Beat (acoustics)
Encyclopedia
In acoustics
, a beat is an interference between two sound
s of slightly different frequencies
, perceived as periodic variations in volume whose rate is the difference
between the two frequencies.
With tuning
instruments that can produce sustained tones, beats can readily be recognized. Tuning two tones to a unison
will present a peculiar effect: when the two tones are close in pitch but not yet identical, the difference in frequency generates the beating. The volume varies like in a tremolo
as the sounds alternately interfere constructively and destructively. When the two tones gradually approach unison, the beating slows down and disappears.
corresponding to the total sound of two strings
, it can be seen that maxima and minima are no longer constant as when a pure note is played, but change over time: when the two waves are nearly 180 degrees out of phase the maxima of each cancel the minima of the other, whereas when they are nearly in phase their maxima sum up, raising the perceived volume.
It can be proven (see List of trigonometric identities) that the successive values of maxima and minima form a wave whose frequency equals the difference between the two starting waves. Let's demonstrate the simplest case, between two sine wave
s of unit amplitude
:
If the two starting frequencies are quite close (usually differences of the order of few hertz
), the frequency of the cosine of the right side of the expression above, that is (f1−f2)/2, is often too slow to be perceived as a pitch
. Instead, it is perceived as a periodic variation of the sine in the expression above (it can be said, the cosine factor is an envelope for the sine wave), whose frequency is (f1 + f2)/2, that is, the average
of the two frequencies. However, because the sine part of the right side function alternates between negative and positive values many times during one period of the cosine part, only the absolute value of the envelope is relevant. Therefore the frequency of the envelope is twice the frequency of the cosine, which means the beat frequency is:
This can be seen on the diagram on the right.
A physical interpretation is that when
equals one, the two waves are in phase and they interfere constructively. When it is zero, they are out of phase and interfere destructively. Beats occur also in more complex sounds, or in sounds of different volumes, though calculating them mathematically is not so easy.
Beating can also be heard between notes that are near to, but not exactly, a harmonic
interval
, due to some harmonic
of the first note beating with a harmonic of the second note. For example, in the case of perfect fifth, the third harmonic (i.e. second overtone) of the bass note beats with the second harmonic (first overtone) of the other note. As well as with out-of tune notes, this can also happen with some correctly tuned equal temperament
intervals, because of the differences between them and the corresponding just intonation
intervals: see Harmonic series (music)#Harmonics and tuning.
, f1 − f2 = 0. As the difference between f1 and f2 increases, the speed increases. Beyond a certain proximity (usu. about 15 Hz), beating becomes undetectable and a roughness is heard instead, after which the two pitches are perceived as separate. If the beating frequency rises to the point that the envelope becomes audible (usually, much more than 20 Hz), it is called a difference tone. The violinist Giuseppe Tartini
was the first to describe it, dubbing it il Terzo Suono (Italian for "the third sound"). Playing pure harmonies (i.e., a frequency pair of a simple proportional relation, like 4/5 or 5/6, as in just intonation
major and minor third respectively) on the two upper strings, such as the C above middle C against an open E-string, will produce a clearly audible C two octaves lower.
is to start from a perfect unison and then very slowly and regularly increase the pitch of one tone. When one tone starts to split out from the former twin-note, a slow rumbling can be heard, gradually increasing into an audible tone.
at the unison
, perfect fifth
, or other simple harmonic intervals. Piano and organ
tuners even use a method involving counting beats, aiming at a particular number for a specific interval.
The composer
Alvin Lucier
has written many pieces that feature interference beats as their main focus. Italian composer Giacinto Scelsi
, whose mature style is grounded on microtonal oscillations of unisons, extensively explored the textural effects of interference beats, particularly in his late works such as the violin solos Xnoybis (1964) and L'âme ailée / L'âme ouverte (1973), which feature them prominently (note that Scelsi treated and notated each string of the instrument as a separate part, so that his violin solos are effectively quartets of one-strings, where different strings of the violin may be simultaneously playing the same note with microtonal shifts, so that the interference patterns are generated). Composer Phill Niblock
's music is entirely based on beating caused by microtonal differences.
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
, a beat is an interference between two sound
Sound
Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
s of slightly different frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
, perceived as periodic variations in volume whose rate is the difference
Difference
Difference may refer to:* Difference , a 2005 power metal album* Difference , a concept in computer science* Difference , any systematic way of distinguishing similar coats of arms belonging to members of the same family* Difference , a statement about the relative size or order of two objects**...
between the two frequencies.
With tuning
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
instruments that can produce sustained tones, beats can readily be recognized. Tuning two tones to a unison
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
will present a peculiar effect: when the two tones are close in pitch but not yet identical, the difference in frequency generates the beating. The volume varies like in a tremolo
Tremolo
Tremolo, or tremolando, is a musical term that describes various trembling effects, falling roughly into two types. The first is a rapid reiteration...
as the sounds alternately interfere constructively and destructively. When the two tones gradually approach unison, the beating slows down and disappears.
Mathematics and physics of beat tones
This phenomenon manifests acoustically. If a graph is drawn to show the functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
corresponding to the total sound of two strings
Strings (music)
A string is the vibrating element that produces sound in string instruments, such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a flexible material kept under tension so that they may vibrate freely, but controllably. Strings may be "plain"...
, it can be seen that maxima and minima are no longer constant as when a pure note is played, but change over time: when the two waves are nearly 180 degrees out of phase the maxima of each cancel the minima of the other, whereas when they are nearly in phase their maxima sum up, raising the perceived volume.
It can be proven (see List of trigonometric identities) that the successive values of maxima and minima form a wave whose frequency equals the difference between the two starting waves. Let's demonstrate the simplest case, between two sine wave
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...
s of unit amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
:
If the two starting frequencies are quite close (usually differences of the order of few hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
), the frequency of the cosine of the right side of the expression above, that is (f1−f2)/2, is often too slow to be perceived as a pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
. Instead, it is perceived as a periodic variation of the sine in the expression above (it can be said, the cosine factor is an envelope for the sine wave), whose frequency is (f1 + f2)/2, that is, the average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
of the two frequencies. However, because the sine part of the right side function alternates between negative and positive values many times during one period of the cosine part, only the absolute value of the envelope is relevant. Therefore the frequency of the envelope is twice the frequency of the cosine, which means the beat frequency is:
This can be seen on the diagram on the right.
A physical interpretation is that when
equals one, the two waves are in phase and they interfere constructively. When it is zero, they are out of phase and interfere destructively. Beats occur also in more complex sounds, or in sounds of different volumes, though calculating them mathematically is not so easy.Beating can also be heard between notes that are near to, but not exactly, a harmonic
Harmony
In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...
interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
, due to some harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
of the first note beating with a harmonic of the second note. For example, in the case of perfect fifth, the third harmonic (i.e. second overtone) of the bass note beats with the second harmonic (first overtone) of the other note. As well as with out-of tune notes, this can also happen with some correctly tuned equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
intervals, because of the differences between them and the corresponding just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
intervals: see Harmonic series (music)#Harmonics and tuning.
Difference tones
Consider the two waves starting in unisonUnison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
, f1 − f2 = 0. As the difference between f1 and f2 increases, the speed increases. Beyond a certain proximity (usu. about 15 Hz), beating becomes undetectable and a roughness is heard instead, after which the two pitches are perceived as separate. If the beating frequency rises to the point that the envelope becomes audible (usually, much more than 20 Hz), it is called a difference tone. The violinist Giuseppe Tartini
Giuseppe Tartini
Giuseppe Tartini was an Italian baroque composer and violinist.-Biography:Tartini was born in Piran, a town on the peninsula of Istria, in the Republic of Venice to Gianantonio – native of Florence – and Caterina Zangrando, a descendant of one of the oldest aristocratic Piranian families.It...
was the first to describe it, dubbing it il Terzo Suono (Italian for "the third sound"). Playing pure harmonies (i.e., a frequency pair of a simple proportional relation, like 4/5 or 5/6, as in just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
major and minor third respectively) on the two upper strings, such as the C above middle C against an open E-string, will produce a clearly audible C two octaves lower.
is to start from a perfect unison and then very slowly and regularly increase the pitch of one tone. When one tone starts to split out from the former twin-note, a slow rumbling can be heard, gradually increasing into an audible tone.
Uses
Musicians commonly use interference beats to objectively check tuningMusical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
at the unison
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
, perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
, or other simple harmonic intervals. Piano and organ
Organ (music)
The organ , is a keyboard instrument of one or more divisions, each played with its own keyboard operated either with the hands or with the feet. The organ is a relatively old musical instrument in the Western musical tradition, dating from the time of Ctesibius of Alexandria who is credited with...
tuners even use a method involving counting beats, aiming at a particular number for a specific interval.
The composer
Composer
A composer is a person who creates music, either by musical notation or oral tradition, for interpretation and performance, or through direct manipulation of sonic material through electronic media...
Alvin Lucier
Alvin Lucier
Alvin Lucier is an American composer of experimental music and sound installations that explore acoustic phenomena and auditory perception. A long-time music professor at Wesleyan University, Lucier was a member of the influential Sonic Arts Union, which included Robert Ashley, David Behrman, and...
has written many pieces that feature interference beats as their main focus. Italian composer Giacinto Scelsi
Giacinto Scelsi
Giacinto Scelsi , Count of Ayala Valva was an Italian composer who also wrote surrealist poetry in French....
, whose mature style is grounded on microtonal oscillations of unisons, extensively explored the textural effects of interference beats, particularly in his late works such as the violin solos Xnoybis (1964) and L'âme ailée / L'âme ouverte (1973), which feature them prominently (note that Scelsi treated and notated each string of the instrument as a separate part, so that his violin solos are effectively quartets of one-strings, where different strings of the violin may be simultaneously playing the same note with microtonal shifts, so that the interference patterns are generated). Composer Phill Niblock
Phill Niblock
Phill Niblock is a composer, filmmaker, videographer, and director of Experimental Intermedia, a foundation for avant-garde music based in New York with a parallel branch in Ghent, Belgium.-Biography:...
's music is entirely based on beating caused by microtonal differences.
Binaural beats
Binaural beats are heard when the right ear listens to a slightly different tone than the left ear. Here, the tones do not interfere physically, but are summed by the brain in the olivary nucleus. This effect is related to the brain's ability to locate sounds in three dimensions.Sample
See also
- Voix célesteVoix célesteThe Voix celeste, [Fr.] is an organ stop consisting of either one or two ranks of pipes slightly out of tune. The term celeste refers to a rank of pipes detuned slightly so as to produce a beating effect when combined with a normally tuned rank...
- Gamelan tuning
- HeterodyneHeterodyneHeterodyning is a radio signal processing technique invented in 1901 by Canadian inventor-engineer Reginald Fessenden where high frequency signals are converted to lower frequencies by combining two frequencies. Heterodyning is useful for frequency shifting information of interest into a useful...
- Interference
- Consonance and dissonanceConsonance and dissonanceIn music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
- Moiré patternMoiré patternIn physics, a moiré pattern is an interference pattern created, for example, when two grids are overlaid at an angle, or when they have slightly different mesh sizes.- Etymology :...
, a form of spatial interference that generates new frequencies.
External links
- Java applet, MIT
- Acoustics and Vibration Animations, D.A. Russell, Kettering University
- A Java applet showing the formation of beats due to the interference of two waves of slightly different frequencies
- Yet another interactive Java applet; also shows equation of combined waves, including phase angle.
- Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings