Soundwaves

Sound Waves Basics: Sound Waves are vibrations that travel through the air. While they travel in the electronic and digital worlds they are commonly shown on a two-dimensional graph with the y-axis defines the amplitude

(or loudness) of a signal and the x-axis defines the frequency (or pitch). The sinewave that is seen above is the most basic waveform in audio. It a single pitch with no harmonics (or overtones). Harmonics are additional pitches that are sounded with a signal. They are higher pitched and based on fractions or ratios of the original signal. For example a guitar string vibrates as a whole. It also vibrates as in halves (one half going one way while the other half does

		the opposite). In addition, it vibrates in thirds, fourths, fifths, and so on. Every time the length of the vibrating material or wave is affected the pitch changes. So, when a string vibrates in halves, it produces a pitch half the wavelength and twice the frequency (or one octave higher) of the original. The series of pitches produced as harmonics may seem

familiar to you. They are the basis of western music, producing octaves, fifths, and thirds. The image on the right shows how different numbered harmonics fall into series with different notes of the scale. If you want to hear these for yourself, there is a simply test that you can do to help your ear notice the different harmonics.

Exercise:

Sit down at an acoustic piano and play a low “C” note. Hold it out and listen for higher pitches in the sound. Stop the note and play one of the harmonics (as listed in the staff such as “G”, “E”, or a higher “C”) immediately afterward. Listen for that harmonic as you play the low “C” note again. Repeat this process for the different harmonics. If you want to make the harmonics sound more apparent, hold down the sustain pedal while playing the fundamental. The harmonics of that note will cause the related note’s strings on the piano to resonate. This is called sympathetic vibration (when a specific item resonates at a specific frequency, other items tuned to the same frequency will begin to resonate as well). Now you should be able to hear many of the harmonics of that low “C” note. See how high in the harmonics series you can hear. You will notice that it gets more difficult as the pitches go higher. Listen to the richness of the sound that is created by having all of those notes sounding as harmonics with the one fundamental note. Finally, try different fundamental pitches and train your ear to listen for harmonics in other instruments as well.

The two above images show the difference between even and odd harmonics. Notice how the even harmonics are more stable ratios such as octaves while odd harmonics are more unstable, like thirds and sevenths. Even and odd harmonics have different characterizstics. Even harmonics are commonly reffered to as being more stable, smoother, and comforting. Odd harmonics are usually described as more jarring, unstable, and sometimes harsh.

Sound Waves Types

Complex waveforms are the combination of hundreds of waves. If a voice sounds bright it is usually because it is producing upper harmonics at higher amplitudes. If it is duller sounding, it has less upper harmonic content. The combination of all of these different simple sound waves at different frequencies and different amplitudes (levels) produces complex waveforms. These variations in complexity and harmonic structure are how we can tell the difference between the sound of a guitar and a trumpet, a knock on metal or wood, and a real human voice verses a synthesized one. There are a handful of basic waves not commonly found in nature, but easily created with electronics. These basic waves are the building blocks of many synthesizers, basic MIDI, and warning systems.

The most basic and simple waveform, a sine wave has only a fundamental and no harmonics. You may recognize it from warning tones and beeps. Listen to it here.

1kHz Sine 500Hz Sine 100Hz Sine

After the sine wave we jump in complexity to the sawtooth wave.

The sawtooth wave has a fundamental with all harmonics present. The second harmonic is quite strong being ½ the amplitude of the fundamental, with the third harmonic at 1/3 the amplitude of the fundamental, and the fourth at ¼ the amplitude. This produces a good deal of harmonic content and therefore a full buzzing sound, which can be heard here.

1kHz Sawtooth 500Hz Sawtooth 100Hz Sawtooth

Now let’s look at the square wave, which differs a bit from the previous two.

The square wave has only odd harmonics. The interesting similarity to the sawtooth wave is that each harmonic decreases in the same manner except the third harmonic is ½ the amplitude of the fundamental, with the fifth harmonic at 1/3 the amplitude of the fundamental, and continuing along in that manner. This harmonic structure gives the square wave a little more bite to the sound, which can be heard here.

1kHz Square 500Hz Square 100Hz Square

Finally we’ll look at the triangle wave, which is very similar to the square wave.

The triangle wave has only odd harmonics like the square wave, but their amplitude is far weaker in comparison to the fundamental. The third harmonic is only 1/9 of the amplitude of the fundamental and progresses in a similar manner from there. The triangle wave sounds more similar to a sine wave, because of its soft harmonic content, but it still shares some characteristics of the square wave having only odd harmonics. Listen here to see what it sounds like.

1kHz Triangle 500Hz Triangle 100Hz Triangle

These main four waves can be seen in most synthesizers, DAWs, and testing equipment. There are some variations that include the sawtooth wave going in opposite directions and the lopsided square wave (being larger on the top and smaller on the bottom or vice versa). Depending on the device these waves can be added together to create complex waveforms, can be used to modulate a signal, or can create the pattern for a pulsating filter sweep.

-Ben Harris

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