Formant: what is a formant?

Definitions like those above are broadly used in acoustics research and industry. In parts of the speech research community, however, 'formant' has come to have other meanings. This page discusses the different usages. A recent publication (Titze et al, 2015) discusses the history of the term and recommends consistent terminology, including the ASA definition above.

After defining formant as "the spectral peaks of the sound spectrum |P(f)|" the speech researcher Gunnar Fant (1960) then defines resonance frequencies of the vocal tract in terms of a gain function T(f) of the vocal tract: 'The frequency location of a maximum' in |T(f)|, i.e., the resonance frequency, is very close to the corresponding maximum in spectrum P(f) of the complete sound.' He then writes: 'Conceptually these should be held apart but in most instances resonance frequency and formant frequency may be used synonymously.' Hence the problem: resonance and formant are indeed conceptually distinct, and their frequencies are only approximately equal. Several examples below make this clear. However, in many circumstances, the formant is the only information that one has about the resonance. For this reason, some writers in voice science use the terms almost interchangeably.

There is even a third meaning in voice research. The acoustics of the vocal tract are often modelled using a mathematical model of a filter (Atal and Hanauer, 1971). The frequencies of the poles of this filter model fall close to those of the formants. As a result, some voice researchers now refer to the frequencies of the poles as formants. So, to some voice researchers, the formant refers to a peak in the spectral envelope (a property of the sound of the voice), to others it refers to a resonance of the vocal tract (a physical property of the tract), while to a third group it refers to the pole in a mathematical filter model (a property of a model).

In the broader field of acoustics, formant retains only its original meaning: a broad peak in the spectral envelope of the sound (of a voice, musical instrument, room etc). When referring to the formant at about 400 Hz in the sound of the French horn, it is obviously a peak in the spectral envelope that is meant, not one of the resonances.

The issue is further complicated because those who use formant to mean resonance often use the term in the acoustical sense as well. For example, virtually all writers use who write about the singers formant and actors or orators formant mean the broad peaks in the spectral envelope occurring around 3 kHz. Some researchers who use formant to mean resonance will also talk about 'formant level', or write that the second formant is 10 dB lower than the first. In these cases, it seems clear that they refer to the amplitude of a peak in the envelope of the sound spectrum and not to a property of the resonance that produced it.

Does it matter? For the voice, a resonance Ri usually gives rise to a spectral envelope maximum Fi and the process may modelled by a filter with a pole Pi. Usually, the three have approximately equal values of frequency. However, as Fant observed, they are conceptually distinct. Let's take some examples:

• Consider a vocal tract with a resonance at 500 Hz, which is being excited by the larynx producing a fundamental frequency of 1 kHz (roughly C6, the high C for sopranos). There is no spectral maximum at 500 Hz. In this case there is a resonance R1 but no corresponding spectral peak F1. Here of course the difference does matter.
• Consider the singers formant or singing formant, a broad band of enhanced power noticed in the spectral envelope of classically trained male singers (and possible others) in a range. Sundberg (1974) attributes this formant to a clustering of the third, fourth and fifth resonances of the vocal tract. Here, where three resonances are thought to give rise to one formant, the distinction between formant and resonance is important.
• Consider a glottal source with a negative spectral slope, input to a vocal tract that (including radiation impedance) has a resonance at R1. The peak in the spectral envelope of the radiated sound in this case has a frequency less than R1. In this case, if one is estimating the spectral peak from the harmonic spectrum of the output voice, the difference between the two is less than the precision of the estimation, so the distinction is usually not important.
• Consider a musical wind instrument, whose bore radiates weakly below some frequency f, and which is excited by a reed or lip valve whose spectral envelope falls with frequency. Here the output sound has a spectral envelope peak that has nothing at all to do with the resonances of the bore.
• Consider this quote*, from Stevens and House (1961): "When resonant frequencies are sufficiently close, however, they are not necessarily identical with the frequencies of the peaks in the spectrum. For example, when two resonances with bandwidths of about 100 cps are about 100 cps apart, the spectrum envelope may show only one prominence: the frequency of the peak will be somewhere between the two resonant frequencies. In the discussion that follows, the the levels of the resonances will be defined to be the levels of the spectral envelope at the frequencies of the resonances (rather than at the spectral peaks)."

What to do? Our preference would be to retain the original meaning for the word formant. We prefer to say "A resonance Ri gives rise to a formant Fi. This may be modelled by a filter with a pole Pi". While acousticians will broadly agree with this use, some members of the speech research community may not. We therefore suggest that, when discussing the voice, the word formant should be defined, to make it clear which meaning is intended. In principle, one could consider abandoning the word. However "broad peak in the spectral envelope" is a long phrase, and we know no other synonym, so it is useful to retain formant for that reason.

A recent paper (Titze et al, 2015) written by Ingo Titze and with 21 other voice researchers as co-authors recommends a consistent terminology for the frequencies, magnitudes and bandwidths of harmonics, resonances and formants and draws attention to the need to explain carefully one's use of these terms.

Whatever one's choice of definition, one should make it clear. And, in literature and in discussions, prepare for some confusion. In a scientific talk, I have heard the sentence: 'Trained sopranos tune the first formant near the note sung, but they usually don't have a strong singer's formant'. When that speaker said 'first formant' he presumably meant 'first resonance' and when he said 'singer's formant' he meant a spectral peak probably due to two or more resonances. So we have the same person using the word in two of its three different meanings in the one sentence.

* It's interesting to rewrite the quote from Stevens and House (1961), substituting 'formant' wherever they write 'resonance': "When formant frequencies are sufficiently close, however, they are not necessarily identical with the frequencies of the formants. For example, when formants with bandwidths of about 100 cps are about 100 cps apart, the spectrum envelope may show only one formant: the formant will be somewhere between the two formants. In the discussion that follows, the levels of the formant will be defined to be the levels of the spectral envelope at the formant frequencies (rather than at the formant frequencies)."

References

• Atal, B. S. and Hanauer, S. L. (1971) "Speech Analysis and Synthesis by Linear Prediction of the Speech Wave", J. Acoust. Soc. Am., 50, 637-655.
• Benade,  A. H. (1976) Fundamentals of musical acoustics, Oxford University Press, London.
• Dowd, A., Smith, J.R. and Wolfe, J. (1997) "Learning to pronounce vowel sounds in a foreign language using acoustic measurements of the vocal tract as feedback in real time". Language and Speech, 41, 1-20.
• Epps, J., Smith, J.R. and Wolfe, J. (1997) "A novel instrument to measure acoustic resonances of the vocal tract during speech" Measurement Science and Technology 8, 1112-1121.
• Fant, G. (1960). Acoustic Theory of Speech Production. Mouton & Co, The Hague, Netherlands.
• Jeans, J.H. (1938) Science & Music, reprinted by Dover, 1968. pp 104, 148.
• Joliveau, E., Smith, J. and Wolfe, J. (2004) "Tuning of vocal tract resonances by sopranos", Nature, 427, 116.
• Joliveau, E., Smith, J. and Wolfe, J. (2004) "Vocal tract resonances in singing: the soprano voice", J. Acoust. Soc. America, 116, 2434-2439.
• Standards Secretariat, Acoustical Society of America, (1994). ANSI S1.1-1994 (R2004) American National Standard Acoustical Terminology, (12.41) Acoustical Society of America, Melville, NY.
• Stevens, K.N., and House, A.S., (1961). An acoustical theory of vowel production and some of its implications, J. Speech & Hearing Research, 4, 303-320.
• Sundberg, J. (1974) “Articulatory interpretation of the ‘singing formant’,” J. Acoust. Soc. America, 55, 838-844.
• Titze, I.R. Baken, RJ, Bozeman, KW, Granqvist, S, Henrich, N, Herbst, CT, Howard, DM, Hunter, EJ, Kaelin, D, Kent, RD, Kreiman, J, Kob, M, Löfqvist, A, McCoy, S, Miller, DG, Noé, H, Scherer, RC, Smith, JR, Story, BH, Švec, JG, Ternström, S and Wolfe, J. (2015) "Toward a consensus on symbolic notation of harmonics, resonances, and formants in vocalization" J. Acoust. Soc. America, 137, 3005-3007. http://dx.doi.org/10.1121/1.4919349