Science Questions with Surprising Answers
Answers provided by
Dr. Christopher S. Baird

What psychological effect makes notes on a piano that are an octave apart sound the same?

Category: Physics      Published: August 15, 2013

vibrating string
Fundamental modes of vibration on a piano string are determined by the string's length. Public Domain Image, source: Christopher S. Baird.

The effect is not psychological. It is physical. Notes on a piano that are separated by an octave are very similar physically. To understand why this is so, you have to understand first the basics of sound. Sound is a waving vibration of air that travels as it oscillates. The pattern of the vibrations in the air (the sound's waveshape) is determined by the vibrating pattern of the object that created it. For a piano, the sound is created by hitting metal strings to get them vibrating. The piano strings then knock into the air and get it  vibrating in the same pattern. The sound is launched from the string, through the air, and into our ears.  If you take a single string of metal and clamp the two ends down, there are only certain ways you can get it vibrating. Let's take a look at the basic components of a string's vibration.

The simplest and strongest vibration a string clamped at both ends can experience (the "fundamental" or "first harmonic") is half of a sine wave (one hump), as demonstrated in the top of the animation. Because the ends are clamped, they cannot move, so the wavelength of the simplest vibration is determined by the distance between the clamps. The next simplest possible vibration is a full sine wave (two humps), shown in the middle of the animation. This vibration has a wavelength equal to half the wavelength of the fundamental vibration. The next simplest possible vibration is one and a half sine waves (three humps), shown in the bottom of the animation. This vibration has a wavelength of one third the fundamental. Hopefully you see the pattern at this point. The next simplest possible vibration (not shown in the animation) would have four humps and so on. All possible simple vibration shapes on a string have a wavelength that is an integer fraction of the wavelength of the fundamental vibration.

When the piano key's hammer actually hits a string, you don't get a perfect sine-wave vibration pattern (you would have to hit it with a giant sine-wave-shaped hammer to get this). Rather, you get a vibration pattern that is a combination off all possible vibrations. In other words, you get a vibration pattern that is a mixture of the various integer-fraction-wavelength sine waves we just talked about. For a piano, the mixture is typically a very strong one-hump vibration (the fundamental), plus weaker one-, two-, and three-hump vibrations (the "higher harmonics" or "overtones"). This combination of many sines waves that are all integer multiples of the fundamental is what gives a piano its distinctive sound. If just the fundamental sine wave vibrated when you pressed a key, it would sound like a cheap alarm clock.

When we experience a sound, its pitch is inversely proportional to its wavelength. Shorter-wavelength vibrations constitute higher-pitched sounds. That is why shorter strings make higher sounds.  When you play the middle C key on a piano, the string's vibration contains a large, pure middle C sine wave sound (the fundamental) plus a smaller sine wave with half the wavelength (which has a pitch of tenor C) plus an even smaller sine wave with one fourth the wavelength (which has a pitch of soprano C), and so on. So when you play middle C, the sound you hear is actually a combination of all higher C notes with middle C dominating. When you play the tenor C key, the sound you hear is a combination of all higher C notes with tenor C dominating. All notes on a piano keyboard separated by octaves make essentially the same sound, just with a different fundamental tone dominating. That is why all notes separated by octaves are labeled with the same letter of the alphabet.  Now, when you play the middle D, the story changes completely. You are vibrating a completely different string with a different length. The middle D string has its own set of harmonics and therefore is physically very similar to all other D notes on the keyboard. In summary, all the notes on a keyboard with the same letter label make sounds that are very similar physically because of the way only integer-fraction and integer-multiple sine wave vibrations can fit on a string.

Topics: acoustics, fundamental, harmonic, octave, overtone, piano, sound, wave, wavelength, waves