Why does an A note sound different across instruments?

This is a really interesting article because the author clearly figured out some stuff but also hasn&#x27;t filled in all the missing pieces or learned the terms yet.<p>They are absolutely correct that the thing that makes different instruments playing the same pitch sound different is additional higher-frequency components.<p>In particular, frequencies that are integer multiples of the lowest <i>fundamental</i> frequency are called <i>harmonics</i>. The set of harmonics and their relative amplitudes determines an instrument&#x27;s <i>timbre</i> (usually prounounce &quot;tamber&quot; in English) or its characteristic sound.<p>There are also <i>inharmonics</i>—frequencies that aren&#x27;t multiples of the fundamental. Those tend to die out quickly because they don&#x27;t form standing waves in the resonating body. These <i>transient</i> sounds form an important part of the very beginning of the sound. Some instruments, like bells, have more or longer-lasting inharmonics.

Spectograms of each note will also make it strikingly clear that there is a dominant frequency invoked and overtones (harmonics) also being invoked that give the sound its full sound-profile. If sound is atmospheric texture, the overtones are irreplaceable grooves in the ether.<p>The simple sine wave is exactly one dominant frequency in a spectrogram, a line. Instruments such as a trumpet will have upwards of 12 overtones, parallel lines, lessening in strength.<p>One interesting idea that came to me last night was trying to reproduce the physical 3D model of an instrument based on its spectrographic fingerprint. With enough samples, this ought be possible, and with a 3D printer one might even be able to create interesting physical instantiations of instruments based on spectrographic fingerprints. One could even create never-before-seen instruments based on a generated spectrogram, in an interesting radar-to-ocean operation (as opposed to ocean-to-radar, how radar normally works). Maybe topography-from-radar is a clearer way to state the same.<p>Generating audio from spectrograms is an open problem and I would love to see more open-source work in this domain.

This fact was confusing for me back in my school’s chorus. I don’t know if it was confusing to anyone else, but it was to me.<p>How does a person match the pitch of the piano? I could hear a few different pitches when one note was played (in a confused way, I would zero in on different parts of the sound), any of which might have been the target pitch to be matched.<p>And was I supposed to make my voice sound more like the piano? Was that part of “matching the note?”<p>Complicating things was the fact that my own voice had different pitches in it. Which part of my voice was supposed to match the note?<p>What a time. Now I know I was noticing the fundamental of the piano note at times and overtones at some others. Also, changing the timbre of your voice can mirror the overtones of the piano better, but that isn’t normally the goal of a singer.

This is a fun exploration, though there&#x27;s a lot of fuzzy usage of terms and missing stuff. Eg. the first 5 paragraphs are apparently trying to define timbre, yet the word appears nowhere. Same with harmonics, etc.<p>I think one of the most interesting things about pitch is that it&#x27;s not well defined, it&#x27;s a psychological phenomenon. If you could extract it from people&#x27;s brains, you would likely get different values from different people. This is compounded by the fact that harmonics produced by real instruments are not exact ratios of each other, yet they affect the perceived pitch.

The title and content don&#x27;t match.<p>The way we distinguish between musical instruments and notes is because of timbre&#x2F;tone color. Which has nothing to do with fourier transforms per se and you can use wavelet for that matter. DFT&#x2F;DTFT are the most common approaches to quantize and convert back to analog and they can be completely left out of the discussion for such a title.

Surprised to see no mention of fundamental frequencies or harmonics

There is a fun program that can display this sort of data as you play notes with your ordinary QWERTY keyboard:<p><a href="https:&#x2F;&#x2F;github.com&#x2F;kevinacahalan&#x2F;piano_waterfall" rel="nofollow">https:&#x2F;&#x2F;github.com&#x2F;kevinacahalan&#x2F;piano_waterfall</a><p>It&#x27;s portable to Linux and Windows at least. It won&#x27;t run well in a virtual machine (including a ChromeBook) because it needs a GPU that can scroll the window fast enough.<p>There are 3 windows. One just shows the selected waveform. The others show an 8192-bucket FFT in red, a 1024-bucket FFT in green, and the active MIDI notes in blue. It&#x27;s live, scrolling up at 93.75 pixels per second.<p>The QWERTY row becomes the white keys, and the number row becomes the black keys. F1 through F4 choose the type of sound. Left and right arrows change the octave in use; your speakers probably don&#x27;t handle the full range very well. The program turns out to be a great speaker test, especially if you change the sound to a sine wave. It&#x27;s also a great keyboard test; see how many keys you can hold down before your keyboard won&#x27;t register any more. Individual colors in either window can be toggled with the 2x3 keypad that has Insert, Delete, Home, End, PgUp, PgDn. (the screenshot has green toggled off)<p>To make a trombone sound, first switch to a type of sound with lots of harmonics, like a sawtooth wave. Pick a low note, then find notes to line up well with the first two harmonics. Switch to the sine wave, and play all three of your chosen notes. For more of a clarinet sound, release the middle of the three that you have selected.

I think it&#x27;s bit of a shame that the time dependence is left as a footnote. ADSR envelope and other expressive dynamics have also huge influence on the perceived sound of different instruments (for at least me..). This has been then explored by electronic musicians by mixing and matching the dynamics and overtone patterns to create all sorts of interesting novel sounds.

I see grey text on white backgrounds more and more. It is infuriating. Please just use #000, it heavily distracts from the content and unecessarily makes the point literally unclear.

All the way into Fourier but no mention of harmonics? This is a little over-academic relative to one of the main factors that influences the texture (&quot;timbre&quot;) of sound - namely its harmonic content.<p>Distortion on a guitar makes an identical phrase sound much richer, and it&#x27;s not just the wavelength limiting - it&#x27;s the much louder harmonics relative to the fundamental.

I’m so glad to see harmonics well discussed in the comments, and was so disappointed to see it not mentioned once in the article. Putting aside percussive differences between instruments or playing style (which also have harmonic effects but aren’t always perceived that way), the difference between a “frequency” that sounds different but measures the same is generally because there are many less prominent frequencies behind the dominant one. That’s why some sounds twang and other sounds thunder. That’s why some coated strings sound dull. That’s why loose drum heads sound loose. That’s why a sawtooth or sine wave sounds synthetic: it is, nothing in real life sounds like that.

The inner ear literally applies a Fourier transform to the incoming waveform, each location containing tiny hairs which each respond to a narrow frequency range. The spectrogram directly reflects what is sensed by the inner ear.

Timbre is not only about harmonic content. There&#x27;s also the envelope of the sound - how it changes over time. Violin pizzicato (plucked) sounds completely different than violin arco (bowed), yet it&#x27;s the same string being excited into vibration. Pizzicato is percussive and decaying, arco is more smooth and sustaining. Same for piano - try to imagine a piano sound that doesn&#x27;t decay and doesn&#x27;t start with a bit of percussive thump, that would sound quite different!

Here is video [1] with example of using physical modeling method `Karplus–Strong string` [2]<p>[1] <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=FOpZYlI-F1g" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=FOpZYlI-F1g</a><p>[2] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Karplus–Strong_string_synthesis" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Karplus–Strong_string_synthesi...</a>

I thought this was going to be a discussion on why different orchestral instruments have different &quot;concert&quot; pitches (i.e. a C on a trumpet is a Bb on a piano and so on), which is an interesting look into the history of European instrument inventors in the 20th century.

Interestingly, exact same phenomenon occurs in speech. What is the difference between the sounds &#x2F;a&#x2F; and &#x2F;o&#x2F;? Turns out it’s timbre and that our vocal cavities changing form changes the timbre while the vocal chords produce the same fundamental frequency.

There&#x27;s a neat thing about a kind of &quot;relative timbre.&quot; E.g., if you listen to a solo piano piece, you make some kind of adjustment to the homogenized timbres and are able to focus in on smaller timbral differences.<p>There&#x27;s a music cognition paper about it somewhere.

The real beauty happens when more than one note is played at the same time. The overlapping harmonics give rise to harmony. Combine various instruments to get all kinds of interesting sounds. This is part of the art of orchestration.

I’d love to hear a synthetic instrument that’s halfway between a piano and a violin.

Here&#x27;s some code on WavTool for trying out overtone combinations: <a href="https:&#x2F;&#x2F;wavtool.com&#x2F;?code=3" rel="nofollow">https:&#x2F;&#x2F;wavtool.com&#x2F;?code=3</a>

In addition to harmonic content, time plays a role too. For instance, harmonics are not the only reason why a snare drum sounds different than an oboe.

Simple answer: different timbres.

Depends on the frequency <a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;A440_(pitch_standard)" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;A440_(pitch_standard)</a>