31:Tcec,I should probably start there... I couldn’t find a definitive source on the internet that answered my questions, but there’s _a lot_ scattered around.

A few things I read that were helpful:

https://music.stackexchange.com/questions/5382/what-is-the-difference-between-a-mode-and-a-scale after seeing this I don’t feel so bad that I don’t understand this stuff. There’s no “aha” answer in there (and lots of confusion).
Search bookmarks for ”music theory”
Wikipedia has _a lot_ of related pages — many that have playable scales, which is great! Unfortunately I found the main one on modes really hard to understand.
I like how this (dated) [Leonard Bernstein presentation](https://www.youtube.com/watch?v=g7oA1-y0EJE) has descriptions of how the modes sound/feel (to him) and plays examples.
https://music.stackexchange.com/a/53522 I found this answer really helpful, especially after experimenting on the piano (next bullet). Most of the stuff out there say modes “start on” a particular note. But it’s really about the key you’re playing in — the [tonic / root / tonal center](https://en.wikipedia.org/wiki/Tonality). I guess I also learned that I didn’t fully understand what "tonal" meant!
https://www.makebettermusic.me/blogs/music-theory/the-difference-between-musical-modes-and-scales at first I didn’t get this, but now I do (which makes it helpful now, but it wasn’t so helpful when I started down this rabbit hole). It did **not** help that the final example — the one I think would have most helped me understand — has the wrong notes for the C Dorian Mode!

After some 🤯, the most helpful way I've found to think about it is **a mode is defined by the sequence of intervals between pitches that divide up an octave**. It’s helpful to think of it as defined by the sequence of _intervals_ rather than the sequence of _notes_: `[2,2,1,2,2,2,1]` instead of `[0=C, 2=D, 4=E, 5=F, 7=G, 9=A, 11=B, 12=C]`. With intervals you can do like I did in the sketch above and get the full set of related modes by rotating the sequence: shift everything in `[2, 2, 1, 2, 2, 2, 1]` to the left, sending the leftmost `2` to the end, to get `[2,1,2,2,2,1,2]`. You can then construct the actual notes by starting with the tonic: - in the first case, `[0,2,4,5,7,9,11,12] = [C,D,E,F,G,A,B,C]`. Those are the notes to hit if you’re playing in C and you want to play the Ionian mode. - in the second case `[0,2,3,5,7,9,10,12] = [C,D,E♭,F,G,A,B♭,C]`. Those are the notes to hit if you’re playing in C and you want to play the Dorian mode. In the absence of any other music, the tonal center (“context”) will be set by the first note in the sequence. But it doesn’t need to be the first — there can be something else going on in the music that sets the tonal center. Then it doesn’t matter what order the notes come in they’ll still sound modal. _Aside: I ended up coming and re-writing this section last, because it took all of the below experimentation to get to where I feel ok about my understanding of what a mode is._32:T5db,✔️ it sounds like the example on the [whole-tone scale](https://en.wikipedia.org/wiki/Whole-tone_scale) Wikipedia page, except more synth-y than piano-y. Listening to this scale, it sounds Zelda-ish to me, though [here’s a video describing scales used in Zelda](https://www.youtube.com/watch?v=L86o-db3XY8) — not whole tone but symmetric. get in touch with Quinton, and ask why he called Zelda “Belzar”. Wut. Aside: on that whole-tone wiki page, I don’t understand (yet) what most of this means: > The composer Olivier Messiaen called the whole-tone scale his first [mode of limited transposition](https://en.wikipedia.org/wiki/Mode_of_limited_transposition). The composer and music theorist George Perle calls the whole-tone scale [interval cycle](https://en.wikipedia.org/wiki/Interval_cycle) 2, or C2. Since there are only two possible whole-tone-scale positions (that is, the whole-tone scale can be transposed only once), it is either C2⁰ or C2¹. For this reason, the whole-tone scale is also [maximally even](https://en.wikipedia.org/wiki/Maximal_evenness) and may be considered a [generated collection](https://en.wikipedia.org/wiki/Generated_collection). Look at all those links to things I don’t know about! This is clearly a very deep rabbit hole that I am just barely even entering. actually try to understand things like “maximal evenness” and “generated collections”.33:T866,function constellationPlot( intervals, { nPitchClasses = 12, size = 90, markers = [] } = {} ) { const checkSum = _.sum(intervals); if (checkSum !== nPitchClasses) { throw Error( `sum of intervals (${checkSum}) must be equal to number of pitch classes (${nPitchClasses})` ); } const strokeWidth = 4; const lightness = 60; const fontSize = 12; const space = 3; const r = size / 2 - strokeWidth / 2 - fontSize - space; const c = size / 2; const indexes = intervalsToIndexes( intervals.slice(0, -1), nPitchClasses, false ); function spoke(i) { return svg` `; } function dotMarker(i) { return svg` `; } function label(i, stroke = "none", fill = "black", fontWeight = "normal") { const radius = r + strokeWidth / 2 + fontSize / 2 + space; const angle = (2 * Math.PI * (i - 3)) / nPitchClasses; return svg`${i}`; } return svg` `; }34:T43c,{ let w = (Math.min(312, width) - 34) / 3; let result = []; for (let nWhole = 6; nWhole >= 0; nWhole--) { const nHalf = 12 - 2 * nWhole; const modes = makeModes(nWhole, 12); result.push( html`

${modes.length} mode${modes.length == 1 ? "" : "s"} with ${nWhole + nHalf} pitches (${nWhole} whole-steps and ${nHalf} half-steps)

` ); let constellations = []; modes.forEach((intervals) => { const key = intervals.toString(); const row = wikipediaLookup[key]; let name = key.replaceAll(",", " "); if (row) { name = row.name; } constellations.push(htl.html`

{ playFreqs(modeToFreqs(intervals, 12), 0.15, true); }} >

${name}

${constellationPlot(intervals, 12, w)}

`); }); result.push(html`

${constellations}

`); } return html`${result}`; }35:T447,There are ${eightPitchRotationGroups.length}. I was hoping to be saying "Aha! These are the transpositionally non-equivalent eight-note sets!" but nope. But even after reading more it does seem like a rotation of the interval sequence means the same thing as transpositional equivalence. I’m missing something, or maybe I have a bug 🐛 (hope not). Or maybe it’s that I’m only allowing half and whole steps? (update: yep, the difference is that I’m only allowing half- and whole-steps) Check out _The Structure of Atonal Music_ by Allen Forte, _Basic Atonal Theory_ by John Rahn, or _Introduction to Post-Tonal Theory_ by Joseph Straus. The first two I’ve seen everywhere talking about this stuff, but [this list of sources](https://www.mta.ca/pc-set/pc-set_new/pages/sources/sources.html) makes me think I’d like the Straus one because it “keeps its musical relevance clear”. Update: ordered a cheap used copy of _The Structure of Atonal Music_, and it got some satisfying “husband/dad got some more weird books” looks from my family 😎.36:T643,function toName(number) { // from https://stackoverflow.com/a/5530230/1431778 var ones = [ "", "one", "two", "three", "four", "five", "six", "seven", "eight", "nine" ]; var tens = [ "", "", "twenty", "thirty", "forty", "fifty", "sixty", "seventy", "eighty", "ninety" ]; var teens = [ "ten", "eleven", "twelve", "thirteen", "fourteen", "fifteen", "sixteen", "seventeen", "eighteen", "nineteen" ]; function convert_millions(num) { if (num >= 1000000) { return ( convert_millions(Math.floor(num / 1000000)) + " million " + convert_thousands(num % 1000000) ); } else { return convert_thousands(num); } } function convert_thousands(num) { if (num >= 1000) { return ( convert_hundreds(Math.floor(num / 1000)) + " thousand " + convert_hundreds(num % 1000) ); } else { return convert_hundreds(num); } } function convert_hundreds(num) { if (num > 99) { return ( ones[Math.floor(num / 100)] + " hundred " + convert_tens(num % 100) ); } else { return convert_tens(num); } } function convert_tens(num) { if (num < 10) return ones[num]; else if (num >= 10 && num < 20) return teens[num - 10]; else { return tens[Math.floor(num / 10)] + " " + ones[num % 10]; } } function convert(num) { if (num == 0) return "zero"; else return convert_millions(num); } return convert(number); }37:T9bb,All right. I’ve gotta wrap this up for now. I understand more about modes now. I think I’d be able to get through explaining to someone what a mode is. By listening to music while doing this exploration and trying things out on the keyboard, I think my ear for hearing intervals and modes improved and I think I’ll better recognize when a piece of music I’m listening to is in a mode. Perhaps most surprisingly, I think I understand better what “tonal” vs “atonal” music is a lot better. I picked up some musically-relevant bits here and there, too. But more than answers, I have _way_ more questions now than when I started. I’m happy about that — analyzing something like music can seem so lame compared to just listening and enjoying, but ending up with a bunch more questions just makes me appreciate musicians and their music even more. There’s a long list of “maybe laters”, and now I’m curious to learn about different temperaments, voice leading, and atonal composition. Also, I came across so many different musical systems that were not Western/European music. How much does the way we hear music have to do with what we have heard? Are particular intervals (either in 12-tone temperament, or different temperaments) always “dissonant” to different people, or does it only really on what we’ve heard and associated? #### Meta-reflections about this notebook - Almost always when I make notebooks like these I’m doing it for myself, with no plan to share. I have a lot of unshared notebooks (on observable but even more in other formats)! Deciding that I’d share it made it so I didn’t quite know the best point of view to write from — it felt a bit awkward and slowed me down since I felt like I needed to write for a reader (and an unclear reader at that — not sure who, if anyone, will actually want to read this). I’d like to see if practicing more will improve that. - I spent a bunch of hours in this, and it wasn’t until near the end that it felt like it clicked for me about what a mode even is. I really needed to make something in order to understand it, rather than just reading/listening. - Making the “maybe later” list was surprisingly helpful. Rather than exploring and taking any turn that I’m curious about, I stayed more focused on answering a smaller set of questions. I do like the random exploration, but it was also nice to get to a stopping point that wasn’t just “I’m bored of this now”.38:T4aa,parseWikipediaResponse = function (response) { const lookup = { Q: 0.5, H: 1, W: 2 }; const stringToNumber = (s) => { let n = 1; let v = s[0]; if (s.length == 2) { n = +s[0]; v = s[1]; } return n * lookup[v]; }; const parser = new DOMParser(); const external = parser.parseFromString( response.parse.text["*"], "text/html" ); const table = document.adoptNode(external.querySelector("table")); const header = table.querySelector("tbody tr"); const rows = table.querySelectorAll("tbody tr:not(:first-child)"); const result = []; [...rows.values()].forEach((r) => { const name = r.querySelector("td:nth-child(1)").textContent.trim(); const intervalsString = r .querySelector("td:nth-child(5)") .textContent.trim() .split("\n")[0] // at least one with multiple versions (Octatonic) .split("-"); const intervals = intervalsString.map((d) => stringToNumber(d)); const nPitchClasses = +r .querySelector("td:nth-child(7)") .textContent.trim(); if (nPitchClasses == intervals.length) { result.push({ name, intervals, nPitchClasses }); } }); return result; }39:T4d9,function maybeLaterList({ selector = "span.maybe-later", title = "Maybe Later", background = "white" } = {}) { const fullSelector = `${selector}:not(.bunga-list ${selector}, ${selector} ${selector})`; return Generators.observe((next) => { let previous = []; let lastCall = 0; function callback() { if (Date.now() - lastCall < 250) { return; } lastCall = Date.now(); const elements = [...document.querySelectorAll(fullSelector)]; if ( elements.length !== previous.length || elements.some((h, i) => previous[i] !== h) || !previous.length ) { previous = elements; next(html`

${title}

${e.innerHTML}

`); } } const observer = new MutationObserver(callback); observer.observe(document.body, { childList: true, subtree: true }); callback(); return () => observer.disconnect(); }); }3a:[{"id":"d05e629621319c428056817340a5699c43fa8cbe93c36373ff40fb3b1042a0b29053518e6c7db40eedad7aaf7acbd16957c1d0ada7f0d4ec302176b8ba5acdc6","href":"https://static.observableusercontent.com/files/d05e629621319c428056817340a5699c43fa8cbe93c36373ff40fb3b1042a0b29053518e6c7db40eedad7aaf7acbd16957c1d0ada7f0d4ec302176b8ba5acdc6","name":"tonic.m4a","type":"audio/x-m4a","lastModified":1699896729436.39,"size":194623},{"id":"caeaeedb645f312b88612c2cfc9d5c5d23578463f59ab4a75abe788cbb984c208f7cf459ef4e456147c3c4af8570ef2610f5f8ded40f3d7475a8e3dbf9501cf9","href":"https://static.observableusercontent.com/files/caeaeedb645f312b88612c2cfc9d5c5d23578463f59ab4a75abe788cbb984c208f7cf459ef4e456147c3c4af8570ef2610f5f8ded40f3d7475a8e3dbf9501cf9","name":"image@2.png","type":"image/png","lastModified":1699389754044.494,"size":206304},{"id":"ada1c456f8eec983b5a086108c5963971a286cfc56ae88b22cb2a7d81b5c15dda82c3f4fd974b36682d978927eac5af729a7886be95babe20ea0d551c9c8d500","href":"https://static.observableusercontent.com/files/ada1c456f8eec983b5a086108c5963971a286cfc56ae88b22cb2a7d81b5c15dda82c3f4fd974b36682d978927eac5af729a7886be95babe20ea0d551c9c8d500","name":"2023-11-05-List_of_musical_scales_and_modes.json","type":"application/json","lastModified":1699235825187.986,"size":204450}] 2f:{"title":"Exploring Musical Modes","cells":[{"id":0,"value":"# Exploring Musical Modes\n\nA notepad for a nerd-bender that started with modes.","pinned":false,"hidden":false,"mode":"md"},{"id":3156,"value":"maybeLaterList({\n selector: \"span.todo\",\n title: \"To Do\",\n background: \"rgba(255,255,0,0.4)\"\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":5362,"value":"## The 🐰🕳️","pinned":false,"hidden":false,"mode":"md"},{"id":1925,"value":"

\n ${await FileAttachment(\"image@2.png\").image()}\n

I jotted down this note while watching this video about scales and modes on the guitar. I’ve learned a little about modes in the past, but didn’t realize that moving from one mode to the next, while staying in the same key, is just rotating the sequence of intervals between each note — e.g. getting to the Dorain from the Ionian is shifting the sequence of seven half- or whole-steps to the left (with the first one going to the end).

\n\n","pinned":false,"hidden":false,"mode":"html"},{"id":1957,"value":"When I learned about modes, I remember it feeling like “memorizing names” and nothing about modes has really stuck with me. But the fact there’s this interesting pattern makes me wonder if I can find a conceptual framework that will help me understand in a different way.\n\nIt also makes me curious: what other modes are there? How many are there? What do they sound like? Are there other patterns like this? Come to think of it, what is a mode?","pinned":false,"hidden":false,"mode":"md"},{"id":4811,"value":"---\n\n## What is a mode?","pinned":false,"hidden":false,"mode":"md"},{"id":4816,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":5525,"value":"---\n\n## Who cares?","pinned":false,"hidden":false,"mode":"md"},{"id":5529,"value":"They make interesting sounds, and have a distinct feel. I think learning more about them will expand my musical palette for both listening and playing.\n\nHere are some songs that heavily use the Dorian mode. They have a vibe.","pinned":false,"hidden":false,"mode":"md"},{"id":5534,"value":"embedSpotify(\n \"https://open.spotify.com/playlist/6FIYyAqgC6PRopR1AyHI7w?si=8eeabab097b44550\"\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2769,"value":"--- \n\n## How many modes are there with only half-or-whole steps?","pinned":false,"hidden":false,"mode":"md"},{"id":1971,"value":"Here’s a function to make all the possible interval sequences that consist of only half-steps or whole-steps:","pinned":false,"hidden":false,"mode":"md"},{"id":238,"value":"function makeModes(nWholesteps, nPitchesPerOctave = 12) {\n const nHalfsteps = nPitchesPerOctave - 2 * nWholesteps;\n const nPitches = nWholesteps + nHalfsteps;\n if (nHalfsteps < 0) {\n return [];\n } else if (nHalfsteps === 0) {\n return [new Array(nPitches).fill(2)];\n }\n const range = new Array(nPitches).fill().map((_, i) => i);\n return _.combinations(range, nHalfsteps).map((indexes) => {\n let result = new Array(nPitches).fill(2);\n indexes.forEach((i) => (result[i] = 1));\n return result;\n });\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":2010,"value":"It is tempting to rewrite this to allow for other intervals (3 half-steps, etc) but I am going to resist for now.\n\nThere should be only one scale with only half-steps: the chromatic scale.","pinned":false,"hidden":false,"mode":"md"},{"id":1996,"value":"makeModes(0)","pinned":true,"hidden":false,"mode":"ojs"},{"id":2016,"value":"How many scales have four whole-steps and four half-steps?","pinned":false,"hidden":false,"mode":"md"},{"id":1999,"value":"makeModes(4)","pinned":true,"hidden":false,"mode":"ojs"},{"id":2205,"value":"How many total modes with only half- and whole-steps?","pinned":false,"hidden":false,"mode":"md"},{"id":364,"value":"nModes = function (...args) {\n return makeModes(...args).length;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"nTotal = nModes(0) + // chromatic (12 notes)\n nModes(1) +\n nModes(2) +\n nModes(3) +\n nModes(4) +\n nModes(5) + // ionian mode, dorian mode, etc in here (5 whole steps, 2 half steps)\n nModes(6) // whole tone scale (6 notes)","pinned":true,"hidden":false,"mode":"ojs"},{"id":171,"value":"There are ${nTotal} different modes that step up only by half-steps or whole-steps in an octave with 12 pitches.","pinned":false,"hidden":false,"mode":"md"},{"id":2209,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":2234,"value":"## What do they sound like?","pinned":false,"hidden":false,"mode":"md"},{"id":2478,"value":"To hear them, we need to convert the sequence of intervals to actual notes.\n\nHere are a couple functions to make the [evenly-spaced frequencies](https://en.m.wikipedia.org/wiki/Equal_temperament) (ratio of subsequent frequencies are equal), and to convert a sequence of intervals into a sequence of those frequencies:\n\nI thought [well-tempered](https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier) was the same as equal-tempered, but they are different. Maybe allow this notebook to play in different temperaments?","pinned":false,"hidden":false,"mode":"md"},{"id":652,"value":"// abbreviating frequencies to \"freqs\" because I keep spelling frequencies wrong\nfunction equalTemperedFreqs(\n nPitchesPerOctave = 12,\n startFrequency = 261.6255653, // start at C4, where A4=440Hz\n nOctaves = 5\n) {\n const n = nPitchesPerOctave;\n const logs = _.range(nOctaves * n + 1).map(\n (i) => Math.log2(startFrequency) + i / n\n );\n return logs.map((d) => 2 ** d);\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":1040,"value":"function modeToFreqs(\n intervals,\n nPitchesPerOctave = 12,\n startFrequency = 261.6255653 // start at C4, where A4=440Hz\n) {\n const notes = equalTemperedFreqs(nPitchesPerOctave, startFrequency);\n\n const result = [];\n let index = 0;\n result.push(notes[index]);\n intervals.forEach((interval) => {\n index += interval;\n result.push(notes[index]);\n });\n return result;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":2480,"value":"Let’s try the combo of these functions on one of the modes:","pinned":false,"hidden":false,"mode":"md"},{"id":2347,"value":"wholeToneFrequencies = modeToFreqs(makeModes(6)[0])","pinned":true,"hidden":false,"mode":"ojs"},{"id":2483,"value":"The numbers aren’t enlightening (but it’s a good check to see that the last note is double the frequency of the first — an octave apart ${wholeToneFrequencies.at(-1) / wholeToneFrequencies.at(0) === 2 ? \"✔️\" : \"WHICH IS NOT TRUE!! YOU HAVE A BUG🐛!\"}). Now to actually play the audio use [tone.js](https://tonejs.github.io/). use the native AudioContext instead of tone.js?","pinned":false,"hidden":false,"mode":"md"},{"id":1053,"value":"function playFreqs(\n freqs,\n duration = 0.25,\n playDescending = false,\n arpeggiate = true\n) {\n const synth = new Tone.PolySynth(Tone.Synth).toDestination();\n const now = Tone.now();\n const n = freqs.length;\n if (arpeggiate) {\n freqs.forEach((f, i) => {\n const dDown = (2 * duration) / 3;\n synth.triggerAttackRelease(f, dDown, now + i * duration);\n if (playDescending && i !== n - 1) {\n let j = 2 * (n - 1) - i;\n synth.triggerAttackRelease(f, dDown, now + j * duration);\n }\n });\n } else {\n freqs.forEach((f, i) => {\n if (i !== freqs.length - 1) {\n synth.triggerAttackRelease(f, duration, now);\n }\n });\n }\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":1013,"value":"function playButton(label, f, ...params) {\n const button = Inputs.button(label);\n button.oninput = () => {\n f(...params);\n };\n return button;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":2461,"value":"playButton(\n \"⏵ Ionian Mode (Major Scale)\",\n playFreqs,\n modeToFreqs([2, 2, 1, 2, 2, 2, 1])\n)","pinned":true,"hidden":false,"mode":"ojs"},{"id":2512,"value":"Sounds like a major scale! How about something different:","pinned":false,"hidden":false,"mode":"md"},{"id":2365,"value":"playButton(\"⏵ Whole Tone Scale\", playFreqs, wholeToneFrequencies)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2405,"value":"$32","pinned":false,"hidden":false,"mode":"md"},{"id":5480,"value":"Here are some other weird ones, the octatonic scales:","pinned":false,"hidden":false,"mode":"md"},{"id":2446,"value":"playButton(\n \"⏵ Octatonic A: starting with half-step\",\n playFreqs,\n modeToFreqs([1, 2, 1, 2, 1, 2, 1, 2])\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2452,"value":"playButton(\n \"⏵ Octatonic B: starting with whole-step\",\n playFreqs,\n modeToFreqs([2, 1, 2, 1, 2, 1, 2, 1])\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":3033,"value":"Both of these sound similar to me.","pinned":false,"hidden":false,"mode":"md"},{"id":2547,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":2550,"value":"## 🔊 Scale / equal-temperament playground","pinned":false,"hidden":false,"mode":"md"},{"id":5486,"value":"Enter in the sequence of intervals and hit the play button to listen to the scale, and explore scales and modes with different subdivisions of the octave than 12-tone equal temperament.","pinned":false,"hidden":false,"mode":"md"},{"id":2590,"value":"viewof rawIntervals = Inputs.form({\n text: Inputs.text({\n label: \"Intervals\",\n placeholder: \"Sequence of steps\",\n value: \"2 1 2 2 2 1 2\"\n }),\n duration: Inputs.range([0.05, 0.5], {\n label: \"Duration\",\n value: 0.15,\n step: 0.01\n }),\n descending: Inputs.toggle({ label: \"Play descending\", value: true }),\n arpeggiate: Inputs.toggle({ label: \"Arpeggiate\", value: true }),\n nPitches: Inputs.range([5, 31], {\n label: \"Pitch classes\",\n value: 12,\n step: 1\n })\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":2596,"value":"intervals = {\n let raw = rawIntervals.text\n .split(/\\W+/)\n .map((d) => +d)\n .filter((d) => d > 0);\n const indexes = intervalsToIndexes(raw);\n const truncated = indexes.filter((d) => d <= rawIntervals.nPitches);\n if (truncated[truncated.length - 1] !== rawIntervals.nPitches) {\n truncated.push(rawIntervals.nPitches);\n }\n return indexesToIntervals(truncated);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1840,"value":"playButton(\n `⏵ Play ${intervals.join(\"-\")} using ${\n rawIntervals.nPitches\n }-tone equal temperament`,\n playFreqs,\n modeToFreqs(intervals, rawIntervals.nPitches),\n rawIntervals.duration,\n rawIntervals.descending,\n rawIntervals.arpeggiate\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2749,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":2763,"value":"## 👀 See all the modes","pinned":false,"hidden":false,"mode":"md"},{"id":3239,"value":"I want to know more about the modes and scales that don’t have names / their own wikipedia page.\n\nAgain, wikipedia brings the inspiration with the concept of a [pitch constellation](https://en.wikipedia.org/wiki/Chromatic_circle#Pitch_constellation) — a radial visualization of a scale. For the 12-tone scale, it’s got the bonus of being just like an analog clock.","pinned":false,"hidden":false,"mode":"md"},{"id":1409,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":1207,"value":"function intervalsToIndexes(intervals) {\n let result = [];\n let index = 0;\n result.push(index);\n intervals.forEach((interval) => {\n index += interval;\n result.push(index);\n });\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":2618,"value":"function indexesToIntervals(indexes) {\n let result = [];\n for (let i = 1; i < indexes.length; i++) {\n result.push(indexes[i] - indexes[i - 1]);\n }\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":3401,"value":"Here’s a visualization of the Ionian mode.","pinned":false,"hidden":false,"mode":"md"},{"id":1422,"value":"constellationPlot([2, 2, 1, 2, 2, 2, 1])","pinned":true,"hidden":false,"mode":"ojs"},{"id":4580,"value":"Here’s the same Ionian mode along with all of the rotations, as you go from left to right the shape is rotated counter-clockwise (there’s a dot on one of the spokes to make it a little easier to tell that this is the same shape).\n\nSo the set of related modes (Ionian, Dorian, ...) are the same shape, but rotated.","pinned":false,"hidden":false,"mode":"md"},{"id":5495,"value":"{\n let w = (Math.min(312, width) - 34) / 3;\n let result = [];\n let i = -1;\n for (let mode of allRotations([2, 2, 1, 2, 2, 2, 1])) {\n i += 1;\n result.push(\n html`

${constellationPlot(mode, {\n size: w,\n markers: [(8 - i) % 7]\n })}

`\n );\n }\n return html`

${result}

`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":5510,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":2846,"value":"And here are the constellation diagrams for all for _all_ of the possible modes that include only half- or whole-steps, where the name is shown if the scale is listed in this [list of musical scales and modes](https://en.wikipedia.org/wiki/List_of_musical_scales_and_modes). If you notice names missing, go edit wikipedia and it will update here :)\n\n_Click on a diagram to hear the scale._","pinned":false,"hidden":false,"mode":"md"},{"id":1689,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":2839,"value":"Pretty fun to click on those to listen and explore!\n\nIt makes sense that the modes with less pitches (e.g. no half-steps, 2 half-steps) have more “named” scales. The different between where the half-steps and whole-steps in those sound a lot more noticeable to me. When there are more half steps, they all start sounding basically like the chromatic scale. But I wonder if that’s true for people with a better ear / more practice listening.\n\nAlso how does `2-2-1-2-1-2-2` not have a wikipedia page, it’s nice! Look at it, it starts with a nice happy whole step. It’s got the fourth and the fifth in it. Its interval sequence is a palindrome (pretty sure that’s what [inversionally symmetrical](https://en.wikipedia.org/wiki/Symmetric_scale) means. If so, that’s easy to see in these visualizations because they have left-right mirror symmetry). This mode is _dope_.\n\n${constellationPlot([2,2,1,2,1,2,2])}","pinned":false,"hidden":false,"mode":"md"},{"id":3426,"value":"playButton(\n '⏵ Maybe this should be named the \"dope\" mode',\n playFreqs,\n modeToFreqs([2, 2, 1, 2, 1, 2, 2], 12, 440), // in A\n 0.2\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":3421,"value":"Somehow it sounds both happy and mysterious to me.","pinned":false,"hidden":false,"mode":"md"},{"id":4608,"value":"Update: this does have name(s) and a [a wikipedia page](https://en.wikipedia.org/wiki/Aeolian_dominant_scale), but it’s not listed in the list page. Here it is in use. You can press the button above to play the mode while the clip is playing, too, to hear it in action.","pinned":false,"hidden":false,"mode":"md"},{"id":5542,"value":"embedSpotify(\n \"https://open.spotify.com/track/4v8vjdYaK6jYmeBN6KZo7P?si=a8cb71bab9d34ab3\"\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":3432,"value":"Anyhow. A few things for the maybe later list:\n- Edit Wikipedia and add the Aeolian dominant scale to the [list page](https://en.wikipedia.org/wiki/List_of_musical_scales_and_modes)\n- I’m getting a bit tired of this synth sound. Change to a piano or something?\n- It would be cool and provide useful feedback if the diagrams animated while the mode played.\n- When I connected this to the inputs above, it got pretty sluggish. I think that could be fixed (but I don’t think I really understand `htl.html`)\n- Modify or redesign the “pitch constellation” visualization to better accommodate more or less pitch classes than 12.","pinned":false,"hidden":false,"mode":"md"},{"id":3243,"value":"---\n\n## Rotations","pinned":false,"hidden":false,"mode":"md"},{"id":3461,"value":"This started with me learning that the Greek-named (diatonic) modes are all the same interval sequence, but rotated. What about the others?\n\nHere’s a function to make all the rotations of an array.","pinned":false,"hidden":false,"mode":"md"},{"id":482,"value":"function* allRotations(array) {\n for (let i = 0; i < array.length; i++) {\n yield [...array];\n //array.unshift(array.pop()); // take the last element and move to front\n array.push(array.shift()); // take the first element and move to end (will do this way because it’s how it was in the sketch above\n }\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":3487,"value":"It works:","pinned":false,"hidden":false,"mode":"md"},{"id":3455,"value":"[...allRotations([1, 2, 3])]","pinned":true,"hidden":false,"mode":"ojs"},{"id":3463,"value":"And here’s a function that groups together all of the scales that are rotations of each other. Is this called a [necklace](https://en.wikipedia.org/wiki/Necklace_(combinatorics)? Explore how the [musical set theory connects with set theory in math](https://en.wikipedia.org/wiki/Set_theory_(music)).","pinned":false,"hidden":false,"mode":"md"},{"id":444,"value":"function groupRotations(scales) {\n // this currently looks at every rotation of every scale that\n // is passed in, which I’m pretty sure is not necessary. It could\n // be made more efficient, but it will do for now.\n\n // keys of incomplete are string\n let incomplete = new Map(scales.map((d) => [d.toString(), d]));\n\n // choose any key as the \"focus\" to start\n let focus = incomplete.keys().next().value;\n let group = [];\n let result = [];\n while (true) {\n // group together all the scales that are rotations of the focus (and remove from incomplete)\n for (let rotation of allRotations(incomplete.get(focus))) {\n const found = incomplete.delete(rotation.toString());\n if (found) {\n group.push(rotation);\n }\n }\n result.push(group);\n\n // choose any key as the next focus, until incomplete is empty\n focus = incomplete.keys().next().value;\n group = [];\n if (focus === undefined) {\n break;\n }\n }\n return result;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":3484,"value":"To test it works here’s an example that should have two groups, because the first array and the second are rotations of each other but the third is different.","pinned":false,"hidden":false,"mode":"md"},{"id":3470,"value":"groupRotations([\n [1, 1, 2],\n [2, 1, 1], // same as first, rotated\n [1, 1, 1] // different\n])","pinned":true,"hidden":false,"mode":"ojs"},{"id":3670,"value":"Let’s try it for the seven-pitch scales:","pinned":false,"hidden":false,"mode":"md"},{"id":3498,"value":"sevenPitchRotationGroups = groupRotations(makeModes(5, 12))","pinned":true,"hidden":false,"mode":"ojs"},{"id":3507,"value":"For seven-pitch scales with only half and whole steps, there are just ${sevenPitchRotationGroups.length} distinct shapes. Rotating these shapes account for most the scales/modes with all the obscure names!","pinned":false,"hidden":false,"mode":"md"},{"id":3384,"value":"// should not be copy pasting this!\n{\n let w = (Math.min(312, width) - 34) / 3;\n let result = [];\n for (let [i, modes] of sevenPitchRotationGroups.entries()) {\n result.push(\n html`

\nShape ${i + 1}\n

`\n );\n\n let constellations = [];\n modes.forEach((intervals) => {\n const key = intervals.toString();\n const row = wikipediaLookup[key];\n let name = key.replaceAll(\",\", \" \");\n if (row) {\n name = row.name;\n }\n constellations.push(htl.html` {\n playFreqs(modeToFreqs(intervals, 12), 0.15, true);\n }}\n>\n

${name}

\n ${constellationPlot(intervals, 12, w)}\n`);\n });\n result.push(html`

${constellations}

`);\n }\n return html`${result}`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":3521,"value":"With these diagrams, it’s a bit easier to see that the ${makeModes(5, 12).length} different scales are three different shapes (interval sequences) and all their rotations.\n\nFactor the code for making these groups of diagrams so this is not copy-pasted.","pinned":false,"hidden":false,"mode":"md"},{"id":3546,"value":"On this [octatonic scale](https://en.wikipedia.org/wiki/Octatonic_scale) page, I noticed this doozy \"there are a total of 43 enharmonically non-equivalent, transpositionally non-equivalent eight-note sets.\" How many different interval sequences are there?","pinned":false,"hidden":false,"mode":"md"},{"id":3530,"value":"eightPitchRotationGroups = groupRotations(makeModes(4, 12)).map((d) => d[0])","pinned":true,"hidden":false,"mode":"ojs"},{"id":3549,"value":"$35","pinned":false,"hidden":false,"mode":"md"},{"id":3591,"value":"So how many of all the 12-note scales are not rotated versions of the others? How many unique interval sequences are there (disregarding rotations)?","pinned":false,"hidden":false,"mode":"md"},{"id":3582,"value":"function nTNE(...args) {\n // TNE = Transpositionally Non-Equivalent, which I am still thinking is the right term\n return groupRotations(makeModes(...args)).length;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":3607,"value":"$36","pinned":false,"hidden":false,"mode":"ojs"},{"id":3575,"value":"nTotalTNE = _.sum(_.range(0, 7).map((nWhole) => nTNE(nWhole)))","pinned":true,"hidden":false,"mode":"ojs"},{"id":3604,"value":"**${toName(31).toUpperCase()}**! ${nTotalTNE} is the answer to life, the universe, and everything (the number of transpositionally non-equivalent pitch class sets with intervals no greater than a whole step).","pinned":false,"hidden":false,"mode":"md"},{"id":4628,"value":"--- ","pinned":false,"hidden":false,"mode":"md"},{"id":5584,"value":"## For all pitch class sets, not just the ones with half- or whole-steps only","pinned":false,"hidden":false,"mode":"md"},{"id":4631,"value":"Bah! Even though I’d listed under “maybe later” I can’t resist figuring out where the 43 comes from, and I’m going to do all the scales instead of just those with half or whole steps.","pinned":false,"hidden":false,"mode":"md"},{"id":3685,"value":"function makeAllScales(n) {\n const allSequences = new Set();\n\n function* divide(...seq) {\n yield seq;\n allSequences.add(seq.toString());\n for (let [i, n] of seq.entries()) {\n for (let m = 1; m < n; m++) {\n const sub = seq\n .slice(0, i)\n .concat([m, n - m])\n .concat(seq.slice(i + 1));\n if (!allSequences.has(sub.toString())) {\n yield* divide(...sub);\n }\n }\n }\n }\n\n return [...divide(n)];\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":4633,"value":"allScales = makeAllScales(12)","pinned":true,"hidden":false,"mode":"ojs"},{"id":4647,"value":"Writing that wasn’t as bad as I was expecting. And, after seeing that the number of scales with different number of tones are powers of two, I probably should have realized that there are ${tex`2^{n - 1}`} different scales, because each pitch can either be in it or not. That suggest another way of writing that function that’s not recursive and just counts in binary then converts it to the sequence of intervals:","pinned":false,"hidden":false,"mode":"md"},{"id":3729,"value":"function* makeAllScalesBinary(n) {\n for (let i = 0; i < 2 ** (n - 1); i++) {\n yield i\n .toString(2) // convert to binary (I’m sure there’s a better way of doing this than casting to string, but 🤷🏻)\n .padStart(n - 1, \"0\") // pad to the correct length\n .split(\"\") // convert back to array\n .map((d) => +d) // convert back to numbers\n .map((d, i) => (d ? i + 1 : null)) // convert to pitch class indexes\n .filter((d) => d !== null) // remove empties\n .concat([n]) // add next octave index\n .map((d, i, a) => d - (i - 1 >= 0 ? a[i - 1] : 0)); // subtract consecutive indexes to get steps\n }\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":4658,"value":"allScales2 = [...makeAllScalesBinary(12)]","pinned":true,"hidden":false,"mode":"ojs"},{"id":4687,"value":"allScales.filter((scale) => !scale.some((step) => step > 2))","pinned":true,"hidden":false,"mode":"ojs"},{"id":4739,"value":"allScales2.filter((scale) => !scale.some((step) => step > 2))","pinned":true,"hidden":false,"mode":"ojs"},{"id":4737,"value":"✔️, filtering to only include scales with half- and whole-steps give the same number of scales as the method all the way up at the top (with both the recursive and non-recursive versions).","pinned":false,"hidden":false,"mode":"md"},{"id":3788,"value":"eightNoteScales = allScales.filter((d) => d.length === 8)","pinned":true,"hidden":false,"mode":"ojs"},{"id":4670,"value":"transpositionallyNonEquivalentEightNoteScales = groupRotations(eightNoteScales)","pinned":true,"hidden":false,"mode":"ojs"},{"id":4675,"value":"[OH YEAH](https://www.youtube.com/watch?v=_fjEViOF4JE)! There it is.\n\nAnd just for good measure:","pinned":false,"hidden":false,"mode":"md"},{"id":4764,"value":"totalShapes = groupRotations(allScales)","pinned":true,"hidden":false,"mode":"ojs"},{"id":4767,"value":"There are ${totalShapes.length} different possible “scale shapes” using twelve pitches.\n\nThis is different than the the 220 “prime forms” listed in _The Structure of Atonal Music_ because these are only equivalent under transposition, not inversion _and_ transposition.\n\nMake something that computes the prime form for a pitch class.","pinned":false,"hidden":false,"mode":"md"},{"id":5623,"value":"embedSpotify(\n \"https://open.spotify.com/track/6Dt6vUL6iKTNWi5OC7kv5Y?si=ad936b19e7e441f7\"\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":5626,"value":"I occasionally like to listen to music like this 👆🏻, but I think if I understand more about the underlying principles of how it’s composed I’ll probably [appreciate it / like it more](https://youtu.be/ZbFM3rn4ldo).","pinned":false,"hidden":false,"mode":"md"},{"id":4747,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":5602,"value":"## Exploring other modes","pinned":false,"hidden":false,"mode":"md"},{"id":4749,"value":"How about all of the five-note scales with all intervals smaller than two whole-steps?","pinned":false,"hidden":false,"mode":"md"},{"id":4753,"value":"sixNoteScales = allScales\n .filter((scale) => !scale.some((step) => step > 3))\n .filter((scale) => scale.length < 6)\n//.filter((scale) => intervalsToIndexes(scale).includes(7)) // only include those that have a major fifth","pinned":true,"hidden":false,"mode":"ojs"},{"id":5600,"value":"_Click on a diagram to play it as a chord._","pinned":false,"hidden":false,"mode":"md"},{"id":3811,"value":"// should not be copy pasting this!\n{\n let w = (Math.min(312, width) - 34) / 3;\n let result = [];\n for (let [i, modes] of groupRotations(sixNoteScales).entries()) {\n result.push(\n html`

\nGroup ${i + 1}\n

${name}

\n ${constellationPlot(intervals, 12, w)}\n`);\n });\n result.push(html`

${constellations}

`);\n }\n return html`${result}`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":3252,"value":"---\n\n## What have I learned so far?","pinned":false,"hidden":false,"mode":"md"},{"id":5607,"value":"$37","pinned":false,"hidden":false,"mode":"md"},{"id":4402,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":2827,"value":"## Get wikipedia table","pinned":false,"hidden":false,"mode":"md"},{"id":2833,"value":"Fetch and parse the wikpedia page with list of modes, falling back to a saved version if the live version doesn’t work.\n\nThe Octatonic scale has two interval sequences [listed as variants](https://en.wikipedia.org/wiki/List_of_musical_scales_and_modes) but this is only parsing one. Fix that so that both are labelled in the diagram.","pinned":false,"hidden":false,"mode":"md"},{"id":2777,"value":"wikipediaLookup = {\n const result = {};\n wikipediaTable.forEach((row) => {\n const key = row.intervals.toString();\n result[key] = row;\n });\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1298,"value":"wikipediaTable = {\n // adapted from https://observablehq.com/@mbostock/working-with-wikipedia-data, but tries to get the live version from wikipedia and falls back to downloaded version if the live version fails for any reason.\n let json;\n let table;\n try {\n let response = await fetch(\n `https://en.wikipedia.org/w/api.php?format=json&origin=*&action=parse&prop=text&page=List_of_musical_scales_and_modes§ion=0`\n );\n if (!response.ok) {\n throw Error(`response not ok: ${response}`);\n }\n json = await response.json();\n // return json // uncomment to return a version for saving\n table = parseWikipediaResponse(json);\n checkWikipediaTable(table);\n } catch (error) {\n const file = FileAttachment(\n \"2023-11-05-List_of_musical_scales_and_modes.json\"\n );\n console.error(`Caught: ${error}, falling back to ${file.name}`);\n table = parseWikipediaResponse(await file.json());\n }\n\n return table;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1389,"value":"function checkWikipediaTable(parsed) {\n if (!Array.isArray(parsed)) {\n throw Error(\"not an array\");\n }\n if (parsed.length == 0) {\n throw Error(\"empty array\");\n }\n let row = parsed[0];\n if (!row.intervals || !row.nPitchClasses || !row.name) {\n console.log(row);\n throw Error(\"does not have intervals, pitch classes or name\");\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1266,"value":"$38","pinned":false,"hidden":false,"mode":"ojs"},{"id":2240,"value":"## Imports","pinned":false,"hidden":false,"mode":"md"},{"id":608,"value":"_ = require(\"lodash\", \"lodash.combinations\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":645,"value":"Tone = require(\"tone\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":5309,"value":"## Spotify","pinned":false,"hidden":false,"mode":"md"},{"id":4901,"value":"function embedSpotify(shareLink, baseURL = \"https://open.spotify.com\") {\n const parsed = new URL(shareLink, baseURL);\n let url = `${parsed.origin}${parsed.pathname}`;\n if (!parsed.pathname.startsWith(\"/embed/\")) {\n url = `${parsed.origin}/embed${parsed.pathname}`;\n }\n return html``;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":4862,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":2093,"value":"## Helpers\n\nSome stuff that helps with the development of the notebook.\n\nMove the `noteList` helper and styling to it’s own notebook and import? This was surprisingly helpful to keep me from getting distracted by everything that I could have done.","pinned":false,"hidden":false,"mode":"md"},{"id":2881,"value":"maybeLaterStyle = {\n return html``;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":3816,"value":"$39","pinned":false,"hidden":false,"mode":"ojs"},{"id":4373,"value":"maybeLaterList()","pinned":false,"hidden":false,"mode":"ojs"},{"id":4410,"value":"
","pinned":false,"hidden":false,"mode":"md"}],"files":"$W3a","stdlib":"1","titling":"auto"}