PHYS341/2024/Project13/Physics of Vibrato

The Physics of Vibrato

This page will discuss the physics of vocal vibrato to investigate what happens to a sustained vowel when vibrato is introduced versus it is not. The background information will discuss what vibrato is and the debate between it being a natural or a learned quality. This page will then look into the physics of vibrato by analyzing a recording of a sustained note with and without vibrato using a real-time-audio analyzer. Finally this page will conclude on the findings.

What is Vibrato?

Vibrato is the periodic undulation of a frequency above and below a central pitch.

Rather than movement between two distinct pitches, as with a trill or a tremolo, Vibrato is a fluctuation in pitch. The term 'vibrato' was introduced fairly recently to music scholarship. Previously, what is now understood as vibrato, was referred to as a 'trill' or 'tremolo' which has caused the historical narrative to debate its true meaning. This study will use the modern definition of vibrato as a periodic frequency change.

An important distinction in this study is between vibrato and a straight tone. A straight tone varies from a vibrato tone in that it does not have an intentional modulation of pitch. Given that a pure tones require precise and consistent breath support that is extremely difficult to obtain, straight tones in singing will have a slight waver in pitch, however it is not as deliberate as in vibrato singing where the singer intentionally fluctuates the pitch.

One of the reasons vibrato is used is to eliminate beats in musical tuning. Beats are an interference pattern that result between simultaneous sounds of slightly different frequencies. By constantly moving around a central frequency, vibrato essentially smooths over the gap that would be created between pure tone fundamentals that are too close together.

Stylistically, vibrato is used to create a richer and fuller sound quality. Vibrato has historically been associated with trained operatic singers for this purpose of ornamental expression. Skilled singers know the amount of vibrato needed to produce an array of resonating harmonics that can be used to express emotional qualities with sound. Conversely, an excess of vibrato has potential to create instability of the perceived pitch heard by listeners. However, it has been found that if a singer's vibrato stays within a certain range of undulations per second (vibrato rate), then it does not reduce the certainty with which the fundamental pitch is perceived. The range of vibrato rate that has been discovered to not affect pitch perception is between between 4Hz and 8Hz.

Is Vibrato Learned or Natural?

The question of whether vibrato is a natural or learned quality is highly debated. To some, vibrato is a natural part of human singing. To others, it is learned through practice. This section will discuss both viewpoints.

From a physiological standpoint, vibrato is a natural vibration in the larynx. The larynx, also known as the vocal box, is the area that connects the throat to the windpipe (trachea) in the human body. The larynx is lined with muscles, called laryngeal muscles. It has been discovered that these muscles are key contributors to pitch control, and consequently, are significant to the production of vibrato. For instance, Hsiao speaks to how the laryngeal muscles, especially the cricothyroid (CT) muscle, affects the fundamental frequency of the voice. The regular contraction of these muscles trigger tension in the vocal folds which ultimately affects vocal pitch.

Do you know that feeling of when you are holding a heavy object and your arm muscles begin to shake under the strain? O'Conner compares this muscular pulsing to the creation of vibrato by the laryngeal muscles. According to O'Conner, a natural rhythmic pulsation occurs in the laryngeal muscles in response to tension. As a protective measure for the vocal folds, pulsations are naturally created, and result in the pitch fluctuations seen in vibrato. Similarly, Titze et al. also contends that vocal vibrato is a "stabilized physiologic tremor in the laryngeal musculature," and adds that this tremor naturally occurs at a rate of 4-6Hz. These study argue that the oscillation of these muscles are activated when singing and therefore naturally result in vibrato.

Others view vibrato as a learned technique used to decorate vocal singing. Central to this argument is the perception that young children most often do not sing with vibrato and develop the tendency as they get older. It has been found that vibrato is naturally occurring in the typical adult voice, but commonly only appears in children after puberty. These findings present vibrato as something that develops as one gets older.

Vibrato is also viewed as an ornament of music that decorates vocal singing, and therefore, is a learned technique. Often vocalists are required to adjust their vibrato to fit with the genre of music being performed. For instance, more vibrato is expected in classical operatic pieces than in popular or country styles. What is clear from the studies is that vibrato can be developed and mastered in a way where singers can manipulate it. As a singer grows in their trade, they are able to adjust the amount of vibrato used in their singing. In their study of the physics of vibrato, Michel and Ruiz discuss how singers "learn to create the circumstances that allow for the vibrato to flourish in its natural state." Similarly, in a study on Vibrato Rate and Extent in College Music Majors, John Nix et al. discusses that vocal training has a substantial impact on the development of vibrato and can be effected by different factors. Some of these factors that can improve the regularity of vibrato include "postural balance, breath management, laryngeal coordination, and vowel shaping."

Synthesizing the above arguments, vibrato is natural, but can be developed in a manner that makes it seem like it is a learned trait. Vibrato occurs naturally through a vibration in the larynx muscles. However, as with most muscles in the human body, the muscles that control vibrato can be strengthened and trained to where vibrato can appear more or less depending on the individual singer.

Method

Figure 1. Straight Tone Spectrogram. Sinusoidal frequencies of fundamental and harmonics without intentional vibrato

Figure 2: Vibrato Tone Spectrogram: Sinusoidal frequencies of fundamental and harmonics with intentional vibrato

To investigate what is happening with vibrato, I took a recording of a person singing a sustained vowel with and without vibrato. I recorded each of these notes in a real-time audio analyzer which gave spectrogram and combined time graphs for each sustained vowel.

Spectrogram

The two sustained vowels were recorded using a 2D spectrogram. Spectrograms show the spectrum of the frequencies of the fundamental and its harmonics over time. According to the Fourier's Theorem, periodic waves are composed of many sinusoidal components including the fundamental and all the resulting harmonics. The spectrogram separates all of these harmonic components into their individual sinusoidal waves. The colors of the spectrogram correspond to the strength of the harmonics. In the vibrato spectrogram (figure 2), eight harmonics are visible, with the fundamental being the brightest and therefore the strongest compared to the higher harmonics.

Combined Time Graphs

The vowels were also recorded using Combined Time Graphs. Each frequency of the fundamental and it's harmonics has a different amplitude and time graph. In a combined time graph, each of these sinusoidal graphs are added together (see figures 3 & 4).

Figure 3. Combined Time Graph for straight tone without intentional vibrato. Green lines show the time segment used in the fundamental calculation (-23ms to 0ms).

Figure 4. Combined Time Graph for the tone with intentional vibrato.

Calculations

Fundamental Calculation

To calculate the fundamental, or perceived pitch, I first estimated the frequency using the Straight Tone Spectrogram. I estimated the fundamental to be around 180 Hz based on where the line falls on the graph. To get a more accurate pitch estimate, I used the straight tone combined wave time graph (see figure 3) to count the number of undulations in a specific amount of time. I used the straight tone graph to calculate the fundamental because it provides consistent oscillations that are not present in the vibrato combined time graph (see figure 4). I also chose to conduct my reading between times -23ms and 0ms because at those times the end of an oscillation corresponded with a fixed point on the graph which ensured a more accurate reading. The time between -23ms and 0ms is shown on the graph with green lines (figure 3). The results were as follows:

Frequency = 4 oscillations/0.023s

Frequency = 173.91 Hz

This calculated frequency of 173.91 Hz is close to the known frequency pitch for F3 (174.61 Hz). Given this difference between the calculated frequency and the F3 standard frequency is only 0.7 Hz, it can be assumed that the singer was intending to perform a perceived pitch of F3.

Figure 5. Illustration of Vibrato Rate and Extent

Vibrato Rate and Extent

Two other important values are Vibrato Rate (VR) and Vibrato Extent (VE).

These values are significant because they have an effect on listener's perception of sound, which impacts the perceived pitch. As vibrato changes vibrations, the resulting sound waves also change, producing different sets of harmonics. The addition of these harmonics effects the perceived pitch heard by the listener.

The first value I will calculate is for is the Vibrato Rate (VR) which refers to the number of pitch pulsations per second. In other words, this is how fast the singer is moving above and below the central frequency each second (see figure 5). Typical values for VR occur between 4.5 and 6.6 Hz. In a study of the mean vibrato rate between male and female vocalists, it was discovered that males tend to have a slower mean vibrato rate of about 5.4Hz as compared to female vocalists with a mean of 5.9Hz. Taking this study into account, I am expecting to find that the male vocal recording used in this study will result in a vibrato rate that is similar to 5.4 Hz.

VR = oscillations/time

In a 1 second interval there are 5 oscillations present on the the vibrato tone spectrogram (figure 2). Therefore VR can be calculated as:

VR = 5 oscillations/ 1 second

VR = 5 Hz

The second value I will calculate is the Vibrato Extent (VE) which measures the depth of vibrato (see figure 5 for illustration). This value measures the amount of a semitone the vibrato moves about the central frequency. This will be expressed either in a band measurement of +/- cents (100 cents = 1 semitone) above/below the perceived frequency, and as a percentage of the semitone.

Figure 6. Linear Spectrogram with vibrato vowel . Black lines represent the mean highest and lowest frequency values of H4. Green lines represent the mean highest and lowest frequency values of H3.

To calculate VE, I converted the vibrato tone spectrogram to a linear format so the frequency (Y value) was a consistent measurement with each tick measuring 50 Hz. I chose the fourth harmonic (H4) to measure because it was the clearest to read on the spectrogram, however I did confirm the value with other harmonics as well which produced similar results (E.g. H5 = ~12%; H3 = ~14%). The two black lines represent the wave height, which is measured from the trough and peak values present in this fourth harmonic. The calculations are as follows:

Estimated H4 Trough from graph: 570Hz

Estimated H4 Peak from graph: 740Hz

H4 average frequency: (740Hz +570Hz) /2 = 655Hz

Wave Height (peak - trough frequency): 740Hz - 570Hz = 170Hz

Vibrato Extent:

To find the VE, we first find the variation about the H4 central frequency by finding half the wave height: 170Hz / 2 = 85Hz.

This value is 12.9% of H4 as calculated by: (+/- 85Hz / 655Hz) x 100% = +/- 12.9%.

The percentage of increase above a harmonic will be consistent for all harmonics. For example, if the central frequency is 100Hz with a peak at 106Hz (+6% semitone increase), then for the second harmonic with a central frequency of 200Hz will also be expected to have a peak frequency +6% of a semitone above it at 212Hz (6% of 200Hz = 12Hz).

Using this logic, it can be estimated that the pitch variation of H4 (+/- 12.9%) can also be applied to the fundamental. Therefore, the fundamental has a pitch variation of +/- 12.9%.

VE is expressed in cents where 100 cents is defined as 1 semitone (5.9% increase). The vibrato has a pitch variation of 12.9%. Given that 100 cents corresponds to 5.9% pitch variation, the vibrato vowel VE is equal to (12.9%/5.9%) x 100 cents = ~219 cents. As a percentage of a semitone, this can be viewed as fundamental VE = 219% of a semitone. This is greater than a whole tone of variation around the central frequency.

Conclusions

The non-vibrato vowel was used as a baseline to be compared to the vibrato vowel. With the non-vibrato vowel, the singer was able to produce a fairly accurate F3 fundamental frequency that had a consistent periodic wave form. The Fundamental frequency, calculated at 173.91, was found to be only 0.7 Hz off of the known 174.61Hz of the F3 frequency. The vowel with intentional vibrato however, did not produce a consistent frequency but rather fell within a range of frequencies over time, as was expected.

The VR shows that the vibrato vowel went through 5 oscillations in one second of time. Although vibrato rate ranges with singers, this value fell below the estimated 5.4Hz averaged by male vocalists but within the 4.5Hz to 6.6Hz range of typical vibratos as discussed above. The recording used for this experiment was made for teaching purposes and therefore had an exaggerated vibrato, which might explain the lower VR which is generally associated with less intensity, control, and steady oscillations.

The VE results were more surprising. It was found that the singer's sustained vibrato vowel had a pitch variation of ~219 cents or 219% of a semitone. This falls into the category of pitch modulation rather than fluctuation of vibrato because the pitch changed by a whole tone rather than remaining within a semitone range. An explanation for this might also be because the recording was taken from an educational video on vibrato where the singer was likely overemphasizing his vibrato to demonstrate the concept. If this was a genuine attempt at vibrato, the singer definitely fell short according to the results found.