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36-454: A room can be said to have formants characteristic of that particular room, due to its resonances, i.e., to the way sound reflects from its walls and objects. Room formants of this nature reinforce themselves by emphasizing specific frequencies and absorbing others, as exploited, for example, by Alvin Lucier in his piece I Am Sitting in a Room . In acoustic digital signal processing , the way

72-411: A π ∑ k = 1 ∞ ( − 1 ) k sin ⁡ ( 2 π k f t ) k {\displaystyle x_{\text{reverse sawtooth}}(t)={\frac {2a}{\pi }}\sum _{k=1}^{\infty }{(-1)}^{k}{\frac {\sin(2\pi kft)}{k}}} In digital synthesis, these series are only summed over k such that

108-484: A ( 1 2 − 1 π ∑ k = 1 ∞ ( − 1 ) k sin ⁡ ( 2 π k f t ) k ) {\displaystyle x_{\text{sawtooth}}(t)=a\left({\frac {1}{2}}-{\frac {1}{\pi }}\sum _{k=1}^{\infty }{(-1)}^{k}{\frac {\sin(2\pi kft)}{k}}\right)} x reverse sawtooth ( t ) = 2

144-427: A 'velar pinch' before the velar and separating from the same 'pinch' as the velar is released; alveolar sounds (English /t/ and /d/ ) cause fewer systematic changes in neighbouring vowel formants, depending partially on exactly which vowel is present. The time course of these changes in vowel formant frequencies are referred to as 'formant transitions'. In normal voiced speech, the underlying vibration produced by

180-671: A 1963 Chamber Chorus concert at New York's Town Hall , Lucier met Gordon Mumma and Robert Ashley , experimental composers who were also directors of the ONCE Festival . A year later, Mumma and Ashley invited the Chamber Chorus to the ONCE Festival in Ann Arbor, Michigan , and in 1966 Lucier reciprocated by inviting Mumma, Ashley, and David Behrman to Brandeis for a concert of their works. The four then embarked on

216-554: A closed or high vowel such as [i] or [u] ; and the second formant F 2 has a higher frequency for a front vowel such as [i] and a lower frequency for a back vowel such as [u] . Vowels will almost always have four or more distinguishable formants, and sometimes more than six. However, the first two formants are the most important in determining vowel quality and are often plotted against each other in vowel diagrams, though this simplification fails to capture some aspects of vowel quality such as rounding. Many writers have addressed

252-399: A collection of formants (such as a room) affects a signal can be represented by an impulse response . In both speech and rooms, formants are characteristic features of the resonances of the space. They are said to be excited by acoustic sources such as the voice, and they shape (filter) the sources' sounds, but they are not sources themselves. From an acoustic point of view, phonetics had

288-741: A formant. Most often the two first formants, F 1 and F 2 , are sufficient to identify the vowel. The relationship between the perceived vowel quality and the first two formant frequencies can be appreciated by listening to "artificial vowels" that are generated by passing a click train (to simulate the glottal pulse train) through a pair of bandpass filters (to simulate vocal tract resonances). Front vowels have higher F 2 , while low vowels have higher F 1 . Lip rounding tends to lower F 1 and F 2 in back vowels and F 2 and F 3 in front vowels. Nasal consonants usually have an additional formant around 2500 Hz. The liquid [l] usually has an extra formant at 1500 Hz, whereas

324-445: A little without altering the character of the vowel. For “long e” ( ee or iy ) for example, the lowest-frequency “formant” may vary from 350 to 440 Hz even in the same person. Formants are distinctive frequency components of the acoustic signal produced by speech, musical instruments or singing . The information that humans require to distinguish between speech sounds can be represented purely quantitatively by specifying peaks in

360-607: A position he held until 1979. Lucier was married to his first wife, Mary, until their divorce in 1972. He then married Wendy Stokes; they had one daughter and remained together until his death. Lucier died at his home in Middletown, Connecticut , on December 1, 2021, at age 90, from complications of a fall. Though Lucier had composed chamber and orchestral works since 1952, the composer and his critics count his 1965 composition Music for Solo Performer as his first mature work. One of Lucier's most important and best-known works

396-461: A reverse (or inverse) sawtooth wave, the wave ramps downward and then sharply rises. It can also be considered the extreme case of an asymmetric triangle wave . The equivalent piecewise linear functions x ( t ) = t − ⌊ t ⌋ {\displaystyle x(t)=t-\lfloor t\rfloor } x ( t ) = t mod 1 {\displaystyle x(t)=t{\bmod {1}}} based on

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432-456: A serious problem with the idea that the effective length of vocal tract changed vowels. Indeed, when the length of the vocal tract changes, all the acoustic resonators formed by mouth cavities are scaled, and so are their resonance frequencies. Therefore, it was unclear how vowels could depend on frequencies when talkers with different vocal tract lengths, for instance bass and soprano singers, can produce sounds that are perceived as belonging to

468-455: A steadily rising sine wave to produce beat frequencies ; the series Still and Moving Lines of Silence in Families of Hyperbolas (1973–74), in which beat frequencies between sine waves and acoustic instruments create "troughs" and "valleys" of sound and silence; and Clocker (1978), which uses biofeedback and a digital delay unit. Lucier was awarded an Honorary Doctorate of Arts by

504-763: A tour of the United States and Europe as the Sonic Arts Group; at Ashley's suggestion, the name was later changed to the Sonic Arts Union . After performing and touring together for a decade, the Sonic Arts Union became inactive in 1976. In 1970, Lucier left Brandeis for Wesleyan University , where he would remain until his retirement. In 1972, Lucier became a musical director of the Viola Farber Dance Company,

540-506: A way to smooth out any irregularities my speech might have,” referring to his own stuttering . Other key pieces from Lucier's oeuvre include North American Time Capsule (1966), which employed a prototype vocoder to manipulate elements of speech; Music On A Long Thin Wire (1977), in which a piano wire is strung across a room and activated by an amplified oscillator and electromagnets ; Crossings (1982), in which tones play across

576-428: Is I Am Sitting in a Room (1969), in which Lucier records himself narrating a text, and then plays the recording back into the room, re-recording it. The new recording is then played back and re-recorded, and this process is repeated. Since every enclosed area has a characteristic resonance (e.g., between a large hall and a small room), the effect is that certain frequencies are gradually emphasized as they resonate in

612-539: Is also known as squillo . Alvin Lucier Alvin Augustus Lucier Jr. (May 14, 1931 – December 1, 2021) was an American experimental composer and sound artist . A long-time music professor at Wesleyan University in Middletown, Connecticut , Lucier was a member of the influential Sonic Arts Union , which also included Robert Ashley , David Behrman , and Gordon Mumma . Much of Lucier's work explores psychoacoustic phenomena and

648-428: Is an essential component of the vocal technique known as overtone singing , in which the performer sings a low fundamental tone, and creates sharp resonances to select upper harmonics , giving the impression of several tones being sung at once. Spectrograms may be used to visualise formants. In spectrograms, it can be hard to distinguish formants from naturally occurring harmonics when one sings. However, one can hear

684-467: Is thought to be associated with one or more of the higher resonances of the vocal tract. It is this increase in energy at 3000 Hz which allows singers to be heard and understood over an orchestra . This formant is actively developed through vocal training , for instance through so-called voce di strega or "witch's voice" exercises and is caused by a part of the vocal tract acting as a resonator . In classical music and vocal pedagogy, this phenomenon

720-473: The ERB-rate scale . Another widely adopted strategy is plotting the difference between F 1 and F 2 rather than F 2 on the horizontal axis. Studies of the frequency spectrum of trained speakers and classical singers , especially male singers, indicate a clear formant around 3000 Hz (between 2800 and 3400 Hz) that is absent in speech or in the spectra of untrained speakers or singers. It

756-539: The English "r" sound ( [ɹ] ) is distinguished by a very low third formant (well below 2000 Hz). Plosives (and, to some degree, fricatives ) modify the placement of formants in the surrounding vowels. Bilabial sounds (such as /b/ and /p/ in "ball" or "sap") cause a lowering of the formants; on spectrograms, velar sounds ( /k/ and /ɡ/ in English) almost always show F 2 and F 3 coming together in

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792-641: The Tanglewood Center. In 1960, he left for Rome on a Fulbright grant , where he befriended American expatriate composer and pianist Frederic Rzewski and witnessed performances by John Cage , Merce Cunningham , and David Tudor , who inspired him to adopt a more experimental style. He returned from Rome in 1962 to take up a position at Brandeis as director of the University Chamber Chorus, which presented classical vocal works alongside modern compositions and new commissions. At

828-472: The University of Plymouth in 2007. Sawtooth wave The sawtooth wave (or saw wave ) is a kind of non-sinusoidal waveform . It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle . A single sawtooth, or an intermittently triggered sawtooth, is called a ramp waveform . The convention is that a sawtooth wave ramps upward and then sharply drops. In

864-432: The floor function of time t is an example of a sawtooth wave with period 1. A more general form, in the range −1 to 1, and with period p , is 2 ( t p − ⌊ 1 2 + t p ⌋ ) {\displaystyle 2\left({\frac {t}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {t}{p}}\right\rfloor \right)} This sawtooth function has

900-425: The frequency spectrum of the sound, using a spectrogram (in the figure) or a spectrum analyzer. However, to estimate the acoustic resonances of the vocal tract (i.e. the speech definition of formants) from a speech recording, one can use linear predictive coding . An intermediate approach consists in extracting the spectral envelope by neutralizing the fundamental frequency, and only then looking for local maxima in

936-420: The slip-stick behavior of the bow drives the strings with a sawtooth-like motion. A sawtooth can be constructed using additive synthesis . For period p and amplitude a , the following infinite Fourier series converge to a sawtooth and a reverse (inverse) sawtooth wave: f = 1 p {\displaystyle f={\frac {1}{p}}} x sawtooth ( t ) =

972-405: The frequency spectrum. Most of these formants are produced by tube and chamber resonance , but a few whistle tones derive from periodic collapse of Venturi effect low-pressure zones. The formant with the lowest frequency is called F 1 , the second F 2 , the third F 3 , and so forth. The fundamental frequency or pitch of the voice is sometimes referred to as F 0 , but it is not

1008-517: The highest harmonic, N max , is less than the Nyquist frequency (half the sampling frequency ). This summation can generally be more efficiently calculated with a fast Fourier transform . If the waveform is digitally created directly in the time domain using a non- bandlimited form, such as y  =  x  −  floor ( x ), infinite harmonics are sampled and the resulting tone contains aliasing distortion. An audio demonstration of

1044-433: The natural formants in a vowel shape through atonal techniques such as vocal fry . Formants, whether they are seen as acoustic resonances of the vocal tract, or as local maxima in the speech spectrum, like band-pass filters , are defined by their frequency and by their spectral width ( bandwidth ). Different methods exist to obtain this information. Formant frequencies, in their acoustic definition, can be estimated from

1080-870: The physical properties of sound. Alvin Augustus Lucier Jr. was born on May 14, 1931, in Nashua, New Hampshire , to Kathryn E. Lemery, a pianist, and Alvin Augustus Lucier Sr., a lawyer and politician who served as mayor of Nashua from 1934 to 1937. He was educated in Nashua public and parochial schools; the Portsmouth Abbey School in Portsmouth, Rhode Island ; Yale University ; and Brandeis University . In 1958 and 1959, Lucier studied under Lukas Foss and Aaron Copland at

1116-618: The problem of finding an optimal alignment of the positions of vowels on formant plots with those on the conventional vowel quadrilateral. The pioneering work of Ladefoged used the Mel scale because this scale was claimed to correspond more closely to the auditory scale of pitch than to the acoustic measure of fundamental frequency expressed in Hertz. Two alternatives to the Mel scale are the Bark scale and

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1152-410: The room, until eventually the words become unintelligible, replaced by the pure resonant harmonies and tones of the room itself. The recited text describes this process in action. It begins, “I am sitting in a room, different from the one you are in now. I am recording the sound of my speaking voice…”, and concludes with “I regard this activity not so much as a demonstration of a physical fact, but more as

1188-437: The same phase as the sine function. While a square wave is constructed from only odd harmonics, a sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd harmonics of the fundamental frequency . Because it contains all the integer harmonics, it is one of the best waveforms to use for subtractive synthesis of musical sounds, particularly bowed string instruments like violins and cellos, since

1224-416: The same phonetic category. There had to be some way to normalize the spectral information underpinning the vowel identity. Hermann suggested a solution to this problem in 1894, coining the term “formant”. A vowel, according to him, is a special acoustic phenomenon, depending on the intermittent production of a special partial, or “formant”, or “characteristique” feature. The frequency of the “formant” may vary

1260-403: The spectral envelope. The first two formants are important in determining the quality of vowels, and are frequently said to correspond to the open/close (or low/high) and front/back dimensions (which have traditionally been associated with the shape and position of the tongue ). Thus the first formant F 1 has a higher frequency for an open or low vowel such as [a] and a lower frequency for

1296-482: The vocal folds resembles a sawtooth wave , rich in harmonic overtones. If the fundamental frequency or (more often) one of the overtones is higher than a resonance frequency of the system, then the resonance will be only weakly excited and the formant usually imparted by that resonance will be mostly lost. This is most apparent in the case of soprano opera singers, who sing at pitches high enough that their vowels become very hard to distinguish. Control of resonances

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