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way of selecting and amplifying the harmonics and have refined their VT motor skills to be able, with great pre- cision in time and frequency, to produce harmonics as melodic sequences. Next, we test the hypotheses that (1) enhancing and selecting a single partial is the result of
VT shapes producing clustering of formants; and (2) that overtone singing is produced with a regular sound source.
Acoustic Theory
Enhancing Single Harmonics
How can formants produce the excessive amplitudes of the single overtones illustrated in Figure 3? Mathemati- cally, the spectral envelope, the function determining the amplitudes of the overtones, is the sum of formant reso- nance curves (which vary with changes in VT shape) and certain factors such as glottal source and radiation char- acteristics (which do not depend on articulatory activity).
The only input to the calculation is the frequencies and bandwidths of the formants. The latter factor generally varies with the former factor in a predictable manner so formant amplitudes need not be specified. Figure 5 illustrates this predictability.
Figure 5 shows three line spectra with the cardinal shapes of the resonance curves for the first and second formants (henceforth F1 and F2). The amplitudes of the partials and their spectral envelopes were derived in accordance
with the standard source-filter model (Fant, 1960). This theory treats the envelope as the sum of formant curves and the constant contributions of source and radiation characteristics, which are not shown in Figure 5. The bandwidths are secondary aspects, being determined mainly by the frequencies of the formants (Fant, 1972).
In Figure 5, F1 was fixed at 600 Hz while F2 was varied. When F1 and F2 approach almost identical frequencies (Figure 5, right), creating, as it were, a double formant, their individual peaks merge into a single maximum, with a significant increase in the amplitude of the clos- est partial. In other words, acoustic theory states that formant amplitudes are predictable and thus suggests an answer to the question asked in the first sentence of this section. Enhancing the amplitude of individual overtones is possible: Move two formants close to each other in fre- quency. Create a double formant!
Measurements and Modeling
Vocal Tract Shapes
As shown above, if the formant frequencies are deter- mined by the shape of the VT, so what was the shape of AMH’s VT? This has actually been documented in another dynamic (MRI) video published by the Freiburg Institute of Musician’s Medicine, Naturtonreihe in Zun- gentechnik (see It was taken when AMH performed overtone singing, enhancing, one by one each overtone of a drone with a fundamen- tal frequency of 270 Hz (pitch about C4), in a rising followed by a descending sequence. Henceforth, the frequencies of the enhanced overtones will be referred to as FE. All overtones, from the 4th, FE = 1,080 Hz, up to the 12th, FE ≈ 3,200 Hz, were enhanced.
The MRI video shows her entire VT in a midsagittal lat- eral profile. Figure 6 shows tracings of the VT for each of the enhanced overtones in the ascending and the descending series.
Voice Source
Is formant clustering an exhaustive explanation of over- tone singing? Fortunately, the transfer function of the VT can be predicted given its formant frequencies. Thus, a vowel spectrum can be analyzed not only with respect to the formant frequencies, which appear as peaks in the spectrum envelope, but also with respect to the voice
  Figure 5. Schematic illustration of the spectrum effects of moving the frequencies of two formants closer together. Vertical lines, partials of a drone with a fundamental frequency of 100 Hz; blue and red curves, first (F1) and second (F2) formants, respectively. Left: F1 = 600 Hz, F2 =
1,400 Hz. Center: F1 = 600 Hz; F2 = 2,150 Hz. Right: F1 = 600 Hz; F2 = 650 Hz, thus creating a “double formant,” that creates a very strong partial (arrow).
 46 Acoustics Today • Spring 2021
source. The trick is simple, inverse filtering!

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