Printer-friendly version

09 January 2004

Sound Frequency and Sound Pitch

John W. Dooley, Physics Department
Millersville University

RESULTS OF LOUDNESS SURVEY:

Before we consider pitch, we have some old business: In the December 12, 2003, issue of the E-Bulletin, readers were asked to listen to pairs of sounds and to identify the pair for which one sound is "twice as loud" as the other. One group of pairs used 500Hz tones, and a second group used white noise. The ratio of amplitudes ranged from 1.1 to 1 up to 10 to 1.

"Twice as loud" is not well defined, but 14 people were willing to give it a try. They were not unanimous on what ratio of amplitudes corresponds to a factor of two in loudness. The number of people who chose each ratio is shown in the charts below.

Results for 500 Hz tone Click images to enlarge	Results for white noise

Some people found the white noise hard to listen to, and one did not submit a noise answer. Most agreed that "twice as loud" corresponded to twice the amplitude. This is consistant with choosing "twice as loud" to be what you hear when you are "twice as close" - that is when you cut the distance between you and the sound source in half. Twice the amplitude corresponds to 4 times the power in the sound wave, and in the usual units for acoustic engineers, this represented as a 6 db change. Jim Hannon used these units in his letter.

FREQUENCY AND PITCH

The sound file called tonestep2.wav is a stereo file, and earphones are recommended, but don't listen to the file just yet. The left signal is a constant frequency of 220 Hz (the A below Middle C). The right signal changes frequency, starting at a frequency below 220Hz and rising to greater than 440 Hz.

If you use GoldWave you can watch the frequency using the graph of signal versus time, as in the first article in this series, or you can use the spectrum analyzer as follows: In the figure below, the GoldWave window button was used to choose "vertical control." The signal-versus-time graphs appear at the left, one above the other. The control panel appears at the right, with buttons at the top and two sub-windows, one above the other. In both cases, the upper graph refers to the left channel, and the lower graph refers to the right channel.

While in GoldWave, right-click in these windows to change what they display. We want "spectrum" for both sub-windows. This window shows a graph of sound intensity versus sound frequency. Having chosen spectrum, right click again, and choose properties, and make two changes: We want a fixed frequency range, from 0 to 2400 Hz, and we want to "show axis."

Now play the file tonestep2.wav, and watch the line move across the time graph to see what part of the file is being sent to the earphones. On the right, you can watch the frequency of the right channel jump up in steps as the file plays. The figure shows the tones being played at 23 seconds. The left channel has remained constant at 220 Hz, so its graph shows a peak at 220 Hz. The right channel shows a peak at 440 Hz. Thanks and best regards,

Finally it is time to listen, but listen to just the right channel first. You will hear increases in pitch that correspond to the jumps in frequency. Sometimes the pitch changes "a lot" and some times the change is so small that it is hard to discern.

Now listen to both channels and play the file. You will hear some strange things as your brain combines the constant tone from the left with the varying tone from the right. Sometimes the right tone is "sour," clashing with the left tone. This condition is called dissonance. At other times the right tone "sounds good." This condition is called consonance. At still other times. you hear a flutter or "beat," when the two channels are almost (but not quite) at the same frequency.

Watching the spectrum graphs, you can see that when the right channel is at the same frequency as the left, the tones are in consonance. The two tones are said to be in unison.

We also hear consonance when the right frequency is twice the left frequency. People do not think of the right channel as having "twice the pitch" as the left, but they do say that the tones are in consonance. In the language of musical scales, the right channel is then one octave above the left channel. The left is the "A below middle C" and the right is the "A above middle C." The tones fit together so well that they are called by the same name.

The "flutter" or beat that you hear makes it easy to tell when the left channel is not quite in consonance with the right. The beat lets us identify very small frequency differences. An old fashioned method of tuning a guitar uses these beats to get the tuning just right. In consonance, the beat disappears.

Finally we explore the Lissajous figure method for visualizing the relation between two pitches. This is an ancient method, developed before oscilloscopes and computers. Because it produces evocative displays, it is still used. For a discussion in terms of oscilloscopes, try the second half of the article at http://www.allaboutcircuits.com/vol_ 2/chpt_12/2.html.

The figure above shows the result of a single change to the GoldWave display. The upper control window was right-clicked, and the "XY-GRAPH" option was chosen. This option plots a graph of the signal on the right channel versus the signal on the left channel. In the figure above, both channels play a 220 Hz tone (and both are "in phase") so that both channels always have the same output signal. The graph is a simple straight line.

The figure at the right shows a photograph of the pattern when the left channel is at twice the frequency of the right channel. The photo is blurred because of the long exposure time required to allow the graph to sample all of its possiblities.

Since the frequency of the left channel is 220 Hz, it takes 1/440 of a second for that signal (plotted on the horizontal axis) to move from minimum to maximum (far left on the graph to far right). Since the right channel frequency is 440Hz, its signal changes at twice the rate. During an interval of 1/440 of a second the right channel (plotted on the vertical axis) can move from minimum to maximum and back to minimum.

Qualitatively, the graph bumps the top twice, while bumping the left only once. That ratio, 2:1 is the same as the frequency ratio.

If you play the entire file, you will see that "consonance" corresponds to a stationary pattern that is fairly simple. "Dissonance" corresonds to a complicated pattern that seems to change chaotically.

As an exercise, search for the consonant sound that corresponds to the Lissajous pattern photographed at the right. The exposure was not quite long enough to capture all of the possible positions. The gap missed a line that rises from left to right through the center of the screen.

This figure bumps the top three times while bumping the left side twice. The ratio of frequencies is 3:2.

Muscians know this frequency ratio or "interval" as a "fifth."

John Williams reminisces about using the "perfect fifth" as the hook for his theme in the movie "Close Encouters of the Third Kind" at this website: http://www.musicweb.uk.net/film/lacejw.htm