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:
http://salfordacoustics.co.uk/sound-waves/standing-waves
Here is a
presentation on "Standing Waves and Natural Frequencies" from
Sam Hokin:
http://www.bsharp.org/physics/guitar
This video illustrates the
modes of vibration of a string:
https://www.youtube.com/watch?v=cnH2ltfW48U
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:
Chrome
Music Lab app
https://musiclab.chromeexperiments.com/Spectrogram/
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)
on to next page back index I index II home