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Comment by munificent | Hacker News Reader

This is a really interesting article because the author clearly figured out some stuff but also hasn't filled in all the missing pieces or learned the terms yet.

They are absolutely correct that the thing that makes different instruments playing the same pitch sound different is additional higher-frequency components.

In particular, frequencies that are integer multiples of the lowest fundamental frequency are called harmonics. The set of harmonics and their relative amplitudes determines an instrument's timbre (usually prounounce "tamber" in English) or its characteristic sound.

There are also inharmonics—frequencies that aren't multiples of the fundamental. Those tend to die out quickly because they don't form standing waves in the resonating body. These transient sounds form an important part of the very beginning of the sound. Some instruments, like bells, have more or longer-lasting inharmonics.

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> There are also inharmonics—frequencies that aren't multiples of the fundamental. Those tend to die out quickly because they don't form standing waves in the resonating body.

This is only true for one-dimensional vibrations on strings and the air inside a long tube. Two and three-dimensional bodies have overtones with arbitrary ratios to the fundamental, depending on the shape of the object. Bells need to be carefully tuned to have harmonic spectrum, but this is an artificial construction based on western music tastes (who wants octaves to not be dissonant). Other percussion instruments (e.g., the indonesian gamelan) are deliberately tuned to a non-harmonic overtone sequence adapted to the local music tastes. You can certainly have inharmonic overtones that don't die out quickly! And they can form standing waves in the resonating body, just like the fundamental.

This helps answer the question I just asked under another comment. Thank you!

Ah, great comment, thanks! I'm still learning too.

I would like to add to this that real instruments are much more complicated in their sound. If you take a look at the frequency histogram of one of those, you'll realize that what you might call a theory of pure frequencies, while perhaps adequate for explaining melody (several notes put together) is nowhere enough for explaining sound. Just like salty/fatty/sweet/sour is nowhere enough for explaining taste.

This is also why it is so incredibly difficult to make a realistic sounding synthesizer.

If you want to play with this, search for a sample pack and try running a couple of the sounds from that through sox to get a histogram.

I'm not saying this is wrong, but the Nyquist-Shannon sampling theorem states that a "theory of pure frequencies" is sufficient to reconstruct a band-limited signal from samples. What's the missing element?

I'd hasard a guess that the problem is not that you can't perfectly reproduce a sound of an instrument being played. However when you simulate it using samples you reduce the sound to a sum of independent recordings, whereas a real instrument will behave differently when you play multiple notes because of the material.

Yes. Plus a huge amount of other stuff.

A naive sampler misses:

1. (as you say) How playing multiple notes at the same time changes the way the instrument responds

2. How playing at different intensities changes how the instrument responds.

3. The many subtle ways your physical interaction with the instrument change the sound

4. The way playing notes in succession at different rates can alter the sound

5. The physical space or choice of amplification can affect the instrument - even an acoustic guitar can "feed back" on itself to some degree

6. A bunch of other things I haven't thought of.

Sophisticated samplers (and sophisticated sample libraries) can simulate some of the above. But physical modelling synths are probably a better way forward.

What is the difference between "a sum of independent recordings" and "reconstructing from samples"?

More than that, the sound is defined by how the overtones change over time, and for acoustic instruments that's defined by various resonant modes and how they're excited.

This article makes it sound as if timbre is static. It isn't. It's the changes that make instruments recognisable.

A static slice of a violin timbre doesn't sound much like a violin.

It's also wrong about having to use a 9ms slice. FFTs apply a windowing function which fades a slice to zero at the edges. Otherwise you get discontinuities which introduce spurious high frequency overtones which don't really exist.

And so on. These are all things that some slightly deeper background reading would have revealed.

I spent hours playing with Friture, which is a continuously-running audio spectrum analyzer. It's really cool to see the characteristics of a musical instrument live and connect the visual cortex to the auditory.

I play several instruments and this is a correct analysis, but I would also add that resonance plays a huge part in the emphasis of certain frequency ranges. An acoustic guitar resonates through the body to emphasize and amplify the vibrations of the guitar strings, and in fact the shape and composition of the body will emphasize different ranges in completely different ways. The strings also cause sympathetic vibrations in each other, further adding to the resonance and contributing to the tonal quality (timbre) of even a single note.

Can you clarify harmonic vs overtone?

All sound waves can be defined as the sum of a (possibly infinite) set of different sine waves. Each such sine wave naturally has a frequency (a pitch), an amplitude, and a phase. If you break a wave into its constituent sine waves like this, each sine wave is known as a PARTIAL.

Sound waves that we perceive as tonal -- as musical -- commonly consist with partials which follow a certain pattern. Namely, there is one lowest partial, and most or all of the higher partials have frequencies which are INTEGER MULTIPLES of the this lowest partial's frequency. For example, if the fundamental is at 200Hz, perhaps the next lowest partial is at 400Hz, and the next one might be at 600 or 800Hz, and so on.

When partials are organized like this, they are known as HARMONICS. The lowest such partial -- the lowest harmonic -- is called the FUNDAMENTAL, and the remaining (higher) harmonics are known as the OVERTONES. It is common, but not always the case, that the fundamental (1) defines the pitch of the sound (2) is the loudest harmonic. Overtones instead tend to add color to a sound.

So all overtones are harmonics. All harmonics except the fundamental are overtones?

Edit: I never realized the fundamental was a harmonic. I thought all harmonics had to be above the fundamental. I thought the overtones were the multiples (e.g. higher octaves) and harmonics were other intervals produced by the instrument (organs being the most extreme example and maybe oboe being the least)

Might be a terminological ambiguity -- like in mathematics you can distinguish "subsets", "proper subsets", and "nontrivial" (or "nonempty") subsets. Or "divisors" and "nontrivial divisors" and "proper divisors".

Even though the terminology can be defined in order to make these distinctions clear, there's an arbitrariness in terms of which definition you prefer (in the past, some mathematicians treated 1 as prime, which makes the definition of a prime simpler, but makes many theorems about primes more complicated to express). And there's a likelihood that even experts will sometimes use the simple term informally when they technically mean a more specific thing (like occasionally saying "divisors" instead of "proper divisors", or something).

I imagine that acoustics experts have a definition available that definitively states whether the fundamental "is" a harmonic, but in certain contexts it intuitively makes sense either to include or exclude it, regardless of that.

A similar case might be "animals"; taxonomically humans are animals (and apes), which is very important sometimes and confusing other times.

The fundamental being in or out of the set is less interesting to me than the (useful, to me in my day to day) distinction between overtones and harmonics. Technical misunderstandings really mess up rehearsals so I try to be as accurate as possible with the objective stuff so I can use it to communicate the subjective stuff.

> So all overtones are harmonics.

Not at all! It's the other way round. Harmonics are those overtones that are integer multiples of the fundamental. You can have other, non-harmonic, overtones.

Just to make sure I've got it: harmonics will always sound like (be?) octaves relative to the fundamental. But all the less-dominant frequencies are overtones, even if I somehow get a tritone or something gross sounding.

Now what happens when I play a chord and get overtones out of the interaction between two strings or a choir? What's happening there?

> Now what happens when I play a chord and get overtones out of the interaction between two strings or a choir? What's happening there?

Nothing. Sound superposition is linear. There's no interference between different frequencies. By playing several strings together you obtain the sum of the sounds played by each of them separately. No new frequencies can appear.

Overly simplistic. You can get beats when you combined two nearby frequencies. The beats have a frequency and are audible. Similarly, sounds can be reinforced or hidden in the interaction between instruments; it might be linear, but that does not say anything about what you actually hear.

Overly confrontational. You can and do get beats when combining nearby frequencies, and it's not at odds with sound superposition being linear.

The beats have a frequency and are audible, but their frequency is not a pitch.

Yes, you got it. Important is that a wave has a frequency and also a shape (e.g. sine, square, etc.). Even if the shape is different and the frequency is the same, the "pitch" is the same. The shape is the "timbre", or what makes the instruments sound different.

You can deconstruct any given sound wave into its partial sine waves. This is typically done using a Fourier Transform algorithm (FT).

Of course, we can also do it the other way around, create a signal out of many sine waves. In practice, this is called additive synthesis.

Instruments not only sound different because of the timbre. Another thing to look at is the loudness over time, e.g. when plucking a string. We usually do this with an ADSR representation (Attack, Decay, Sustain, Release).

Knowing these chracteristics is enough to recreate an instrument with a synthesizer. Of course, it gets more complex when there are inharmonics and if they are irregular (different depending on each tone). That's why synthesizing instruments realistically is a pretty time consuming science.

...and why guys like Martin Galway were absolute genius in the music they could get out of the C64 SID.

Not as I understand it. A harmonic is an integer multiple of the original frequency. An octave is a power of two. So the second and fourth harmonics are octaves. The third is not, though still on a normal western musical scale. The fifth is not even on the musical scale. The 12 note "well tempered" scale was and remains a fix to try to put some kind of order into all of this.

> harmonics will always sound like (be?) octaves relative to the fundamental.

Since you mention "octave" here I want to point out that this is a common misconception. Harmonics are not just octaves of the fundamental. The octave scale is logarithmic but harmonics are linear and include all integer multiples.

If your fundamental is 100, the octaves are 200, 400, 800, 1600, .... But the harmonics are 200, 300, 400, 500, 600, ... There are many extra harmonics that aren't overtones.

This is important for many reasons, but a fun one is that you can use this in sound design by relying on a clever thing our brains can do. We are so good at doing frequency analysis in our heads that we can figure out what fundamental must be present even when it isn't. If you play sine waves at 200, 300, 400, 500, 600, etc. your brain can figure out that those would all be multiples of a 100-Hz fundamental, even though that fundamental isn't present [1].

This lets you do a neat trick where you hi-pass a sound to remove some of the lowest frequencies in order to make room in the mix for other bass sounds. Even though the sound loses its fundamental, listeners will still hear it as "functionally" having a bass register. (This is also why when you listen to music on a crappy tiny speaker, you still hear the bass as bass even though it's actually quite tinny and high-frequency.)

[1]: https://en.wikipedia.org/wiki/Missing_fundamental

The fundamental is harmonic #1. This is the case for essentially all additive synthesizers for example.

Overtones are any partials other than the fundamental. While overtones CAN be non-harmonic (notably in bells), I think it's fair to say that most of them, or at least the most important ones, are usually harmonics. This is because the terms "fundamental" and "overtone" are historically music terms, and are generally applied to sounds we perceive as tonal or musical: and such sounds are largely composed of harmonics.

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