25:["$","title",null,{"children":"Euclidean rhythms | Ped Ram | Observable"}] 49:T585,md` The _Euclidean rhythm_ in music was discovered by [Godfried Toussaint](https://en.wikipedia.org/wiki/Godfried_Toussaint) in 2004 and is described in the paper [The Euclidean Algorithm Generates Traditional Musical Rhythms](http://cgm.cs.mcgill.ca/~godfried/publications/banff.pdf) from 2005. In his paper Toussaint describes an [algorithm developed by Eric A. Bjorklund](https://web.archive.org/web/20161222131610/https://ics-web.sns.ornl.gov/timing/Rep-Rate%20Tech%20Note.pdf) in connection with the operation of certain components [...] of spallation neutron source (SNS) accelerators used in nuclear physics:

[A] gate is to be enabled by a timing system that generates pulses [...]. The problem for a given number ${tex`n`} of time intervals, and another given number ${tex`k < n`} of pulses, is to distribute the pulses as evenly as possible among these intervals. [...] The problem [...] reduces to the following: construct a binary sequence of ${tex`n`} bits with ${tex`k`} one’s, such that the ${tex`k`} one’s are distributed as evenly as possible among the zero’s. [...] The problem of primary interest is when ${tex`k`} and ${tex`n`} are relatively prime numbers, i.e., when ${tex`k`} and ${tex`n`} are evenly divisible only by ${tex`1`}.

Toussaint goes on by following one of Bjorklund's examples. Let's go through his instructions step by step.`4a:T450,function drawSequencer(patterns) { const w = Math.floor(width / 3 * 2); const p = w / 20; const r1 = w / 100; const r2 = w / 60; const h = (r2 + 3) * 2 * patterns.length + 10; const svg = d3.create("svg") .attr("width", w) .attr("height", h) .attr("viewBox", `0 0 ${w} ${h}`); const y = d3.scalePoint() .domain(d3.range(0, patterns.length)) .range([5, h - 10]) .padding(0.25); const maxSteps = d3.max(patterns, d => d.steps); const xWidth = Math.min(w - p * 2, (r2 + 3) * 2 * maxSteps); patterns.forEach((pattern, i) => { const rhythm = rotate(E(pattern.pulses, pattern.steps), pattern.rotation).split(''); const data = rhythm.map((d, j) => ({ ...pattern, pulse: +d, index: j, color: colors[i % colors.length] })); const x = d3.scalePoint() .domain(d3.range(0, maxSteps)) .range([p, xWidth]); const g = svg.append("g") .attr("transform", `translate(0 ${y(i)})`); g.selectAll("circles") .data(data) .join("circle") .attr("cx", (_, i) => x(i)) }); return svg; }4b:T1ab4,examples = [ { name: "E(2,3)", pulses: 2, steps: 3, pattern: "x.x", description: "_E(2,3)_ = [x . x] is a common Afro-Cuban drum pattern. For example, it is the conga rhythm of the (6/8)-time _Swing Tumbao_. It is also common in Latin American music, as for example in the _Cueca_." }, { name: "E(2,5)", pulses: 2, steps: 5, pattern: "x.x..", description: "_E(2,5)_ = [x . x . .] is a thirteenth century Persian rhythm called _Khafif-e-ramal_. It is also the metric pattern of the second movement of Tchaikovsky’s _Symphony No. 6_. When it is started on the second onset ([x . . x .]) it is the metric pattern of Dave Brubeck’s _Take Five_ as well as _Mars from The Planets_ by Gustav Holst." }, { name: "E(3,4)", pulses: 3, steps: 4, pattern: "x.xx", description: "_E(3,4)_ = [x . x x] is the archetypal pattern of the _Cumbia_ from Colombia, as well as a _Calypso_ rhythm from Trinidad. It is also a thirteenth century Persian rhythm called _Khalif-e-saghil_, as well as the _trochoid choreic_ rhythmic pattern of ancient Greece." }, { name: "E(3,5)", pulses: 3, steps: 5, pattern: "x.x.x", description: "_E(3,5)_ = [x . x . x], when started on the second onset, is another thirteenth century Persian rhythm by the name of _Khafif-e-ramal_, as well as a Rumanian folk-dance rhythm." }, { name: "E(3,7)", pulses: 3, steps: 7, pattern: "x.x.x..", description: "_E(3,7)_ = [x . x . x . .] is a _Ruchenitza_ rhythm used in a Bulgarian folk-dance. It is also the metric pattern of Pink Floyd’s _Money_." }, { name: "E(3,8)", pulses: 3, steps: 8, pattern: "x..x..x.", description: "The Euclidean rhythm _E(3, 8)_ [...] is none other than one of the most famous on the planet. In Cuba it goes by the name of the _tresillo_ and in the USA is often called the _Habanera_ rhythm used in hundreds of rockabilly songs during the 1950’s. It can often be heard in early rock-and-roll hits in the left-hand patterns of the piano, or played on the string bass or saxophone. A good example is the bass rhythm in Elvis Presley’s _Hound Dog_. The _tresillo_ pattern is also found widely in West African traditional music. For example, it is played on the _atoke_ bell in the _Sohu_, an _Ewe_ dance from Ghana." }, { name: "E(4,7)", pulses: 4, steps: 7, pattern: "x.x.x.x", description: "_E(4,7)_ = [x . x . x . x] is another _Ruchenitza_ Bulgarian folk-dance rhythm." }, { name: "E(4,9)", pulses: 4, steps: 9, pattern: "x.x.x.x..", description: "_E(4,9)_ = [x . x . x . x . .] is the _Aksak_ rhythm of Turkey [6]. It is also the metric pattern used by Dave Brubeck in his piece _Rondo a la Turk_." }, { name: "E(4,11)", pulses: 4, steps: 11, pattern: "x..x..x..x.", description: "_E(4,11)_ = [x . . x . . x . . x .] is the metric pattern used by Frank Zappa in his piece titled _Outside Now_." }, { name: "E(5,6)", pulses: 5, steps: 6, pattern: "x.xxxx", description: "_E(5,6)_ = [x . x x x x] yields the _York-Samai_ pattern, a popular Arab rhythm, when started on the second onset." }, { name: "E(5,7)", pulses: 5, steps: 7, pattern: "x.xx.xx", description: "_E(5,7)_ = [x . x x . x x] is the _Nawakhat_ pattern, another popular Arab rhythm." }, { name: "E(5,8)", pulses: 5, steps: 8, pattern: "x.xx.xx.", description: "_E(5,8)_ = [x . x x . x x .] is the Cuban _cinquillo_ pattern [...]. When it is started on the second onset it is also the _Spanish Tango_ and a thirteenth century Persian rhythm, the _Al-saghilal-sani_." }, { name: "E(5,9)", pulses: 5, steps: 9, pattern: "x.x.x.x.x", description: "_E(5,9)_ = [x . x . x . x . x] is a popular Arab rhythm called _Agsag-Samai_. When started on the second onset, it is a drum pattern used by the _Venda_ in South Africa, as well as a Rumanian folk-dance rhythm." }, { name: "E(5,11)", pulses: 5, steps: 11, pattern: "x.x.x.x.x..", description: "_E(5,11)_ = [x . x . x . x . x . .] is the metric pattern used by Moussorgsky in _Pictures at an Exhibition_." }, { name: "E(5,12)", pulses: 5, steps: 12, pattern: "x..x.x..x.x.", description: "_E(5,12)_ = [x . . x . x . . x . x .] is the _Venda_ clapping pattern of a South African children’s song." }, { name: "E(5,16)", pulses: 5, steps: 16, pattern: "x..x..x..x..x...", description: "_E(5,16)_ = [x . . x . . x . . x . . x . . . ] is the _Bossa-Nova_ rhythm necklace of Brazil. The actual _Bossa-Nova_ rhythm usually starts on the third onset [...]. However, there are other starting places as well, as for example [x . . x . . x . . x . . . x . .]." }, { name: "E(7,8)", pulses: 7, steps: 8, pattern: "x.xxxxxx", description: "_E(7,8)_ = [x . x x x x x x] is a typical rhythm played on the _Bendir_ (frame drum), and used in the accompaniment of songs of the _Tuareg_ people of Libya." }, { name: "E(7,12)", pulses: 7, steps: 12, pattern: "x.xx.x.xx.x.", description: "_E(7,12)_ = [x . x x . x . x x . x .] is a common West African bell pattern. For example, it is used in the _Mpre_ rhythm of the _Ashanti_ people of Ghana." }, { name: "E(7,16)", pulses: 7, steps: 16, pattern: "x..x.x.x..x.x.x.", description: "_E(7,16)_ = [x . . x . x . x . . x . x . x .] is a _Samba_ rhythm necklace from Brazil. The actual Samba rhythm is [x . x . . x . x . x . . x . x .] obtained by starting _E(7,16)_ on the last onset. When _E(7,16)_ is started on the fifth onset it is a clapping pattern from Ghana." }, { name: "E(9,16)", pulses: 9, steps: 16, pattern: "x.xx.x.x.xx.x.x.", description: "_E(9,16)_ = [x . x x . x . x . x x . x . x .] is a rhythm necklace used in the Central African Republic. When it is started on the fourth onset it is a rhythm played in West and Central Africa, as well as a cow-bell pattern in the Brazilian _samba_. When it is started on the penultimate onset it is the bell pattern of the _Ngbaka-Maibo_ rhythms of the Central African Republic." }, { name: "E(11, 24)", pulses: 11, steps: 24, pattern: "x..x.x.x.x.x..x.x.x.x.x.", description: "_E(11,24)_ = [x . . x . x . x . x . x . . x . x . x . x . x .] is a rhythm necklace of the _Aka_ Pygmies of Central Africa. It is usually started on the seventh onset." }, { name: "E(13,24)", pulses: 13, steps: 24, pattern: "x.xx.x.x.x.x.xx.x.x.x.x.", description: "_E(13,24)_ = [x . x x . x . x . x . x . x x . x . x . x . x .] is another rhythm necklace of the _Aka_ Pygmies of the upper _Sangha_. It is usually started on the fourth onset." } ]4c:[{"id":"5b2a35ee88796fd63c047280f19bb17bdee8690473c93a70b1dc7a4c7afc8d72efe657e2cd15fcd483cc7ad26642dba4419ee539e26b5c64777db3a7d6a444c2","href":"https://static.observableusercontent.com/files/5b2a35ee88796fd63c047280f19bb17bdee8690473c93a70b1dc7a4c7afc8d72efe657e2cd15fcd483cc7ad26642dba4419ee539e26b5c64777db3a7d6a444c2","name":"Kick.wav","type":"audio/vnd.wave","lastModified":1679562954648.59,"size":30988},{"id":"3f55efcd9e49866a867ce332a82ed65fa62ce27f49ab9239f10201c995b650587fafcf9952e9b6048138372147ce19a06591efd35233bdb9aebcab5049ba9aae","href":"https://static.observableusercontent.com/files/3f55efcd9e49866a867ce332a82ed65fa62ce27f49ab9239f10201c995b650587fafcf9952e9b6048138372147ce19a06591efd35233bdb9aebcab5049ba9aae","name":"Rim.wav","type":"audio/vnd.wave","lastModified":1679562954648.59,"size":26570},{"id":"88932362e4452dd55ed367c658a1989a70ae77e85c035096f030cca900549e0099701cfc6a04ab37a21bbc481e2bc62f09dc96d6466ed435e78f10760620bf8f","href":"https://static.observableusercontent.com/files/88932362e4452dd55ed367c658a1989a70ae77e85c035096f030cca900549e0099701cfc6a04ab37a21bbc481e2bc62f09dc96d6466ed435e78f10760620bf8f","name":"HiHat.wav","type":"audio/vnd.wave","lastModified":1679562954648.59,"size":90002},{"id":"3880e86ce7c87462d65318ef994fb4dd2bd5ce87bd98f9c028ec9d399595d56556817d1bb99c47ec05c483b9926d02fe11aa4952b72798555b1c18ffc63c8359","href":"https://static.observableusercontent.com/files/3880e86ce7c87462d65318ef994fb4dd2bd5ce87bd98f9c028ec9d399595d56556817d1bb99c47ec05c483b9926d02fe11aa4952b72798555b1c18ffc63c8359","name":"Clap.wav","type":"audio/vnd.wave","lastModified":1679562954648.59,"size":47144},{"id":"037a4213b14c36d221083fcc3d752f2e43c8fe252bc83b95e94b3a7caa451e83a07c08ff18fc096c4ab0d6bff0bd67834ae11161ce708decc5f09d8b2dbf2243","href":"https://static.observableusercontent.com/files/037a4213b14c36d221083fcc3d752f2e43c8fe252bc83b95e94b3a7caa451e83a07c08ff18fc096c4ab0d6bff0bd67834ae11161ce708decc5f09d8b2dbf2243","name":"Snare.wav","type":"audio/vnd.wave","lastModified":1679562954648.59,"size":52340}] 29:{"id":"a1205b125d06cad0","slug":null,"title":"Euclidean rhythms","thumbnail":"3765702fb53ac408b9cd63640873cc96f15a11f7c38c67f2d0842b81988c3664","access_level":"public","create_time":"$D2023-03-23T09:15:54.648Z","update_time":"$D2023-03-23T09:20:14.228Z","view_time":null,"first_public_time":"$D2023-03-23T09:15:54.648Z","has_unpublished_edits":null,"fork":{"id":"6249ebd58377d556","slug":"euclidean-rhythms","title":"Euclidean rhythms","owner":{"id":"03836a98946da550","login":"toja","name":"Torben Jansen"}},"stars":0,"starred":false,"owner":{"type":"team","id":"b0bbcf444e46bf02","login":"pedahd","name":"Ped Ram","avatar_url":"https://avatars-next.observableusercontent.com/ef13a6f0f32bd05288b47eb131f8eb1c5f9160b8e901276508df258390e51492","tier":"starter_2024","bio":"","followed_by_you":false,"follows_you":false},"version":3211,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"5456960e585c7bbb","login":"pedahd","name":"Ped Ram","avatar_url":"https://avatars-next.observableusercontent.com/ef13a6f0f32bd05288b47eb131f8eb1c5f9160b8e901276508df258390e51492","tier":null,"bio":"","followed_by_you":false,"follows_you":false}],"members":[{"type":"individual","id":"5456960e585c7bbb","login":"pedahd","name":"Ped Ram","avatar_url":"https://avatars-next.observableusercontent.com/ef13a6f0f32bd05288b47eb131f8eb1c5f9160b8e901276508df258390e51492","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Euclidean rhythms","cells":[{"id":0,"value":"# Euclidean rhythms","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1316,"value":"md`

In this notebook we'll exlore what _Euclidean rhythms_ are and implement an algorithm to create them from scratch, following the instructions from Godfried Toussaint's paper _The Euclidean Algorithm Generates Traditional Musical Rhythms_.

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1327,"value":"md`---\n\n## Euclidean what?!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":877,"value":"md`

\"What do African rhythms, spallation neutron source (SNS) accelerators in nuclear physics, string theory (stringology) in computer science, and an ancient algorithm described by Euclid have in common? The short answer is: patterns distributed as evenly as possible\". Godfried Toussaint

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2253,"value":"md`Over the last couple of years [Euclidean rhythms](https://en.wikipedia.org/wiki/Euclidean_rhythm) became relatively popular in (electronic) music creation and production. Essentially they are a way of spacing out ${tex`k`} events (or _pulses_) across ${tex`n`} positions (or _steps_ or _time intervals_) as evenly possible. If you space out, for example, _4 pulses_ across _16 steps_, the result is 4 evenly spaced pulses.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2313,"value":"viewof fouronthefloor = { \n const form = html`\n\n`\n let play = false;\n \n form.play.onclick = () => {\n play = !play;\n form.play.value = play ? \"Stop\" : \"Play\";\n form.dispatchEvent(new Event(\"input\", { bubbles: true }));\n }\n \n form.oninput = () => form.value = play;\n form.oninput();\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2257,"value":"sequencer({\n patterns: [{ pulses: 4, steps: 16, color: \"#2274a5\", play: rim }],\n run: fouronthefloor\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":612,"value":"md`If the number of onsets and the number of steps are [Coprime integers](https://en.wikipedia.org/wiki/Coprime_integers) (that is the only positive integer that is a divisor of both of them is 1), the resulting pattern is more interesting. In fact, almost all of the most important world music rhythms are _Euclidean rhythms_.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2147,"value":"viewof example = { \n const form = html`\n\n`\n let play = false;\n \n form.play.onclick = () => {\n play = !play;\n form.play.value = play ? \"Stop\" : \"Play\";\n form.dispatchEvent(new Event(\"input\", { bubbles: true }));\n }\n \n form.oninput = () => {\n form.value = { play, index: form.example.selectedIndex }\n }\n \n form.oninput();\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2159,"value":"sequencer({\n patterns: [{ ...examples[example.index], color: \"#2274a5\", play: rim }],\n run: example.play\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2150,"value":"md`${examples[example.index].description}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2334,"value":"md`And if that's not already cool enough, you can combine different _Euclidean rhythms_ to create your on drum patterns.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2523,"value":"viewof preset = { \n const form = html`\n\n`\n let play = false;\n \n form.play.onclick = () => {\n play = !play;\n form.play.value = play ? \"Stop\" : \"Play\";\n form.dispatchEvent(new Event(\"input\", { bubbles: true }));\n }\n \n form.oninput = () => {\n form.value = { play, index: form.preset.selectedIndex }\n }\n \n form.oninput();\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2132,"value":"sequencer({ ...presets[preset.index], run: preset.play })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2512,"value":"presets = [\n {\n name: \"Love is in the air (Intro)\",\n patterns: [\n { pulses: 6, steps: 8, play: kick },\n { pulses: 1, steps: 8, rotation: 4, play: clap },\n { pulses: 5, steps: 8, rotation: 4, play: hihat },\n ],\n bpm: 12,\n timeDivision: 4\n }\n \n]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":844,"value":"md`---\n## So, how does it work?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":833,"value":"$49","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":616,"value":"md`\n#### Step 0\n\n

Consider a sequence with n = 13 and k = 5. Since 13 − 5 = 8, we start by considering a sequence consisting of 5 one’s followed by\n8 zero’s which should be thought of as 13 sequences of one bit each at this stage:

\n\n${tex.block`${getTex(step_0)}`}\n\nAll we do here is to initialize our variables. The variables ${tex`a`} and ${tex`b`} are used to represent the bit patterns stored in ${tex`k`} and ${tex`m`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":655,"value":"step_0 = ({ a: \"1\", b: \"0\", k: 5, m: 8 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":630,"value":"md`\n#### Step 1\n\n

We begin moving zero’s by placing a zero after each one, to produce five sequences of two bits each, with\nthree zero’s remaining:

\n\n${tex.block`${getTex(step_1)}`}\n\nTake a closer look and examine what has changed here. For the condition ${tex`k <= m`} only the values of ${tex`a`} and ${tex`m`} have changed. That is the first case we've to cover in our algorithm.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":633,"value":"step_1 = {\n let { a, b, k, m } = step_0;\n \n a += b;\n m -= k;\n \n return { a, b, k, m }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":635,"value":"md`\n#### Step 2\n\n

Next we distribute the three remaining zeros in a similar manner, by placing a [0] sequence after each [10]\nsequence to obtain:

\n\n${tex.block`${getTex(step_2)}`}\n\nAgain, take a minute and reflect what has changed. On first sight: everything! But on second sight only the condition ${tex`k > m`} in the result of _step 1_ makes a difference here. That is the second of the two cases we've to cover in our algorithm. \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":637,"value":"step_2 = {\n let { a, b, k, m } = step_1; \n let cpy, tmp;\n\n cpy = a;\n a += b;\n b = cpy;\n \n tmp = k;\n k = m;\n m = tmp - m;\n \n return { a, b, k, m }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":679,"value":"md`\n#### Step 3\n\n

Now we have three sequences of three bits each, and a remainder of two sequences of two bits each.\nTherefore we continue in the same manner, by placing a [10] sequence after each [100] sequence to obtain:

\n\n${tex.block`${getTex(step_3)}`}\n\nAt this point the procedure should have become clear:\n\n~~~javascript\nif (k <= m) {\n // do step 1\n} else {\n // do step 2\n}\n~~~\n\nSo we do:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":683,"value":"step_3 = {\n let { a, b, k, m } = step_2; \n let cpy, tmp;\n\n cpy = a;\n a += b;\n b = cpy;\n \n tmp = k;\n k = m;\n m = tmp - m;\n \n return { a, b, k, m }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":691,"value":"md`\n#### Step 4\n\n

The process stops when the remainder consists of only one sequence (in this case the sequence [100]), or\nwe run out of zero’s. The final sequence is thus the concatenation of [10010], [10010], and [100]:

\n\n${tex.block`[${step_4}]`}\n\n

Note that one could proceed one step further in this process by inserting [100] into [10010] [10010]. However, Bjorklund argues that since the sequence is cyclic it does not matter (hence his stopping rule).

\n\nTranslated to code this is:\n\n~~~javascript\ndo {\n // step 1 or step 2\n} while (m > 1 & k > 1)\n~~~\n\nOnce the loop has ended, the concatenation is easily done:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":695,"value":"step_4 = {\n let { a, b, k, m } = step_3;\n \n let rhythm = \"\";\n\t\n while (k > 0) (rhythm += a, --k);\n\twhile (m > 0) (rhythm += b, --m);\n\n return rhythm;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":769,"value":"md`And that's it, we've reproduced Toussaint's algorithm for creating _Euclidean rhythms_. Now let's put all the pieces together into a nice function.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":502,"value":"function E(pulses, steps) {\n \n\tif (pulses > steps) pulses = steps;\n\n // step 0\n let a = \"1\";\n let b = \"0\"; \n let k = pulses;\n let m = steps - pulses;\n\n let cpy, tmp;\n\n\tdo {\t\n\t\tcpy = a;\n a += b;\n\t\tif (k <= m) { // step 1 \n m -= k;\n\t\t} else { // step 2, 3\n b = cpy;\n tmp = k;\n k = m;\n\t\t\tm = tmp - m;\n\t\t}\n\t} while (m > 1 & k > 1);\n\n // step 4\n\tlet rhythm = \"\";\n\twhile (k > 0) (rhythm += a, --k);\n\twhile (m > 0) (rhythm += b, --m);\n\t\t\n\treturn rhythm;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1982,"value":"md`---\n## And what has Euclid to do with all of that?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":778,"value":"md`\n[Euclid](https://en.wikipedia.org/wiki/Euclid) was a Greek mathematician, often referred to as the \"founder of geometry\" or the \"father of geometry\". In his _Elements_, one of the most influential works in the history of mathematics, he also describes an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that divides them both without a remainder.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":23,"value":"function euclid(m, k) {\n if (k === 0) return m;\n else return euclid(k, m % k);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":3168,"value":"In his paper, Toussaint argues that the _Euclidean algorithm_ has the same structure as that of Bjorklund’s algorithm if ${tex`m`} and ${tex`k`} are equal to the number of zero’s and one’s, respectively, in a binary sequence (with ${tex`n = m + k`}). Looking at the code this is a little bit counter-intuitive on first sight. But in Euclid's original version of the algorithm, the quotient and remainder are found by repeated subtraction:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":3158,"value":"function eucildOriginal(m, k) {\n while (m !== k) {\n if (m > k) {\n m -= k;\n } else {\n k -= m;\n }\n }\n return m;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":3131,"value":"md`This looks a lot more like in our function from above to generate, very efficiently, a large family of rhythms used as timelines (ostinatos). Therefore these rhythms are dubbed as _Euclidean rhythms_. They have the property that their onset patterns are distributed as evenly as possible. And in fact, almost all of the most important world music rhythms are _Euclidean rhythms_.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":361,"value":"md`---\n## Appendix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1407,"value":"kick = await sample(await FileAttachment(\"Kick.wav\").arrayBuffer())","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1501,"value":"rim = await sample(await FileAttachment(\"Rim.wav\").arrayBuffer())","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1461,"value":"hihat = await sample(await FileAttachment(\"HiHat.wav\").arrayBuffer())","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1503,"value":"clap = await sample(await FileAttachment(\"Clap.wav\").arrayBuffer())","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1476,"value":"snare = await sample(await FileAttachment(\"Snare.wav\").arrayBuffer())","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1401,"value":"audioContext = new AudioContext();","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1394,"value":"async function sample(buffer) {\n webaudioPolyfill; // Safari\n const sampleData = await audioContext.decodeAudioData(buffer);\n \n const panNode = audioContext.createStereoPanner();\n panNode.connect(audioContext.destination);\n \n const gainNode = audioContext.createGain();\n gainNode.connect(panNode);\n \n function play({ gain = 0.8, pan = 0 } = {}) {\n const source = audioContext.createBufferSource();\n source.buffer = sampleData; \n source.connect(gainNode); \n \n gainNode.gain.value = gain;\n panNode.pan.value = pan;\n\n audioContext.resume().then(() => {\n source.start();\n })\n }\n\n return play\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1987,"value":"function rotate(str, chars = 0) {\n const n = -chars % str.length;\n return str.slice(n) + str.slice(0, n);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":718,"value":"function getTex({ a, b, k, m }) {\n const K = Array.from({ length: k }, (_, i) => `[${a}]`).join(\"\\\\thinspace\");\n const M = Array.from({ length: m }, (_, i) => `[${b}]`).join(\"\\\\thinspace\");\n return `\\\\underbrace{${K}}_k \\\\space \\\\underbrace{${M}}_m`\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1124,"value":"sequencer = async function* ({ patterns, bpm = 100, timeDivision = 4, run = false }) {\n\n const w = Math.floor(width / 3 * 2);\n const r1 = w / 100;\n const r2 = w / 60;\n \n const svg = drawSequencer(patterns);\n const dots = svg.selectAll(\"g\").selectAll(\"circle\");\n\n let step = -1;\n\n function active(d, i) {\n return i === step % d.steps;\n }\n \n function getRadius(d, i) {\n return d.pulse && active(d, i) ? r2 : r1;\n }\n\n function getColor(d, i) {\n if (d.pulse) return d.color;\n if (active(d, i)) return \"grey\";\n return \"lightgrey\";\n }\n\n async function play(selection) {\n selection.data().forEach((d, i) => {\n if (d.pulse && active(d, d.index)) d.play();\n });\n }\n\n // milliseconds\n const tickInterval = Math.max(20, Math.round(60000 / (+bpm || 1) / timeDivision));\n\n while (run || step < 0) {\n \n dots\n .attr(\"r\", getRadius)\n .attr(\"fill\", getColor)\n .call(play);\n \n step++;\n \n yield svg.node();\n await Promises.tick(tickInterval);\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2969,"value":"$4a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2958,"value":"colors = [ \"#2274a5\", \"#ffbf00\", \"#32936f\", \"#e83f6f\", \"#31bac4\"];","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2137,"value":"$4b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2570,"value":"test = {\n const o = { 0: \".\", 1: \"x\" }\n function runTest({ pulses, steps, pattern }) {\n return E(pulses, steps).replace(/0|1/gi, key => o[key]) === pattern;\n }\n return examples.map(runTest)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2408,"value":"import { webaudioPolyfill } from \"@tomlarkworthy/audio-inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W4c","stdlib":"1"}} 24:null