Self-organization is the spontaneous formation of structures in space and/or time in systems composed of multiple components. Some form of overall order arises from local interactions between sub-system. The process occurs when sufficient energy is available in the system and nonlinearity exists in system components. No external control is needed. Unpredictable fluctuations in internal parameters, usually arising from nonlinearity, are amplified by positive feedback to bring about this organization. Chaos theory describes self-organization in terms of islands of predictability (attractors) in an otherwise unpredictable system.
We ask if the production of subharmonics falls into the category of self-organization. First, there is ample nonlinearity in the responses of sub-components. Glottal airflow is not proportional to transglottal pressure. It is more of a quadratic relation when the glottis is open. Furthermore, airflow can be clipped by vocal fold collision. Nonlinearity also exists in tissue mechanics. Tissue displacement is not proportional to applied force, especially when vocal fold collision occurs.
Interaction (coupling) with energy transfer between multiple sub-components exists. Vocal fold tissues, with their connections to cartilage boundaries, produce spatial tissue modes of vibration with specific natural frequencies. In the vocal tract, acoustic resonances also have their own natural frequencies. Some of their energy couples to pressures in the glottis.
The combined system is “attracted“ to stable states if there is sufficient energy transfer between the subsystems. If natural frequencies are in proximitry, they can be entrained in (m:n) integer ratios. If a harmonic has a frequency designated with the integer m=1, then the subharmonic frequencies will be 1/2, 1/3, 1/4, etc. As an example, for a fundamental frequency of 100 Hz, the subharmonic frequencies would be 50 Hz, 33.3 Hz, 25 Hz, 12.5 Hz, ….Stability with entrainment may not be reached or maintained, however, resulting in chaotic vibrations (known as a strange attractor).
Bifurcation is the term used for sudden changes of attractors. Bifurcations are observed in human phonation, animal vocalization, and simulation models, especially when a high-energy harmonic is within 1-2 bandwidths of a low formant (F1 or F2). The bifurcations can be sudden pitch jumps, squelched oscillation, subharmonics, or chaos.
Figure 1 is a spectrogram of a pitch glide (high-low-high) produced by a computer simulation of a female voice (Titze, 2008). Similar pitch glides have also been reported from human vocalizations (Titze, Riede, and Popolo, 2008; Maxfield et al, 2016), but the analysis is more complete with simulation. The simulation was 8 seconds long. At the beginning, we see two black lines representing the fundamental frequency (760 Hz) and a second harmonic (1520 Hz). Around 1.0 seconds, a bifurcation occurs to a ¼ subharmonic (190 Hz), which now carries an entire series of its own harmonics (integer multiples of 190 Hz).
Immediately after the subharmonic regime, a sudden pitch jump occurs, followed by a continuation of the downward glide with a single harmonic series. At 3.4 seconds, a bifurcation into chaos occurs. There appears to be little, if any, harmonic structure for about a second at this low pitch. Roughness would be perceived in this second. At 4.6 seconds, the harmonic structure resumes, but now there is a ½ fo subharmonic that lasts until 5.2 seconds. Finally, at 7.6 seconds, there is an upward jump in fundamental frequency.
The white lines in Figure 1 are the acoustic reactance curves of the airways (supraglottal to the left and subglottal to the right). Bifurcations occur when a harmonic enters or leaves the region of sudden change in the reactance. That happens at about 1 second into the utterance, when the fundamental (fo) is near 600 Hz (the first supraglottal resonance). It happens again at 3.5 seconds, when the second harmonic (2fo) is near the first resonance. Interactions with the second resonances near 1800 Hz contribute less to the bifurcations.
In summary, subharmonic generation meets the criteria for self-organization in vocalization. There is strong interaction (energy transfer between source and vocal tract), a sudden change in an internal parameter (acoustic reactance), and nonlinearity in vocal fold mechanics.
Reference
Titze, I.R. (2008) Nonlinear source-filter coupling in phonation: Theory. J. Acoust. Soc. Am.123(5), 2733-2749.
Titze, I.R., Riede, T., and Popolo, P. (2008). Nonlinear source-filter coupling in phonation: VocalExercises. J. Acoust. Soc. Am. 123(4) 1902-1915.
Maxfield, L., Palaparthi, A. and Titze,I.R. (2016). New evidence that nonlinear source-filter coupling affects harmonic intensity and fo stability during instances of harmonics crossing formants. J Voice, Aug 5.
How to Cite
Titze, I.R. (2023),
Can the Production of Subharmonics in Vocalization be Considered a Form of Self-Organization?. NCVS Insights, Vol 1(3), pp. 1-2. DOI: https://doi.org/10.62736/ncvs135000