wave is composed of a fundamental and only odd harmonics (f, 3f, 5f, 7f, ...), the summed wave is half-wave symmetric; it can be inverted and phase shifted and be exactly the same. If the wave has any even harmonics (0f, 2f, 4f, 6f, ...), it is asymmetrical; the top half is not a mirror image of the bottom. Conversely, a system that changes the shape of the wave (beyond simple scaling or shifting) creates additional harmonics (harmonic distortion). This is called a non-linear system. If it affects the wave symmetrically, the harmonics produced are all odd. If it affects the harmonics asymmetrically, at least one even harmonic is produced (and probably also odd harmonics). Harmony If two notes are simultaneously played, with frequency ratios that are simple fractions (e.g. 2/1, 3/2 or 5/4), the composite wave is still periodic, with a short period—and the combination sounds consonant. For instance, a note vibrating at 200 Hz and a note vibrating at 300 Hz (a perfect fifth, or 3/2 ratio, above 200 Hz) add together to make a wave that repeats at 100 Hz: every 1/100 of a second, the 300 Hz wave repeats three times and the 200 Hz wave repeats twice. Note that the total wave repeats at 100 Hz, but there is no actual 100 Hz sinusoidal component. Additionally, the two notes have many of the same partials. For instance, a note with a fundamental frequency of 200 Hz has harmonics at: :(200,) 400, 600, 800, 1000, 1200, … A note with fundamental frequency of 300 Hz has harmonics at: :(300,) 600, 900, 1200, 1500, … The two notes share harmonics at 600 and 1200 Hz, and more coincide further up the series. The combination of composite waves with short fundamental frequencies and shared or closely related partials is what causes the sensation of harmony. When two frequencies are near to a simple fraction, but not exact, the composite wave cycles slowly enough to hear the cancellation of the waves as a steady pulsing instead of a tone. This is called beating, and is considered unpleasant, or dissonant. The frequency of beating is calculated as the difference between the frequencies of the two notes. For the example above, |200 Hz - 300 Hz| = 100 Hz. As another example, a combination of 3425 Hz and 3426 Hz would beat once per second (|3425 Hz - 3426 Hz| =