What you need to know
Every sound has a frequency we can study thanks to its spectral representation.
First, there is the spectrum which brings out features. Every spectrum has at least, one feature which is called the fundamental frequency. Then, spectrums are different by their number of features which represent harmonics / overtones. As you have seen in the first part, these harmonics characterize the timbre of a sound.
A sound wave has also its own representation which brings out a signal. The signal of a sound is generally periodic. That means that the signal repeats the same pattern over time. The duration of the pattern is called the period, noted T and expressed in an unit of time. Mathematically, the signal is represented by a periodic function:
F=1/T
The frequency F is expressed in hertz (Hz).
The period T is expressed in second (s).
Thanks to a microphone linked to a computer with the software Logger pro, we recorded the spectrum and so the frequency of an "A" (=La) note played by a diapason. Then, we did this experiment again and again by replacing the diapason by a violin and Mona's voice. The only thing that did not change was that these three instruments played an "A" note.
Here is the experience:
From this experiment, we can notice that there are two kinds of familiar sounds :
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There are pure sounds whose signal is sinusoidal. Their spectrum has only one harmonic which is called the fundamental frequency.
For example, the sound signal of the diapason is pure.
Here are the results of the diapason. The signal is sinusoidal and we notice only one harmonic which is the fundamental frequency.
In the video above, you can see that we got two harmonics instead of one when we recorded the frequency of a La played by a diapason. This mistake was owed to the bustle and noises in the room.
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There are complex sounds whose signal is the sum of several sinusoidal signals (click on the button for a demonstation to understand it). We can decompose all those signals; it is the sum of harmonics also known as the Fourier series. Their spectrum has one fundamental frequency and at least one harmonic frequency.
Instruments' sound signals are nearly always complex.
Here are the results of a "La" note played by a violin. The signal is not sinusoidal anymore as there is an addition of sinusoidal signals. Indeed, there are few harmonics.
Here are the results of La sung by Mona. The signal is not either sinusoidal and we distinguish several harmonics.
Why instruments sound different
Here's a conundrum. If a violin and a piano make sound waves with the same amplitude and frequency, why do they sound so different? If the waves are identical, why don't the two instruments sound exactly the same? The answer is that the waves aren't identical! An instrument (or a human voice, for that matter) produces a whole mixture of different waves at the same time. There's a basic wave with a certain amplitude and pitch, called the fundamental, and on top of that there are lots of higher-pitched sounds called harmonics or overtones. Each harmonic has a frequency that's exactly two, three, four, or however many times higher than the fundamental. Every instrument produces a unique pattern of a fundamental frequency and harmonics, called timbre (or sound quality). All these waves add together to give a unique shape to the sound wave produced by different instruments, and that's one reason why they sound different. The other reason is that the amplitude of the waves made by a particular instrument changes in a unique way as the seconds tick by. Flute sounds are immediate and die quickly, while piano sounds take longer to build up and die out more slowly as well.
The fundamental frequency is always the same but, the number and the intensity of its respective harmonics are different and ears can distinguish the two instruments.
What is called a "dissonance" ?
Most of sound sources produce complex sounds which are composed of one fundamental frequency and of harmonics. Overtones are integer multiples of the fundamental frequency f.
We distinguish :
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Even harmonics : 2f, 4f, 6f, 8f….
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Odd harmonics : 3f, 5f, 7f, 9f…
To our ear, odd overtones are not pleasant. That is why they are called anti-musical harmonics.
As an example, if the frequency is f = 1kHz, then the pleasant harmonics are 2kHz, 4kHz, 6kHz, .. and the unpleasant harmonics are 3kHz, 5kHz, 7kHz, ..
The human ear is sensitive to the ratio between the frequencies of two notes played simultaneously.
When an instrument emits an "A3" (La3), of fundamental frequency f1 = 440 Hz, the ear perceives a sound of frequency f1 and, according to the instrument, harmonious of frequency f2 = 2 f1, f3 = 3 f1, ..
When a second note is played, an A4 of fundamental frequency f '1 = 880 Hz, the ear already hears this sound in the harmonics of A3's; it's the same for the second harmonics of an A4 of frequency f '2 = 2f '1=1760 Hz = 4*440 = 4f1.
The more the harmonics of two notes have common frequencies, the more these notes are harmonious to the ear.
Harmony between two notes is when the ratio of the frequencies of their fundamental is "simple". The simplest ratio is the one which has for value 2. Both notes are called "the octave". Played simultaneously, these two notes seem to make only one.