In trying to understand how sound travels through space it is important to consider the three-dimensional nature of its movement. I used to visualize sound waves as the waves on a beach or the vibration of a string. However, these metaphors are limited because they are two-dimensional.
A better way to think about the movement of waves is to visualize them as patterns of lower- and higher-density areas that travel through space. The areas of higher density is said to be in a state of compression, whereas the areas of lower density are in state of refraction.
It is important to always keep in mind that sound is actually composed of discrete particles, which have wave-like properties (this is one of the many things we’ve learned from quantum physics). Check out this interactive applet that features an interesting 3D visualization of sound waves.
Sinusoids (Sine Wave)
The sine wave, also known as sinusoid, is a wave the features a smooth repetitive oscillation. It is an important concept for music, mathematics and physics because sine waves are the only waves that retain their waveshape when added to another wave of the same frequency but at another phase.
Why is this important? Because sine waves can be used as building blocks for any other type of wave. This is what Joseph Fourier identified, when he discovered that multiple sine waves can be used to describe any type of periodic wave form, such as square, triable and sawtooth waves. Miller Puckette stated that “sinusoids, and combinations of sinusoids, can be used to achieve many musical effects.” Link to wikipedia Sine Wave page.
Frequency refers to the number of occurrences of a repeating event per a unit of time – it is also referred to as temporal frequency. When talking about sound waves, frequency is measured in Hertz. One hertz (1Hz) is equivalent to one repeating event per second. a kilohertz (1kHz) is equivalent to a thousand repeating events per second.
Therefore, the greater the frequency the smaller the wavelength or period. These terms refer to the duration of one cycle in a repeating event, so they are inversely proportional to the frequency.
Humans are able to hear frequencies between 20Hz and 20,000Hz. As we grow older our ability to hear frequencies around the higher end of this spectrum decreases considerably. Animals, such as dogs, can often hear frequencies of up to 60,000Hz. Link to wikipedia Frequency page.
Amplitude is the magnitude of change in an oscillating system (such as a sound wave) within each oscillation cycle. For instance, sound waves are oscillations in atmospheric pressure. Their amplitudes are proportional to the change in pressure during one oscillation cycle.
If the variable undergoes regular oscillations, and a graph of the system is drawn with the oscillating variable as the vertical axis and time as the horizontal axis, the amplitude is visually represented by the vertical distance between the highest and lowest points on the y-axis (peak-to-peak amplitude), or from the center to the highest point on the y-axis (peak amplitude).
The phase of an oscillation or wave refers to the location within the cycle of a wave at specified point in time. If two waves or oscillators that have the same frequency are at different locations in their cycle at a given point in time then they are said to have a phase difference, or to be out of phase with each other. The amount by which such oscillators are out of step with each other can be expressed in degrees from 0° to 360°, or in radians from 0 to 2π.
The sample rate is an important attribute of digital recordings that refers to the number of discrete samples per second that are used to create a continuous-like sound. Sampling rate is measured in Hertz (parts per second), like frequency. CD-quality audio features a 44kHz sampling rate.
Harmonics and the Overtone Series
A partial is any of the sine waves by which a complex tone is described. A harmonic (or a harmonic partial) is any of a set of partials that are whole number multiples of a common fundamental frequency. This set includes the fundamental, which is a whole number multiple of itself (1 times itself).n
An overtone is any partial except the lowest. Overtone does not imply harmonicity or inharmonicity and has no other special meaning other than to exclude the fundamental. This can lead to numbering confusion when comparing overtones to partials; the first overtone is the second partial.
In most pitched musical instruments, the fundamental (first harmonic) is accompanied by other, higher-frequency, shorter-wavelength, harmonics. These waves occur with varying prominence and give each instrument its characteristic tone quality.
The fact that a string is fixed at each end means that the longest allowed wavelength on the string, the fundamental, is twice the length of the string – one round trip, with a half cycle fitting between the nodes at the two ends. Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, etc. times that of the fundamental (take a look at the picture below).
The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f, 5×f, …). In terms of frequency (measured in cycles per second), the difference between consecutive harmonics is therefore constant and equal to the fundamental.
On the other hand, the octave series is a geometric progression (2×f, 4×f, 8×f, 16×f, …), and we hear these distances as “the same” in the sense of musical interval. In terms of what we hear, each octave in the harmonic series is divided into increasingly “smaller” and more numerous intervals. Link to wikipedia page about Harmonics.
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon which has been referred to as the “basic miracle of music,” the use of which is “common in most musical systems.” It may be derived from the harmonic series as the interval between the first and second harmonics.
octaves of a note occur at 2n times the frequency of that note (where n is an integer), such as 2, 4, 8, 16, etc. and the reciprocal of that series. For example, 50 Hz and 400 Hz are one and two octaves away from 100 Hz because they are ½ (or 2 −1) and 4 (or 22) times the frequency, respectively.
Octave got its name because on many scales it includes 8 notes, the original note plus 7 additional ones. Though scales exist that include different numbers of notes (such as 12, 17, 19, and 24). Link to wikipedia entry on Octave.
Consonance and Dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance — considered unstable (or temporary, transitional). These concepts have been understood and heard differently throughout history. What I mean to say is that some sounds that were considered consonant are now considered dissonant, and vice versa. These classifications also vary between cultures at any given point in time. Link to wikipedia entry on Consonance & Dissonance.
Linear and Logarithmic Scales
When working with sound it is important to consider that human perception often follows a logarithmic rather than linear scale. For example, Decibels (dB) provide a logarithmic scale for measurements of amplitude. Also, our ears respond to sound nonlinearly so we perceive higher harmonics as “closer together” than lower ones. Link to wikipedia page about Decibels.
MIDI itself is an old hardware protocol which has unfortunately insinuated itself into a great deal of software design. In hardware, MIDI allows only integer pitches between 0 and 127. “MIDI pitch” uses the standard Western scale, dividing octaves into 12 semitones – each MIDI pitch refers to a different semitone. The pitch 69 is assigned to a frequency of 440 cycles per second—the A above middle C. The MIDI pitch scale cannot, however, describe frequencies less than or equal to zero cycles per second.