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Let's Learn About Waveforms (surge.sh)
989 points by tomduncalf on Feb 14, 2018 | hide | past | favorite | 128 comments

Very cool site. Just one nitpick: "Frequency" denotes the technical speed of vibration, as said in the article.

"Pitch" is the human perception of how high or low a tone is. Pitch is a "quale", it is purely perceptual, and not measured in Hz. We perceive one pitch for a guitar tone, even though it is made up of many frequencies (like the sawtooth example).

A related technical term is "Fundamental Frequency" or "Base Frequency", which is the lowest frequency in a harmonic tone complex.

In most cases, pitch correlates with the fundamental frequency. But not always. For example, pianos are not strictly harmonic, their higher harmonics are slightly higher than integer multiples of the fundamental. Yet we perceive a single pitch. Another example: If we cut off the lowest harmonics and fundamental using a filter (like in some telephones or cheap loudspeakers), we still perceive the pitch as that missing harmonic. Another example: If you play different harmonics on different ears, we still perceive it as one homogenous tone. Another one: You can play two tones with offset fundamental frequencies such that we should hear a common fundamental below either tone. But we don't, because our brain somehow separates co-modulated tones into distinct auditory objects.

It is endlessly fascinating how human perception works, and "pitch" is surprisingly nuanced and complex if you look into it deeply.

There's an analogy in vision. "Wavelength" is the measurable physical aspect of light. "Color" is the complex human perception formed by various wavelengths hitting the retina and subsequent processing in the brain.

> we still perceive the pitch as that missing harmonic

I think you mean missing fundamental here.

But aside from that, you make some very good points. The human brain has some amazing abilities to understand what physical phenomenon might have caused the frequencies it's picking up, even if it isn't getting all of them.

Yes! Thank you for the correction.

Interesting! I hadn't thought of it that way. Those examples are wild.

And yeah, human perception of sound was something I had wanted to explore more, but I didn't want it to be even longer... might be interesting as a follow-up (although I don't know exactly how to visualize it...

Thanks for the feedback :)

I just want to thank you maker. It's things like these that really make the internet a special place, with so much power that can be leveraged. It's great to see someone leveraging it to create rather than try to get people addicted on stuff here.

Thanks for the kind words :D

This may be the best example of seemless pedagogy I've ever seen on this topic.

Now if you pushed it too the limit and went on to digital sampling and compression codecs and ended with the Fourier transform... That would be something legendary indeed.

Some of that was covered by a couple of videos Xiph put out [0]. I was really hoping they'd put out a bunch of them since I found them quite informative, but they seemed to stop. It's too bad since I, like you, would've liked to have seen some coverage on some of the more advanced topics.

[0] https://xiph.org/video/

I too very much appreciate this, as soon as I saw it I shared it with a couple students I mentor. Interestingly we're talking about synthesizers as an instrument and this is a great visual tool! Keep it up! I'd be happy to send a small donation for continued write-ups.

Well, we want them to get addicted to self-directed learning on technical topics ;-)

It's about time we get high on this.

Yeah, I was thinking as I was reading it how dry this would have been without the visual and aural accompaniment (in fact, I think I probably did try to read something similar when I was in college). I do wish there was a way to pause the animations, though, they got distracting as I was trying to read.

I really, _really_ love this kind of visual guide. Do you guys know of anything similar for other (unrelated) learning subjects?

Edit: just saw this one in the footer of the Waveforms page. http://www.r2d3.us/visual-intro-to-machine-learning-part-1/

There's one about programming the hexagonal grids (eg. for the turn-based strategy games):


Creator here, also keen to know if anyone has suggestions. The r2d3 one was (obviously) a huge inspiration, it's such a nice way to learn things.

I can't seem to find it now, but either the NY Times or the Washington Post built something similar, for the 2018 US house/senate races. Lots of graphs as you scroll, with scroll-based events, IIRC.

EDIT: ah, how could I forget Nicky Case's explorabl.es! Was a huge inspiration for me as well.

Mike Bostock is the creator of d3.js and does, or did, a bunch of visualizations for NY Times. Check his website, it's amazing.


Nice work!

I have a suggestion about the chord:

With a single chord like this I think it sounds better to tune it with "just intonation". For a major chord it is [ f * 1, f * 1.25, f * 1.5 ]. With these frequencies the chord will not wobble.

Thanks for the feedback; I saw this further down the comments as well. Will give it a shot!

EDIT: Yep, sounds way better. Will definitely have to look more into this! Thanks again.

A little wobble can be nice though. When I write synthesizer patches, I almost always detune the oscillators a few cents.

On a synth in a musical piece, absolutely.

But in a technical piece like this, I prefer if it isn't.

there's the https://pudding.cool site that kind of does this stuff?

Here's a whole page of them: http://explorabl.es/

Wow, thanks for sharing. That site and all the sites it links to restore my faith in humanity.

Here's a fantastic one for complex numbers. It visually motivates the "imaginary" axis, how complex numbers describe rotation and motion, and why they're useful.


I love these kind of things too and link to them frequently from my audio-themed newsletter (http://webaudioweekly.com). I'm particularly fond of Tero Parviainen's interactive explorations of generative music, for example: https://teropa.info/blog/2017/01/23/terry-rileys-in-c.html

Hi Chris! Creator here, I'm a subscriber to Web Audio Weekly. Appreciate the work you do, always excited when I receive it :)

Thanks Josh! This is going in the next issue, of course :)

Bret Victor has been a major protagonist of interactive visualization: http://worrydream.com/

"The Ladder of Abstraction" is close to this explicit document style, but if you're interested in the history and philosophy of interactive visualization, you can lose days on that site.

Here's a ML one (linked to by the above site):


Distill has some great ML articles that are kind of similar:


Tensorflow Playground, more of a self-driven place to play with machine learning: http://playground.tensorflow.org/

Please checkout 3Blue1Brown.

3Blue1Brown, by Grant Sanderson, is some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations and for difficult problems to be made simple with changes in perspective.

Here's a favorite of mine that lets you explore Fourier synthesis. Be sure to left-click for more settings.


Edit: Oh, it's down. But I'll leave this here in case it's a temporary problem.

Ah. Here's one that was inspired by that one.


> Do you guys know of anything similar for other (unrelated) learning subjects?

Raft consensus http://thesecretlivesofdata.com/raft/

There's the 4D toybox by the person who's making Meigakure: https://www.youtube.com/watch?v=0t4aKJuKP0Q

Check out https://idyll-lang.org/ , a language that aspires to make making these easy.

Here's another good link, similar in message & medium:

Seeing Circles, Sines, and Signals: A Compact Primer on Digital Signal Prcoessing, by Jack Schaedler


I was going to link this one if someone did not already. It is a natural followup from this article and must-read for getting intuition on DFT and its limitations, many of which are just limitations of sampling.

As far as progressing even further into DSP, especially for audio processing, I've found https://www.dsprelated.com/freebooks/filters/ to be a good map of the territory. But as someone with a CS degree rather than an engineering or math degree, it's all pretty overwhelming.

I've found Steven Smith's DSP guide to be quite good. http://www.dspguide.com/

That said it's all about the underlying math. If you want something with lots of source code to start playing around with you won't find it here. But since understanding the math is critical to understanding what the source is doing I think both sides are necessary.

Also this is yet another area of CS/engineering where C rules the roost. If you're not comfortable with C you'll have a lot of trouble using actual DSP hardware.

Likely DFT abbreviates discrete Fourier transforms. By now the main interest in DFT is the FFT -- fast Fourier transform, for positive integer n points, n / log(n) times faster.

The FFT was reinvented by J. Tukey, at Princeton and Bell Labs, at a US Presidential Science Advisors meeting, while Tukey was taking meeting notes with one hand and doing Fourier derivations with the other, from a query from R. Garwin, at IBM's Watson lab. Garwin said he was using too much computer time calculating Fourier transforms, so Tukey showed him the FFT. Later Cooley at IBM programmed the FFT, and Cooley and Tukey published the work. The FFT was revolutionary for signal processing, for sonar, radar, molecular spectroscopy, etc.

The sampling issue is: Suppose have a periodic waveform with highest frequency 20 KHz (kilo Hertz, 1000 cycles per second). Then there is the canonical theorem of interpolation theory that says that can reproduce the wave form exactly from the values of the waveform at equally spaced points 40,000+ times a second, or some such (there is a detail right at the 40,000). There is a cute pseudo-proof based just on pictures!

So, for DFT/FFT, we start with those sampled values, and that's the "discrete" part. E.g., the reason audio CDs use 44 KHz or some such is that they want to be good for music up to 22 KHz. If the music really does have significant power at 22 KHz and we sample at only, say, 15 KHz, then we will have under sampled and end up with distorted music.

A computer sound card or chip has to reverse the discrete sampling and generate a continuous waveform for the audio system and speakers, that is, to analog.

There is a good start on Fourier series (for periodic waveforms) with the math done carefully in W. Rudin, Principles of Mathematical Analysis. If a waveform is not periodic but defined for all time, then we can do the closely related Fourier transform, and there is a nice treatment of the math in W. Rudin, Real and Complex Analysis.

Suppose we have waveforms x(t) and y(t) where t is time and x(t) and y(t) are real numbers. Suppose we also have real numbers a and b. Suppose we are in, say, a concert hall and musicians are playing x(t) and y(t). Suppose due to the concert hall, for function h what we hear from x is h(x). Then we hope and believe that

h( ax + by ) = ah(x) + bh(y)

that is, the the concert hall effect h is a linear operator. If in addition h doesn't change over time, say, yesterday to tomorrow, then, presto, bingo: All h and the concert hall can do to x and y is adjust the volume of the harmonics! Or, a time invariant linear system is a linear operator and, from Fourier theory, a convolution. Here I am simplifying somewhat.

Well, the world is awash in linear systems, especially for sonar and radar. So, one of the big, early uses of DFT/FFT was in analyzing the acoustic signals from reflections from subsurface layers from small explosions at the surface, all for looking for oil.

Good... except that the reason for 44.1KHz isn't to reproduce signals up to 22.05KHz, it's to reproduce signals up to 20KHz reliably. It is possible, though improbable, to hit nothing but zero-crossing values if you sample at exactly twice the highest desired frequency. Sampling just a little more often than 2X eliminates that error; you'll always be forced to non-zero values from which you can reconstruct the original signal.

The nonzero transition band of real antialias filters is typically a way bigger concern than the strict equality of the Nyquist theorem.

> It is possible, though improbable, to hit nothing but zero-crossing values if you sample at exactly twice the highest desired frequency.

Yup! You just filled in what I called a "detail"!!!

Yeah, but... have you read the (fine) article?

Just kidding, graycat, I'm a fan! How's your project going? any news?

Sure, this time I went through the article! Again, I would have liked to have some such and more if I ever teach trigonometry again.

For the project: The work for the project is as usual fast, fun, and easy but gets delayed by random, external nonsense. But have to expect such random stuff. Between now and going live is all just routine.

The current random nonsense is my development computer got sick: Apparently there was a motherboard problem giving serious data corruption.

I was working up orders for parts for two new computers when the old computer finally seriously quit.

So, I rushed out and just -- horrors -- actually BOUGHT a computer, not just parts but an actual computer. I got an HP laptop with Windows 10 Home Edition.

Then I ordered the parts. They have come now and are ready to plug together.

I deliberately selected parts about one generation out of date.


(1) My startup software makes some use, actually light, of SQL Server; it insists on ECC (error correcting coding) main memory, and I like it, too.

Apparently ECC main memory needs all of the memory, motherboard, and processor to support ECC. Well, now the easy way to get that is to buy some old parts.

(2) I'm happy using a computer with a good BIOS but don't want to get involved in the newer Unified Extensible Firmware Interface (UEFI).

Why against UEFI? No visible upside for me. Likely big downsides of lots of new architecture I don't need, lots of new complexity I don't want to have to work with, maybe some new bugs, and no doubt rather poor documentation, and less good technical information on the Internet.

Computer Industry: If you don't work hard to document your work, then I'll work hard to avoid your work.

(3) Older parts tend to have fewer bugs and better technical information.

For the case I got a

Thermaltake V3 VL80001W2Z (Black)

It's big, with a 120 mm fan and lots of air holes and places for more fans.

For a motherboard, I considered the

Asus m5a97 R2.0

but gave up as apparently they are now out of production and super hard to find in stock.

So, I settled on the

Asus m5a78l-m-usb3

It's a cute little thing, micro-ATX, a BIOS but no Unified Extensible Firmware Interface (UEFI). It supports ECC (error correcting coding) main memory and has an AM3+ socket for the AMD FX series of processors. I got two of the motherboards from AVADirect in Cleveland.

For main memory the motherboard has four DDR3 slots and can take a total of 32 GB. I was able to find and got 4 DIMMs of DDR3, 4 GB per DIMM, at 1333 MHz with ECC.

The processor is an AMD FX-8350, 64 bit addressing, with 8 cores with a standard clock of 4.0 GHz. Yes it can consume 125 W of power; thus, I will have maybe more than one fan in the case.

For the needs of my startup, that processor, motherboard, and main memory will do a LOT of computing. E.g., my old computer that died had a single core processor at 1.8 GHz, and it had my Web site pages showing on the screen before I could get my finger off the Enter key. My actual software timings indicate that this new FX-8350 computer will have plenty of capacity to do very nicely as a first server.

For disks, I got two Western Digital drives, SATA at 3.0 Gbps, and 500 GB each.

And I will bring over the SATA drives from my old computer.

And I got lots of fans, cables, etc.

Long ago I got tired of wrestling with file system drive letters. So, I have some directories DATA01, DATA02, DATA03, DATA05, PROG01, PROG02, PROG03, and they have all MY data, and my software works fine no matter what drive letters those directories are on or larger trees they are in. I will be using this little approach again.

I have a thing, goal, I want:

In my past work with Windows for this project, too often I had to reinstall Windows and all my other software, and those reinstalls, even with all the practice I got, were a lot of work. So my goal is to solve this problem of reinstalling.

So, I want:

(1) More than one instance of a bootable operating system.

(2) Each installed instance on its own hard disk partition with little or nothing else important on such a partition.

(3) In case a hard disk with a bootable partition fails, bootable partitions on at least two hard disks.

(4) Good backups of the bootable partitions.

So, if something seems wrong with a bootable partition, say, C:, then I will boot another bootable partition, say, D:, and restore C: from a backup.

I want the restore to be fast, easy, and bit-perfect.

So, how to get such bootable partition backups and do such restores? Well, maybe Acronis True Image or the Western Digital Data Lifeguard Tools will be sufficient -- I have fully legal copies of both. In the past I had some really bad times with Microsoft's NTBACKUP, but I'm willing to try again.

But for Acronis, Lifeguard, NTBACKUP, I accept I will have to run some experiments. So, as soon as the computer is running, I will do the experiments and get saving and restoring a bootable partition fast, easy, reliable, and well documented.

Yes, apparently backup and restore of the files for SQL Server requires some special considerations.

So, backup/restore needs are for (A) instances of installed operating systems, (B) files used by SQL Server, and (C) everything else, that is my data.

For (C) my long used solution is just using Microsoft's XCOPY with some carefully selected options and called from one of several scripts I have in Rexx. I have some scripts for full backup of a file system directory sub-tree and also incremental backups.

For the experiments in backup and restore of bootable partitions, I may just use my old legal copy, that before the experiments are over, I will have to reinstall several times, of Windows XP SP2 Professional.

When I get those experiments done I will install a legal copy I have of Windows 7 64 bit Professional, on bootable partitions on at least two bootable drives, get the updates, and backup up the partitions.

Then I will do software installations and do backups again.

Next get SQL Server running again.

Then try to get my old SQL Server data base working again.

My experience doing such things in the past has me suspect that before I get this far I will have to make use of the results of my experiments in restoring bootable partitions.

Then install the .NET Framework and get my startup software running again.

That will be a good day!

I have a few tweaks in mind for the software.

For the code for the crucial core applied math, I want to check that a third time. The last time I checked that code, I got out a lot of trace data, programmed the calculations again independently, and confirmed that on the trace data both codes gave the same results. I want to do that again.

Then I will do some more data collection.

Then I will give a critical review and consider revisions.

Then comes alpha test, beta test, going live, ....

In short, between now and then, it's all just routine work.

Wow, thanks for the update! Good luck!

Oh wow, this is awesome! Thanks for sharing.

I never learned any music. My only understanding of music came from the physics labs at school, messing around with signal generators. My wife was trained in piano, I've asked her countless times, so what frequency are you playing when you hit this key? Why is this key, called middle C, yet C repeats further down the keyboard? Doesn't a C map to a unique frequency? I've been asking her for a translation of music into physical terms for years -- to no avail (she's a doctor). I'm looking for a "for dummies" type book, but instead, it would be "Music for Engineers with no music background." Also why are there pure C notes and C chords... I want that book!

It's called Middle C because it's the C in the middle of the keyboard.

All the other Cs are called C because we perceive them as "the same note."

Ask her to play a tune, and then play it again an octave higher and then lower. The pitch is different, but there's also a sense that all three versions are the same.

Mathematically, pitch is like a helix. Points with the same rotation - which happen to be a frequency ratio of powers of 2 - sound very closely related.

Instead of giving every single note a different name, it's useful to give notes the same repeating set of 12 names and then add a separate "height" variable - called the octave - to describe which turn of the helix they're on.

If you play pitches with the same rotation at the same time they blend well together. In fact you can barely tell them apart.

Of course you can rotate by less than a full circle. If you play the same tune with a different pitch offset - say one note instead of twelve, which is the smallest rotation you can do on a conventional keyboard - you'll hear it's still the same shape, but there's a jarring effect between the different versions.

Some rotations sound good together, but not as good as the identity rotation. Others clash and sound bad.

As a guide, simple divisions of the pitch circle - like 3:2 - sound smoother than the more complicated divisions.

The twelve notes approximate the rotations that sound best together. You get a good mix of smooth and not so smooth blends, and you can build all kinds of combinations with them.

You use approximations and not exact whole number ratios because you want to be able to write music that isn't stuck with a single starting/reference note. You can play the same tune at different starting pitches and the relationships between the notes doesn't change.

If you use whole number ratios a tune that sounds good starting on one note sounds weird and sour starting on others. The pitch spiral turns into an oval corkscrew, and the neat rotations get distorted.

And yes, all the As, Bs, Cs have their own specific frequencies. By convention A = 440Hz, and everything else is tuned around that. All the As are powers of 2 either side of 440Hz - from 27.5Hz, which is the lowest note on a piano, to 3520Hz for the highest A.

Wow. Thank you for this! Your writing here is making clear what's been the issue. Music people don't speak Hz, they have their own specific terms that I can't parse because they layer many physical concepts.

So true. Communication often breaks down when people who think or speak at different levels of abstraction meet, and their backgrounds collide.

Worst case: they argue in vain. Best case: the above happens

In most Western music there are 12 tones in an octave. So if you play a note with a fundamental frequency of 440 Hz, it will repeat itself (one octave higher) 12 steps later at 880 Hz. Notes one or more octaves apart all have the same name!

Diving deeper: a 'C' on a piano is actually made up of -multiple- frequencies. The 'same' 'C' on any other instrument is made up of a -different- set of frequencies. See: overtones.

About that specific question, the term you're looking for is "pitch class".

I made an interactive "lab" for tuning and customizing my modem. The lab has a spectrograph so you can see how the audio content changes as you tune it. Sorry for the plug, but I thought it might be interesting for people who are into this stuff :)



This is very well done. As a synthesist, I'd suggest (since you got so close) that you add one detail towards the end:

the amplitude and frequencies of the overtones are ALL that gives (apart from attack/decay and modulations) any instrument (physical or electronic) playing a single note its individual character.

It'd be really cool if you could demo that.

Interesting idea! I think that, as another commenter said, there's a lot in that "apart from..." - I feel like I risk oversimplifying it (plus, as I'm sure you know, convincing physical instruments are notoriously hard to reproduce with oscillators and envelopes and filters alone).

But yeah, maybe I could do more to show just how wide the application of this concept is... it's not just limited to abstract waveforms, it's the fundamental rules for all instruments. Will give this some thought (and if you have ideas for how to demo it, let me know!)

This would be a great 'part 2' if you had time. You could also consider demonstrating some vowel synthesis [1].

Great work by the way!

[1] https://www.youtube.com/watch?v=sx5KNQYY_cg

> (apart from attack/decay and modulations)

This is a much bigger factor

if you loop steady state parts of classical instruments it becomes very difficult to differentiate them.

Well, sort of. Depends on how experienced the listener is (and/or is willing to become - many/most listeners CRAVE instant familiarity).

But that importance depends on how different the instruments are already. The saxophone and the clarinet are quite different 'steady state'. Bell and piano are quite different. So I proposed a 'gateway concept'. Else it takes a long time to grok it all.

Eventually the differences all add up to very complex waveforms (I didn't even mention register, which is important for differentiating the classical string families). Personally, I'm glad of this, since the timbre space explored by conventional instruments is a VERY VERY small part of the sonic universe.

Very nice! If you want to take it into the next step of waveform/signal processing, Kevin Reid (kpreid) built an amazing interactive lesson a few years back. Left / Right arrows to navigate the slides, click drag the graph if you like:


FYI: You need to use the arrow keys to navigate this website

I've thought that I needed some such when I last taught trigonometry. Sure, it's a graphical introduction to the applied math of Fourier series.

The OP started saying that the horizontal axis was time but later discussed 180 degrees without connecting degrees with time.

Why the harmonic frequencies are distinct positive whole number multiples of the fundamental frequency? Because the sine waves at those frequencies form an orthogonal coordinate system with all the advantages of such a system in 2 and 3 dimensions we know well. Similarly for linear algebra with positive whole number dimensions. So, right, the periodic waves are in a vector space of countably infinite dimensions. Usually in practice we can get good approximations by considering only the first dozen or two harmonics.

The waves travel according to the wave equation. It would be good to connect with that.

The lecture might have discussed the effect of tone controls and linear systems and linear filtering more generally.

The wrong frequencies on the frequency axis really aren't necessary. It doesn't make it more complex if you put the real frequency values the and it is less confusing to people who know that sound below 20hz roughly is never audible.

Otherwise, great site! I think it is great that Fourier was basically explained without mentioning his name.

The biggest problem was that the waveforms are animated in some parts, and they actually move in-speed (so 1Hz does 1 cycle a second). If it was, say, 50Hz, the inaccuracy would shift so that it was moving at 1/50th speed, and that seems harder to explain (even here, I'm not sure if my explanation makes sense!)

And yeah, hah. My goal was to not mention his name anywhere, since "fourier transforms" just sounds scary and jargony.

This reminded me of Jim Blinn's mathematics work (https://www.youtube.com/watch?v=iO0t1IPk8KA). Very well done. Next up, the FFT :-)

Awesome. Stuff I found interesting: Triangle and Square waves have only "odd" harmonics: 3x, 5x, 7x of the base frequency and so on... Sawtooth has even & odd harmonics. Sound "brightness". Everyone speaks about it but it's something that you learn to recognize by example. For instance it is my impression that sines sound "brighter" than triangles (triangles reminding me of my Atari 2600...) The convergence slider was an excellent addition. Chords have recognizable shapes. Never thought of that. Makes sense.


The new tool Observable is a pretty friendly / straight-forward way to make this type of thing, for folks who want to get started on making interactive explanations.

Here’s one in the same general area as the link under discussion: https://beta.observablehq.com/@freedmand/sounds

(Observable has the additional benefit that readers are treated as peers/authors and can play with the code directly.)

Oh wow, awesome! I've never heard of this, but it seems really great.

Best thing to do from this is to grab a free synthesizer and play around. This is an "easy" on written by college students in Berlin. Has 3 oscillators and 3 filters and can build a ton of sounds from a wave sound.


If you want to go really deep down the rabbit hole, Oppenheim's MIT OCW lecture/text are basically the signals bible. Oppemheim's text is clear and lucid -- and he was Bose's student to boot!


come for the wisdom, stay for the 80s polyester ties.

For years I've been using paper and pen to explain waveforms, noise cancellation, and the psycho-acoustic model to a few of my musician friends. Every time their eyes have glazed over, and I've thought "there's gotta be some animated web page that explains this better." And here it is - exactly as imagined! Thanks for creating it.

Nicely done.

That major chord at the end sounds a little wobbly. Not sure if it's out of tune or tuned exactly to equal temperement and it's just more obviously out of tune with respect to the just intervals because of the timbre.

It would be interesting to have a just major chord (i.e. the 4th, 5th, and 6th harmonics of some note) and compare with the equal tempered triad.

When using sine waves in a single chord the tuning is much more obvious.

I looked at the code and for the major chord it uses:

[f * 1, f * 1.2599388379204892, f * 1.4984709480122322]

With f = 440Hz it will be [440.000, 554.373, 659.327].

Equal temperament (12-TET) would be:

[f * 1, f * 1,25992104989487, f * 1,49830707687668]

With f = 440Hz it will be [440.000, 554.365, 659.255].

Since it isn't pure it will wobbly a little bit.

In this case it would be better to use "just intonation":

[f * 1, f * 1.25, f * 1.5]

With f = 440Hz it will be [ 440, 550, 660 ].

(creator here)

Thanks for this! To be honest, my music theory isn't that deep; I had never heard of equal temperament or just intonation. I think I just did the math based on a note-to-frequency table I googled. Clearly I have some research to do!

I did find the chord a little "off", but I thought that was just me. Originally, the audible frequencies for the other bits were completely outside the western music scale, and so the chord sounded _really_ off.

For what it's worth, it's easy to generate the 12-tone equal tempered scale by starting with 440 and repeatedly multiplying by the 12th root of 2.

For a just scale, it's a bit more complicated because there are sometimes multiple options. For instance, a minor seventh might be 9/5, 16/9, 12/7, or 7/4 depending on context.

In general, a pretty good "default" 12-note chromatic scale if you want to use JI in a way that works with most normal western music is 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, and 15/8. (Just multiply the fraction by the frequency of the root note to get the frequency.) 2/1 would be an octave above the root.

Just major chords are three tones that make a 4:5:6 ratio, such as 4/3, 5/3, and 2/1. You can think of a major chord as the 4th, 5th, and 6th harmonic of a note two octaves below the root. (1:3:5 or other octave-equivalents are also major chords.)

If you add on the 7th harmonic to make a 4:5:6:7 chord (and for this you'll need at least one note not on the scale I provided above), you get a barbershop dominant seventh chord.

Just minor chords make a ratio of 10:12:15, which you could also think of as 4/4 : 4/5 : 4/6.

The advantages of just tuning are that it sounds very stable and allows precise distinctions between very similar notes. Equal temperament is an approximation of just intervals and has been dominant for the last few hundred years largely because it allows instruments with a moderate number of fixed pitch notes to play in any arbitrary key and with each other.

Great, you found my comment!

That explains why I didn't find what tuning you where using.

By the way, there is no contact information on the page (if there is, I didn't see it when I was looking for it.)

Yeah, at the very bottom there's a link to my twitter, but you're right, it's not the most approachable way to collect feedback. Sorry about that!

I like the format of this. It has supplementary "slides" and animations (and sound!), but I can read the material at my own pace. I frickin' hate the recent trend of "documentation through talk". Want to know this weird trick in Kubernetes? Here, watch this talk from the last KuberConf. What? No.

Good work! It takes a lot of effort to make this kind of content.

I would love it if in the diagrams of air molecules, they weren't all lined up vertically. This gives a misleading impression. How difficult a change would it be to draw the dots in spatially random positions, but such that the _density_, not the the positions form vertical bands?

E.g. Like this: https://qph.ec.quoracdn.net/main-qimg-e041242ac2954be7e84765...

While this page is mostly about sound waves, here's a great talk using similar tools to explain RF: http://www.youtube.com/watch?v=DUGr_Z04SKs&t=12m31s

Here's the interactive slides to play with: http://visual-dsp.switchb.org/

This is brilliant! Person who made this: thank you!

Would it be possible to make this available for translation into other languages, as a community project (all hosted in the same place)? This would be awesome for high school physics in any language.

FYI, I think I spotted a tiny mistake:

"Try adjusting the convergence slider to see how the phase of a waveform affects how loud the resulting wave is."

I think this paragraph should say "phase slider"?

Hey there! Creator here.

I actually did set this up with i18n in the back of my mind; all of the copy lives in 1 file, so I imagine it wouldn't be _too_ much work!

If there are volunteers to copy it into other languages (the only other one I know is French and I wouldn't trust myself to get the details right), I can spend some time working on getting it to support it.

EDIT: Also, not a typo :) I wanted to make sure users try out both sliders in that section, since you get a lot of interesting effects. When the convergence is 100%, you get phase cancellation. When it's 50%, you get some really groovy looking waveforms.

This is amazing, I would love to something similar to this in Augmented Reality, here is a snapchat AR experience where you visualize several waveforms:


I was absolutely mind blown by this. Then I found out it was made in React, I was always on the fence about using it, now I’m in love ️

Happy Valentine’s Day

Glad to hear that! :D

I've been working with React almost exclusively for 2-3 years (I do occasionally experiment with Vue, or work with Backbone). Once you get used to thinking in React, it makes projects like this really easy; the data flow never got complicated, even though I'm not using any other "state management" library (don't believe the "React is just the V in MVC" saying!).

If you're interested in getting started with React, I wrote a "no fluff" blog post: https://hackernoon.com/simple-react-development-in-2017-113b...

My goal with the post was to cut out everything unnecessary, and focus on getting the user to build something with React with as little time/effort as possible. React now comes with an amazing build system that abstracts away all the configuration, but a lot of folks starting out don't know that, and wind up spending a week tinkering with Webpack and Babel and build config, and installing and learning dependencies they don't need.

Thanks a lot will definitely check out your post on React.

Very good educational experience. Even if you already know the concepts, the animations help with understanding it all intuitively.

This page reminds me of the interactive demos you'd find in a science museum. Those demos were very influential in my understanding of how basic phenonena worked, though I think they often raised more questions than they answered. Nice job to the creator for making something that, I think, would really get a kid's curiosity going.

There is a nice book on waves called the Wavewatcher's Companion. It's not a mathematical description though.

You guys and gals should check out audiokit https://github.com/AudioKit/AudioKit your one stop shop for all your audio needs (as long as these needs happen on a platform ending with “OS”)

>> WARNING: The slider lets you add up to 75 additional harmonics, but it's computationally intensive to calculate and render all these waves! If you're on a slower device, it may make the page slow / unresponsive if you climb up too high.

ugh just use a (i)FFT, dammit.

How would that help? The IFFT would left you quickly synthesize the total waveform From the frequency spectrum, but the whole point of the visualization is to be able to show all the frequency components and interpolate between the individual components and the resulting waveform. The IFFT doesn't help you there.

No matter how you do it, rendering 75 independent waves will take more computation than rendering 2.

The complexity of a (i)FFT scales with O(n·log(n)), whereas a naive, direct fourier transform scales with O(n·m) where `n` is the number of samples and m is the number of channels. Therefore for any `m > log(n)` the direct method wastes a lot of computational power.

The base of the logarithm depends on the radix of the FFT implementation, but in the end it comes out as a constant factor anyway, and for most FFTs out there it's a base in the range of 2…5, so it's at most a factor of about log(5)/log(2) =~= 2.32

Say you're synthezising a waveform of 1024 samples, then the break even between a fast against a direct fourier transform comes it at around 10 channels. For small numbers of channels the direct method is better. Prime example: Quadratur demodulators where you operate on just a single frequency with the real and imaginary parts.

But when you want to do simple additive synthesis, just put your amplitudes and phases at the right places in the spectrum and to a Inverse fourier transform. It's almost always going to beat any other method. Also it's numerically more stable (although the amplitudes in audio synthesis are never going to be so spread out that precision is going to be a problem).

I understand the complexity and function of the FFT. However, the sluggishness of this demo is not due to the actual synthesis of the final wave, it is due to calculating and render each individual waves. I agree you could easily synthesize the final wave with the IFFT, but the point of the demo is to display all the components. The IFFT wouldn't help with this.

I've been looking for an approachable read on waves and so I like this very much. A nice touch is the retro visitor counter at the bottom of the page. Made me nostalgic. When I got to it it said ~1600. I reloaded the page and it was almost at 3000. The power of HN I guess.

Hah, thanks! I open-sourced the counter separately (although beware, it's a React component): https://github.com/joshwcomeau/react-retro-hit-counter

Also, for any folks who immediately scroll down to see it: the hit-counter only loads a few seconds after the page initializes. This was done intentionally to avoid adding yet another thing to the busy period of initial mount. It means visits of less than a few seconds aren't counted, but -shrugs-.

Another highly recommended presentation and an on-topic followup to waveforms, namely, how waveforms are digitally sampled ...


My eyes just can't stop staring at the animated wave form. I can't read the text on the right side. But the content seems to be very nice.

Not to mention the text fading in is extremely annoying.

(creator here)

Sorry it bugged you! I learned when showing this to people that not everyone focuses their eyes on the same part of the screen, and so the focusing thing might not align to where you like to read.

I needed some way to make it clear which "step" the user is on, since the stuff the waveform does changes from step to step.

Open to suggestions for future audio things I do if you have ideas for how to align them without faded text!

Wow - I _love_ this site! Such a great use of the web - both the interactivity and nicely laid out text lessons.

On a pedagogic level I also love how it builds up too

Well done!

Thank you! The pedagogy was super important to me (I work at Khan Academy, so I suppose that was a given, haha). I really tried to make it not just a gimmick, but have the interactions/animations help clarify what's difficult for words/static images to explain.

I think that kinda stuff should be complimentary for physics lessons in school. Interactive, quick to grasp, the only thing I want is more!

Very nice. Few people play with this using analog equipment any more. Now JavaScript is fast enough to do synthesis in real time.

too much text.

tl;dr: fiddle with knobs to modulate linear sound.

This is great in principle! Computer music fueled my understanding of signals; Just as drawing is instrumental to understanding linear algebra. Even, the two are isomorphic upto dimension, Einsteinian relativity, etc..

In case anyone missed it, the most genius description of the Fourier Transform was published recently by the 3Blue1Brown YouTube site (https://www.youtube.com/watch?v=spUNpyF58BY). It is absolutely brilliant.

Fantastic experience, nothing more to add. Except, well, thanks for this marvel !

i thought the waveform generally represents air pressure not displacement. maybe they end up being the same, or analogous?

The two are related :) I chose displacement because it's a simpler thing to explain visually, and didn't want to cover both.

In the bit with the air molecule grid, the areas where the molecules are clustered together are high-pressure, while the areas with large spaces between them are low. This is caused by the staggered nature of the displacement vibrations.

(I'm sure there's a more sophisticated way to explain this, but that's as I understand it)

Does wave animation has to be so slow on mobile



pitch is a function of frequency AND loudness

This is correct. Although I would say amplitude or intensity, rather than loudness: for loudness itself is a function of intensity and frequency.

At least for pure tones that aren't too high, most listeners perceive a pitch drop at higher intensities.

Both in the Journal of the Acoustical Society of America:

S.S. Stevens, "The relation of pitch to intensity", Vol. 6, 1935, pp. 150-154. http://asa.scitation.org/doi/10.1121/1.1915715

W.B. Snow, "Changes of pitch with loudness at low frequencies", Vol. 8, 1936, pp. 14-19. http://asa.scitation.org/doi/10.1121/1.1915846

When you start seeing cycles or cps instead of Hz in the references you know you have reached the good stuff! x)

Pitch is the perceptual correlate of fundamental frequency. When you make acoustic measurements, you measure frequency, not pitch.

Sorry, as a musician, I've never heard pitch defined this way. Pitch is essentially another word for frequency (though perhaps used more commonly in the context of a musical instrument or a human voice, and most musical instruments emit harmonics, as described in the article, and are thus not a singular frequency).

Merriam-Webster corroborates this:

> the property of a sound and especially a musical tone that is determined by the frequency of the waves producing it : highness or lowness of sound

Wiktionary as well:

> The perceived frequency of a sound or note.


> The quality of a sound governed by the rate of vibrations producing it; the degree of highness or lowness of a tone.

Usually for music, pitch is another term for the frequency of the fundamental tone. So, a sine wave at 440 Hz will seem to the ear to have the same pitch as the open A string on a violin or the first A above middle C on a piano, and the same for other frequencies, pitches, and musical instruments. E.g., we can tune a whole orchestra with just a single tuning fork for A4 at 440 Hz, and when the whole orchestra plays A it sounds good, that is, without beats from the fundamental frequencies of some of the instruments being a few Hz away from a whole number multiple or fraction of 440 Hz.

However, IIRC, this simple relationship between frequency of the fundamental and pitch to the ear does not hold well for all possible periodic sounds.

>The perceived frequency of a sound or note.

I think the claim that pitch is a partial function of loudness is about perception. If pitch is defined as the 'percieved' frequency, then it includes whatever synthesis of sensory data the brain uses.

The most obvious, is a comparison to concurrent or recently heard sounds. A b-flat preceeded by a c really does sound different than one preceeded by a g. (At least, it really does for me).

Edit: But my understanding of pitch has also, always been pitch = frequency.

Amplitude (loudness) has nothing to do with frequency, and changes in amplitude will not affect the frequency perceived. Pitch == frequency.

Consulting the online experts, I see pitch as being a function solely of frequency. Loudness is another attribute of what one hears. There might be some psychoacoustic magic going on that could cause one to interpret a single frequency to have a variable pitch based upon the amplitude of the sound but that's a artifact of the listener, not the sound. Look at it this way. If you stike middle A on a piano with different intensities and monitor the sound using a frequency analyzer, the analzyer will report a sound at approximately 440 Hz regardless of how soft or hard the key is struck.

I think it's okay in a text like this to say frequency = pitch. Frequency is the primary factor in how we perceive pitch.

Since "frequency" has a straightforward definition and a straightforward unit of measure, "pitch" is a fancier word reserved for alternate units (e.g. MIDI note numbers) or slightly modified concepts (e.g. pitch classes, not frequency classes).

this is much better than the manual for my analog synth

Can you _stop_ with asking the reader to mute/unmute/play with the tool?

It is a) patronising and b) super annoying.

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