Acoustics is a very important branch of physics which deals with mechanical waves in any form of matter. The word "acoustic" comes from the Greek word "akoustikos," meaning "of or for hearing," which comes from "akoustos," meaning "heard." The Latin synonym is "sonic," which today we know of as being a part of words like "ultrasonic" or "supersonic" [1] . Musical acoustics is a subfield of acoustics which deals with the physics of music. More specifically, how sounds are employed to make music. This includes everything from musical instruments, the human voice, computer analysis of melody, and even the clinical use of music in music therapy. This paper will examine a few important topics in this subject.
The fundamental issue that musical acoustics is concerned with is sound, which physically speaking, is a vibration that propagates as a wave of pressure, through some form of matter (sound cannot be transmitted through a vacuum). In gases, plasma, and liquids, sound is a longitudinal wave, which means that it is made up of oscillating pressure deviations from the equilibrium pressure [1] . This will cause localized regions of compression. In solids, sound can be either a longitudinal wave or a transverse wave. Transverse waves include an oscillating shear stress perpendicular to the propagation direction. Humans perceive these waves as sounds, but we are only able to hear waves when their frequency lies between 20 Hz and 20 kHz. This range varies among different animal species, however [1] .
Whenever multiple pitches are played simultaneously, much like any other wave, the sound waves interact with each other, forming a new wave. This means that any sound wave can be modeled as a sum of sine waves with different frequencies and amplitudes, otherwise known as a frequency spectrum. Under the right conditions, sounds can have a harmonic frequency spectrum. The lowest frequency is called the fundamental, which happens to be the frequency that the entire wave vibrates. The overtones are frequencies which are integer multiples of the fundamental [2] .
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II History of Acoustics
The history of acoustics can be traced back to the sixth century BCE to Pythagoras. The fundamental question that he wanted to answer was why various sounds in music seemed more pleasing than others. Of course, what he was referring to was the harmonic series. He made the observation that musical tones on vibrating strings will be most harmonious when the ratios between the string lengths are integers. He also noted that smaller integer scales produced even more harmony [1] . Today, we know that if a string is plucked, it will make a note that is an octave lower than a string of half its length.
Fast forward to the first century BCE, where the Roman architect Vitruvius was building the foundations for architectural acoustics. He wrote a treatise on interference, echo, and reverberation, as it pertains to theaters. In Book V of his De architectura (The Ten Books of Architecture), Vitruvius talks about some of the wave properties of sound. He described sound as a wave which propagates in three dimensions. This wave would travel back and interfere with other waves upon obstruction. His writings also described how the seats in theaters were designed to prevent this deterioration of sound. Because of his work, Roman theaters begun to have bronze vessels strategically placed in them [1] . This was so that they would resonate with the most desirable notes.
Another major development in acoustic history was during the Islamic Golden Age. In the first and second century CE, Abu ̄ Rayha ̄n al-B ̄ıru ̄n ̄ı is credited with the realization that the speed of sound is in fact considerably slower than that of light (today we know that this is obviously correct) [1] .
The Scientific Revolution ushered in a plethora of new insights in the subject. The two main catalysts of this were Galileo Galilei and Marin Mersenne. These two people (independently from each other) discovered the laws of vibrating strings. Galileo wrote “Waves are produced by the vibrations of a sonorous body, which spread through the air, bringing to the tympanum of the ear a stimulus which the mind interprets as sound.” This was the beginning of the notion that the brain interprets sound. During this time, many
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experiments were done to calculate the speed of sound in air, many of which were done by Mersenne. Around the same time, the famous Isaac Newton was able to derive the relationship for the velocity of waves in solids [1] .
The eighteenth and nineteenth centuries was another time period that included a lot of advancements in acoustics. Many of these had to with using calculus to elaborate theories of sound wave propagation. Hermann von Helmholtz and Lord Rayleigh were two major play- ers in this area. Helmholtz made advancements in mathematical acoustics, while Rayleigh worked with physiological acoustics. Rayleigh eventually released his book called The The- ory of Sound (1877). Another development at this time was the electricity/acoustics analogy. This was made by Wheatstone, Ohm, and Henry [1].
This innovation gave rise to the developments of twentieth century, which included a large number of technological advancements. During World War I, underwater acoustics played a huge role in the detection of submarines. Sound recording led to the telephone which, as we know, has completely changed the world. Today, there are countless applications of relatively recent innovations in acoustics in everyday life [1] .
III Harmonics
The frequency at which the entire wave vibrates is known as the “fundamental.” All of the other sinusoidal components that are found at frequencies above the fundamental are known as “overtones.” Combining these two components gives you the total waveform, or the “partials,” which together form the harmonic series. When the overtones are perfect integer multiples of the fundamental, they are harmonics. However, if an overtone is near an integer multiple, but not exact, it is a harmonic partial [2] .
If a wave is made up of a fundamental and only odd harmonics, then the resultant wave is half-wave symmetric, meaning that if it were to be inverted and phase shifted, it would be the same. However, if the wave includes any even harmonics, it is an asymmetrical wave.
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This to say that the bottom half is not a reflection of the top. Alternatively, something which alters the shape of the wave creates more harmonics, also known as harmonic distortion. If it produces a symmetrical alteration, all of the resultant harmonics are odd. However, if it is asymmetric, at least one even harmonic is produced.