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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?

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Not always -- go high up enough to get the equivalent of a semitone harmonic and you may not like the result. – Carl Witthoft Dec 3 '13 at 12:49
    
@CarlWitthoft If you keep the harmonics in between, I have no issue with at least the 15th harmonic. Well, it sounds very metallic, but not false by any mean. Just playing a low note and it’s 15th harmonic is weird, likely because, well, the 15th harmonics of any instrument is quite low and there is nearly four octaves between the fundamental and it’s 15th harmonic, so our ears naturally distinguish them. – Édouard Dec 3 '13 at 13:24
    
I was being humorous, but typically us musicians would allow the 4th octave to be played against the 15th harmonic (rather than the base note) . – Carl Witthoft Dec 3 '13 at 14:03

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.

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So it is cultural, then. I’ll have a closer look to that book. – Édouard Dec 3 '13 at 13:22
    
@Édouard well, the instruments may be cultural, but anyone can put together a bunch of badly-out-of-tune percussion instruments, which is part of a gamelan ensemble. Gamelan, IMHO is more based on tradition than on any conscious (aka scientific) plan to produce certain harmonies or clashes. – Carl Witthoft Dec 3 '13 at 14:08
    
What's exactly do you mean when you say, "partial" ? – tjt263 May 25 at 19:17

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".

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We craft the guitars so that the partials are (close to) harmonics. By choosing souple strings, e.g. — stiff strings can be inharmonic, as it’s the case for the piano high register. Now for guitar, it’s not obvious (an everyday string already is quite harmonic) ; but the reason the bars of a vibraphone are thinner in the centre is to have harmonic partials. – Édouard Dec 3 '13 at 13:20

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).

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I don't think this is true: Neurons are excited by a bending wave of the basilar membrane in the cochlea. Different frequencies are simply exciting different areas on the membrane and there is nothing inherently different about harmonic or in-harmonic excitation. Neither one results in equal spacing of the excitation points. – Hilmar Dec 3 '13 at 12:59
    
Yes, different areas on the membrane and different sensosry neurons, but then signals are aggregated by intermediate neurons (excitment propagates further in the brain and we can already "feel" something about it) where signal clash occurs to make us feel pain. It is simplification, e.g. to be more detailed: incoming audio information in the "head" is splited in two ways and goes to amygdala to "feel & react" (ASAP) and to neocortex areas for more precise and highlevel analysis. Sound perception is very interweaved with consciousness (as well as other perceptions and their interpretations) – rook Dec 3 '13 at 13:29
    
I had to add "and complex process" but had no space left for it :) – rook Dec 3 '13 at 13:45

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.

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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.

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I’m not sure this is an answer to the question. If it is, I certainly don’t get it: I read it as “Sounds with more partial sound richer”, am I wrong? – Édouard May 27 at 1:02
    
Essentially, yes. It's a difficult question to answer because, in my opinion, the question is subjective and I'm trying to elaborate it's definition in an effort to answer it. A richer cake might taste better because it has deeper flavors. Why does it taste good? Because it does, potentially. There's no right answer here, in this sense. – hailthemelody May 27 at 4:35
    
I really believe you did not understand the question, then. I am wondering why (my possibly restraint view of) music tend to favor integer ratios between frequencies. I’m also wondering if you don’t mix up partials and harmonics (I know I used to). I don’t have a good equivalent for your cake analogy, but let’s say I’m wondering why berries go well with chocolate and not with buckling. Is it cultural? Are there physical or physiological reasons behind it? – Édouard May 27 at 4:48
    
It's possible I don't understand your question, though I am a pro. Your question felt more theory-based in nature and given the other technical responses I wanted to balance it out with a more 'natural' response, however this doesn't seem to be what you wanted. Perhaps you'd be interested if I expanded my thoughts on the Law of Sound / Harmonic Series? I'd be happy to but don't want to commit the time if that's not what you're interested in. – hailthemelody May 27 at 5:08
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This law of sound? In which case, I would be interested in justification as to why “Notice that all natural sound can be treated as composed by a linear combination of harmonic waves with frequencies related to the fundamental one, f_0, as f_n = n*f_0 for integer n (= 1, 2, 3, ...). All these harmonic components are considered as consonant with the fundamental and the relative weight of these harmonics is a characteristic of the sound source, which frequencies spectrum defines one of the most important sound quality, the timbre.” is true. – Édouard May 27 at 10:28

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