2c:null 2d:["$","title",null,{"children":"Modes of the Diatonic scale in 12EDO | Carlos Scheidegger | Observable"}] 32:T985,## Making the modulations "more natural" Moving modes while keeping the same tonic is a little weird; it's not very common in the music I listen to. I do hear stuff like switching to the parallel minor, but that's "three modes down". A common modulation is to go to the same mode, one fifth above or below. So I wondered if we could get through the entire table by moving in fifths. I thought I would be able to, but I haven't, not exactly. ## 2023-08-21 12:46PM PDT Edit My comment on the original post made it seem like there was a simple solution going only by the circle of fifths. I was wrong, sorry! I can't seem to be able to edit or delete my comment at the original post, but I don't think what I suggested in the comment works. I apologize for the noise! When we switch modes, our scales are seemingly off by more notes than one. ## 2023-08-21 2PM PDT Edit However, I seem to have found a sequence that has the property of: - Always moving by a half-step from one mode to another - Visiting each enharmonic mode exactly once - Moving the root along the circle of fifths, _almost always_ The way to describe this sequence is that when we move from one mode to the next, we move the root _twice_ in the circle of fifths. Intuitively, the mode is getting darker by a semi-tone, and moving by the circle of fifths sharpens the scale by a semitone. In order to keep moving ahead "one semi-tone at a time", we need to move the root twice. This makes sense to me. We move from C# Lydian to Eb Ionian, ``` C# Lydian: C# D# E# F## G# A# B# C# Eb Ionian: Eb F G Ab Bb C D ``` The half-step modulation took C# to D, and the other notes match by enharmonicity. The strange property here is that the note to be sharped is _always_ the C# (or Db)! I don't understand why. In Ionian, then, we start the loop through the circle of fifths "one step ahead" of Lydian, in Eb (or D#). In Mixolydian, we start in Bb/A#. And so on. This seems to work. Musically, the modulation in fifths within a mode seems pretty natural to me. But I've never seen a song that modulates (eg) from Ab Ionian to Bb Mixolydian. It's also worth noting that the very last step in this sequence still moves a single note (still Db) by a single half-step, but now the root moves up by a _half-step_ rather than a whole step. Specifically, we have: ``` G Locrian: G Ab Bb C Db Eb F G Ab Lydian: Ab Bb C D Eb F G ``` I don't really understand why.33:T415,function scale(fullMode) { let { tonic, mode } = fullMode; while (mode < 0) mode += 7; while (tonic < 0) tonic += 12; mode = mode % 7; tonic = tonic % 12; tonic = tonicName[tonic]; mode = modeTranspositions[mode]; const baseNote = tonic[0]; const allNotes = "CDEFGABCDEFGAB"; const allIntervals = [2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1]; const ix = allNotes.indexOf(baseNote); const notes = allNotes.slice(ix, ix+7); const noteIntervals = [0, ...allIntervals.slice(ix, ix + 7)]; const modeIntervals = [0, ...allIntervals.slice(mode, mode + 7)]; const result = []; let runningAccidental = isFlat(tonic) ? -1 : isSharp(tonic) ? 1 : 0; for (let i = 0; i < 7; ++i) { runningAccidental += modeIntervals[i] - noteIntervals[i]; const accidental = // you monster (runningAccidental === 0 ? "" :runningAccidental < 0 ? "♭".repeat(Math.abs(runningAccidental)) :"♯".repeat(runningAccidental)); result.push((notes[i] + accidental).replaceAll("♯♯", "𝄪")); } return result; }34:[] 30:{"id":"8e2de2d02a1af477","slug":"modes-of-the-diatonic-scale-in-12edo","title":"Modes of the Diatonic scale in 12EDO","thumbnail":"f44b0dcb3b0e4e9dc03ce5ba0f24ac778f9e7b554520888274755213a30854e9","access_level":"public","create_time":"$D2023-08-21T17:26:47.010Z","update_time":"$D2023-08-21T22:04:54.568Z","view_time":null,"first_public_time":"$D2023-08-21T17:26:47.010Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"be2e428cd8e75a20","login":"cscheid","name":"Carlos Scheidegger","avatar_url":"https://avatars-next.observableusercontent.com/e56dd8fa390d748e094247eea197dbf71a9be34a3e47541e7d93e61956803cd9","tier":"starter_2024"},"version":216,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"79e836b269fc6922","login":"cscheid","name":"Carlos Scheidegger","avatar_url":"https://avatars-next.observableusercontent.com/e56dd8fa390d748e094247eea197dbf71a9be34a3e47541e7d93e61956803cd9","tier":null}],"members":[{"type":"individual","id":"79e836b269fc6922","login":"cscheid","name":"Carlos Scheidegger","avatar_url":"https://avatars-next.observableusercontent.com/e56dd8fa390d748e094247eea197dbf71a9be34a3e47541e7d93e61956803cd9","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Modes of the Diatonic scale in 12EDO","cells":[{"id":0,"value":"# Modes of the Diatonic scale in 12EDO\n\nLet's riff off of . The idea there is that there is a way to move along all modes of the diatonic scale in all 12 notes of the chromatic scale (12EDO) by always changing exactly one note, and always moving it down a half-step.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":144,"value":"## Their big table\n\nThis is the [table built by Wyrd Smithe](https://logosconcarne.com/2023/08/19/musical-scale-modes/), sans coloring:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":101,"value":"{\n let result = [];\n for (let tonic = 0; tonic >= -11; --tonic) {\n let tp = tonic;\n while (tp < 0) tp += 12;\n for (let mode = 0; mode < 7; ++mode) {\n result.push(`${tonicName[tp]} ${modeName[mode]}` + scale({tonic, mode}).map(s => `${s}`).join(\"\") + \"\");\n }\n }\n return html`${result}

`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":170,"value":"# Appendix: Code","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":6,"value":"function makeMode(tonic, mode) {\n return { tonic, mode }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"// we're preferring some enharmonics here, unavoidable; we're choosing \"shortest path of fourths of fifths from C, tiebreaker goes to flat\"\ntonicName = [\"C\", \"C#\", \"D\", \"Eb\", \"E\", \"F\", \"F#\", \"G\", \"Ab\", \"A\", \"Bb\", \"B\"]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":142,"value":"modeName = [\"Lydian\", \"Ionian\", \"Mixolydian\", \"Dorian\", \"Aeolian\", \"Phrygian\", \"Locrian\"]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":18,"value":"function isSharp(note) {\n return note[note.length - 1] === '#';\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"function isFlat(note) {\n return note[note.length - 1] === 'b';\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":28,"value":"modeTranspositions = [3, 0, 4, 1, 5, 2, 6];\n ","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"$33","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":31,"value":"scale({tonic: 4, mode: 1})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":72,"value":"function makeModeFromNumber(number) {\n while (number < 0) number += 84;\n const tonic = number % 12;\n const mode = ~~(number / 12) % 7;\n return { tonic, mode };\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":103,"value":"makeScaleFromNumber = x => scale(makeModeFromNumber(x))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W34","stdlib":"1"}} 31:"$30:body"