8. Fundamental frequency and harmonics

     We know that what we hear as a single sound or pitch when someone is speaking (for example, 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]) plus a whole series of harmonics 泛音 or overtones 倍音 (these two terms do not mean exactly the same thing, but we will use them interchangeably for now).

      The harmonics are multiples of the fundamental frequency. So if the fundamental frequency is 100 Hz, the 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), Bb''' (1792 Hz), and so on. (Actually, your piano is tuned somewhat differently, because it uses 'equal temperament'. [Listen to the difference between 'pure' and equal temperament at this site.] 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. It's explained clearly in this page by the University of Manitoba:

http://www.umanitoba.ca/faculties/arts/linguistics/russell/138/sec4/source.htm

     Sometimes it helps to read the same material presented by someone else from a slightly different perspective. This page explains the same thing again from a guitarist's rather than a phonetician's point of view. You don't have to read the whole thing in detail; just pick out the parts that help you understand the nature of harmonics better:

http://www.harmony-central.com/Guitar/harmonics.html

      At this site, entitled "Sines and Wonders" [for non-native English speakers: this is a play on words; "signs and wonders'' is a phrase from the Bible], you can see animated demonstrations of the different modes of vibration of a string, along with some technical but not too difficult explanations:

http://test.cty.jhu.edu/imagine/sines.htm

     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.

     To see the harmonics of your own voice, make a short recording on WASP, 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 in real time and in color, then try downloading the Frequency Analyzer you can get at the link below. It's freeware, from a company called Reliable Software. Go to this URL and click on Frequency Analyzer to download:

http://www.relisoft.com/freeware/index.html

     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. Whistling does not produce much in the way of overtones, so you should see just a single line this time.

      WASP can do most of the same things that the Frequency Analyzer does and more, but the Frequency Analyzer is kind of fun and is a cool gadget 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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