8. Fundamental frequency and harmonics

     We know that what we hear as a single sound or pitch when someone is speaking (for example, when making the sound [i]) is really a fundamental frequency 基頻 (determined by how many times the vocal folds vibrate in one second, and measured in cycles per second [cps], or hertz 赫 [Hz]; named after the German physicist Heinrich Rudolf Hertz), plus a whole series of harmonics 諧音 (often called 泛音 on stringed instruments) or overtones 倍音. These terms overlap in meaning, but: "harmonic" includes the fundamental frequency and all of the overtones above it, while "overtones" include all frequencies greater than the fundamental frequency.

     The harmonics are multiples of the fundamental frequency. So if the fundamental frequency is 100 Hz, the higher harmonics will be 200 Hz, 300 Hz, 400 Hz, 500 Hz, and so on. If the fundamental frequency were 220 Hz, the harmonics would be 440 Hz, 660 Hz, 880 Hz, and so on. In terms of intervals on the scale, we hear a base tone, its octave (eight notes up), then a note that is a twelfth up, i.e. a perfect fifth above the octave above the starting pitch, then a note two octaves up from the starting pitch, then one that is a major third above that, and on and on. If the starting pitch is middle C (C'; 256 Hz), the overtones are C" (512 Hz), G" (768 Hz): C''' (1024 Hz), E''' (1280 Hz), G''' (1536 Hz), B♭''' (1792 Hz), and so on. (Actually, your piano is tuned somewhat differently, because it uses "equal temperament". (The difference between "just intonation" and equal temperament are demonstrated in this video. But that's another story!)

    We normally don't hear the harmonics as separate tones, first of all because they have an increasingly lower amplitude than the fundamental frequency the higher up they go. The harmonics are nevertheless present in the sound, and they add a lot of richness to the sound of a human voice, a musical instrument, and many other kinds of sounds. Without them a voice would sound thin and uninteresting.

     But where do the harmonics come from, or more precisely, how are they produced? If you play the guitar, you are probably familiar with harmonics and how to produce them, even if you don't fully understand how they work. A guitar string works something like the vocal folds when it vibrates, and is a little easier to illustrate and visualize. So we will first look at how a guitar string vibrates in order to understand by analogy how the vocal folds do. Look at the animations at the bottom of this page, from the University of Salford in Manchester, UK, under Standing waves:


     Here is a presentation on "Standing Waves and Natural Frequencies" from Sam Hokin:


      This video illustrates the modes of vibration of a string:


     Think of how an eel moves in water. Imagine each of the little ripples that glides across its body as smaller movements of the vocal folds while they are vibrating. Each of these little 'peaks' in the wave is hitting the air at its own (faster) rate and producing its own little sound at the same time as the whole eel-like flaps of the vocal folds are producing a lower sound from the 'biggest' wave that rolls over the flaps from end to end.

     We will use the lowest string of a guitar (which is tuned to E, 82 Hz; the whole string is 66 cm long) in class to demonstrate the overtones. (This page can help you calculate the values of the overtones in Hz; this page will show the vibration modes of a string for the first seven harmonics.)

1. E (full length; 66 cm)

2. E' (1/2 of length; one octave up; 164 Hz; 33 cm; touching the string in the middle suppress the fundamental or first harmonic so you hear this pitch)

3. B' (1/3 of length; an octave and a fifth up; 246 Hz; 22 cm; touching the string here suppresses the second harmonic so you hear this pitch)

4. E'' (1/4 of length; two octaves up; 328 Hz; 16.5 cm; touching the string here suppresses the third harmonic so you hear this pitch)

5. G''' (1/5 of length; two octaves and a third up; 410 Hz; 13.2 cm; touching the string here suppresses the fourth harmonic so you hear this pitch; in fact a tempered G''' is 415.305 Hz, so the tuner will register this note as "sharp" if you touch the string right above the fret)

     To see the harmonics of your own voice, make a short recording on Praat, and choose the narrowband spectrogram display. You will see a series of evenly-spaced horizontal black lines (for now don't worry about the thick dark bands, or formants 共振峰; we'll talk about these later). These are the overtones of the fundamental frequency of the vibration of your vocal folds.

     If you'd like to see the overtones as presented by a spectrograph in real time and in color, then try this:

Music Lab app

     Click on the microphone icon and try saying a pure vowel, like [i] or [e], into your computer microphone while the frequency analyzer is running, and note the evenly spaced lines you see. Next try singing a scale (do, re, mi...), and watch the lines rise, still maintaining their equal spacing from each other. As a contrast, try whistling or click on the whistling icon. Whistling does not produce much in the way of overtones, so you should see just a single line this time.

    Praat can do most of the same things that the Frequency Analyzer does and much, much more, but the Chrome Music Lab spectrogram is kind of fun and is a cool page to show your roommates if they ever ask about what you do in phonetics class!

Next: Vowels and Formants: Resonance (with soda bottle demonstration)

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