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euclidean-rhythms.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Euclidean rhythms</title>
<meta name="description" content="Using Euclidean rhythms to make music">
<style>
:root {
--border-code: #764b3b;
}
body {
margin: 4em 2em;
line-height: 1.5;
max-width: 40em;
}
@media (width < 40em) {
body {
margin: 1em 0em;
}
}
body {
margin-block-end: 5em;
}
@media (prefers-color-scheme: dark) {
body {
background-color: oklch(31.14% 0.021 285.75);
color: oklch(90% 0.008 286.75);
}
a {
color: #a2cffe;
}
a:visited {
color: #ca9bf7;
}
}
h2, h3, p, ol {
margin-inline: 3rem;
}
p, li {
margin-block: 2.25em;
}
h3 {
margin-block-start: 5rem;
}
h2 + h3 {
margin-block-start: 2.25rem;
}
ol {
padding-inline-start: 1em;
}
li {
padding-inline-start: 1ch;
}
pre {
margin-inline: 1rem;
padding: 2.5rem;
border: 1px solid var(--border-code);
overflow-x: auto;
}
@media (width < 40em) {
h2, h3, p, ol {
margin-inline: 1rem;
}
pre {
margin-inline: 0;
padding-inline: 1rem;
border-inline: 0;
}
}
blockquote {
border-inline-start: 2px solid #bbb;
margin-inline-start: 1rem;
margin-block: 3rem;
opacity: 0.8;
}
blockquote p {
margin-block: 1.5rem;
margin-inline-start: 2rem;
}
@media (width < 40em) {
blockquote p {
margin-inline-start: 1rem;
}
}
button {
cursor: pointer;
background-color: transparent;
padding: 1ch 2ch;
font-size: 1rem;
min-width: 5rem;
border: 1px solid var(--border-code);
color: inherit;
border-radius: 3px;
}
.playing {
border-color: tomato;
}
footer {
margin-block: 20svh;
padding-block-start: 2.25em;
}
</style>
<style>
.beats {
margin-inline: 3em;
}
@media (width < 40em) {
.beats {
margin-inline: 1rem;
}
}
.beats {
height: 100px;
margin-block-start: 2rem;
margin-block-end: 1.5rem;
display: flex;
gap: 18px;
@media (width < 400px) {
gap: min((100% - 12 * 10px) / 11, 18px);
}
& > div {
width: 10px;
border: 1px solid tomato;
box-sizing: border-box;
}
}
.beats > div {
visibility: hidden;
}
.beats > div[data-state] {
visibility: visible;
}
.beats > div[data-state="on"] {
background-color: tomato;
}
.beats > div[data-state="accent"] {
background-color: lawngreen;
}
.beats.demo-2 {
height: 40px;
justify-content: space-between;
& > div {
width: 20px;
}
}
.beats.demo-3 {
height: 10px;
justify-content: space-between;
& > div {
width: 10px;
border-radius: 5px;
}
}
</style>
<script type="module">
import { beep } from "./js/javascript-audio.ts";
import { E } from "./js/euclid.ts";
const e38 = E(3, 8);
const e78 = E(7, 8);
const initialCycleState = {
k: 3 /* The `k` in `E(k, n)` */,
n: 4 /* The `n` in `E(k, n)` */,
p: 0 /** The phase of (i.e. the offset into) the current E(k, n) */,
};
const cycleTick = ({ k, n, p }) => {
p = p + 1;
if (p === n) {
p = 0;
k = k + 1;
if (k === n) {
k = n < 5 ? 1 : 3;
n = n + 1;
if (n === 13) {
n = 4;
}
}
}
return { k, n, p };
};
// Demo 1 --- E(3, 8)
const b1 = document.querySelector("button.demo-1");
const ticker1 = { i: 0, intervalId: undefined };
b1.addEventListener("click", () => {
if (ticker1.intervalId) {
ticker1.intervalId = (clearInterval(ticker1.intervalId), undefined);
} else {
ticker1.intervalId = setInterval(() => {
ticker1.i = (ticker1.i + 1) % e38.length;
update1();
}, 1000 / 7);
}
update1();
});
const update1 = () => {
const { intervalId, i } = ticker1;
const divs = document.querySelectorAll(".demo-1 > div")
for (const [j, v] of e38.entries()) {
const on = intervalId && v && i == j;
if (on) beep(0.01);
divs[j].setAttribute("data-state", on ? "on" : "off");
}
intervalId ? b1.classList.add("playing") : b1.classList.remove("playing");
}
update1();
// Demo 2 --- Cycle
const b2 = document.querySelector("button.demo-2");
const ticker2 = { state: { ...initialCycleState }, intervalId: undefined };
b2.addEventListener("click", () => {
if (ticker2.intervalId) {
ticker2.intervalId = (clearInterval(ticker2.intervalId), undefined);
} else {
ticker2.intervalId = setInterval(() => {
ticker2.state = cycleTick(ticker2.state);
update2();
}, 1000 / 7);
}
update2();
});
const update2 = () => {
const { intervalId, state } = ticker2;
const divs = document.querySelectorAll(".demo-2 > div")
const { k, n, p } = state;
for (const div of divs) div.removeAttribute("data-state");
for (const [j, v] of E(k, n).entries()) {
const on = intervalId && v && p == j;
if (on) beep(0.01);
divs[j].setAttribute("data-state", on ? "on" : "off");
}
intervalId ? b2.classList.add("playing") : b2.classList.remove("playing");
}
update2();
// Demo 3 --- Modulate
const onset = E(9, 15);
const accent = E(4, 15);
const b3 = document.querySelector("button.demo-3");
const ticker3 = {
/* phase (indexes into both onset and accent rhythms) */
p: 0,
intervalId: undefined,
};
b3.addEventListener("click", () => {
if (ticker3.intervalId) {
ticker3.intervalId = (clearInterval(ticker3.intervalId), undefined);
} else {
ticker3.intervalId = setInterval(() => {
ticker3.p = (ticker3.p + 1) % onset.length;
update3();
}, 1000 / 7);
}
update3();
});
const update3 = () => {
const { intervalId, p } = ticker3;
const divs = document.querySelectorAll(".demo-3 > div")
for (const [j, v] of onset.entries()) {
const on = intervalId && v && p == j;
const an = on && accent[j];
if (on) {
if (an) {
beep(0.02, 0.01, 0.1);
} else {
beep(0.08, 0.001, 0.1);
}
}
divs[j].setAttribute("data-state", on ? (an ? "accent" : "on") : "off");
}
intervalId ? b3.classList.add("playing") : b3.classList.remove("playing");
}
update3();
// Demo 4 --- Everything
const b4 = document.querySelector("button.demo-4");
const ticker4 = { state: { ...initialCycleState }, intervalId: undefined };
b4.addEventListener("click", () => {
if (ticker4.intervalId) {
ticker4.intervalId = (clearInterval(ticker4.intervalId), undefined);
} else {
ticker4.intervalId = setInterval(() => {
ticker4.state = cycleTick(ticker4.state);
update4();
}, 1000 / 7);
}
update4();
});
const update4 = () => {
const { intervalId, state } = ticker4;
const { k, n, p } = state;
const p8 = p % 8;
const divsA = document.querySelectorAll(".demo-4a > div")
for (const [j, v] of e38.entries()) {
const on = intervalId && v && p8 == j;
if (on) beep(0.01, 0.001, 0.1, 660);
divsA[j].setAttribute("data-state", on ? "on" : "off");
}
const divsB = document.querySelectorAll(".demo-4b > div")
for (const div of divsB) div.removeAttribute("data-state");
for (const [j, v] of E(k, n).entries()) {
const on = intervalId && v && p == j;
if (on) {
beep(0.01);
beep(0.005, 0.001, 0.02, 660);
}
divsB[j].setAttribute("data-state", on ? "on" : "off");
}
const divsC = document.querySelectorAll(".demo-4c > div")
for (const [j, v] of e78.entries()) {
const on = intervalId && v && p8 == j;
if (on) beep(0.1, 0.001, 0.1, 110, 0.05);
divsC[j].setAttribute("data-state", on ? "on" : "off");
}
intervalId ? b4.classList.add("playing") : b4.classList.remove("playing");
}
update4();
</script>
</head>
<body>
<h2>Using Euclidean rhythms to make music</h2>
<h3>Euclid's algorithm</h3>
<p>
Sometimes proclaimed as the oldest algorithm put down on paper, Euclid's
algorithm is enthralling in its recursive simplicity.
</p>
<p>
Euclid's version used a minus instead of a modulo (<code>%</code>), but
otherwise here it is in its full glory.
</p>
<pre><code>gcd = (n, m) => m ? gcd(m, n % m) : n
</pre></code>
<h3>Euclidean rhythms</h3>
<p>
At the surface level it is calculating the GCD, the largest number that divides
both <code>n</code> and <code>m</code>. But really it is unravelling a number pair.
</p>
<p>
In 2005, Godfried Toussaint realized that Euclid's algorithm encodes many drum
beats. For example, here is how <code>E(3, 8)</code> sounds.
</p>
<div class="beats demo-1">
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
</div>
<p><button class="demo-1">Play / Pause</button></p>
<h3>Code</h3>
<p>Here is an implementation of the algorithm described by Toussaint.</p>
<pre><code>const E = (k, n) => {
let s = Array(n).fill()
.map((_, i) => (i < k ? [1] : [0]))
let d = n - k
n = Math.max(k, d)
k = Math.min(k, d)
let z = d
while (z > 0 || k > 1) {
for (let i = 0; i < k; i++)
s[i].push(...s[s.length - 1 - i])
s.splice(-k)
z = z - k
d = n - k
n = Math.max(k, d)
k = Math.min(k, d)
}
return s.flat()
};
</pre></code>
<p>
At each step, we "fold" the end of the sequence into the beginning. How much to
fold is given by the value of <code>gcd(n, m)</code> at that step.
</p>
<p>
When we call this function with two numbers (<em>the number of 1's</em>, and
<em>the length of the sequence</em>), it returns a sequence where the 1's are
maximally displaced. e.g. if you copy paste this code to your browser's
developer tools, and then call <code>E(3, 8)</code>, you'll get
</p>
<pre><code>> E(3, 8)
[1, 0, 0, 1, 0, 0, 1, 0]
</pre></code>
<p>
Surprisingly, this function is all we need to replicate many real world rhythms.
All we need is a way to convert the <em>ones</em> into sound, like how we did
with the player above, which produced a beep each time it found a one in the
<code>E(3, 8)</code> sequence.
</p>
<blockquote>
<p>
See <a href="https://github.com/mnvr/mrmr.io/blob/main/js/er.js">the source code
of this site</a> for a version of this function with comments and more examples.
</p>
</blockquote>
<h3>Cycling with Euclid</h3>
<p>Here is a player that cycles through various Euclidean rhythms.</p>
<p>I have listened to this for hours.</p>
<div class="beats demo-2">
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
</div>
<p><button class="demo-2">Cycle</button></p>
<h3>Not just beats</h3>
<p>
It is customary to use Euclidean rhythms to trigger onsets of beats, but there
is nothing stopping us from using it for other purposes, as a generic organic
pattern.
</p>
<p>Here we use it to accent certain notes by using a different attack and release.</p>
<div class="beats demo-3">
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
</div>
<p><button class="demo-3">Modulate</button></p>
<h3>Everything at once</h3>
<p>
And we can do all these at once. The integral framework provided by Euclidean
rhythms ensures that the mismash doesn't sound as chaotic as mixing arbitrary
patterns would've. It might even sound nice to some.
</p>
<div>
<div class="beats demo-4a">
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
</div>
<div class="beats demo-4b">
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
</div>
<div class="beats demo-4c">
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
<div></div>
</div>
</div>
<p><button class="demo-4">Everthing at once</button></p>
<footer>
<small>
<p>
There is also a <a href="https://mnvr.github.io/gm1k/e/">standalone single HTML
file</a> version of this song with no dependencies.
</p>
<p>
I also wrote a small tutorial on how to make <a href="/javascript-audio">sounds
using JavaScript</a> that might help with understanding the code used to convert
the output of Euclid's algorithm into sound in your browser.
</p>
</small>
</footer>
</body>
</html>