing was the source, the medium was air, and the receiver was the human’s ear.  This is the basic definition of sound, and if it is to be followed then the age-old question of if a tree falls in a forest and no one is around to hear it, does it make sound, and the answer would be no, since there isn’t a receiver. 
Pythagoras is believed to be the first to have done experiments with sound.  Other physicist’s have also done experiments, including Galileo and Newton.  They were all fans of music, and they all had various theories about it, many of which where similar to their other theories.  They also believed that the motions of the planets were related to music.  Many big names in science and philosophy played major roles in the development of music.  
The famous Zeno’s paradox can be related to music as well.  It goes as follows; A hare can run twice as fast as a tortoise and lets the tortoise start halfway to the finish.  The hare never catches up however; this can be shown if you think of the course as a string and put points on the string that are half the distance of the previous point.  By the time the hare reaches point one the tortoise will have reached point two, etc...  (Maconie 164)  When I discuss octaves you will see how this is related to music. 
It is believed that Pythagoras first became fascinated with sound when he was walking by the local blacksmith one day.  He noticed that when the blacksmith struck metal with hammers of varying sizes, it produced different pitches.  He believed that the intervals of these pitches where related to the mass of the hammers.  
As time progressed music started to take shape, and different cultures had different styles of music.  Different cultures also had different intervals, which made up their notes.  Western music consists of twelve notes: A, A#, B, C, C#, D, D#, E, F, F#, G, and G#.  The intervals for the frequencies of these notes have varied over time, but are relatively the same as they were originally.  
Note that I don’t have any flat notes and only sharp notes.  This is because now there isn’t any distinction between the two.  D sharp is the same as E flat.  The reason why the two different notations exist is that before the advent of tempered tuning, which I will discuss later, there was a difference.  The difference was very subtle and now we have a blending of the two original notes.  Only someone with an extremely good ear can tell the difference between the two notes anyways.  Besides twelve notes are a lot easier to work with than seventeen.
Below is a table of the frequencies of the twelve notes of western music.

Note	Frequency(Hz)	Note 	Frequency(Hz)	
A	440	D#	622.25	
A#	466.16	E	659.26	
B	493.88	F	698.46	
C	523.25	F#	739.97	
C#	554.37	G	783.99	
D	587.33	G#	830.61	

(Johnston 78)
       This is just one octave of notes.  You’ll notice that as the frequencies get higher the notes get higher and vice versa.  The notes repeat themselves after G#; this is called an octave.  To find the octave of any note all you have to do is multiply or divide the original notes' frequency by two.  For example to get the A octave above the one listed in the table, just multiply 440 hertz by two, and you will get 880 hertz.  If we call the A at 440 hertz A4, then in order to find the frequency of A3, all we have to do is divide by two.  Stringed players do the same thing when they fret a note that is at the midpoint of a string.  They are playing a note at an octave above what that note would be if the string was left open, or untouched.  
Frequencies were first measured in Galileo’s time.  They were first measured at this time because this was around the same time mechanical clocks were invented.  Before this, clocks weren’t accurate enough to measure the frequencies of notes.  This was also right around the baroque period, which saw some great music.  
There are three basic laws for stretched strings and music.  The first is, vibration is inversely proportional to the length of the string.  It is directly proportional to the square root of the tension on the string.  Finally, the vibration is inversely proportional to the square root of the mass of the string.  There is an equation that explains this:

(Johnston 71)
Where length is the length of the string and measured in meters(m).  Tension is how tight the string is tuned and is measured in newtons(N).  Line density is the density of the string and is measured in kilograms per meter cubed (kg/m^3).  And finally frequency is measured in hertz (Hz).  This is true for all strings, and also doesn’t depend on how hard the string is plucked.  This is because varying the strength of how hard a string is plucked only changes the amplitude of the wave and not the frequency.  All amplitude changes is the intensity, or how loud the note is.  A sound wave looks like a sine curve, where the period is the frequency.  As we all know when you change the amplitude of the sine function it does nothing to the frequency.   
It has been found that a column of air behaves very much like a string.  The same rules apply for air also, if you cut the column of air in half you will get a note an octave above the original note.  To find the frequency of a column of air you can use a formula that is very similar to the formula used for a stretched string;

(Johnston 71)
Where length is measured in meters(m), and is the length of the column of air inside the tube it is being played in.  Pressure is the pressure of the air outside, and is measured in pascals (Pa).  Density is the density of the air inside the tube and is measured in kilograms per meter cubed(kg/m^3).  Since the variable’s inside the radical don’t change much, and will be the same for all instruments in a particular area, the frequency of an air column mainly depends on the length of the column.  This equation also tells us that since the density of air is related to the temperature of the air, then the hotter an instrument becomes the higher the frequency.  This is a problem because, towards the end of a concert brass instruments will be hotter than at the beginning, so they tend to go sharp. 
    Pitches in brass instruments are changed much like stringed instruments.  By making the column of air, which can be thought of as a string, shorter you can raise the pitches of notes.  This is similar to fretting the note of a stringed instrument and making that string shorter, thus raising the pitch.  
The tuning of notes to different frequencies has evolved over the years.  Like I said earlier however notes have stayed at relatively the same frequencies throughout time.  Nowadays, tuning is tempered, this makes it easy for people to play together, even if they are all on different instruments.  It also makes it easier to switch keys.
When I say it makes it easier to switch keys, I’m not talking about keys on a piano.  I’m talking about the key of a song.  The key of a song is a little hard to explain, but it can be thought of as the center of the song.  That which governs the song, and tells it which notes would be best for a certain situation.  When you hear a song title such as Bach’s “Fugue in D minor” it is stating that key of the song is in D minor.  This tells the musician to use scales in D minor.  Also some songs have formulas, and most of the time the keys of these are the note or chord that the song is to be started on.
An extremely common and known song type is the blues.  Most blues songs adhere to the same I-IV-V format.  This is known as 12-bar blues.  Where the first four bars are the I chord, bars five and six are the IV chord, seven and eight are I again, nine is V, ten is IV, and the remaining two are I.  I is the chord that this format starts on, and is also the key of the song.  Not all formats start on the I chord.  IV is the chord that is a fourth from the I chord, and V is a perfect fifth.  So if you wanted to do a blues song in the key of A, the I chord would be A, the IV chord would be D, and the V chord would be E.  
This doesn’t mean however that all blues songs are in this particular format.  In fact in the early days of the blues the musicians didn’t even realize that they were following a formula.  It wasn’t until music theorist’s discovered that most blues songs go by this formula.  
If you have the key of the song, you can also get scales to be used.  A scale is a set of notes that are often used for playing single note lines, or harmonies.  Since the key of the song is A you could use the A blues scale.  Which will go along very nicely with this song.  This particular scale consists of six notes;  A, C, D, D#, E, and G.  This scale would most likely be used to construct riffs, fills, and solos.  You don’t particularly have to use this scale, in fact some people would argue that it doesn’t matter what notes you play, or if it makes musical sense as long as it sounds good.     
This might lead one to believe that they could construct a song with random notes as long as it stayed within these parameters.  While you could construct a song, it wouldn’t necessarily sound good.  If this were so we could make computer programs that follows a set of parameters found in popular music, and it could write an infinite amount of great songs consisting of random notes.  This has in fact been tried and at best the songs sounded average.  The reasons why is for one the computer doesn’t have any emotions, like a human does.  This is very crucial for music, most of the best songs are based on some emotion whether it be happy or sad.  Another reason is that music is very opinionated in the sense that something sounds good only in the view of a certain listener.  What sounds good to me, might sound like complete garbage to someone else.  
I have done a lot of discussing about notes, but I haven’t really talked about how we came up with these notes.  In music’s beginnings, right before the classical period, notes where very different depending on the region that you were in.  The notes were also different depending on which key you were playing in.  Nowadays tuning is tempered.
In order to find a note you can take a set frequency, such as A which is 440 Hz, and multiply it by the twelfth root of two, or 1.0595.  This is how the frequencies in the chart were found.  They are tempered, and a musician with a very good ear will tell you that they are a little off.  Overall though they sound right and don’t require retuning of an instrument when someone decides to change keys.  So this tempered tuning is really a compromise.
If it weren’t for tempered tuning our world would be a much different place.  Well maybe not that much different, but musically speaking, it wouldn’t be that great.  Like I said before, tempered tuning makes it much easier to change keys.
When you hear a note played you not only hear its fundamental frequency but you also hear various overtones.  In some cases, for instance if you are listening to the radio you may not even hear the fundamental frequency, but only the overtones.  Overtones depend on the note that is played as well as the instrument.  Different instruments produce different overtones, and this is what some people believe gives instruments distinct sounds, and makes them sound different from each other.  That has not been proven however, and is debated to this day.  Below is a chart of some common overtones of A2(110 Hz).

Note	Frequency(Hz)	
A2	110	
A3	220	
E3	330	
A4	440	
C#5	550	
E5	660	

(Johnston 50)
  Where A2 is the fundamental frequency, or the note being played, and the other notes are all overtones.  When you hear an A being played you hear not only the original A, but also certain overtones of it, as I said earlier, the overtones you hear depend on the instrument, or better yet the source.  I like to think of overtones as almost being subliminal notes.    
You’ll notice in the above chart that the overtones are only three actual notes just at different octaves.  These three notes make up what is called a triad, which is the term used in music when three different notes are played together.  This also makes up a chord, since by definition a chord is made up of three distinct notes.  Most chords are accompanied by various octaves of the notes in the chord.  The three notes are A, E, and C#.  These are what make up and A major chord.  I found this very interesting when I noticed this.  Although I’m not sure if it’s true, I think that maybe when musicians initially thought up chords they took this into consideration when making up the major chords.
Another thing you’ll notice is that the C# in the above chart is different than the C# given in the table that goes over tempered tuning.  This is a flaw in tempered tuning that we all must suffer through.  The difference in frequencies however is only a little more than four hertz, and the effect is minimal(it’s less than a one- percent difference).  
There are times when you hear only the overtones and not the fundamental frequency.  Instead you are tricked into thinking you heard the fundamental frequency.  You are tricked in the same way that your eyes will trick you into seeing gray when black is surrounded by white.  Most often this is used in electronic production of music, such as when you are listening to the radio.  If you are listening to a radio with a small set of speakers, they will not be able to reproduce the lowest bass notes.  But instead they only produce overtones of those notes.  You will think you heard the fundamental frequency but in fact all you heard were the overtones.  
Scientist’s believed that overtones where what gave an instrument its tone.  This was found to be false, but overtones different instruments produce have a lot to do with it.  They found this out when they first tried to develop synthesizers.  They found however when they told a synthesizer to produce certain overtones that it didn’t give the same effect as the original instrument.  This is mainly because the material of the instrument is what gives it, its overtones.  
Work is still being done, but certain instruments have been replicated quite well.  Electronic instruments are becoming more and more popular, and I believe will one day be a class of there own, like brass and stringed instruments.  With the digital components getting cheaper, and the ease of maintenance they are becoming more and more practical.  Many people believed that they would kill music and that we wouldn’t have any need for using real musical instruments anymore.  This has since been proven wrong, however I would welcome the change.  
Music has been pretty stagnant recently.  Most of it is recycled old music that is given a new flavor.  But when you really listen to it you’ll notice that nothing new is being done.  Even in the sixties with the advent of rock n’ roll it sounded so new and fresh, but really it was just blues and country music.  But I like to listen to it so I don’t mind all that much at all.  You never know what could happen, however, one day all of a sudden someone could invent a new style of music and forever change the world.

Works Cited
Johnston, Ian.  Measured Tones: The Interplay of Physics and Music.  New York:  IOP Publishing Ltd, 1989.
Maconie, Robin  The Science of Music.  New York:  Oxford University Press, 1997.

Works Consulted
Bell, Michael S.  “Thee Effects of Rate of Devition and Musical Context on Intonation Perception in Homophonic Four-Part 	Chorales.”  Diss. Florida State University, 	1994.
 Janes, E.T. The Physical Basis of Music. Missouri: 1993.
Jeans, James.  Science and Music.  New York: The Macmillan Company, 1937
Roederer, Juan G.  Introduction to the Physics and Psychophysics of Music.  London: The English Universities Press Ltd. 	1974.
Rossing, Thomas D.  Science of Percussion Instruments. Singapore: World Scientific Publishing Co., 2000.
Smith, Robert.  Harmonics.  New York: Da Capo Press, 1966.