Glossary of Musical and Mathematical Terms
- Acoustics is the study of physical sound as it behaves in the world.
- 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.
- 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.
- Band
- Bandwidth
- 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.
- 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.
- 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.
- Domain
- 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.
- 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.
- Frequency
- Angular
Frequency - Rate
- \(f\)
- \(\omega\)
- 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.
- 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
\(\operatorname{triangle}(f t)\).
- 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.
- 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.
- Interference
- 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.
- Linear
- Nonlinear
(Effects or
systems)
- 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.
- 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.
- 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.
- 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°.
- 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.
- 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.
- Sinc
- \(\operatorname{sinc}(t + \tau)\)
- 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:
\(\operatorname{sinc}(x) = \frac{\sin(x)}{x}\).
- Sine
- Cosine
- Sinusoidal
- \(A \sin(\omega t + \phi)\)
- \(A \cos(\omega t + \phi)\)
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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): \(\lambda = v T\).
- Xor
- Exclusive Or
- \(\oplus\)
- 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.”
