Why do harmonics sound good?

Question

Harmonics, as in multiple of a fundamental frequency, sound good to our ears. A few examples:

• Octaves, i.e. doubling the frequency. We assimilates pitches separated by octaves to a same pitch class. The octave of a B♭ still is a B♭ ; the octave of a G still is a G.
• Choosing the right temperament to preserve as many perfect fifths (multiplying the frequency by 3) without throwing the third away (multiplying the frequency by 5) as possible is an endless debate.
• We try to avoid inharmonicity in most instruments, i.e., we want partials to be as close to harmonics as possible.

This is probably the most basic fact I acknowledge when I’m thinking about music. But it also is a fact I’ve always taken for granted. If I ever were asked to justify that fact, I couldn’t really.

What is the underlying reason why harmonics sound good?

My best guess would be “there probably is a physiological reason”. After all, simple oscillating systems often are quite harmonics, and our auditory sensors could very well be. Or maybe it is cultural: most simple, readily available oscillating systems are quite harmonics, thus even the first instruments must have been. The voice, I think, is too.

Does anyone have any insight on the question?

Answer 1

Music: a Mathematical Offering by Dave Benson has a lot to say about this. The gist of it is, it's not precisely harmonics (in the sense of integer multiples of the fundamental frequency) that sound good; more important are matching partials (and the avoidance of nearly-matching partials). A relevant excerpt:

For pure sine waves, the ear detects nothing special about a pair of signals exactly an octave apart, and a mistuned octave does not sound unpleasant. Interval recognition among trained musicians is a factor being deliberately ignored here. On the other hand, a pair of pure sine waves whose frequencies only differ slightly give rise to an unpleasant sound. Moreover, it is possible to synthesize musical sounding tones for which the exact octave sounds unpleasant, while an interval of slightly more than an octave sounds pleasant. This is done by stretching the spectrum from what would be produced by a natural instrument. ...

The origin of the consonance of the octave turns out to be the instruments we play. Stringed and wind instruments naturally produce a sound that consists of exact integer multiples of a fundamental frequency. If our instruments were different, our musical scale would no longer be appropriate. For example, in the Indonesian gamelan, the instruments ... do not produce exact integer multiples of a fundamental.... So the western scale is inappropriate, and indeed not used, for gamelan music.

Even in Western music, this often becomes relevant. For example, pianos (especially smaller pianos) are slightly inharmonic, because of the stiffness of their strings. As a result, piano octaves are tuned to be slightly wide, so that the fundamental of one note will match the slightly-off "second harmonic" of the note an octave below.

Answer 2

Harmonics are very common in nature and musical instruments. If you plug the low A string on a a guitar, it will products 110Hz, 220Hz, 330 Hz, 440 Hz. That's simply physics and reflects the way the string moves. See for example http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html, third picture down.

This true for all musical instruments. Basic physics results in all fundamental notes always coming together with their harmonics (unless you play sonata for sine wave generators).

Now if you play two notes at the same time. Say an A and E, you get the fundamentals and the harmonics of both notes. That's a lot of frequencies and can get messy real quick. Now if the two notes are harmonically related, there harmonics are often the same. The third harmonic of the A is the same frequency as the second harmonic of the E etc.

As a result if the fundamentals are harmonically related, you have a lot less frequencies in the overall combination than if they are different. Hence the spectrum is "cleaner" and "less cluttered".

It also minimizes "beating". Beating occurs when two frequencies are close together: you will get a modulation that occurs with the difference of the two frequencies which sounds "harsh" and "unmelodic".

Answer 3

When you add harmonics to a fundamental sine wave, the periodicity of the result remains the same: whenever the waveform of the fundamental repeats, that of the harmonics repeat as well.

There are no beatings or artifacts with a frequency lower than that of the fundamental. That means that you get a tone quality that is as constant as that of the fundamental.

Answer 4

It has primarily biological basis. Simply speaking - sounds transformed to the neuron impulses and when it goes simultaneously it feels good (for brain and the rest of body) otherwise it is a signal/chemical mess, stubby flow and we feel a bit of pain. So to go simultaneously we just need to solve some harmonic equations (where the math physics helps us) to find proper frequencies which will produce oscillations in whole numbers - to coincide and resonate instead of trying damp each other. Such air flow will be perceived as considerable. So not so magic.. about tone (but not timbre).

Answer 5

This may be an oversimplification, but there are plenty of accurate technical responses to this question already in this post so I'll take a more abstract approach:

To my ears, more harmonics means more sonic information. In other words, additional timbre and 'dirt' gives a sound additional characteristics.

Further, most sounds (excluding perfect waveform emulation like a sine wave) tend to have higher sounds as part of their base sound (aka - harmonics). These higher sounds are based on the fundamental, and are naturally linked according to the law of sound (see 'harmonic series'), and further the experience and behavior of the fundamental. This is similar to 'bootstrapping' in computer programming.