Hi again,
the even harmonics are tolerable and sometimes pleasing to the ear...
whereas the odd harmonics are grating.
Not always odd harmonics are grating. In clarinet and bassoon, for example, odd-order harmonics are dominant and are essential for giving them their characteristic sound.
Several odd-order harmonics are consonant; speaking in terms of musical intervals:
- the 3rd harmonic is an octave plus a perfect fifth above the fundamental,
- the 5th harmonic is two octaves plus a major third above the fundamental.
The 7th harmonic is dissonant.
The even harmonics contribute to create a triangle wave.
Odd harmonics create a square wave.
Triange wave has only odd-order harmonics too. Think that it can be obtained passing the square wave by an integrator, so the two waveforms have the same frequencies, but with different relative amplitudes.
Square wave is a rather extreme case, as it is too rich in odd-order dissonant harmonics (7th, 11th, 13th. etc.)
In square wave, the relative amplitudes fall off as 1/n, while in triangle wave they fall off as 1/n^2, i.e. the relative amplitudes are:
WAVE......|.fundamental...3rd...5th...7th....etc
Square....|......1........1/3...1/5...1/7....
Triangle..|......1........1/9...1/25..1/49...
This lower harmonic content makes triangular wave much softer.
Regards
Z