34:["$","title",null,{"children":"Using Bitwise Operators for Music Theory | travistingey | Observable"}] 3c:T5e3,piano_template = ` `3d:[] 38:{"id":"8b421e46c36d8f33","slug":"using-bitwise-operators-for-music-theory","title":"Using Bitwise Operators for Music Theory","thumbnail":"84454c14556d281823d54507accfa9eaf6c8819470211b38538fe3bc82aa1c7b","access_level":"public","create_time":"$D2021-02-15T01:30:14.932Z","update_time":"$D2025-06-26T17:57:08.765Z","view_time":null,"first_public_time":"$D2021-02-16T20:55:37.542Z","has_unpublished_edits":null,"fork":false,"stars":4,"starred":false,"owner":{"type":"team","id":"ebe2f2222861d30c","login":"travistingey","name":"","avatar_url":"https://avatars-next.observableusercontent.com/f6b1c1a85c172841a109bf5c86c22431e37905fd0d859039d7f12f07d41c64ba","tier":"starter_2024","bio":"","followed_by_you":false,"follows_you":false},"version":1825,"publish_version":1825,"role":null,"authored":false,"authors":[{"type":"individual","id":"212eec5edb545995","login":"travistingey","name":"","avatar_url":"https://avatars-next.observableusercontent.com/f6b1c1a85c172841a109bf5c86c22431e37905fd0d859039d7f12f07d41c64ba","tier":null,"bio":"","followed_by_you":false,"follows_you":false}],"members":[{"type":"individual","id":"212eec5edb545995","login":"travistingey","name":"","avatar_url":"https://avatars-next.observableusercontent.com/f6b1c1a85c172841a109bf5c86c22431e37905fd0d859039d7f12f07d41c64ba","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":3,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Using Bitwise Operators for Music Theory","cells":[{"id":0,"value":"md`# Using Bitwise Operators for Music Theory \n\nIdea for this notebook came from thinking of using bits and bitwise operators to identify interval patterns in a group of notes. If given 12 bits, we can represent a collection of notes to analyze intervals that make up scales and chords with a single number. Bitwise operators provide an easy means to compare and analyze these representations.\n\nI'm leveraging this notebook means of prototyping a smarter Eurorack quantizer that not only locks notes into a certain scale, but also analyzes note frequency to identify most relevant chord harmonies.\n\n## Identifying Notes\nThis is based on the Eurorack convention of utilizing 1 volt / octave. We're going to assume 12-Tone Equal Temperament. The note/voltage relationship is relative, but we're are going to assume a voltage of 0 is equal to C.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"temperament = 1/12","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":185,"value":"md`\n## Scales, Chords, and Intervals\nSo, they're all the same thing right? My thinking is that bitwise operators can offer a convenient method of analyzing collections of notes to determine which group of intervals will fit. This notebook defines intervals as an array of degrees starting at 0.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":260,"value":"Scales = [{name: \"Chromatic\", intervals: [0,1,2,3,4,5,6,7,8,9,10,11]},{name: \"Modal\", intervals:[0,2,4,5,7,9,11]},{name: \"Pentatonic\", intervals: [0,3,5,7,10]}]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1799,"value":"IntervalsToBitMask(Scales[1].intervals)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1228,"value":"PianoRoll(scaleSelect.intervals.map(d=>(d+Transpose)),Transpose,true)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":663,"value":"viewof scaleSelect = Select(Scales,{label: \"Select Scale\", format: d=>d.name, value: Scales[1]})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1237,"value":"viewof Transpose = Range([0, 11], {step: 1, label: \"Transpose\", value: 0})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":211,"value":"md`## Bits\nCouple functions to take our interval arrays and convert them to bit masks and vice versa. Remember that ascending intervals will read from right to left when visualized as binary.\n`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":107,"value":"IntervalsToBitMask = (scale) => {\n if(scale.length == 0){\n return 0\n } else {\n return scale.reduce((a,c)=>(1 << c) | a, 1 << scale[0])\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1822,"value":"IntervalsToBitMask([0])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1773,"value":"IntervalsToBitMask([0,4,7])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":144,"value":"BitMaskToIntervals = (mask) => {\nlet bitCount = mask.toString(2).length\nlet result = []\n\nfor(let i = 0; i 0) result.push(i)\n}\n\nreturn result\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":207,"value":"md`## Quantizing Notes\n\nCurrent function takes a normalized scale of 1 octave tune voltages to the nearest scale degree within its current octave. I'm curious about the challenge of quantizing scales that span beyond an octave or even have different ascending or decending scales... but I'm too dumb to make use of that musically.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":35,"value":"function Quantize(volt,scale,offset){\n offset = offset || 0\n let note = (volt - Math.floor(volt)) / temperament\n let s = scale.map(i=>(i+offset)%12)\n let quant = s.reduce((acc,curr)=>Math.abs(curr-note) < Math.abs(acc - note)?curr:acc)\n \n return {degree: quant, voltage: Math.floor(volt) + quant * temperament, note:Math.floor(volt) * 12\n + quant}\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":252,"value":"md`So, let's generate some random values between 0 and 5 volts and quantize them`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"viewof btn1 = Button(\"Generate\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1279,"value":"PianoRoll(quantized_notes.map(d=>d.note), Transpose)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":581,"value":"viewof quantized_notes = {\n \n \n let freq = NoteFrequency(quant_result.map(n=>n.note)) \n let data = quant_result.map(d=>Object.assign({},d))\n data = data.map(d=>{\n d.interval = nts(d.degree,Transpose)\n d.octave = Math.floor(d.voltage).toString()\n d.name = itn(d.degree)\n d.value = d.note\n d['semitones'] = (d.degree > 0)? \"+\" + d.degree : 0\n\n return d\n })\n return Table(data, {columns:['name','interval','octave','semitones','voltage','value'],sort: 'value'})\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":269,"value":"quant_result = {\n let data = random_voltages\n return data.map((v)=>Quantize(v,scaleSelect.intervals,Transpose))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1545,"value":"random_voltages = {\n btn1;btn2;btn3;\n let n =[]\n for(let i=0;i<8;i++){\n n.push(Math.random() * 5)\n }\n \n return n\n\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":284,"value":"md`## Identifying Intervals\nThe curious thing with random voltages is that despite quantizing to a standard scale, it doesnt necessarily refect the sound of the mode. It will come down to the frequency of certain intervals, which can provide some insight as to what chords can be used to harmonize.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1426,"value":"{\n let result = found_table.reduce( \n (a,c)=>a.concat(c.interval.intervals\n .map(d=>(c.root + d)%12)),\n found_table[0].interval.intervals.map(d=>(found_table[0].root + d)%12))\n \n \n return PianoRoll(result,Transpose,true)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1557,"value":"viewof btn2 = Button(\"Generate\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1666,"value":"md`This table shows full results of found intervals. Scoring is the sum of the the occurance of each note in the interval. \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":498,"value":"viewof found_table = {\n let freq = NoteFrequency(quant_result.map(n=>n.note))\n \n let data = found_intervals.map(d=>Object.assign({},d))\n data.map(format_interval_table)\n \n //Scoring is the sum of the frequency table\n data.map(d=>{\n d.score = d.interval.intervals.reduce((a,c)=>a+=freq[(c + d.root)%12],freq[d.root])\n })\n \n return Table(data, {columns: ['note','voicing','extensions','intervals','degree','score'],sort: 'score', reverse: true})\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":832,"value":"found_intervals = {\n let notes = quant_result.map(d=>d.note % 12) \n \n return FindChords(notes)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":904,"value":"NoteFrequency = (notes)=>{\n let freq = new Array(12).fill(0)\n let result = notes.map(n=>{\n let base = n%12\n freq[base]++\n return base\n })\n return freq\n\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":395,"value":"Intervals = [\n\n{name: \"major \", intervals: [0,4,7]},\n{name: \"minor\", intervals: [0,3,7]},\n{name: \"°\", intervals: [0,3,6]},\n{name: \"+\", intervals: [0,4,8]},\n{name: \"sus4\", intervals: [0,5,7]},\n\n{name: \"maj7 shell\", intervals: [0,4,11]},\n{name: \"7 shell\", intervals: [0,4,10]},\n{name: \"m7 shell\", intervals: [0,3,10]},\n{name: \"m6 shell\", intervals: [0,3,9]},\n\n{name: \"maj7\", intervals: [0,4,7,11]},\n{name: \"7\", intervals: [0,4,7,10]},\n{name: \"m7\", intervals: [0,3,7,10]},\n{name: \"m6\", intervals: [0,3,7,9]},\n{name: \"ø\", intervals: [0,3,6,10]},\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":324,"value":"FindIntervals = (notes, interval,root) => {\n let note_mask = IntervalsToBitMask(notes)\n let found = []\n let interval_mask = IntervalsToBitMask(interval.intervals)\n \n for(let i = 0; i< 12; i++){\n if((note_mask & interval_mask) == interval_mask) {\n let ext = BitMaskToIntervals(note_mask ^ interval_mask)\n\n ext = ext.filter(e=>e != 11 && e != 10 && e != 3 && e != 4)\n \n found.push({\n note_mask: note_mask,\n interval_mask: interval_mask,\n root: i,\n ext: ext,\n interval\n })\n }\n note_mask = (note_mask >>> 1) | ((note_mask & 1) << 11)\n }\n return found.sort((a,b)=>{\n return b.intervals - a.intervals\n })\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1805,"value":"FindChords = function(notes){\nlet result =[]\n \n for(let i in Intervals){\n FindIntervals(notes, Intervals[i],Transpose).forEach(d=>{result.push(d)})\n }\n \n return result\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1670,"value":"md`## Collapsing Results\nResults from the found intervals are collapsed if two results share a root note when the one interval contains the other's. For example, if a complete A major 7 chord is identified, the major triad and the maj 7 shell voicing are discarded.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1712,"value":"{\n let result = collapsed_table.reduce( \n (a,c)=>a.concat(c.interval.intervals\n .map(d=>(c.root + d)%12)),\n collapsed_table[0].interval.intervals.map(d=>(collapsed_table[0].root + d)%12))\n \n \n return PianoRoll(result,Transpose,true)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1743,"value":"viewof btn3 = Button(\"Generate\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1519,"value":"viewof collapsed_table = { \n let data = found_table\n\n data= data.filter((d,i)=>{\n for(let j = 0; j< data.length; j++){\n if(i == j) continue;\n\n let a = d\n let b = data[j]\n \n if(a.root == b.root){\n let test = a.interval_mask | b.interval_mask\n let test_a = test == a.interval_mask\n let test_b = test == b.interval_mask\n \n // 1 4 7 11, 1 5 7 | 1, 7\n if(test == a.interval_mask){\n return true\n } else if(test == b.interval_mask){\n return false\n }\n }\n }\n return true\n })\n \n //data = data.map(format_interval_table)\n \n return Table(data, {columns: ['note','voicing','extensions','intervals','degree','score'],sort: 'score', reverse: true})\n\n\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1719,"value":"md`\nWe now have some weighted harmonies to choose from! Now the question is, how does this analysis work when the random notes are quantized as a stream?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":578,"value":"md`## Appendix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":991,"value":"PianoRoll = (notes,offset,normalized)=>{\n const svg = d3.create('svg')\n \n \n if(normalized){\n svg.attr('viewBox','0 0 1600 200')\n svg.append('g').html(piano_template)\n } else {\n svg.attr('viewBox','-50 0 1600 200')\n for(let octave = 0; octave < 5; octave++){\n let roll = svg.append('g').attr('class','o-'+octave).html(piano_template)\n roll.attr('transform','translate(' + 300 * octave + ' 0)')\n }\n }\n \n \n svg.selectAll('.note')\n .style('stroke','#aaa')\n .style('stroke-width','2px')\n .style('fill','#fff')\n\n if(normalized){\n notes.forEach(n=>{\n svg.selectAll('.note.' + itn(n))\n .style('fill', colorScale( (n - offset >= 0)? n - offset : n - offset + 12 ) )\n })\n } else { \n notes.forEach(n=>{\n let oct= Math.floor(n/12)\n \n svg.selectAll('.o-'+ oct +' .note.' + itn(n)).style('fill',colorScale( (n%12 - offset >= 0)? n%12 - offset : n%12 - offset + 12 ) )\n })\n }\n \n \n return svg.node()\n \n \n \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":806,"value":"itn = (n)=>{\n \n let notes =['C','Db','D','Eb','E','F','Gb','G','Ab','A','Bb','B']\n return notes[n%12]\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1503,"value":"itc = (n)=>{\n let def = ['R','b9','9','b3','3','11', 'b5', '5', 'b13','13','7','maj7']\n \n return def[n]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":804,"value":"nts = (note,root) =>{ \n let intervals=['I','bII','II','bIII','III','IV','bV','V','bVI','VI','bVII','VII']\n\n return intervals[(12 - root + note) % 12]\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1233,"value":"colorScale = d3.scaleSequential(d3.interpolateSpectral).domain([0, 11])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1586,"value":"format_interval_table = (d) => {\n d.note = itn(d.root)\n d.degree = nts(d.root,Transpose)\n d.voicing = d.interval.name\n d.intervals = d.interval.intervals.map(d=>nts(d,0))\n d.extensions = d.ext.map(a=>itc(a))\n \n return d\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":958,"value":"$3c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":949,"value":"md`## Imports`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":955,"value":"d3 = require(\"d3@^5.8\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":723,"value":"import {Button, Checkbox, Toggle, Radio, Range, Select, Text, Search, Table} from \"@observablehq/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W3d","stdlib":"1"}} 33:null