Tutorial 5 - Making Music

 Contents

The artist in the picture below is the composer and musician Mike Oldfield in a performance on 1st December 2006.  Oldfield made a name for himself as a young composer in the nineteen-seventies, with albums such as Tubular Bells.

Alexander Schweigert, Wikimedia commons

Musical instruments make their sounds by standing waves.  Standing waves can be set up by:

• plucking a tightened string, or scraping it with a bow, or striking it with a hammer;

• blowing a raspberry (an embouchure) to a mouth-piece;

• vibrating air using a reed or a whistle.

Different notes can be made by changing the length of the string, or the length of the pipe.  The volume is changed by altering the amplitude of the wave.

Musical notes have the following properties:

• The frequency or pitch;

• The amplitude or volume;

• The quality, or timbre.

 The pitch is NOT the volume.

Stringed Instruments

We can show standing waves on a string with Melde’s Apparatus.

If we start the frequency of the vibration at a low level, increasing it slowly, we see little of significance until at a certain value, a single large vibration loop is seen.  This is due to resonance and is called the fundamental frequency or the first harmonic.  The second harmonic has two vibration loops.  It is twice the fundamental frequency.  We will study resonance in Physics Unit 4.  We have discussed the fundamental frequency (first harmonic) in the previous tutorial.

The frequency at which resonance happens depends on:

• The tension

• The length

• The mass per unit length (how thick the string is).

Although this is NOT on the AQA AS level syllabus, the fundamental frequency can be calculated using the formula:

The Physics Codes are:

• f0 - the fundamental frequency;

• L - the length of the string (m);

• T - the tension if the string (N);

• m - the mass per unit length of the string (kg m-1).

We have done this in Tutorial 4.

If you look inside a piano, the bass strings are not much longer than the middle note strings, but they are much thicker.

Picture: Frost Nova, Wikimedia Commons.

 How do you think these principles apply to musical instruments?  Explain how the instrument can be tuned.

A string at the fundamental frequency looks like this:

You can see that it forms one resonance loop, which is 1/2 of one wavelength (l/2).  If we increase the frequency, no pattern is observed until we reach a frequency of twice the fundamental frequency (2f0).  The distance between the two fixed ends is one wavelength.  Each resonance loop is 1/2 wavelength.  This is called the second harmonic, or first overtone.

And if we increase the number of resonance loops to 3, we have a frequency of 3f0.  This is the third harmonic, or second overtone.

Each whole number change in fundamental frequency represents a change in note of 1 octave.  An octave is formed by 8 notes on the scale, shown by the keys on a piano:

The different notes are achieved by having strings (or pipes) of different lengths.

Middle C is, as the name suggests, in the middle of a piano keyboard.

Frequencies (Extension)

In an orchestra, all instruments have to be tuned to a particular frequency, otherwise the resulting din would be unpleasant  to listen to.  By convention, the note A (above Middle C) is sounded at a frequency of 440 Hz.  The table below shows the frequencies that are used by instruments tuned to International Pitch Notation.  The table shows the notes in the octave between Middle C and the C above middle C.

 Note Solfège Frequency / Hz C4 Do (Ut) 261.626 C#4 Do# 277.183 D4 Re 293.665 D#4 Re# 311.127 E4 Mi 329.628 F4 Fa 349.228 F#4 Fa# 369.994 G4 Sol 391.995 G#4 Sol# 415.305 A4 Fa 440 A#4 Fa# 466.164 B4 Si 493.883 C5 Do 523.251

Notes:

• The Solfège-do system of naming notes is used in many countries such as France, Spain, and Italy.  It was invented by Guido of Arezzo (991 - 1033), and developed by other musical theorists.  The names came from the first syllables to lines from a hymn in Latin to St John the Baptist (Ut queant laxis, Resonare fibris, Miragestorum, Famuli tuorum...).

• In Britain, Germany, and Sweden the note letters are used.

• In the table, I have missed out the flat notes.  The note G-flat (Gb) is the same note on the piano as F-sharp (F#).

• In some musical scores, notes like E and B are made sharp.  E-sharp is played on the note F and B-sharp is played on the note C.  Similarly, F-flat is played on the note E.

• The subscript number 4 shows that it's the fourth octave on the piano.

Image from Wikimedia Commons.  Author: Alwaysangry

The frequency doubles for every octaveThe note A below middle C has a frequency of 220 Hz.  There is a formula that is used to work out the frequency.  Note number 1 on a piano keyboard is A0.  Note number 88 is C8. For the nth key:

 Worked example Calculate the frequency of note number 32 on the piano. Answer f(n) = 232-49/12 × 440 Hz = 2-1.417 × 440 Hz = 164.814 Hz   This is Note E in the third octave (E3).

This formula is not on the syllabus, but it may come up in one of those question that introduces a new concept and assesses how you tackle it.

Musical scores for most instruments use the same key as the piano.  However, some instruments like the clarinet use a score that is written in a different key.  If the score for a clarinet is written in C, the piano score has to be written in B-flat.  If the piano score was written in C, the two instrument would be out of tune.

But this is a tutorial in Physics, not Musical Theory!

Wind Instruments

Wind instruments make sounds by making columns of air resonate.  A church organ such as the instrument in this picture has many hundreds of pipes that give the instrument the ability to make a wide range of different sounds.

Air columns as in an organ make longitudinal standing waves, which we can show with a Kundt's tube.   (August Kundt (1839 - 1894))

 Be careful how you pronounce "Kundt's"; it is best said "Kunst".

We should note that:

• The wave is longitudinal so that all the particles vibrate parallel to the tube.

• The amplitude is at a maximum at the open end of the tube, so there must be an antinode.

• The amplitude at the closed end is zero; there is a node.

• All molecules between nodes vibrate in phase.

• All molecules either side of a node vibrate in antiphase.

• Adjacent nodes are half a wavelength apart.

In a church organ a motor blows air into a reservoir, which in turn supplies a chest where the valves that sound the notes are.  The keys (which are like those on a piano) operate air valves to allow air into the pipe.  A whistle arrangement sets the air vibrating at the fundamental frequency.  Large instruments will have three keyboards, so that the organist can mix the sounds that the instrument can make.  There is also a keyboard operated by the feet, which plays the bass notes.  The organist has a variety of stops which enables her to choose the sounds that she wants for the music she is going to play.

In the old days (and in churches where there is no electricity supply) somebody had the task of pumping the air into the organ.  If the music got loud, the person doing the pumping would have to pump hard, or the instrument would run out of puff, and the music would fade away.  The classical composer, Felix Mendelssohn, played the organ at St Paul's Cathedral in London.  His playing was so enjoyed by the congregations after church services that they stayed on to listen.  But the vergers wanted them to get out of the church so they could lock it up.  So the people who were pumping the organ were told to stop, leaving the young Mendelssohn playing on silent keys.

At fundamental frequency, a closed organ pipe has half a vibration loop. Note that we are representing the longitudinal wave graphically.

Since this is ¼ of a wavelength, the organ pipe sounds a note whose wavelength is 4 times its length.  The antinode is formed by air passing a whistle arrangement.  Very deep bass notes have large stopped pipes.

A couple of rules when looking at air columns:

• At the open end, there is always an antinode.

• At the closed end, there is always a node.

For the closed pipe, the pattern for the next harmonic looks like this:

This gives a second harmonic in which the frequency is 3f0.  The next harmonic is 5f0.

 Question 2 The diagram shows a closed organ pipe being played at its fundamental frequency. What do you think the harmonics will be like?  What happens to the frequency?  Draw diagrams to illustrate your answer.

Organs have open pipes as well.  The fundamental wave pattern looks like this:

In this case the wavelength of the note is twice the length of the pipe.

The next harmonic happens at 2f0:

The harmonics work like those on a string, i.e. 2f0, 3f0, 4f0, and so on.

 What happens if the pipe is open at both ends at fundamental frequency?  What harmonics do you get?  Draw a diagram to illustrate your answer. In music, the frequency of a note doubles if you go up an octave.  Some organs have very deep bass notes.  A deep bass note A has a frequency of 22.5 Hz.  (a)    How many octaves is this below the note A of concert pitch 440 Hz. (b)    If sound has a speed of 340 m/s, what is the wavelength? (c)    What is the minimum size of pipe needed?  What form should it take?  Explain your answer.

This has important implications for wind instruments.    The picture shows a trumpeter blowing into his embouchure:

Jose Manuel, Wikimedia Commons

A brass instrument like a trombone has an antinode made by the player pursing his lips and vibrating them (an embouchure).  The mouthpiece of the trumpet is called an embouchure as well.  There is an antinode at the bell of the instrument, and a node half way down.

The fundamental frequency can be changed by altering the length of the tube.  In a trombone, the player moves part of the tube in and out.  A trumpeter changes the length by pressing in valves.  The trumpeter can go up octaves by blowing harder, setting up standing waves in the second or third harmonic.  However the trumpet cannot play bass notes below its lowest fundamental frequency.

We can do an experiment to work with standing sound waves using apparatus like this:

We carry out the experiment using the following method:

• Adjust the water level so that l is about 25 cm.

• Switch on the signal generator and set the frequency is about 250 Hz.

• Adjust the volume control so that the tone can just be heard.

• Increase the frequency carefully until there is a sharp increase in volume.

We note the frequency at which the increase of volume occurs.  We know that there is a node at the water surface and an antinode at the end of the glass tube.

 The length of air column in the experiment was found to be 27 cm when the increase in volume occurred.  What was the frequency, assuming the speed of air to be 340 m s-1?

The next maximum amplitude will be at a frequency of 3 time the fundamental frequency.

Quality of Sound

The principles we have discussed have modelled the sound as sine waves (or pure tones).  These are very dull to listen to, and musical instruments make much more complex sound patterns, which give them their distinctive sounds.  An organ playing the note D will sound different to a trumpet playing the same note.  This is often referred to as the quality (or timbre) of the sound.  Playing a musical instrument into a CRO will show this.  You can set the visualisation function on your PC music player to scope and see the music as it would be displayed on the CRO.

Complex waveforms like those in musical instruments are discussed further in Physics 6 Tutorial 6.

Music is a truly international language, that is there to be enjoyed by everyone.  It acts at a far deeper psychological level than words.  You do not have to be a professional musician to write or make music.  The classical composer, Alexander Borodin, was little known as a composer when he was alive; he was the Chemistry professor at the University of St Petersburg, whose research gave us much of organic chemistry as we understand it today.  Nowadays, he is better known as a composer than a chemist, although organic chemists use his results every day.

On the other hand, you would not want to hear my piano-playing.  I do well if I get 80 % of the notes right (and not necessarily in the right order, or the right time).  There is an orchestra for those who are as musically challenged as I am - see link.

Music can unite, but also divide.  Someone's taste in music may disgust somebody else.  (And for most people at the age of about 35, something dreadful seems to happen to pop music.)  Another of my favourite composers at the time the picture below was taken was Rick Wakeman.  He now appears on the BBC's series Grumpy Old Men.  He is also the resident grumpy for Watchdog  on BBC 1.

The record on the record deck is one of Mike Oldfield's compositions, Ommadawn,  I think.  His boxed set is certainly there; I still have it.