Category: physics

Vowel Cube Formant Chart

Diagram of articulatory cube of vowels projected onto a formant frequency chart.

Characteristics of vowels, such as those of the International Phonetic Alphabet, are typically plotted on a flat chart by their first and second formant frequencies or by qualities such as front, central, back, close, mid and open along the axes. The two dimensional charts with articulatory features as axes typically show the vowels in a diagram with vowels having one of the articulatory features in common being joined by a straight line, usually such that the space of vowels is bound by a quadrilateral or a roughly triangular shape. Sometimes the distinctions between the vowels are displayed in a three dimensional plot with the acoustic properties of the first, second, and third formants as axes or by articulatory features with the three axes as frontness, height, and roundness. While the plots in two dimensions by the first two formant frequencies are often compared to those in which height and frontness are the two axes, comparison of articulatory and acoustic plots in three dimensions are not so usual as a method of teaching phonetics.

On the Base Dozen Forum there is presented a portrayal of vowels in a three dimensional plot by articulatory features of high versus low, back versus front, and rounded versus unrounded as the three axes. The space of vowels is let be a cube with extreme vowels at its corners. For example, the most high and rounded back vowel is one of the corners, and the lowest and furthest back unrounded vowel is another corner. Another corner is allocated to the most high and front unrounded vowel. These are three of the standard cardinal vowels proposed by the phonetician Daniel Jones. In the three dimensional space that is here proposed, these three cardinal vowels form the vertices of an equilateral triangle in the cube. To the other corners are assigned vowels according to the resulting articulatory dimensions of the axes aligned with the edges of the cube. Apart from vowels at the corners of the cube, other vowels of intermediate qualities are plotted between the corners, either along the edges or on the faces of the cube. At the centre of the cube is a schwa.

To compare this cube of vowels by their articulatory features to their acoustic properties on a formant chart, the cube is transformed by rotational orientation and scaling before it is projected onto the planar formant chart. The result is that the positions of the projected vowels agree extraordinarily well to their positions by their measured formant frequencies. Using the outlines of the edges of the projected cube, it is possible to predict the effects of changes in articulation on the acoustic qualities of the vowel. This should make this proposal more useful for understanding what a vowel will sound like in relation to known vowels than an ordinary chart in which unrounded and rounded vowels having the same features in other respects are merely placed beside each other in pairs without indicating how their formants differ, although it is expected that rounding applied to front vowels or unrounding applied to back vowels will centre them acoustically.

Diagram of articulatory cube of vowels projected onto a formant frequency chart.
Diagram of articulatory cube of vowels projected onto a formant frequency chart.

References:

https://dozenal.forumotion.com/t87-vowel-cube

Disk Drive Frequencies

This morning in the newspaper Irish Independent on Saturday, I read the article entitled “Laptops can be crashed by Janet Jackson song, Microsoft says”, by Josie Ensor on pages 24 to 25. This prompted me to recount an experience I had and wrote about some years ago:

“Monday 8th March 2010. Yesterday I was using the compact disk […] that I bought on 10th January 2009. On my computer, I used the […] software on the disc to record and analyse […] in order to measure the frequencies […]. Almost immediately after I recorded […] there was an error on my computer and I was forced to close the software and I restarted the computer. Then I recorded the set […] instead. When I listened to the recording through earphones I noticed a descending glissando note […]. When I analysed the spectrogram of the recording, I saw what appeared to be like a formant of linearly decreasing frequency […]. By choosing spectrogram controls in the software program and setting the contrast at a maximum and brightness quite low and increasing the “Analysis window length” to a maximum I was able to see what appeared to be overtones or higher formants of the first glissando note. The other frequencies also decreased in pitch linearly with respect to time but with larger slope of the oblique lines as the formant increased in pitch. I noted particular frequencies of the formants at the arbitrary time 1.161 seconds from after the beginning of the recording. At that time in the recording, the fundamental frequency of the glissando note was 1549 Hertz; the frequency of the second formant or first overtone was 2268 Hertz; the third was 3139 Hertz; the fourth was 3888 Hertz; the fifth was 4597 Hertz; and the sixth was 5367 Hertz. I examined the ratios of these frequencies and found that they were as ratios of simple whole numbers. Setting the frequency of the first formant or fundamental frequency as one unit, the frequencies of the notes or formants are: f1 = 1; f2 = 3/2; f3 = 2; f4 = 5/2; and f6 = 7/2. From these values, where f_n = (n + 1)/2, it can be seen that the sequence of the frequencies forms an arithmetic sequence. Since I had noted these simple rational frequencies at an arbitrary time, I reasoned that the ratios would be maintained at all times during the glissando. I took more data points of the frequencies at different times and plotted the results in the software […] to determine the equation of the fundamental frequency trendline. I found the linear regression equation Frequency (in Hertz) = -1382.5t + 3134.7 where t is the time in seconds from the beginning of the recording and the frequency is that of the fundamental formant. I plotted the trend lines for the other formants by multiplying the above equation by the ratio of that formant frequency to the fundamental frequency and superimposed this graph upon the picture of the spectrogram. I found a better fit to the upper formants by changing the intercept by translation from 3134.7 to 3157 without changing the slope very much from -1382.5 to -1383. The graph according to the equations fit the spectrogram perfectly. I do not know what the source of the glissando note was, although since I was making the recording while running the […] software directly from the compact disc drive of my computer, the source may have been the disc drive spinning.”

I then went on to speculate about the source of the observed ratios of the overtones, comparing them to musical intervals. I continued:

“Thus, if the frequencies that were produced simultaneously were of similar intensities, which they were not, then the chord produced would be a major chord in root position or uninverted, with the fundamental frequency as the root and base of the chord.”

I then went on to eliminate various types of sources of sound:

“[…] a vibrating string is supposed to produce overtones at whole number ratios to the fundamental frequency, […] about overtones that are produced in cylindrical pipes, both open and closed at one end, […] in a pipe that is open at both ends, […]. Since the frequency of a wave is inversely proportional to its wavelength by the velocity or speed of propagation of the wave through the medium as the constant of proportionality, […]. Thus, the ratios of harmonics produced by a pipe that is open at both ends would be the same as for those produced by a vibrating string. In contrast, for a pipe closed at one end, […]. Thus, the ratio of the nth harmonic frequency to the frequency of the first harmonic in a pipe that is closed at one end is 2n-1. These ratios increase arithmetically as odd whole numbers rather than half integer numbers, therefore the pipe closed at one end could not be the type of source of the glissando note that I recorded yesterday. The source could not be a vibrating string or pipe that is open at both ends either, since in both of those sources the ratios of harmonics to the frequency of the fundamental frequency increase arithmetically by a difference of whole numbers rather than half integer numbers, unless what I assumed to be the fundamental frequency of the glissando note is not really the fundamental note. This frequency was by far the most intense of the formants of the glissando note. However, the fundamental frequency is not necessarily the most intense harmonic. I have read, for example, that in the clarinet the note that is heard to be the one intended and played by the instrumentalist is a higher harmonic than the fundamental frequency of the first harmonic. However, even if the fundamental frequency is not the most intense harmonic of a composite note, we would still expect the fundamental frequency to appear with some intensity comparable to the higher harmonics. On the spectrogram of the glissando note, I could not find any evidence of a formant or harmonic below the one that I attributed to be the fundamental frequency. I plotted an oblique line of a formant with one arithmetic sequence frequency difference below the attributed fundamental so as to show where the possible expected hypothetical alternative fundamental frequency harmonic could be that could allow the glissando note to have been produced by a source like a vibrating string or an open pipe. Since there were no signals along the expected line for a lower fundamental frequency, either there is no lower harmonic of the glissando note, or for some strange reason the real fundamental frequency or first harmonic was not produced or is virtual.”

I then argued to eliminate causes for the glissando, debating speed of propagation, energy of the wave, temperature and cooling, density, and pressure of the medium, the action of pistons on the volume of cylindrical pipes and the ratios of frequencies in compartments closed at all ends, and changes in the mass or temperature by mixing of air by an air pump. Having eliminated these, I proceeding towards a conclusion:

“Tuesday 9th March 2010. […] The remaining possibility could be a change in the energy of the source of the wave, by change in the energy of vibration of a tremulous source. I think that the most likely explanation for the glissando would be a decrease of the angular speed of some rotor or wheel at a constant rate, until being stopped at the minimum speed by application of a brake, such that the frequency did not appear to reach zero. I think that the most likely wheel or rotor to be the source was a disc drive of the computer, especially the compact disc or digital video drive. Another less likely possibility is the fan of the computer that is used for cooling, as this would revolve and is not always turned on. […] I do not know what pattern of ratios or sort of overtones that [sic] a spinning disc could produce. […] it might be possible for a spinning disk to have fractional revolutions that produce overtones. […] I think that a more likely explanation would be that the disk wobbles or precesses at different frequencies to its frequency of revolution.”

I next debated precession, resonance, and coupling.

“Thursday 11th March 2010. […] Half-integer ratios between frequencies could be caused by resonance or by coupling. […] a spinning disk can have any frequency because it can rotate at any speed, and this explains how the glissando note could have arisen.”

Next, I described tidal coupling of planetary periods.

“Friday 12th March 2010. […] Perhaps then analogously, there could be coupling between different periods of precession or spin in a single rotating object, such as a disk. This coupling would give rise to fixed simple ratios of frequencies of the overtones. If the fundamental frequency were also fixed, then all the frequencies would be quantised. […]”

The arguments spanned some pages and were longer and more detailed than I would reproduce in full here. Later, I noted the following:

“Sunday 4th July 2010. Sometime after Saturday 13th March 2010 I looked at a book called “Genius, Richard Feynman and modern physics”, by author James Gleick because I wanted to read about the ancient scripts or writing system […]. I noticed the following which may be of relevance to what I wrote about between Monday 8th March 2010 and Saturday 13th March 2010: “A few days later he was eating in the student cafeteria when someone tossed a dinner plate into the air — a Cornell cafeteria plate with the university seal imprinted on one rim — and in the instant of its flight he experienced what he long afterward considered an epiphany. As the plate spun, it wobbled. Because of the insignia he could see that the spin and wobble were not quite in synchrony. Yet just in that instant it seemed to him — or was it his physicist’s intuition? — that the two rotations were related. […] he tried to work the problem out on paper. It was surprisingly complicated, but he used a Lagrangian, least-action approach and found a two-to-one ratio in the relationship of wobble and spin. […]” […]”