Why do instruments sound different despite playing t/he same notes?
When we listen to music, we are exposed to various different instruments. Choice of instrumentation is one of the main factors that contribute to the overall feeling of a song. Even though major and minor chords and scales can be played across most instruments, the instrument that is chosen to play the note plays a large part in what the musician wants to express. For instance, a double bass playing an E may come across as serious, but a flute playing the same note may come across as cheerful. I aim to investigate why such is the case.

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Across all instruments, notes are produced by creating standing wave, which will be discussed below. It is these standing waves that cause the air surrounding the object to vibrate, causing a sound wave to spread out. The main factor that causes different sounds in an instrument is the harmonic frequencies and overtones that an instrument outputs on notes, with other factors such as material affecting this factor.
A vibrating string does not produce a single frequency, but a mixture of fundamental frequencies and overtones. Say that that an A note is being played on a violin string. If just the fundamental harmonic is heard, it would sound dull. It would also sound similar to other instruments playing the same note in the same pitch, provided only the fundamental frequency is being heard. However, when the string is bowed, multiple harmonic frequencies are produced at the same time. You cannot necessarily hear each harmonic note being played, as all these harmonics blend in to produce the overall sound you are hearing. This image shows the harmonic frequencies, also known as the harmonic scale, that are involved when an A note is played on a violin. These are found by stopping the vibrating string at certain intervals. Pythagoras noticed this when he stopped a vibrating string halfway along its length, which brought the pitch to an octave higher. He did this every half interval of the previous half interval, and found that the pitch consistently became an octave higher. He also found that when the string was stopped a third way through, an octave and a fifth was produced, which also produced increasing pitches in octave intervals. To examine the formula relating the wavelengths, we need to understand how waves are formed on a string.
A standing wave is produced when a driver transfers energy to the medium. Energy is transferred down the string, and as it is trapped between two points, reflects of one end and superimposes with waves coming in the opposite direction. However, standing waves6 do not occur at any frequency. Only at specific frequencies do standing waves occur. When transferring energy to the medium at the right frequency, the fundamental frequency is produced. In the case of a violin, the bow that bows the string is the driver, and the string is the medium. Let L be the length of the string, and λ be the length of the wave. Let F be the frequency, and velocity of the string be V. As velocity is constant, and v = F λ, F is inversely proportional to λ. At the fundamental frequency5, λ0 = (2/1)L and F0 = (v/ λ0) , as depicted in this image
http://www.chemistry.wustl.edu/~coursedev/Online%20tutorials/Waves.htm
As the string is stopped at certain intervals, as Pythagoras did, nodes and antinodes are produced. This is a result of constructive and destructive interference occurring, nodes being points of displacement where destructive interference occurs, due to a π phase difference, and antinodes being where constructive interference occurs, due to a 2π phase difference, as shown in this image.
http://www.chemistry.wustl.edu/~coursedev/Online%20tutorials/Waves.htm
This causes λ1 = (2/2) L, F1 = (v/ λ1)
Considering the fact that the first harmonic causes a relation of λ0 =2L, we can see from this image that the wavelength has halved. When the string is stopped a third of the length through, such an image occurs.
This produces λ2 = (2/3)L, F2 = (v/ λ2). The pattern is consistent for the fourth, fifth, sixth and so on harmonic. From here a formula can be formed relating the wavelength of the harmonic to the length of the string. For the first harmonic, λ0 = (2/1)L and F0 = (v/ λ0). In the second, the denominator of coefficient of L increases by one, whilst F varies according to the wave being produced, in the relationship of Fn ∝ (1/ λn), and the same thing occurs for the third harmonic. So, for the nth harmonic, n being a natural number, this formula shows what the wavelength will be; λn =(2/n)L.
As discussed, previously, a note produced on an instrument creates various harmonics. Therefore, by taking the sum of the amplitudes of each harmonic, we can find the shape of the wave produced when a note is played. This is depicted below.
2
A fundamental note of, say, 100 Hz
2
A second harmonic of, say, 200 Hz
2
Adding both waves together produces the resultant patterns above. Just from this very simple example above, we can already visually see how mixtures of harmonics and overtones create interesting waves. From this we can see that different instruments harmonic scales must have different properties. The various harmonics on each instrument do not necessarily have the same strength. For instance, a clarinet is strong in the odd numbered harmonics, but weaker in the even numbered harmonics, whilst a flute is stronger the other way round. 1
As a real life example, this graph shows all the frequencies that are produced when a violinist bows a D note, at 294 Hz.

http://www.nagyvaryviolins.com/tonequality.html
From close inspection we can see that the first harmonic occurs at around 300Hz. The second harmonic occurs around 600Hz, and the next harmonic at around 900Hz. This fits in with the relationship of Fn ∝ (1/ λn). In each harmonic step, the wavelength decreases, but the frequency increases which is the case above. From the first to second harmonic the wavelength goes from 2L to L, which is a decrease of scale factor ½. Yet the frequency has increase by a scale factor of 2, which fits the relationship.
This graph shows the frequencies produced when a vocalist produces the same note, at the same frequency.
http://www.nagyvaryviolins.com/tonequality.html
You can see by comparison that there are similarities in the shape of each graph, but with subtle differences. Although the harmonics occur in the same pattern as above, their peaks are slightly different, and at frequencies beyond 5000 Hz the frequencies outputted by the vocalist have a much lower dB than the violin. Therefore we can conclude that the harmonic scales, as shown on the first page, must vary for each instrument. The various harmonics1 on each instrument do not necessarily have the same strength. For instance, a clarinet is strong in the odd numbered harmonics, but weaker in the even numbered harmonics, whilst a flute is stronger the other way round.
However, even the same instruments have certain characteristics that distinguish them from other instruments in the same category. For example, a Gibson Les Paul, a type of guitar, produces a much heavier tone than a Fender Stratocaster, and you can even tell the difference between a cheap Les Paul and a custom shop Les Paul if you listen closely. This is down to the materials used to create the instrument, and the dimensions chosen. This is explained below, using the example of a violin.
The belly and back plates of a violins4 body are designed to easily resonate. To identify these frequencies, the Authors of this page mechanically drove isolated violin bellies. The force applied is the driving force, and to see the frequency response an accelerometer was used. The acceleration was then monitored, enabling them to plot the ratio of force to acceleration against frequency. Chlandi patterns7 were then used to identify which frequencies the plates resonated at most easily. Chlandi patterns are symmetrical patterns formed when a standing wave is formed on the plate. To see these patterns, granules of sand are placed on the plate, much like iron fillings are used to show the magnetic field of a magnet. The most important frequencies patterns were placed on the graph below.

We can see that the resonant frequency is 163Hz as it is forms the most symmetrical pattern, and is the first symmetrical pattern to occur. Resonant frequency is found using the formula6 F = 1/2π sqrt (k/m), where K is the spring constant and m is the mass. Therefore the closer the frequency is to this frequency, the closer it is to the resonant frequency. Therefore k and m vary according to dimensions and materials used, the resonant frequency varies between instruments, even of the same make.
Gibson Les Paul guitars3 are made from mahogany, a dense wood, whilst Fender Stratocaster guitars are made from either ash or alder wood. These types of wood are lighter and less dense than mahogany, which results in the brighter tone Fender Stratocasters are known for, and the heavier and darker tone Gibson Les Pauls are known for. It is the way that the wood responds to the vibrations that travel through it when a note is played that causes this. As discussed above, the body of a violin has certain frequencies it resonates best at. It can be applied here, and can be deduced that the denser material does not pronounce higher frequencies with the same clarity as the less dense material, therefore causing this difference in tone.
To conclude, it seems that the main factor that affects the quality of a note produced on an instrument is the harmonics that an instrument produces. The fundamental frequency and harmonics that are expressed the most depends on the dimensions of the instrument and the materials. Even in the cases of the same instrument they can sound different depending on the skill of the maker. It is fascinating to realise that the reason why there are so many instruments in the world and why we are able to experience all these different sounds and feelings are essentially down to the physics of standing waves and resonance.
Sources:

http://homepage.ntlworld.com/terence.dwyer/The%20Harmonic%20Series%20Explained.pdf
http://www.phys.uconn.edu/~gibson/Notes/Section4_2/Sec4_2.htm

Difference Between Gibson and Fender

http://newt.phys.unsw.edu.au/jw/violintro.html
http://www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics
Adams S. and Allday J., Advanced Physics, Oxford, Oxford University press, pg

Physics demonstrations: Chladni patterns