Glossary of Musical and Mathematical Terms

Acoustics is the study of physical sound as it behaves in the world.
\(a + b\)
In sound, addition is a fundamental operation where you layer one sound on top of another, also known as mixing. While technically addition preserves the spectrum of a sound (it does not create new frequencies), addition does create virtual beat frequencies, and will alter the spectrum if nonlinearities are present.
See multiplication.
The amplitude of a wave is how big or loud it is. Waveforms are generally described in terms of their amplitude as a function of time. While during each cycle a wave will pass through many amplitudes, it can also be useful to talk about the overall amplitude of a wave, since a wave can be amplified or attenuated without changing its overall shape, phase, or frequency. In the expression \(A \sin(\omega t + \phi)\), \(A\) is the amplitude.
Logical and. True if and only if both its inputs (on either side of it) are true.
See multiplication.
Pass Band
Stop Band
A band is a particular range of frequencies. The difference between the highest and lowest frequency in that band is called the bandwidth. In a filter, the pass band is the range of frequencies whose amplitude is relatively unaffected, whereas the stop band is the range of frequencies which are attenuated.
A beat frequency is a virtual frequency that appears from the changing constructive and destructive interference from two frequencies as their phases shift relative to each other.
Complex Numbers
Imaginary Numbers
Although most of this series will try and do without them, imaginary numbers are multiples of the square root of negative one, denoted as \(i\). Complex numbers include other numbers as well, for example \(4.5 + 6.2i\). For a symmetrical waveform, inverting the waveform, or multiplying by \(-1\), is the same as shifting the phase by 180 degrees, or one half circle. You can think of \(i\) as a number which, when you multiply by it, it will give you a waveform shifted 90 degrees. The four quarters of a circle, 90, 180, 270, 360, become \(\times{}i\), \(\times{}i\times{}i = \times{}-1\), \(\times{}i\times{}i\times{}i = \times{}-i\), and \(\times{}i\times{}i\times{}i\times{}i = \times{}1\). A complex waveform includes a waveform in both regular numbers and imaginary numbers, and would be written \(A\cos(\omega t + \phi) + i A\sin(\omega t + \phi)\) which is more often given as the equivalent \(A\operatorname{e}^{i(\omega t + \phi)}\).
Correlation is a measure of the independence of two signals. With two completely correlated signals, when one signal rises, the other will always also rise, and when one falls, the other will always also fall. With uncorrelated signals, whether one signal is rising or falling does not in any way affect the likelihood that the other signal will be rising or falling. If two signals are anticorrelated, their movements are correlated but in opposite directions, so when one falls the other rises and vice versa.
The cutoff frequency of a filter is the frequency at which the pass band transitions to the stop band. It is commonly defined as the frequency for which the filter attenuates the signal by \(1/\sqrt{2}\) or 70.7%. This is approximately -3dB. With resonance, however, the cutoff frequency may be boosted beyond 100%.
A decibel, abbreviated dB, is a logarithmic measure of magnitude commonly used with sound. It is defined as \(20\log_{10}(x/x_0)\), where \(x_0\) is a reference value. Different reference values give different scales. For example in air decibels are generally measured relative to the threshold of hearing, in digital audio dBFS is measured relative to the maximum value (and so is generally negative), and in pro audio dBu is measured relative to 0.77 volts. Note that 6dB is approximately equal to double and -6dB is approximately equal to one half. Generally when a multiple of six is used, this is just a short hand for the corresponding power of two, so -12dB is one quarter of the reference value, -18dB one eighth, -24dB one sixteenth, etc.
Total Harmonic
Distortion (THD)
In the broadest sense, distortion is a measurement of how much a sound changes as it passes through something. This is the sense in which distortion is meant in total harmonic distortion. Typically this measurement is broken up into distortion or THD (the new components of the sound that are correlated with the original sound) and noise (the new components that are not correlated with the original sound). These measurements are often given together as THD + noise.
In the narrower sense, distortion is the result of nonlinearities that change the shape of the wave depending on its instantaneous amplitude. This is also known as waveshaping. Distortion can create new harmonic frequencies. When distortion occurs in such a way that a rise in amplitude past a certain point reverses direction, perhaps more than once, it is called wavefolding.
Time Domain
Frequency Domain
In general, the domain of some piece of information is the set of things that information is about. In sound, we generally only talk about two domains: the time domain and the frequency domain. In the time domain, given an instant in time, we can tell you the amplitude of a sound or signal at that instant. In the frequency domain, given a particular frequency, we can tell you the amplitude and phase of the spectrum at that point. Trying to analyze a signal according to both domains at once always involves compromises.
Duty Cycle
See square wave.
Exclusive Or
See xor.
Exponential motion will move a constant proportion of the total remaining distance to a target value during a given time. When viewed on an absolute scale, exponential motion is faster when it is further from its target, and slows down as it gets closer, making a characteristic exponential curve. Exponential motion is given by the equation \(x_0 + (x - x_0)\operatorname{e}^{t/T}\), where \(x_0\) is the initial value, \(x\) is the target, \(t\) is time, and \(T\) determines how quickly or slowly the equation approaches the target value.
While sometimes “filter” refers to anything that alters a sound, in the narrower sense, a filter is a linear device which selectively attenuates certain frequencies and boosts others.
The frequency of a wave, denoted as \(f\), is how often that wave repeats during a given time period. Frequency is typically measured in Hertz (abbreviated as Hz), where one Hertz is one cycle per second. This unit is often modified by SI prefixes as kHz (1000 Hz), MHz (1000000 Hz), or more rarely, mHz (0.001Hz). Older texts might use the equivalent “CPS.” When phase is measured in radians, it can make sense to measure the frequency as radians per second. This is known as angular frequency and is denoted by \(\omega\) (a lower case Greek omega), where \(\omega = 2 \pi f\). Frequency is related to period as \(f = 1/T\). See also pitch. In the expression \(A \sin(\omega t + \phi)\), \(\omega\) is the angular frequency.
Frequency Domain
See domain.
A function is anything which relates one or more input values (the values within the parentheses) to one output value. Kind of like a module. Here we will mainly be dealing with functions of time, such as \(\sin(\omega t)\) or \(\operatorname{triangle}(f t)\).
Fundamental Frequency
See harmonics.
Harmony is the relationships between different sounds happening in the same time. Thinking of the way music is written on a staff, harmony describes the vertical structure of a piece of music. Contrast melody.
Strictly speaking, a harmonic sound has its harmonics spaced at integer multiples of the fundamental, as \(f, 2f, 3f, 4f...\). An inharmonic sound will have fractional multiples, such as \(f, 2.2f, 3.4f, 4.8f...\). A harmonic sound is always pitched, while an inharmonic sound may be either pitched or unpitched.
When we analyze the spectrum of a pitched waveform, we find that it generally consists of a fundamental frequency plus a set of frequencies spaced in regular intervals from the fundamental frequency. We call these frequencies the harmonics of the waveform, and number them starting with the first harmonic, which is the fundamental.
Inclusive Or
See or.
An integer is a whole number, including negative numbers and zero, such as \(1, -5, 0, 12\). A fractional number lies between two whole numbers, such as \(-1.4, 2.8, 5/6\). Sometimes fractional is used in the narrower sense as a number between zero and one. A rational number is either an integer or a fractional number. While technically there are real numbers which are not rational, for our purposes these terms are usually interchangeable. Usually the letter \(n\) denotes an integer.
Constructive Interference
Destructive Interference
When two waves are added together, if both have the same polarity the overall amplitude of the result is larger and we call this constructive interference. If they have opposite polarities, the result is smaller and we call this destructive interference. Note that the relative polarities of two waves will change if they have a different relative phase.
See amplitude.
(vs. Exponential)
See exponential.
(Effects or
Anything in between a sound source and a listener—air, oil, microphones, amplifiers, filters, etc.—can affect the sound in linear or nonlinear ways. Linear effects attenuate or amplify certain frequencies or the signal as a whole, but they don't otherwise change its shape or introduce new frequencies. Nonlinear effects change the shape and thus introduce new frequencies. Waveshapers are nonlinear devices that deliberately change the shape of a wave in order to change its frequency content.
See amplitude.
Melody is the temporal relationship between different sounds happening at different times. Thinking of the way music is written on a staff, melody describes the horizontal structure of a piece of music. Contrast harmony.
Modulation is any alteration of one signal based on another signal. Since in theory any parameter of a signal can be modulated, there are many kinds of modulation. The most common are amplitude modulation, frequency modulation, and phase modulation. Unfortunately, phase modulation is often referred to as frequency modulation, even though there are differences between these. Without qualification, modulation generally refers to amplitude modulation. Note that amplitude modulation is the same as multiplication.
See addition.
\(a b\)
\(a \times{} b\)
Multiplying one signal by another changes the relative magnitude of a signal. Attenuation and amplification are the same thing as multiplication, but they usually imply a constant value, a fractional value in the case of attenuation and a value greater than one in the case of amplification. VCAs and ring modulators both multiply, but VCAs will stick at 0 when their control input goes below 0, whereas ring modulators will invert the signal when the control input goes below zero.
In the broadest sense, noise is any component of a signal which is not intended. In a narrower sense, noise is a randomly fluctuating signal. There are several kinds of noise, usually named after colors and based on the distribution of frequencies in the spectrum. Often noise refers to white noise, in which all frequencies are equally present and there is no correlation between the past and present amplitudes.
Logical Not. True if and only if its input (to the right of it) is false.
An octave is an interval of two frequencies such that one is twice the frequency of the other.
Inclusive Or
Logical or. True if and only if one or both of its inputs (on either side of it) are true. It differs from an exclusive or by including the case where both inputs are true.
Most commonly used of filters, in which a first order filter attenuates the stop band by \(1/2\) for every octave, a second order by \(1/4\) for every octave, a third order by \(1/8\), etc. More generally, a first order system or effect can be completely characterized by the amplitude and slew rate (rate of change in amplitude) of its input. A second order system also depends on the acceleration (the rate of change of the slew rate); a third order system also depends on the jerk (the rate of change of the acceleration), and so on.
The period of a wave is the length of time for one cycle. It is the reciprocal of frequency. That is, \(T = 1/f\). Period is also sometimes used in the broader sense of period of time, without particular reference to a wave.
Given a waveform that repeats after a certain period, the term phase describes the relationship between that period and some other temporal reference. Given an absolute time \(t\), the absolute phase (or just phase) of a wave at \(t\) tells us which part of the waveform is happening at the instant of \(t\). Given a reference wave, the relative phase of another wave tells us which part of the other wave is happening at the beginning of the reference wave. That is, relative phase tells us how two waves are aligned with each other in time. Phase is generally measured as a portion of the period, rather than in seconds. It can be expressed as a percentage or fraction, but it is often given as an angle, where the entire wave is one complete revolution. The relationship between a fraction, a percentage, and an angle expressed in degrees or radians can be given as \(frac = pct/100\% = deg/360\degree = rad/(2\pi)\). In the expression \(A \sin(\omega t + \phi)\), \(\phi\) is the phase.
Pitch is the overall frequency of a sound. While we might say that a given sound consists of many frequencies added together (its spectrum), we usually describe a sound as having only one pitch. When we are not thinking about spectra, pitch and frequency are generally the same thing. However, sometimes a particular spectrum has no clear overall frequency, and hence no pitch. These sounds are said to be unpitched. Pitch can also be used as a perceptual term, and sometimes the perceived pitch differs from the overall frequency. We might also refer to this as the apparent pitch.
Compared to a reference waveform, a waveform with a reversed polarity has a lower amplitude when the reference has a higher amplitude, and a higher amplitude when the reference has a lower amplitude. It has been “flipped” or “inverted.” For symmetrical waveforms, reversing the polarity is the same as shifting the phase by 180°.
Pulse Wave
See square wave.
When a system exhibits constructive interference at certain frequencies, those frequencies are boosted in amplitude. A filter can exhibit similar effects. Though we won't use it much here, Q factor is a common way of designating a filter's resonance.
Sawtooth Wave
\(A\ \operatorname{saw}(ft + p)\)
A sawtooth wave is a waveform which decreases at a constant linear rate, reaches a minimum, then immediately begins again from the maximum value. It is not symmetrical and contains both even and odd harmonics. Mathematically, its phase is generally measured as a fraction rather than as an angle (and so we use \(f\) rather than \(\omega\) and \(p\) rather than \(\phi\)), although a phase shift may still be informally expressed as an angle. If you subtract a phase shifted saw from another saw at the same frequency, you will produce a pulse wave.
See Wave
\(\operatorname{sinc}(t + \tau)\)
The sinc function, not to be confused with sine, is a function which is equal to one at zero and has a decaying sinusoidal signal on either side. It occasionally appears elsewhere, but it is primarily found in the sampling theorem, where, in theory, a perfectly sampled signal can be perfectly reconstructed as a function of time \(t\) with the formula \(\sum_{n=-\infty}^{\infty}x[n]\ \operatorname{sinc}(\frac{t-nT}{T})\), where \(x[n]\) is sample number n, and T is the sample period (see summation). In this context usually the normalized sinc function is used, which is related to sine as \(\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \). In other contexts, typically the unnormalized sinc function is used: \(\operatorname{sinc}(x) = \frac{\sin(x)}{x}\).
\(A \sin(\omega t + \phi)\)
\(A \cos(\omega t + \phi)\)
A sine wave is generally considered to be the purest wave, with a spectrum that includes only one single frequency, \(\omega\). A cosine is offset in phase 90 degrees from a sine, such that \(\cos(\omega t) = \sin(\omega t + 90\degree)\). A sinusoidal signal will have the general shape of a sine wave, without being an exact sine wave. For example, because sine waves continue forever, a decaying sine wave is more properly referred to as sinusoidal.
Slew Limiting
The slew of a signal is the rate at which it moves. The maximum slew can be limited, in which case the output will never move faster than a set rate, regardless of how quickly the input signal moves. Slew is closely related to frequency, as higher frequency signals will have to move faster to reach the same amplitude in a smaller amount of time.
pl. Spectra
We can think about the different types of waves as different shapes. But we can arrive at these same shapes by adding together simpler waveforms of many different frequencies. We call those many frequencies the spectrum of the wave. A spectrum is completely described by a set of frequencies, the amplitudes of the simple waves at those frequencies, and the phases of the waves at those frequencies. However, we often ignore phase information and express the spectrum as amplitude as a function of frequency. It can be useful to plot this visually as a spectrogram.
Square Wave
Rectangle Wave
Pulse Wave
Duty Cycle
\(A\ \operatorname{square}(ft + p)\)
\(A\ \operatorname{pulse}(ft + p, d)\)
A square wave is a waveform which holds a constant value for exactly half of its cycle, and holds the inverse of that value for the other half. A square wave is symmetrical and contains only odd harmonics. However, the term “square wave” is sometimes used informally to refer to the broader category of pulse or rectangle waves, in which one constant value is held for some portion of the period, and another constant value is held for the remainder. The portion of the period for which the higher value is held is called the duty cycle and is usually measured as a percentage. The square wave, properly speaking, is a special case where the duty cycle is 50%. Unless they are square waves, pulse waves are not symmetrical and contain both even and odd harmonics. Mathematically, the phase of these waveforms is generally measured as a fraction rather than as an angle (and so we use \(f\) rather than \(\omega\) and \(p\) rather than \(\phi\)), although a phase shift may still be informally expressed as an angle. If you subtract a phase shifted saw from another saw at the same frequency, you will produce a pulse wave. If the phase shift is 90°, you will produce a square wave.
\(\sum_{n=a}^{b} ... \)
This is a shorthand way to write the sum of many terms. On this site it is generally used to describe a spectrum, for example: \(\sum_{n=1}^{\infty} \sin(n \omega t)/n \) \( = \sin(\omega t)/1 \) \( + \sin(2 \omega t)/2 \) \( + \sin(3 \omega t)/3 ... \) which is the spectrum of a sawtooth waveform. The subscript, \(n=a\), usually \(n=0\) or \(n=1\), will tell you the first value of \(n\). The superscript, \(b\), will tell you the final value, or \(\infty\) to continue indefinitely. The summation is given by substituting subsequent values for the variable n in the expression to the right, and then adding all the resulting terms together. Note that the letter \(n\) is arbitrary and could be replaced with another variable; \(k\) or \(i\) are common choices.
A waveform is symmetrical if each movement in one direction is matched by an identical movement in the other direction, such that the wave has the same shape above and below zero, and corresponding upward and downward slopes. For example, a triangle wave and a sine wave are symmetrical, whereas a sawtooth wave is not. A waveform is symmetrical when it has only odd harmonics.
Time Domain
See domain.
A signal which is time invariant will produce the same output for the same input, no matter at what time that input arrives. For example, a clean delay will always produce echoes of a sound delayed by a certain amount of time, no matter when those sounds arrive. On the other hand, if you modulate the delay time, then the length of the delay depends on what the modulator is doing, and so it depends on the time at which the signal arrives.
A term is an element of a mathematical expression that will be added to another element. For example, in the expression \(2 a^2 + b c + 4\), \(2 a^2\) is the first term, \(b c\) the second, and \(4\) the third.
Triangle Wave
\(A\ \operatorname{triangle}(ft + p)\)
A triangle wave is a simple waveform which increases at a constant linear rate, reaches a maximum, then decreases at the same rate. It is symmetrical and contains only odd harmonics. Mathematically, its phase is generally measured as a fraction rather than as an angle (and so we use \(f\) rather than \(\omega\) and \(p\) rather than \(\phi\)), although a phase shift may still be informally expressed as an angle. It can be produced from a parabolic wave by subtracting the 90° phase shifted version of that wave.
Wave shape
A wave is any signal which repeats, such as sound. It has a particular shape, or follows a particular path during its cycle, which is known as the waveform or wave shape.
Wavelength is how much physical distance a complete waveform occupies. It is denoted by \(\lambda\) and depends on the period and the speed of wave propagation in a given medium (for example, the speed of sound in air): \(\lambda = v T\).
See distortion.
Exclusive Or
Logical exclusive or. True if and only if one, but not both, of its inputs (to either side of it) are true. Exclusive means that it excludes the case where both its inputs are true. It differs in this way from an inclusive or, usually just written “or.”
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