Physics of Sound: The Spectrogram (or what the heck am I looking at part 2)

I probably should give a little lesson on how to read a spectrogram, since my next post will feature spectrograms rather heavily.  I made all these spectrograms on PRAAT, which is free and downloadable if you wish to play with it.  (I know the website looks a little sketch, but I had to get it for my classes using the site I linked to and it's totally safe for your computer.)  PRAAT is a lovely piece of software that will record a sound and then give you both a spectrogram and a waveform of that sound.  As you look at the images below, the waveform is the image on the top with the thick black band and blue vertical lines, and the spectrogram is the grey-scale mess below that waveform.  So, on to the important part of this post!

How to read a spectrogram:

The x-axis (horizontal) is time, the y-axis (vertical) is frequency, and the grey-scale shows amplitude.  So a spectrogram can show three dimensions, time, frequency, and amplitude vs. a waveform that shows only two, time and frequency.  On fancier programs, the amplitude is sometimes shown in color, like having blue be the softest sounds and red being the loudest, but in PRAAT, the darker the band, the higher the amplitude.  In terms of the frequencies, I set the spectrograms to show from 0 Hz to 7000 Hz.  PRAAT can display up to 20,000 Hz, but then the formant bands I want to focus on get too squished together.  If you click and make the image bigger, you can see a dotted red line with a frequency number off to the left.  I set those lines there just to give you some idea of where the upper formant lies in terms of Hz.  And remember from the last post that the formant will be somewhere around this frequency, not right at the single frequency itself.

So this is what a typical spectrogram will look like with the upper frequency set at 7000 Hz.  (I think PRAAT's default setting is usually 5000 Hz.):

The spectrogram above is me sustaining the vowel /a/ with my speaking voice.  You can clearly see five dark bands going horizontally across the image, but the bottom two dark bands are the darkest, indicating that those are the highest amplitude formants.

Sustained-speech of an /a/ vowel with formants marked.

This is the same spectrogram as the one above it, but I've set PRAAT to show me the first five formants, which it does by adding in those red lines.  The software is simply determining where the highest amplitudes are and sticking bands in there.  I'm not controlling where those thick red lines go.

Sustained, spoken /i/ vowel, no formants marked in.

Here's me sustaining an /i/ vowel with my speaking voice.  Note the wide distance between the first and second formants, which is just what the /i/ vowel does.  Oh /i/, you so crazy!

Sustained, spoken /i/ vowel, first five formants marked in red.

Above is the same spectrogram again, but with PRAAT marking the first five formants in red.

Spoken phrase:  "One, two, three, go," no formants marked.

 And there's a spectrogram of me speaking the phrase, "one, two, three, go.'  Here, you can see the movement of the formants as I go through those words and the "white space" between the words.  (Those areas where there's a thick blue vertical band on the waveform is where the /t/ and the "th" sound of "two" and "three" are.  And, you can see the antiformants present in the /n/ sound right at the end of the first word "one."  Pretty cool, huh?)  (Scroll to the bottom of page 2 on that antiformant link to read more about them.) And here's the same phrase with the formants marked in:

"One, two, three, go," with formants marked in red.

Now, some super cool people can actually read spectrograms like they're reading words off the page.  I'm not quite that awesome yet, but if you tell me what the phrase is, I can pick out where each specific word is using my knowledge of vowel formants and consonant frequencies.  It'd be cool to become that person who can just read them, though!

Now the reason I kept setting the spectrogram to 7000 Hz instead of 5000 is two-fold:  First, I wanted to make sure the upper formant wasn't cut off since that formant does occasionally go higher than 5000 Hz, and second, I wanted you to see that there actually is a thick band of amplitude above the 5000 Hz mark, which you can see in the spectrogram above.  So there are more "formants" above that 5000 Hz mark...we just don't really regard frequencies higher than 5000 when discussing speech or singing very much.  (Although, this article does!)  Heck, PRAAT doesn't even mark in any formants above the 5000 Hz area...usually the fifth formant area.  But, I wanted to make sure you know that it's not like formants and harmonics just disappear above 5000 Hz.  Mathematically speaking, harmonics would just keep on going higher and higher, and so would formants.  However, the amplitude lessens the higher you go, so vocal harmonics and formants do dampen out eventually...just not at 5000 Hz.

Up next:  The singer's formant!  I'mma gonna break apart a common misconception in the hopes that it clarifies what is we're actually doing when we carry over that orchestra.

Physics of Sound Series: Formants, formants, and more formants

According to Raphael et al., the source-filter theory of speech production states that the source of vocal sound, i.e. the vocal folds, is filtered through the air spaces in the vocal tract (p. 330).*  This is a fairly simplistic model of vocal production, but it is very useful just because of its simplicity.  Other models of speech production out there get a lot more detailed, but for a general, conceptual knowledge of the relationship between the vocal folds and vocal tract in terms of acoustic output, I think the the source-filter model can't really be beat.

So what does this have to do with formants?  Well, on the last physics post, I left off by stating that the vocal tract can change it's shape and configuration to filter out different harmonics from the same sound source.  The shape of the vocal tract will also amplify certain harmonic frequencies, while dampening others.  The resulting "peaks" in amplitude at specific frequency ranges are what we call formants.  One important thing to note here is that formants are not the same thing as harmonics.  You can think of formants as being a certain specific collection of harmonics, so the first formant is not the same as the first harmonic.  The idea of a harmonic is that it is one particular sine wave that is related, mathematically, to the fundamental, but the formants are collections of these sine waves.  The language you typically see is that the first formant is around a specific frequency.  So while you might read about the singer's formant being somewhere around 3000 Hz, the formant isn't actually only at 3000 Hz, it's just a collection of frequencies centered somewhere around 3000 Hz.  I think the semantics might get a little fuzzy there for a lot of people, but what seems like a little, unimportant detail actually makes a big difference when discussing harmonics vs. formants.  If you use those terms interchangeably, you'll just confuse the folks who know they're different things and then you'll get confused that they're confused and yadda yadda yadda...

Think of it like this:  Let's say you have a collection of all the Star Trek episodes from every Star Trek series, even the crappy ones.  If you consider the first series, the original Star Trek, as the fundamental, the first "harmonic" would then be Star Trek:  The Next Generation, the second would be Deep Space Nine, the third Voyager, etc.  However, it's possible that if these "harmonics" get filtered into formants, the first formant could consist of the first five seasons of The Next Generation, with the last two seasons filtered down to really low amplitude.  The second formant could be the last four seasons of Deep Space Nine, with the first three seasons of DS9 being filtered down.  The third formant could be the last five seasons of Voyager with the first two seasons filtered down, etc.  See the difference?  So harmonics are the building blocks of formants, but harmonics come from the resonance of the vocal folds themselves and formants come from the resonance of the acoustic filter or vocal tract.

What's great about formants is that they happen to be the way we distinguish vowels during speech.  In fact, the relationship between vocal tract shape and the acoustic output (vocal sound once it exits the mouth) is so interrelated, we are able to classify vowels by both the vocal tract shape and the acoustic output, depending on what we're talking about.  I.e.:  Talking about articulation?  You'll be talking about the shape of the vocal tract made by the articulators (tongue, soft palate, etc.).

If you happened to click over to that Wikipedia article on vowels, you probably noticed there's a section on articulation and a separate section on acoustics.  The position of the tongue in the mouth happens to make the biggest difference to the overall shape of the vocal tract, and so, a lot of vowels can be categorized by place of tongue articulation during production.  For example:  An /i/ ("ee") vowel is categorized as a high, front vowel because the tongue is positioned very high near the roof of the mouth, but it is also positioned quite forward in the mouth and is, therefore, a high-front vowel.  A high-back vowel, such as /u/, has the tongue positioned as a "hump" near the back of the mouth, so it's high, but in the back.  A low vowel, such as /a/, doesn't involve the tongue in a raised position at all, and is closer to a neutral vowel position, of which the schwa sound is considered the most neutral.  (I know a lot of singers consider /a/ as the most neutral vowel, but linguists and speech scientists have researched tongue positions, and schwa is indeed the most neutral.  I think the reason singers like the focus on /a/ so much more is that we don't tend to sing schwa very often, and if we do, we don't sustain a sound on schwa.  So schwa gets kinda a bad-rap in the singing world, but it is an important little vowel in spoken language.)  

Because a larger space will resonant at lower frequencies, and a smaller one at higher frequencies, the formants are a result of the size of the pharyngeal space and/or oral space as determined by the tongue position, primarily.  A good example of this is if you tap on a glass with some water in it, then tap again after drinking the water, the second tap will be a lower pitch than the first tap because there is more air inside the glass after the water is gone to resonant the sound.  Or a better example:  A cello is bigger than a violin.  So...there you go.  Therefore, in a simplified sense, these tongue positions all correspond to the formant frequencies of each vowel.  The /i/ vowel is known for having a low first formant (more pharyngeal space created by the high tongue position) and a high second formant (small oral space created by tongue position,) and in fact, this vowel has the widest space between the first and second formant as it's trademark sound.  The /u/ vowel has a low first formant (from the high tongue position creating more pharyngeal space), but also has a low second formant (from the tongue position being near the back of the mouth, creating more space in the oral cavity).  Once again, this is a very simplified way of looking at this, but it's an easy way to understand the basic idea.  Just be aware that the science of acoustics can get pretty darn complicated in this area.


*Raphel, L. J., Borden, G. J., Harris, K. S. (2007).  Speech science primer:  Physiology, acoustics, perception of speech (5th ed.).  Philadelphia, PA:  Lippincott Williams & Williams.

Physics of Sound Series: Harmonics and fixed strings and open-closed tubes, oh my!

So, we've got the vocal folds acting like a fixed string with multiple resonant frequencies called harmonics, but before we go further, I realized there was some terminology that I should go over.  Remember how the actual sound wave produced from vocal fold vibration is far more complicated than that of a single string?  That's mainly due to the motion of the lamina propria, but even single strings can produce complex sound waves.  This is a wave pattern that is created from the interaction of multiple frequencies known as harmonics.  If you have several, or even thousands, of simple sine waves that are harmonically related, you've got a complex periodic sound wave.  Periodic means it still has a predictable pattern, as opposed to complex aperiodic waves, like white noise.  But the linguistic signal contains both periodic and aperiodic waves.  Periodic would be vocal sounds, like vowels; aperiodic would be like unvoiced consonants like /s/, and a combination of periodic and aperiodic happens during a lot of voiced consonants, like /z/.  We're just going to focus on complex periodic waves for the purposes of understanding resonance more fully.  Complex periodic waves can be broken down into their simple sine wave components by using something called a Fourier transform.  We certainly won't go through the math for all of that, but just know that all complex periodic waves are made up of simple sine waves, and even though the vocal folds create a more complicated sound than a string does, we're going to continue with the string comparison cause I think it makes everything so much easier to visualize.


The thing about vibrating strings fixed at both ends is that they have a fundamental resonance of 2 times the length of the string.  This just means that one-half of a wave can "fit" on a string at any given pass along the string.  It also means that the string will vibrate at both even and odd harmonics of the fundamental.  This can be represented mathematically if we use our knowledge of frequency = velocity divided by wavelength.  In this basic case, we're going to consider velocity to be the speed of the wave being produced, and in the human voice, that speed is determined by the tension and mass of the vocal folds.  So when the folds elongate, tension increases and mass decreases resulting in a high frequency of vibration (cool, huh?).  And the wavelength, thanks to the half-wave resonator we're working with here, will be two times the length of the string (or folds).  Using this equation and setting our speed at 340 meters per second, the approximate speed of sound at sea level, we can figure out the harmonics of a 1 meter string.  The fundamental frequency would be 170 Hz, the first harmonic would be 340 Hz, the second harmonic would be 510 Hz, the third 680 Hz, the fourth at 850 Hz etc.  This sound wave (up to the fourth harmonic) would sound like this:

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File made with Audacity

Now that we've compared the vocal folds to strings, what do we have to compare the vocal tract to?  An open-closed tube!  ...which is not that exciting at all.  But what the vocal tract does is pretty darn exciting.  Of course, the vocal tract itself can change it's shape for communication and such, but the open-closed tube is a good simplification of what the basic function of the vocal tract is.  An open-closed tube is a quarter-wave resonator, as opposed to the half-wave resonator that the string up there is.  So what does that mean?  A quarter-wave resonator means that only a quarter of the wave can "fit" during one pass through the tube.  So this resonator only vibrates at odd harmonics of the fundamental frequency.  So, if we look at that 170 Hz frequency produced from that meter-long string up there, The open-closed tube resonating at this fundamental 170 Hz would have a length of 0.5 meters and the first harmonic would be at 510 Hz, the second at 850 Hz, etc.  Notice something there?  This tube is only resonating at even frequencies of the string up there.  So what happens to the sound wave produced by that string as it passes through this tube?  Well, it'll sound something like this:

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File made with Audacity

Where did those other harmonics go?  The tube ate them.  No really!  Well...it kinda-sorta did.  See, the tube acts as a filter for that sound wave.  Those missing frequencies, the ones that the tube won't resonate, are going to be filtered out due to destructive interference, while the frequencies the tube vibrates at are going to constructively interfere and exit the tube for us to hear.  Yup, that's right.  Without resonance, we wouldn't hear our own speech, much less a singer singing over an orchestra.  Your voice is always resonating all of the time; it's just that opera singing requires a difference resonance than your speaking voice...obviously.  We don't really sound like we're talking when we're singing, do we?  

Now, I don't know about you, but I personally find the second audio file a little more pleasing than the first.  The first one is objectively "richer," in the sense that it has more harmonics, but the second one subjectively sounds "richer" to me.  I'm not really sure why, but I suspect it has something to do with the fact that I am physiology wired to find the sound of the human voice important, as are you, and so perhaps I also find sounds from an open-closed tube more pleasing?  (And if you didn't find this to be true, you're really messed up!  Just kidding.)  And where this "pleasing" association would occur in the brain, I'm not sure.  But I know my brain is associating the second file with a richer sound that I happen to find more pleasing, because the first sound has more harmonics in it for sure...I would know; I inputted the frequencies myself!  But if you played those two tones for me without my knowing about the harmonic structure, I would assume the second one has more harmonics.  The brain sure is one crazy organ, amirite?  Of course, I digress, but this is an example of some of the stuff people are trying to figure out in terms of how we listen, pick out, and associate the speech signal into meaning in our lives all day long.  It's some cool stuff, for sure.  Perhaps I'll learn an answer to that soon and will update you guys.

Now, in a stationary tube, the harmonics are pretty fixed, but lucky for us, our vocal tract can change shape, length and configuration to produce a lot of different sounds.  By changing it's shape, the vocal tract filters the same sound source differently, producing all of the different sounds we make in our languages and then some.  Conveniently for us, it seems to do this pretty much on auto-pilot most of the time, like when we're speaking, or how the vocal tract lengthens when our voice drops in pitch (the larger cavity will resonant at lower frequencies and shorter at higher).  The shape the tract takes determines which frequencies are amplified and which ones are dampened out.  And this sets us up quite nicely to talk about formants next time, doesn't it?

Resources: 

http://www.physicsclassroom.com/

Raphel, L. J., Borden, G. J., Harris, K. S. (2007).  Speech science primer:  Physiology, acoustics, perception of speech (5th ed.).  Philadelphia, PA:  Lippincott Williams & Williams.

The Physics of Sound: Resonance and Standing Waves

So what does happen when two sound waves are in phase with one another?  The two waves constructively interfere with one another to result in one wave that this double the amplitude of the two waves.  Basically, they both add up like some awesome crime-fighting team...and they...help people hear and stuff.  (Yeah...I don't know where I was going with that metaphor.)  Anyway, to better understand this, let's talk a little more about the phenomenon of interference.

Interference is when two waves sorta "line up" together.  Depending on how they "line up," the two waves combine to form one wave that is either of lesser or greater amplitude than the two waves were just on their own.  Think of it like this:  If one wave is going along with an amplitude of, let's say, 2 dB, and it meets up with another wave that's out of phase with this first wave, and the second wave's amplitude is 1.5 dB, then the resultant effect will be the 1.5 dB wave "canceling out" some of the amplitude of the first wave.  So you'd get a net result of a 0.5 dB sound wave.  If, however, the 2 dB sound wave meets up with a wave that's totally in phase with it, and this wave is going along at 2 dB, the resultant wave will be 4 dB.  So, yes, interference is very much like when you hang out with that soul-sucking person you really shouldn't be around (destructive) or that person who just makes you feel great (constructive).  (That's a super-basic way to represent interference mathematically, and the real math is much, much more detailed and complex, but it's just there to give you an idea.  So please don't go around thinking it's just addition and subtraction when scientists are figuring out interference.  It'd be a bit like those people who think a graduate degree in vocal performance just means you sing karaoke all day and get a degree for it.)

Sound waves travel along just fine until they hit a boundary.  When that happens, the waves bounce off the boundary and become reflected waves.  The initial wave, called the incident wave, can meet up with the reflected wave where the two waves interfere with one another to form a new wave that is the sum of the other two waves.  This is called the principle of superposition.  (I know I'm getting a bit redundant, but hang with me here.)  If, during superposition, two waves meet up that are completely in phase, the result is a standing wave.

As you can see above, one type of standing wave doesn't travel anywhere.  It stays in the same place constantly.  This results in areas where the displacement is zero, called nodes (shown by the red dots above), and areas of maximum displacement called antinodes (the tall peaks and valleys above).  So the constructive interference of an initial wave meeting up with a reflective wave to form a standing wave looks something like this:

The red and blue waves meet up to form the standing wave in black.  Other cool animations can be found here and here. 

But how do standing waves that don't go anywhere contribute to a singer's resonance?  Well, that question is kinda jumping a bit farther ahead than where we are now.  For now, just think about standing waves on a medium that is fixed on both ends, like a string.  Ever played with a string or  necklace where you bounce the string up and down?  If you have, you actually formed a standing wave at the string's first resonant frequency, called the fundamental frequency.  But the string does have other frequencies it could resonant at, called overtones.

First fundamental and first six overtones of a string

But why am I talking about strings?  What do strings have to do with vocal resonance?  Think about it a second:  What acts like vibrating strings with fixed ends when we speak or sing?  Yup, the vocal folds.  But, the vocal folds vibrate in patterns that are much more complex than just a single string.  Remember how the air opens them from the bottom to the top, due to subglottal air pressure, and then the folds get sucked in laterally because of the Bernoulli effect?  The resulting wave pattern is very intricate,which results in a complex waveform (multiple simple sine waves going out at once) being produced at the level of the vocal folds.  The fundamental frequency and all of the overtones of the human voice originate at the level of the vocal folds.

*That last bit is very, very important, and it seems to be where a lot of singers get very confused...usually not due to any fault of their own.  The vocal tract absolutely cannot create sound waves or overtones to those sound waves:  Not the singer's formant, not the harmonics, not any of it.  All of the frequencies picked up by a spectrograph originate from vocal fold vibration.  The vocal tract only acts as a filter for the frequencies sent out by the vibrational pattern of the vocal folds.  And that's where we'll pick up next time!

Raphel, L. J., Borden, G. J., Harris, K. S. (2007).  Speech science primer:  Physiology, acoustics, perception of speech (5th ed.).  Philadelphia, PA:  Lippincott Williams & Williams.

Physics of Sound Series: The Waveform (or, What the Heck am I Looking At?)

We now know that a sound wave is made up of moments of compression and rarefaction, and we know a little bit about a waveform as well.  But there are other parts to a wave that we need to know about before we get into just what resonance really is.  Those parts are:  Period, frequency, amplitude, phase, and wavelength.

No doubt, you've heard of some of these before.  Frequency and amplitude, in particular, get a lot of attention in the music world.  A lot of times, we talk about frequency and amplitude as synonymous with pitch and loudness, and for most purposes they are.  However, when I talk about frequency and amplitude, I'm going to be referring to the actual physical properties of the sound wave (or waves), meaning the parts of the wave that can be measured with proper instrumentation and then studied.  Therefore, frequency and amplitude are objective measurements.  Pitch and loudness are usually associated with the perceptual properties of the wave, i.e. just how loud or how high/low a person perceives that sound differs from person to person (and from ear to ear, for that matter); so pitch and loudness cannot be measured per se, but rather discussed subjectively.  (And remember from my previous post:  Anytime I'm talking about perception, I'm talking about the interaction from a sensory signal with a person's higher cognitive functions and life experiences.  Therefore, perception is always subjective.)

Yup, those are the same sine and cosine functions from trigonometry that you see on your calculator.

When you look at the waveform of a basic sine wave (the red line) shown above, you'll see that it has a repeating pattern.  The number of repeats of this pattern in a given amount of time is called the frequency of the wave.  This is usually given in Hertz, but can also be stated as the number of cycles per second of the wave pattern.  (See, it was originally called cycles per second (cps), but then the International Electrotechnical Commission (IEC) decided to honor Heinrich Hertz's contribution to the field of electromagnetism, so they gave him a unit of measurement, cps, and called it Hertz (Hz).  Scientists are always re-naming units to honor the great contributors to the field.  Sorta like how medical terminology is also littered with the names of big anatomy contributors, etc.)  So, for the famous A440 that orchestra's (supposedly) tune to, the frequency is 440 cycles per second, or 440 Hz.  In math terms, frequency is shown by:  frequency = velocity over wavelength (f = v/λ).  Sounds pretty fancy, but the reason I'm putting this here is because the period of a wave is related to the frequency.  The period is how long it takes for one cycle of the wave pattern to complete itself.  So if the frequency could be shown as: 1/period, then the period is shown by:  1/frequency.  Seems like we're talking about the same thing, but in general, the frequency refers to how often the wave pattern is repeating itself where the period refers to how long it takes for one pattern of the wave to complete itself.  Why do we bother with this distinction?  Well, because it comes in really handy for mathematical analysis of wave patterns.  Why should singers bother to know about this?  Because...well...I'll get there for ya.  (Besides, if your ever playing around with PRAAT or some other spectrograph software, you'll probably see options for period or frequency change, and you might want to know what you're changing out as you play around.)

Wavelength corresponds to the distance one cycle of the wave travels.  Slower frequencies have longer wavelengths, so one cycle of A440 travels double the distance through the atmosphere than A880.  (In the above equations, wavelength is represented by that funny-looking symbol, which turns out to be the Greek letter lambda.  So now, when you poke around wikipedia and see frequency equations, you'll know some of what you're looking at.)

The amplitude of a wave corresponds to it's perceptual loudness, and is related to the amount of displacement the air particles go through in the sound wave.  Because it has to do with how far each particle is being "pushed," amplitude represents the atmospheric pressure of a sound wave, and is measured in decibels (dB).  In the waveform shown above, amplitude is represented on the vertical axis.  So if the sine wave had a higher amplitude, it would have taller peeks and lower valleys, going past the 1.00 marked above.  Even though amplitude of sound is represented by this vertical displacement on the waveform, in longitudinal waves, the displacement in the real world is happening horizontally.  This is different for other wave types, like light, but the math and the graphical representations are the same.  (I just want to point that out because it's easy to misinterpret the sound waves from your mouth as looking just like the waveform representation, but if we could see the sound wave, it would like more like the animation here.  That's kinda important to remember once we get to the anatomy of the ear and the role the ear drum plays in hearing.  And, oh yeah!  I'm going to get into the anatomy of the ear and how it plays a role in resonance as well, for both the audience and the singer!)

Phase is where we start to get into some important stuff when it comes to understanding resonance, and especially the phenomenon of standing waves.  The phase of the sine wave in the above picture basically is where in the cycle the wave starts when you're looking at the vertical axis.  Let's look at it more closely:

See how the sine wave is passing through the vertical axis where the horizontal line equals 0?  Now look at the cosine wave (blue, dotted line).  Cosine is passing through that vertical axis where a horizontal line equals 1.  So the phase of the sine wave is not the same as the phase of the cosine wave.  The fancy way of saying that is that cosine has a different phase shift than sine.  In fact, that's actually the main difference between sine and cosine:  The phase shift between the two.

Phase is really, really important because if you have two sound waves that are out-of-shift like this:

Look at the three middle waves to see the phase difference.

You'll see how when one wave has a peek in its amplitude, the other wave has a valley, or a negative amplitude.  This means that when one wave is in it's period of compression, the other is in rarefaction.  The result is that these two waves actually cancel each other out, because if the atmospheric pressure is equally positive in one wave while the pressure is equally negative from the other wave, the two pressure differences cancel each other out.  1-1=0, right?  Crazy, huh?

But what happens when two sound waves are perfectly in phase?  What if you've got 1 + 1 instead of 1 -1?  That's where the phenomenon of standing waves comes in, and that's what we'll start up with next time.  Stay tuned!

Raphel, L. J., Borden, G. J., Harris, K. S. (2007).  Speech science primer:  Physiology, acoustics, perception of speech (5th ed.).  Philadelphia, PA:  Lippincott Williams & Williams.

Physics of Sound Series: The Acoustic Wave

I always have the hardest time starting up these series, because I spend a lot of time trying to figure out where to start.  I always know what the ending conclusion should be, but what's the beginning?  What's basic without being too basic?  So I'm going to start out where I think it should start out, but if I'm not being basic enough, please feel free to post any questions you may have.

Everybody always talks about resonance in the singing world.  Resonance, resonance, resonance.  Let's face it, as opera singers, we're pretty obsessed about it.  And why wouldn't we be?  It is, after all, the key to how opera singing works.  It is exactly how we are able to sing over an orchestra for hours at a time without hurting our voices.  The only issue I have with all this resonance talk is that it is painfully obvious that (some) singers have absolutely no clue what resonance really is.  It often gets talked about as a subjective thing that changes from person to person.  This is understandable given that so much of the sensation of singing is subjective, and therefore, how we teach singing is subjective.  It only makes sense that singers would start to think everything about singing is subjective somehow.  However, when we take something from the hard sciences, like resonance, and think of it as something that acts differently from person to person, as if it doesn't follow the laws of nature, we kinda sound like fools.  The other issue with all this resonance-as-subjective talk is that it makes what could be very clear pedagogy very fuzzy and confusing.  So, in order to fully understand what resonance is and how it can help us sing better, let's start with how a single sound wave works.

There are a lot of things in nature that function like waves:  Light, sound, the water in your bathtub...(okay, fine, ocean water too), but what exactly does that mean for sound to have a wave-like pattern of behavior?  Well, here's the definition of a wave from physics:  "a disturbance (an oscillation) that travels through space and time, accompanied by a transfer of energy...often with no permanent displacement of the particles of the medium (Wikipedia)."  Sounds pretty fancy, am I right?  But it does make a lot of sense.  If you drop a rock straight down into a body of still water, the rock disturbs the water's stillness causing a rippling of waves that travel out to the edges of that body of water.  Energy was transferred from the rock to the water which then traveled out to the edges of the body of water.  The water itself, though, will return to being still, i.e. it doesn't just keep traveling away from the rock until there's no water left, so there wasn't a permanent displacement of the particles of that water (you know, H₂O).

So what's the "medium" for sound waves?  Air particles!  All the lovely little air particles that make up our atmosphere is the medium for all the sound waves we hear, and the ones we don't hear too (i.e. ultrasound, infrasound, etc).  For our purposes, we'll think of a sound wave as beginning with air particles at rest.  An external force then comes along and sets those particles in motion (like when the electric slide is played at a wedding...sorry, couldn't resist.)  Anyways, let's imagine those particles are all lined up nicely next to one another.  The particles in row A, the ones closest to the external force, then get "pushed" up towards the particles in row B.  This is where we say the row A particles are "compressed" against row B, which then gets pushed up against row C, and so on.  (Anyone who's ever seen elementary-school kids line up for recess knows what I'm talking about here.)  So while each row is going into it's period of compression with the particles in front of it, the rows that have already been compressed then go into a period of rarefaction.  This would be when row A, after compressing with row B, swings back towards it's resting position.  But instead of landing at rest, row A actually over-shoots its resting position and ends up being spaced out farther from the row B particles.  If we want to get even more specific here, the property of inertia for those particles causes row A to compress with row B, then the property of elasticity over-takes row A's inertia, sending the particles back towards resting.  However, the property of inertia for that row of particles then over-takes elasticity and causes row A to over-shoot it's resting position.  But, don't fear, cause elasticity will over-take inertia and send row A back to towards resting.  This process will repeat itself until row A is again completely at rest.  Sounds complicated, but if you've ever set a pendulum into motion and watched until it came to rest again, you've seen this same action at work.  (*Edit to add:  This pattern of motion is called simple harmonic motion and is actually what pretty much everything in nature can be reduced to.)

This pattern of compression and rarefaction makes up what we call the sound wave.  This is why sound waves are sometimes called compression waves, but more commonly, they are called longitudinal waves.  (If you clicked on the link I had above on "compressed," you probably saw that coming.)  I encourage you to go ahead over to the link for longitudinal waves, because there are some very nice animations over there for you to see these waves in action.

One last thing before I sign off:  This pattern of compression and rarefaction is often graphically represented as a waveform.  Typically, waveforms are set on a typical Cartesian coordinate system (the graphs with x and y from math class), with the y, or vertical, axis representing the amount of displacement, which also happens to be the amplitude of the sound wave, and the horizontal axis representing the amount of time the wave has traveled.  We'll go over this all a bit more later, but I wanted to introduce it here for you just to get you more familiar with the terminology I'll be using.

Raphel, L. J., Borden, G. J., Harris, K. S. (2007).  Speech science primer:  Physiology, acoustics, perception of speech (5th ed.).  Philadelphia, PA:  Lippincott Williams & Williams.

Physics of Sound Series (Part I): Why do I need to know this stuff?

Because the necessity to resonant over the sound of an orchestra is dependent on vocal tract, and therefore resonance, adjustments.  However, many singers either do not understand resonance, formants, or harmonics well enough, or don't understand how physics relates to physiology well enough, that many misconceptions develop that can greatly hinder vocal progress during training.

Now, I don't mean to generalize, but I do know a lot of singers who roll their eyes at words like "physics" and "math."  In fact, I have had so, so many conversations with musicians about these topics now that I'm taking math and physics courses.  They usually go something like this:  "I can't meet then because my physics class is at that time.  Can you do Monday?"  "Physics?  Why on Earth are you taking Physics?"  "Well, I want to have more detailed knowledge of how the physics of sound and air pressures work so I can understand certain areas of SLP research better."  "Well, good for you.  I know I would never take those classes.  My brain just doesn't work that way."  It is that last sentence that I take the most issue with.  Why, oh why do we as musicians have to demean ourselves when it comes to the potential our brains have to understand something?  Do we even realize the message we're sending out?  I mean, we are the people who learn multiple languages for our roles, we learn some anatomy and physiology of the voice, and we are supposed to have at least some foundational knowledge in harmonics and formants when it comes to resonance.  This is all in addition to music theory, history, performance practice, etc.  Why do we pass off math and physics like it's "over our heads?"  Or maybe we want the world to recognize that we're plenty smart in our own right and should be respected for that (which is true).  Maybe we think that in order for our field to be respected as art we have to separate so thoroughly from science that we must turn our noses up at it.  Maybe we're sick of people in the sciences turning their noses up at us...(I know I'm sick of that).  Maybe we don't want to have to add more stuff to our already extensive list of stuff to know.  Either way, I do wish my fellow musicians would stop looking at me like I've grown a second head when I say I'm enjoying learning calculus and calculus-based physics.  But I digress...

Perhaps most of the issues with math and physics for singers, or just most people in general, comes from the fact that these subjects are very rarely taught well in high school (in the US,) and even in college, for that matter.  Much of the time, teachers in these subjects see the class as some sort of grand IQ test in which student's successes or failures have no bearing on the teacher's ability, or inability, to effectively teach the material.  That's a common fallacy of certain hard-science classes.  (Personally, I liked my calculus's professors take on it:  Success in her class, as far as she was concerned, was totally up to the student's dedication and motivation to keep up with the homework (practice) and get help when needed.)  So we've relegated the teaching of these concepts to a month or so during a vocal pedagogy class.  But maybe, just maybe, voice teachers trying to teach these concepts don't quite give the right amount of time or clarity to these concepts either.  I mean, if you're knowledge doesn't have a strong foundation, it is really easy to get confused when, a few years after your pedagogy class, you've been swamped with new information, new ideas, new research, new teachers, new coaches, etc.  I know I did!

I thought I got plenty of this stuff in my vocal ped. courses.  I thought I had a very good understanding of harmonics, resonance, formants, etc. because I was one of the few in my pedagogy class that was not confused by the lectures or book chapters on it.  I now know I was mistaken.  My mistake came from not having enough of a base-level of understanding in physics to be able to apply these concepts effectively to understanding my own vocal training, and to not get confused a year or two down the road.  There is a huge interaction between the physiology of the voice and how the physics of vocal resonance, as well as the physics of air pressure to breath support, work.  Those connections were simply missing from my pedagogy classes, and, from what I can gather from other conversations with singers, I think it's missing from many singers' academic training as well.

I had such simple misconceptions that I would be embarrassed to admit to in front of anyone with basic physics knowledge, now that I know better.  I see a lot of musicians saying some of these same misconceptions quite frequently, and I really, really want us to stop sounding like complete fools in regards to basic math and basic physics to a large portion of the general population (and not just those in hard sciences, either).  And I know a lot of singers who would really, really like to not sound like fools, but it's just never been explained well enough, or thoroughly enough, to avoid it.  Even if your interest in this might just be cursory, a more thorough understanding of the physics-physiology connection really does help to understand the science behind how the voice, and operatic singing, works and how to apply that knowledge to long-term training.  


So here's how this series is going to work:  I'm not going to get all up in calculus, cause I'm not interested in making this a math course, but I will present some basic algebraic equations.  I will also thoroughly explain these equations so that you can see how the equation is a working representation of how your vocal tract shapes sound.  We'll start with the basics and move up from there, but I'm also going to do my best to detail the interaction between physics and physiology...even if I can't get to that interaction until I get a little further down the series.  If you've ever been confused looking at a spectrogram of your singing, like in PRAAT, then this series should help you out a lot.  It shouldn't be as long as the anatomy and physiology series, so I hope you can hang in here with me.  And ultimately, just like the A&P series, I want this to be a reference tool for singers and teachers to be used whenever you need it.