It would be good for completeness sake to know about the modern physics of sounds. We would just briefly touch on some of the issues relating to the modern study of sounds, called Acoustics.

Well, what is sound? Very simply, sound is the vibration of any substance. The substance can be air, water, wood, or any other material, and in fact the only place in which sound cannot travel is a vacuum. When these substances vibrate, or rapidly move back and forth, they produce sound. Our ears gather these vibrations and allow us to interpret them as sounds.

Sound waves are often depicted in graphs like the one below, where the x-axis is time and the y-axis is pressure or the density of the medium through which the sound is traveling.

There are four main parts to a sound wave: wavelength, period, amplitude, and frequency. We will not discuss all of them here as not all of it are related to this porject. We will only talk about pitch and its relation to the frequency of a sound wave.

**Frequency**

Every cycle of sound has one condensation, a region of increased pressure, and one rarefaction, a region where air pressure is slightly less than normal. The frequency of a sound wave is measured in hertz. Hertz (Hz) indicate the number of cycles per second that pass a given location. For example, if a speaker's diaphragm is vibrating back and forth at a frequency of 900 Hz, then 900 condensations are generated every second, each followed by a rarefaction, forming a sound wave whose frequency is 900 Hz.

**Pitch**

How the brain interprets the frequency of an emitted sound is called the pitch. We already know that the number of sound waves passing a point per second is the frequency. The faster the vibrations the emitted sound makes (or the higher the frequency), the higher the pitch. Therefore, when the frequency is low, the sound is lower.

This translates to the different notes we hear on the musical scale. Each note has a unique frequency to it so that it sounds different from the other notes.

To complete this discussion of the modern physics behind music, we will lastly touch on the acoustics of instruments. The acoustics for different musical instruments are varied. For simplicity's sake, we will just look at the acoustics for stringed instruments here.

**Acoustics
of Strings**

Music can be loosely defined as a series of sounds. We know that sounds can actually be understood using sine waves, and sine waves can actually be used in the discussion of the perception of pitch using the differential equation for simple harmonic motion. To put it briefly, the solution to the differential equation is

The above differential equation represents what happens when an object is subjected to a force towards an equilibrium position, the magnitude of this forcebeing proportional to the distance from the equilibrium.

Consider a vibrating string anchored at both ends. Suppose at first that the string has a heavy bead attached to the middle of it, so that the mass of the bead M is much greater than the mass of the strong. The string exerts a force F on the bead towards the equilibrium position and whose magnitude, at least for small displacements, is proportional to the distance y from the equilibrium position, and F= -ky., where k is a constant.

Following from the differential equation above, we get

whose solutuions are the functions

y = A cos ( k/mt) +B sin ( k/mt)

where the constants A and B are determined by the initial position and velocity of the string.

If the mass of the string is uniformly distributed, then more vibrational "modes" are possible. For example, the midpoint of the string can remain stationary while the two halves vibrate with opposite phases. On a guitar, this can be achieved by touching the midpoint of he string while plucking and then immediately releasing. The effect will be a sound exactly an octave above the natural pitch of the string or exactly twice the frequency. The use of harmonics in this way is a common device among guitar players. If each half is vibrating with a pure sine wave, then the motion of a point other than the midpoint will be described by the function

y= A cos (2 k/mt) + B sin (2 k/mt)

In general, a plucked string will vibrate with a mixture of all the modes described by multiples of the natural frequency, with various amplitudes. The amplitudes involved depend on the exact manner in which the string is plucked or struck. For example, a string struck by a hammer, as happens in a piano, will have a different set of amplitudes than that of a plucked string. The general equation of motion of a typical point on the string will be

y= ( An cos (n k/mt) +Bn sin (n k/mt)

As a generalisation, the frequency of a vibrating stretched string is inversely proportional to its length, directly proportional to the square root of its tension, and incersely proportional to the square root of its linear density.

It is based on this understading of the physics behind the acoustics of strings that we use to reproduce the sounds of the Chinese scale in this project.