Balancing Odd and Even Harmonics in the Source Spectrum

Ingo Titze

The pulse-like airflow in the glottis is the primary source of sound for vocalization. It is produced in the process of vocal fold vibration. When the vocal folds collide, airflow is suddenly interrupted. The sudden change from a pulse of airflow to little or no airflow generates a spectrum of harmonic frequencies. For a typical flow pulse, both odd and even harmonics are produced. The amplitudes of these harmonics decrease uniformly with increasing harmonic number. In other words, higher harmonics have progressively lower amplitudes. This harmonic amplitude decay (also known as spectral roll-off, slope, tilt) is often quantified in decibels per octave (dB/oct). A typical harmonic amplitude decay for normal voice registration is on the order of -10 to -15 dB/oct, with the negative sign indicating the loss of harmonic amplitude.The questions arise: Can the glottal airflow exhibit a harmonic spectrum for which the roll-off is not evenly distributed across the harmonics? Can there be peaks and valleys in the source spectrum? More specifically, can the odd harmonic series be different from the even harmonic series? Interestingly, the often sketched caricature of a glottal flow waveshape answers these questions dramatically. A truncated sinusoid, for which the first half of the flow cycle is positive (a smooth rise and fall) and the second half is zero (glottal closure), contains no odd harmonics other than first (Figure 1). The third, fifth, seventh, etc. harmonics are all missing. The cause of this nonexistence of even harmonics is the perfect symmetry in the waveshape. There is symmetry in the rise and fall of the flow pulse, and there is symmetry in the duration of the flow pulse (half of the time "on" and half of the time "off").Two metrics have been defined for these symmetry features. The first is known as the open quotient (Q^sub o^), defined as the ratio of the time the glottis is open (there is glottal flow) to the total duration of the cycle. For the waveform in Figure 1, the open quotient is 0.5. The second metric is known as the skewing quotient, defined as the ratio of the time the flow rises to the time the flow falls. For the waveform in Figure 1, the skewing quotient is 1.0. The flow rises from 0-1.25 ms and falls from 1.25-2.5 ms. The loss of odd harmonics is a direct consequence of these exact symmetry ratios, Q^sub o^ = 0.5 and Q^sub s^ = 1.0.Odd harmonics appear when either Q^sub o^ moves away from 0.5 or Q^sub s^ moves away from 1.0. Figure 2 shows a glottal waveform and its spectrum for Q^sub o^ = 0.4 and the skewing quotient Q^sub s^ remaining at 1.0. Note the appearance of all even harmonics (not labeled). The full harmonic spectrum of the fundamental frequency (200 Hz) is now represented. A similar spectrum appears when Q^sub o^ is raised to 0.6 (Figure 3).

Balancing Odd and Even Harmonics in the Source Spectrum

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