Vibrations

6 Spectral content

Vibrating systems and overtones

Vibrating systems- strings, tuning forks, metal bars, drum heads and so- come in many shapes and types. Not surprisingly, how a system vibrates affects how it sounds. What do the vibrations of different string instruments have in common? How are vibrations from a string different than those of a tuning fork or a chime?

Almost all musical instruments make sounds that are made of many frequencies happening simultaneously. Each musical note you hear is made up of a fundamental frequency and overtones. How overtones are related to the fundamental frequency is what makes a string different than a chime.

Overtones and musical instruments

A single musical note usually contains many frequencies at once- the fundamental frequency and overtones. Many systems (like the human voice, most string instruments and most wind instruments) make overtones that are whole number multiples of the fundamental. For instance, when a cello plays a note with a fundamental frequency of 110 Hz, the note contains overtones at 220 Hz, 330 Hz, 440 Hz, 550 Hz and so on. Overtones that are whole number multiples of the fundamental frequency are called harmonics. The word derives from the Greek word meaning “agreement.” Western music theory is built on the math of whole number ratios. The figure below shows how harmonics relate to (Western) musical intervals. (If you’re not a musician, skip this figure).

Musical staff shows first five harmonics of C2, with musical intervals and frequencies
First five harmonics of the note C2 and their frequencies.

Other instruments, like bells, chimes and tuned drums, make overtones that are not harmonics. For these instruments, overtones are also multiples of the fundamental, but the multiples are irrational numbers[1]. For example, a chime’s tone includes overtones at 2.76 times the fundamental, 5.40 times the fundamental, 8.93 times the fundamental (as well as other higher frequencies). Since bells and chimes have overtones that are not harmonics, they can sound “out of tune” to Western ears. Such “oddball” overtones are usually just called overtones, though some books call them anharmonic overtones.

A side note about terminology: Harmonics are a special type of overtone, just like squares are a special type of rectangle. Saying “anharmonic overtone” is a bit like saying “rectangle with unequal sides.”

Which overtones a musical instrument produces are a result of how certain parts of the instrument vibrate. String and (most) wind instruments produce harmonics because of the way strings and columns of air vibrate. (How strings and air columns vibrate is covered in detail later in this book). Metal bars, bells and most tunable drums vibrate in ways that cause overtones that aren’t harmonics.  The math that explains how these systems vibrate is too complicated for this book.

Predicting overtones

Every vibrating system has its own “overtone signature.” For instance, string instruments (and many wind instruments) produce harmonics. Expressed mathematically, the frequencies produced by these instruments are (f, 2f, 3f, 4f…) where the letter f represents the fundamental frequency. Frequency lists expressed this way are sometimes called spectral content. Some winds (like slide whistles and organ pipes that are closed at one end) only produce a spectral content of (f, 3f, 5f, 7f…). In other words, the only overtones slide whistles and open-closed organ pipes produce are odd harmonics. Instruments with anharmonic overtones also have spectral content, even though the multiples are not integers. For example, the spectral content of chimes can be expressed as (f, 2.76f, 5.40f…). 

Stop to think 1

Which of these vibrations might have been made by a string instrument? a slide whistle? a chime?

  • Vibration A has a fundamental of 100 Hz and overtones at 300 Hz, 500 Hz, 700 Hz and so on
  • Vibration B has a fundamental of 200 Hz and overtones at 400 Hz, 600 Hz, 800 Hz and so on
  • Vibration C has a fundamental of 300 Hz and overtones at 600 Hz, 900 Hz, 1200 Hz and so on

You can calculate a a musical sound’s overtones if you know the fundamental frequency and what instrument produced the sound. For instance, what overtones does a 200 Hz note on a slide whistle have? Since the spectral content of slide whistles is (f, 3f, 5f, 7f…), the 200 Hz note has overtones of 600 Hz, 1000 Hz, 1400 Hz and so on. Likewise, if you know what overtones one note on an instrument has, you can calculate the overtones of any other note on that same instrument, even if you don’t know the instrument is. That’s because, if you know the overtones and the fundamental frequency of one note, you can figure out the instrument’s spectral content.

Stop to think 2

One vibrating metal bar on a glockenspiel has a fundamental of 200 Hz and overtones at 520 Hz and 1080 Hz. Does the glockenspiel have the same spectral content as a chime?

Stop to think 3

One vibrating metal bar on a glockenspiel has a fundamental of 200 Hz and overtones at 520 Hz and 1080 Hz. What overtones would the 400 Hz metal bar on the glockenspiel have?

Reading spectral content from graphs

Frequency domain graphs are best for determining spectral content- FFTs show which frequencies are present in a vibration (as well as how much of each frequency is present).  To determine the spectral content, read off the list of frequencies that are present and divide each frequency by the fundamental. It is customary to ignore the heights of the FFT peaks when comparing spectral content of real-world sound vibrations. Which frequencies are present is usually more important than how much of each overtone is present.

The FFT graph shows two different notes with the same spectral content, but different fundamental frequencies. Notice that none of the frequencies on the upper graph match the lower graph. In this example, the fundamental in the lower graph is double that of the upper graph. Notice that every frequency in the vibration doubles- not just the fundamental. In order for the spectral content to remain the same, every frequency in a vibration must increase by the same percentage (or ratio). The FFT has the same structure as before, but the peaks are more spread out.

FFTs of two vibrations with the same spectral content, but different fundamental frequency

Spectral content is nearly impossible to determine from time domain graphs. That’s because time domain graphs do not show information about overtones directly. To make things even more difficult, two vibrations with similar spectral content can have very different looking time domain graphs.

In the (very rare) event that two time domain graphs have the same repeating pattern but different frequencies, you can conclude that the two sounds have identical spectral content. (About the only context you will see this is with electronic synthesizers).

Time domain graph of two vibrations with identical spectral content
Time domain graph of two vibrations with identical spectral content

 

 

Stop to think answers

  1. Vibrations B and C could have made by a string instrument (but not by a chime or a slide whistle), because the spectral content is (f, 2f, 3f, 4f…); Vibration A could be made by a slide whistle (but not a chime and probably not by a string). The spectral content contains only odd harmonics: (f, 3f, 5f, 7f…).
  2. Divide each of the overtone frequencies by the fundamental to get the spectral content: (520 Hz)/(200 Hz) = 2.76 and (1080 Hz)/(200 Hz) = 5.4. This is the same as for chimes.
  3. You can find the spectral content from the first note (see Stop to Think 2) and apply that to the 400 Hz note. An easier way is to recognize that the overtones of the 400 Hz bar will be double the overtones of the 200 Hz bar: 520 Hz * 2= 1140 Hz and 1080 Hz * 2 = 2160 Hz.

Image credits

  1. First five harmonics of C2 and their frequencies. Created by Abbott.
  2. FFTs of two vibrations with the same spectral content, but different fundamental frequency. Created by Abbott using desmos.com.
  3. Time domain graph of two vibrations with identical spectral content. Created by Abbott using desmos.com.

  1. Irrational numbers are numbers (like pi and Napier's e) that cannot be expressed as simple fractions.

License

Icon for the Creative Commons Attribution 4.0 International License

Understanding Sound by abbottds is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

Share This Book