Click or poke the dots to play them. The stronger the color, the closer, harmonically speaking, that tone is to the root tone, circled in black.

Click ? (top left) for more info, or ⚙ (top right) to tinker.

What is this thing?

It's a keyboard for playing music in an unusual tuning system, called just intonation. I call it Jintone.

Why would I want to play music in this "just intonation" tuning?

Because it sounds more harmonious, or as music nerds would say, consonant (the opposite of dissonant), than the tuning system that all modern music uses. In a sense, all modern music is out of tune, because the tuning system itself that it's all based on is a bit off.

Wait, what?! Are you saying that all modern instruments are tuned wrong?

Well, not wrong, but it's not the most consonant way of tuning your instruments, it's imperfect. Let me explain.

We are used to the idea that all music is built out of tones like C or A♯. But what exactly is a C? A C played by a violin or a Casio synth sound very different, but what they have in common is that they have the same pitch. And the pitch of a sound is just the frequency of oscillations in that sound: Higher pitch, higher frequency. So that's the definition of, say, middle C: It's sound the oscillates at the frequency of 261.626 Hz.

So if you want to build yourself a brand new tuning system, you need to choose a set of frequencies that you define to be your tones, and then music in that tuning system will consist of playing those tones in various ways. In the common Western tuning system we choose 12 tones within each octave, and give them the names C, C♯, D, etc., but you could pick some different number of tones and name them however you want. The point is to have a set of fixed frequencies that are the building blocks of your music.

Here's one way we might build a tuning system. Start by fixing one tone, call it A, to have a frequency of 440Hz. That will be the root tone, the harmonic centre of the tuning system, and we'll choose the other frequencies in relation to this one.

Next, go looking for other frequencies that sound pleasant to the human ear when played together with the A of 440Hz. (After all, we presumably want the music we play to sound pretty.) You'll most probably quickly land on 220Hz and 880Hz, so double and half of the root frequency. In fact, to the human ear these sound so naturally resonant together with the A at 440Hz, and they too, are usually called A, but just one octave below (220Hz) and above (880Hz) the original one. So the interval of an octave just means that the frequencies of two tones are related by a ratio of ²⁄₁.

Once you've added those to your tuning system, and go look for more frequencies that ring nicely and consonantly with the ones you already have, you'll find that all the ones that are related to the root tone of 440Hz by simple ratios stand out. ³⁄₂ · 440Hz = 660Hz and ⁴⁄₃ · 440Hz ≈ 587Hz are usually called the perfect fifth and perfect fourth of the A at 440Hz, because they resonate so beautifully with it. Note that whether the frequency expressed in Hz is a "round number" has nothing to do with this: What matters is that the ratio of the frequencies is something simple, like ³⁄₂.

The list of simple ratios goes on: ⁵⁄₄ gives the interval called major third, ⁶⁄₅ is a minor third, etc. The more complicated the ratios get, the less consonant they start to sound. For instance, two tones with frequencies related by a ratio of ⁶⁵⁄₅₃ probably aren't going to sound especially harmonious together.

You could give some of these tones familiar names, like if A is 440Hz then many people would call the tone at ³⁄₂ · 440Hz = 660Hz E. I personally prefer to just label the tones by the ratios, so call call them ¹⁄₁ and ³⁄₂ instead of A and E.

So this way, starting from a single tone with a fixed frequency, for instance our A at 440Hz, you can generate a whole tuning system by including all the tones whose frequencies are of the form of a some simple ratio times 440Hz. What exactly counts as simple, is in the ear of the listener. You most probably want ²⁄₁, ³⁄₂, and ⁴⁄₃, as well as inverses and multiples of theirs, like ½ and ⁸⁄₃. Most people would probably include ⁶⁄₅ and ⁵⁄₄ too. Maybe also for instance ⁹⁄₈ and ¹⁶⁄₉. If you are adventurous, you can try ⁷⁄₄ or ¹⁶⁄₁₁ as well. Whatever pleases you ears.

A tuning system like this, is called a just intonation tuning.

I'm not sure I understand. You are multiplying all these ratios by 440Hz to get these frequencies you say sound harmonious together, but what's the importance of 440Hz?

Nothing at all! 440Hz happens to be the frequency to which most musicians are used to tuning their middle A, but it's an entirely arbitrary number. You could just as well choose 123Hz or 666Hz. The magic is all in the ratios. See, unless you are one of the very few people with with so called "absolute pitch" ability, humans can't really tell whether the frequency of some tone they hear is 440Hz or 450Hz or something else in the same ballpark. But what they can tell with great accuracy, is the relation between two frequencies that they hear at the same time, or in close succession.

So the root frequency that we chose to be 440 Hz is arbitrary, but the special status of tones related by simple fractions comes directly from some simple physics of sound waves: When you play at the same time two frequencies related by a a ratio like ³⁄₂, the two sound waves interact in a particular way (keywords: resonance, interference), so that the result sounds like more than the sum of its parts, and the combined sound has special qualities that people find beautiful and interesting. This special nature of simple fractions is literally what all of harmony is about!

Okay, that makes sense. But this ratio-based just intonation isn't the way we usually tune our instruments then?

Not at all! The usual (Western) tuning system that's probably used by all music you've ever heard works like this: Take the octave, the ratio of ²⁄₁, and split it into 12 equal parts, and those are your tones. So, starting again from the A at 440Hz, you would include A♯ at 2^¹⁄₁₂ · 440Hz, B at 2^²⁄₁₂ · 440Hz, etc., all the way to G♯ at 2^{¹¹⁄₁₂} · 440Hz and the next A at 2^{¹²⁄₁₂} · 440Hz = 880Hz. This system is called the 12-EDO (Equal Division of Octave) or 12-TET (12-Tone Equal Temperament) tuning.

But why on Earth does that work at all? You just said that to get tones that sound nice and consonant together you need the frequencies to be related by simple ratios, but numbers like 2^³⁄₁₂ are not simple ratios at all!

That's true, but the magic of the 12-EDO tuning system is that they happen to be pretty good approximations to the simple ratios. For instance, 2^⁷⁄₁₂ ≈ 1.498 is very close ³⁄₂ = 1.5. In practice, the human ear usually can't hear a difference that small. A little less perfect is the 12-EDO idea of what counts as a major third: 2^⁴⁄₁₂ ≈ 1.26, where as the exact ratio that one uses in just intonation is ⁵⁄₄ = 1.25. That's a difference you can easily hear already, although it's not too jarring (dissonant) to bother most people, especially since we are all very used to it.

Note that it's very important for this work that you divide your octave into 12 equal parts, and not say 11 or 13. Essentially, it's just a very fortunate coincidence that numbers like 2^⁷⁄₁₂ and 2^⁴⁄₁₂ happen to be very close to the simple ratios that sound consonant to us, so that the division of the octave to 12 equally spaced pitches produces tones that sound acceptably pleasant together. Wikipedia has a chart that illustrates this coincidence really well.

If the usual 12-EDO tuning system is just an approximation to just intonation, why are we using it? Why isn't everybody using just intonation?

There are a few reasons for that. The 12-EDO tuning is extremely versatile, flexible, and fail-safe. For instance, all pairs of neighbouring tones in 12-EDO are always related to each other in exactly the same way: By a ratio of 2^¹⁄₁₂. This means that you can easily switch keys without retuning your instruments, and the instrument sounds exactly the same in any key. This isn't true at all in just intonation: In the example above, we built our tuning around the A at 440Hz. If you try to use the same set of tones generated from that to play music that is centered around some other tone (i.e. to play in another key), it's not going to work well.

12-EDO also never has any really jarring intervals in it, but just intonation may. For instance, consider going up by three perfect fifths and a minor third, and then down by two octaves. In 12-EDO, you end up back where you started, for example C₁ → G₁ → D₂ → A₂ → C₃ → C₁. In just intonation, this means going to a tone with a frequency ³⁄₂ · ³⁄₂ · ³⁄₂ · ⁶⁄₅ · ½ · ½ = ⁸¹⁄₈₀ = 1.0125 times the original one you started from. That's almost 1.0, which would mean coming back to where you started, but off by just enough that the human ear can tell, and it sounds very, very dissonant and out of tune. (These kinds of problematic intervals are sometimes called "commas" when talking about tuning systems.)

One final disadvantage of just intonation is that for many types of music you might want to play, you just need a lot of tones in your just intonation system. For instance, if you go up by two major thirds, you end up with a ratio of ²⁵⁄₁₆ ≈ 1.56; up a major sixth is ⁸⁄₅ = 1.6; and up a perfect fourth and down a major second is ¹²⁸⁄₈₁ ≈ 1.58. In 12-EDO all of these are approximated by the same tone with an interval of 2^⁸⁄₁₂ ≈ 1.59. That may be inaccurate in a sense, but from the point of view of building an actual instrument, it's quite a lot easier, when you don't have to build so many keys or frets (though this is something I hope technology like Jintone can overcome). It also allows for all kinds of harmonic transitions and tricks, when you don't have to commit to saying whether the interval you are playing really is ⁸⁄₅ or ²⁵⁄₁₆, and can make use of this ambiguity.

So the traditional 12-EDO tuning system isn't wrong, but it's a compromise. It gives up purity of harmony, that quality of being perfectly in tune, for many practical benefits.

Okay, enough numbers and theory. How do I play this thing?

Just click on any of the dots to play that tone. Poking them on a touch screen should also work, although some devices and browsers are better at handling this than others.

On the left you can see three buttons, Drone (blue), Sustain (yellow), and Move (red), as well as their keyboard shortcuts. Sustain works like the sustain pedal of a piano: Trying holding it while playing. While holding the Drone button, you can click tones to turn them on, and they'll keep playing even if you let go of the Drone button. You need to hold Drone and click on them again to turn them off. Finally, the Move button activates the panning mode: Hold it down and click-and-drag to move around your view of the keyboard. The Settings tab also has sliders for zooming in and out.

At the bottom, there's also a traditional 12-EDO piano keyboard, which may help orient yourself, and see how just intonation and 12-EDO compare. You can play it too, by clicking on the keys.

What's the logic and meaning of how the dots/tones are laid out?

The left-right axis corresponds to pitch, exactly like on a piano keyboard. So tones on the left are low pitch, the right is high pitch. If a dot is exactly above some piano key, then they have exactly the same pitch too. The grey vertical lines and the notches at the tops of the piano keys mark the exact positions.

There's in principle an infinite number of tones in the just intonation system, even within one octave, so you can't just lay all of them next to each other like you do on a piano, which only has 12. So what I've done is I've spread them out along the up-down axis. This is done in such a way, that one interval always corresponds to making the same kind of step on the screen regardless of where you start that interval from. So for example, if you find the triangle formed by the three tones in a major chord, e.g. ¹⁄₁, ⁵⁄₄ and ³⁄₂, then you can be certain that any triplet of notes anywhere on the screen that form a triangle of the same shape are also a major chord. And the same holds for any type of chord, with any number of tones: Same shape, same chord.

More specifically, in the default layout (there are other layouts available in Settings), each band of tones that runs from bottom-left to top-right corresponds to moving up and down by perfect fifths (³⁄₂) and fourths (⁴⁄₃) only. Jumping to the band below means going up by a minor third (⁶⁄₅) and jumping to the band above corresponds to going up by a major third (⁵⁄₄).

Finally, there's also the colours of the tones. The stronger the color, the simpler the harmonic relation between the root tone (¹⁄₁) and the given tone. So ²⁄₁ (an octave) is a very strong red, ¹²⁸⁄₈₁ is barely visible.

Where should I start if I want to witness the difference between just intonation and 12-EDO?

Try the following: Hold down Shift (or touch the Sustain bar if you don't have keyboard), and click on ¹⁄₁, ⁵⁄₄ (a bit above and right), and ³⁄₂ (further to the right). The three tones should now be playing at the same time, and they form the (just intonation) C major chord. Listen to its beautiful sound and vibration. Now release Shift and press it down again, and click on the corresponding piano keys directly below the tones you just played: C, E, and G. Now you're listening to the 12-EDO version of the same C major. Notice how it sounds a bit wobbly or muddy in comparison? That's the difference between pure harmony and the 12-EDO approximation to it.

You can also click on the ⁵⁄₄ by itself and then on the E on the piano keys below it. These are the just intonation and 12-EDO versions of the middle tone in the previous chord, and you can probably easily hear the difference. If you play them at the same time, you can hear the beating between them.

If you get into it and spend a while studying the various interesting pure chords and intervals you can play, at some point I would recommend going to Settings, scrolling down to the section called Tones, and trying the basic 7-limit preset. This adds more tones to the keyboard, and many of these are such that they just can't be even decently approximated in the 12-EDO system. Try for instance playing a chord with ¹⁄₁, ⁷⁄₆, and ³⁄₂. It's a beautiful sound, but one that is impossible to recreate on a piano, and simply absent from modern Western music.

You mentioned the Settings tab a couple of times. What's that?

It's the thing that pops out when you click on the ⚙ in the top-right corner. You'll find some simple things there, such as

• vertical and horizontal zoom
• hiding/showing features like the 12-EDO keyboard
• how you want the tones to be labeled (fractions vs letters)
• presets for how the tones should be laid out

And if you dig deeper you'll find that you can customise almost anything:

• add 7-based intervals to your tuning system, or 23-based if you want
• recolor the whole keyboard with the most distasteful neon shades you dare use
• change how harmonically-far-away tones are shown
• change the way the tones our laid out
• etc.

All the changes you make are written into the URL, so if you bookmark the page after you make changes, or copy-paste the URL, you can keep all the settings.

This theory of tuning systems stuff seems pretty cool, I'm intrigued. How do I learn more?

Wikipedia should obviously be your first stop, maybe read the articles on EDO tunings (there are more than just 12-EDO) and just intonation to get started, and follow the links from there. Beyond that, I was greatly enlightened and inspired by the books of W. A. Mathieu: Bridge of Waves is a paperback popular audience intro to the topic, and Harmonic Experience is the hard cover hard core epic that you can spend years studying.

Who made this?

Jintone was made by me, Markus Hauru. If you have any feedback, questions, or feature requests, feel free to email markus@mhauru.org.

Inspiration for Jintone came largely from talking to my friend Ismo Tähtinen, and from the books of W. A. Mathieu, which Ismo got me to read. He, and another friend of mine, Henri Seijo, also provided some great feedback.

The implementation is all in Javascript, and available, free, on GitHub: github.com/mhauru/Jintone. A special shout-out to the libraries RxJS, Tone.js, and SVG.js, which Jintone makes heavy use of.

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