2e:null 2f:["$","title",null,{"children":"Play In Tune! | K Lindberg | Observable"}] 34:T520,style = { const size = 8; const huge = 1023; const cursor = "cursor: ew-resize;"; const background = "background-color: lightgray;"; return html``; }35:T69c,function frequencyInput(initialFrequency, pureFrequency, mode, tuningAid) { const step = 0.5; const initial = Math.round(initialFrequency/step)*step; const min = 0; const max = pureFrequency * 5/4; const result = html` `; const range = result.range; const number = result.number; switch (mode) { case Mode.numeric: range.setAttribute("disabled", true); break; case Mode.readonly: range.setAttribute("disabled", true); number.setAttribute("disabled", true); number.setAttribute("readonly", "readonly"); break; } function update(value) { result.value = { tuned: value, pure: pureFrequency }; } update(initialFrequency); range.value = initial; number.value = initial; const note = new Pizzicato.Sound({ source: "wave", options: { frequency: initial } }); function link(from, to) { from.addEventListener("input", () => { const value = from.valueAsNumber; to.value = value; update(value); note.frequency = Math.max(value, 0.000000001); }); } link(range, number); link(number, range); control(range, note); result.note = note; if (tuningAid) { updateResult(result, tuningAid); range.addEventListener("input", () => updateResult(result, tuningAid)); number.addEventListener("input", () => updateResult(result, tuningAid)); } return result; }36:T4ca,function tuner(baseFrequency, initialFrequency, pureFrequency, playPure, tuningAid, mode) { const result = frequencyInput(initialFrequency, pureFrequency, mode, tuningAid); result.prepend(showNoteName(pureFrequency)); function showNoteName(frequency) { return html`${noteName(frequency)}`; } result.appendChild(html`

`); const base = frequencyInput(baseFrequency, pureFrequency, Mode.readonly, false); base.prepend(html`${showNoteName(baseFrequency)}`); result.appendChild(base); result.append(playButton()); const button = result.button; control(button, result.note); control(result.range, base.note); control(button, base.note); const pure = frequencyInput(pureFrequency, pureFrequency, Mode.readonly, false); if (playPure) { control(result.range, pure.note); control(button, pure.note); } if (!tuningAid) { button.addEventListener("mousedown", () => updateResult(result, tuningAid)); result.range.addEventListener("input", () => result.result.innerHTML = ""); result.number.addEventListener("input", () => result.result.innerHTML = ""); } result.pureFrequency = pureFrequency; return result; }37:T48b,function interval(baseNotePitch, tunedNotePitch, sameColor, clef = "treble") { const result = html`

`; const VF = Vex.Flow; const renderer = new VF.Renderer(result, VF.Renderer.Backends.SVG); renderer.resize(200, 110); const context = renderer.getContext(); const stave = new VF.Stave(0, 0, 100); stave.setClef(clef); stave.setContext(context); stave.draw(); function createNote(pitch) { const result = new VF.StaveNote({ clef: clef, keys: [pitch], duration: "w" }); if (pitch.includes("#")) { result.addAccidental(0, new VF.Accidental("#")); } return result; } const tunedNote = createNote(tunedNotePitch); if (!sameColor) tunedNote.setStyle({fillStyle: tunedColor}); const voice1 = new VF.Voice(); voice1.addTickables([tunedNote]); const baseNote = createNote(baseNotePitch); if (!sameColor) baseNote.setStyle({fillStyle: baseColor}); const voice2 = new VF.Voice(); voice2.addTickables([baseNote]); const voices = [voice1, voice2]; const formatter = new VF.Formatter().joinVoices(voices).format(voices, 0); voice1.draw(context, stave); voice2.draw(context, stave); return result; }38:T459,function melody(timeSignature, rightHandNotes, leftHandNotes, frequencies) { const result = html``; const button = playButton(); result.appendChild(button); const VF = Vex.Flow; var vf = new VF.Factory({ renderer: {elementId: result.notation, width: 150, height: 220} }); var score = vf.EasyScore(); score.set({time: timeSignature}); var system = vf.System(); system.addStave({ voices: [ score.voice(score.notes(rightHandNotes)) ] }).addClef('treble').addTimeSignature(timeSignature); score.set({clef: 'bass'}); system.addStave({ voices: [ score.voice(score.notes(leftHandNotes)) ] }).addClef('bass').addTimeSignature(timeSignature); system.addConnector(); vf.draw(); const notes = frequencies.map(frequency => new Pizzicato.Sound({ source: "wave", options: { frequency: frequency } })); const group = new Pizzicato.Group(notes); button.addEventListener("mousedown", () => group.play()); button.addEventListener("mouseup", () => group.stop()); return result; }39:[] 32:{"id":"8910fdaa76484a3d","slug":"play-in-tune","title":"Play In Tune!","thumbnail":"dc88d481539aa67f633d380f98d6f64acd4e03083e9a45c97d595095c6ab3516","access_level":"public","create_time":"$D2018-05-20T20:10:00.955Z","update_time":"$D2018-05-20T20:10:00.955Z","view_time":null,"first_public_time":"$D2018-05-20T20:10:04.623Z","has_unpublished_edits":null,"fork":{"id":"8fe0dde959564732","slug":"play-in-tune","title":"Play In Tune!","owner":{"id":"26479aff057bfddc","login":"thsoft","name":"Dénes Harmath"}},"stars":1,"starred":false,"owner":{"type":"team","id":"63f5a83fdb550e46","login":"essingen123","name":"K Lindberg","avatar_url":"https://avatars-next.observableusercontent.com/f32fa345052e3b48114aa58703cb58d8886d6a48bd89a61b2892d012f73b0b53","tier":"starter_2024"},"version":2140,"publish_version":2140,"role":null,"authored":false,"authors":[{"type":"individual","id":"dca00f83ab125ba4","login":"essingen123","name":"K Lindberg","avatar_url":"https://avatars-next.observableusercontent.com/f32fa345052e3b48114aa58703cb58d8886d6a48bd89a61b2892d012f73b0b53","tier":null}],"members":[{"type":"individual","id":"dca00f83ab125ba4","login":"essingen123","name":"K Lindberg","avatar_url":"https://avatars-next.observableusercontent.com/f32fa345052e3b48114aa58703cb58d8886d6a48bd89a61b2892d012f73b0b53","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Play In Tune!","cells":[{"id":1173,"value":"md`# Play In Tune!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":0,"value":"html`

An Explorable Explanation About Musical Temperament

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2018,"value":"md`__Technical note: Unfortunately this notebook works only in desktop browsers. I apologize for the inconvenience.__`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":445,"value":"md`## Uncle Tim's present`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":32,"value":"md`It's just one week till your Mom's birthday! This year, you want to surprise her with a really special gift! You plan to perform *Happy Birthday* in a family collaboration: you play the piano accompanying Uncle Tim who has a really beautiful tenor voice.\n\nYou start the preparations eagerly. You call Uncle Tim and explain your plan. \"Great! I also have an awesome arrangement in C major\", he says, \"but I and my sophisticated ear insist that the piano be perfectly in tune! Last Christmas it was such a mess, I could barely sing 'Twinkle Twinkle' to the end!\" Uh-oh! The family's good ol' piano is in fact horribly out of tune. You'll have to tune it note for note... This gets even more intimidating when Tim sends you the sheet music:

\n\nThis is going to be though! But you take a deep breath, dust the family piano and grab a tuner...`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":456,"value":"md`### Your first pure note`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":458,"value":"md`You take the first note, C3 as a base note (it is a bit flat, but Uncle Tim doesn't have absolute pitch and doesn't like to sing that high anyway), and want to tune C4 first (displayed in ${tunedColor} below). Drag the ${tunedColor} slider to change the pitch of the note until it is in tune.\n\nFortunately, you have a tuner handy, which indicates visually whether you should tune the note higher or lower, or it is already in tune. Hold the play button to hear the interval.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":818,"value":"interval(\"c/3\", \"c/4\", false, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":394,"value":"tuner(c3, c4Impure, c4, false, true, Mode.interactive)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":793,"value":"md`Congratulations for your first pure note! The pitch of the note is measured as its *frequency* which is displayed in the numeric field. Its unit is Hertz. The higher or lower the note, the greater or smaller its frequency respectively.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":881,"value":"md`### Hearing the beating`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":883,"value":"md`While tuning the next note (E4 to C4), you discover a new feature on your tuner: during tuning, it can also play the pure note that you want to achieve.\n\nThis way, you can also hear easily whether the note to tune is pure: if not, you can hear an interesting phenomenon called *beating* which means that instead of the pure note and the note to tune, you hear one note whose volume is fluctuating.\n\nThe closer you are to the pure note, the slower the beating. Pay attention to it while tuning!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":919,"value":"interval(\"c/4\", \"e/4\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":924,"value":"tuner(c4, e4Impure, e4, true, true, Mode.interactive)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":938,"value":"md`### Your first pure chord`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":940,"value":"md`Aw, snap! Now the visual tuning aid on your tuner has gone out of order. You have to tune the next note solely by ear based on the beating. Press the play button to check the result. Good luck!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":943,"value":"interval(\"c/4\", \"g/4\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":952,"value":"tuner(c4, g4Impure, g4, true, false, Mode.interactive)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1070,"value":"md`Wow, you completed tuning the pure I chord in C major! Now you can play the first measure of Happy Birthday! Click and hold the play button to enjoy pure harmony!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1098,"value":"melody(\"3/4\", \"(C4 E4 G4)/h.\", \"C3/h.\", [c3, c4, e4, g4])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1313,"value":"md`### Calculating the frequency`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1352,"value":"md`On the other hand, this process took so much time that you doubt you'll finish tuning the notes necessary to perform your Mom's favorite birthday song in time. There must be a rule so that you don't have to carefully adjust the pitch... Let's tune G3 and try to find it!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1354,"value":"interval(\"c/3\", \"g/3\", false, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1356,"value":"tuner(c3, g3Impure, g3, true, false, Mode.interactive)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1360,"value":"md`Now we have C3 and C4 as well as G3 and G4. What could be their relationship? Let's visualize their frequencies!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1370,"value":"interval(\"c/3\", \"c/4\", true, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1364,"value":"tuner(c3, c4, c4, false, true, Mode.readonly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1372,"value":"interval(\"g/3\", \"g/4\", true)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1368,"value":"tuner(g3, g4, g4, false, true, Mode.readonly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1389,"value":"md`Look at the length of the ${baseColor} bars and notice: *a note that is an octave higher than another note has a frequency which is twice the frequency of the other note!* This can be written in a more concise way in the language of mathematics, if you prefer:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1419,"value":"tex`f_{note + octave} = f_{note} \\cdot 2`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1428,"value":"md`### Intervals as frequency ratios`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1431,"value":"md`Next day, you hear angry knocking on the door. You open it and encounter your neighbour who seems mad as hell. \"You are driving me crazy with this unbearable cacophony! One more of that horrible caterwauling effect and I'll prosecute you!\", he yells and shuts the door before you could say anything. Oh no! That means you can't tune any more by ear, you have to get the pitch right without trial and error. Fortunately, we just learned a way of calculating the frequency. Let's verify whether it works also with perfect fifths! You already tuned G3 to C3.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1436,"value":"interval(\"c/3\", \"g/3\", true, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1438,"value":"tuner(c3, g3, g3, false, true, Mode.readonly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1443,"value":"md`Let's calculate the frequency of D4 related to G3! Just like in the case of C3-C4 and G3-G4 octaves, we suspect that if there is the same interval between two notes, then their frequencies are related in the same way. Again, look at the bar lengths! At a rough estimate, the longer one looks like it is 1.5 times long as the shorter one. If our assumption is true, the frequency of D4 can also be calculated as 1.5 (or 3/2) times G3's frequency:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1563,"value":"tex`f_{D4} = f_{G3} \\cdot \\frac{3}{2}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1560,"value":"md`Enter the result of this calculation in the numeric field, listen to it and hear whether the note is pure!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1459,"value":"interval(\"g/3\", \"d/4\", false, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1461,"value":"tuner(g3, 0, d4, true, false, Mode.numeric)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1536,"value":"md`This is also true in general:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1527,"value":"tex`f_{note + fifth} = f_{note} \\cdot \\frac{3}{2}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1525,"value":"md`It turned out that *intervals can be defined as frequency ratios*! And curiously, the intervals we encountered had quite \"beautiful\" ratios — that is, rational numbers, not like ${tex`\\pi`}, for example. Is this the case for others as well?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1588,"value":"md`### Calculating the frequency ratio`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1591,"value":"md`Let's continue with B3! It is a major third higher than G3 whose frequency we already know. And we also know the frequencies of two other notes a major third apart:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1594,"value":"interval(\"c/4\", \"e/4\", true)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1596,"value":"tuner(c4, e4, e4, false, true, Mode.readonly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1609,"value":"md`Calculate the frequency ratio of the major third from the frequencies of C4 and E4, then enter the result below in the already known formula. B3 will be automatically tuned based on it.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1602,"value":"viewof majorThird = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1613,"value":"interval(\"g/3\", \"b/3\", false, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1621,"value":"tuner(g3, addInterval(g3, majorThird.value), b3, false, false, Mode.readonly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1690,"value":"md`Great, you are not only done with your second pure chord (C major V) which you can play confidently, but you also discovered useful relationships with which you can tune the remaining notes much faster and without driving your neighbours to nuts!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1692,"value":"melody(\"3/4\", \"(B3 D4 G4)/h.\", \"G3/h.\", [g3, b3, d4, g4])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1698,"value":"md`### Inverted intervals`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1700,"value":"md`You only need one more note for the next measure, B2. The simplest way to derive it is to tune an octave *down* from B3. So if the higher note of an octave is twice the frequency of the lower one, how do we calculate the lower note from the higher one? Enter the answer in our case.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1711,"value":"interval(\"b/3\", \"b/2\", false, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1713,"value":"tuner(b3, 0, b2, true, false, Mode.numeric)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1734,"value":"md`Correct! Tuning an interval *upwards* means *multiplying* by its frequency ratio, tuning *downwards* means *dividing* by it!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1738,"value":"md`### Finding the point of reference`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1741,"value":"md`We reached a point where we know the frequencies of many pure notes as well as the frequency ratios of many friendly intervals, so we can calculate the frequencies of many new notes based on these numbers. For example, E3 is easy: we just take C3 and go up a major third. F3 is a bit trickier. Find the already familiar note and interval with which it can be derived!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1745,"value":"f3Tuning = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1776,"value":"tex`f_{F3} = f_{${noteName(f3Base)}} ${f3Direction == \"up\" ? \"\\\\cdot\" : \":\"} ${f3Interval}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1840,"value":"tuner(f3Base, addInterval(f3Base, f3Direction == \"up\" ? f3Interval : 1/f3Interval), f3, false, false, Mode.readonly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1866,"value":"md`Well done! The only remaining note is A3, which you quickly and easily tune by tuning a major third upwards on F3.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1868,"value":"md`### The birthday`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1897,"value":"md`The birthday party is quickly approaching. You finish the tuning just before the guests start to arrive. When Uncle Tim enters, he asks: \"Did you tune the piano appropriately?\" \"Yes, of course\", you respond, although you are a bit nervous when you sit down to the piano. But soon, sounds of sweet harmony emerge...`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1917,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1948,"value":"md`

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1913,"value":"md`\"I can't believe it! You did a miracle to your piano!\", says Uncle Tim proudly after your performance, tapping your shoulders. But it is Mom who is the happiest: \"This was truly the most beautiful birthday present ever!\"\n\nTinkering with the piano was really worth the hard work! 😌 And what's more, you can play any song in C major that uses the I, IV and V chords — which is an awful lot of songs! 😉`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1934,"value":"md`## Aunt Alice's surprise`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1939,"value":"md`Later during the party, Aunt Alice approaches you: \"This song was so inspiring that I took a liking for singing it.\" \"But you are an alto!\", you object. \"No problem, I got another arrangement in G major! Could we sing it right away?\"\n\n

\n\nUnfortunately you have to tune some more notes, but you make a deal that tomorrow you'll rehearse it together.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1963,"value":"md`### An unexpected dischord`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1965,"value":"md`Soon you realize the only note left to tune is F#. You routinely tune it:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1976,"value":"interval(\"d/4\", \"f#/4\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1967,"value":"tuner(d4, fis4Impure, fis4, true, false, Mode.interactive)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1992,"value":"md`Then you start the rehearsal:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1989,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2026,"value":"md`Aunt Alice stops singing abruptly when she hears the V chord in G major. \"You said you had tuned the piano...\", she says indignantly. You respond: \"I swear!\" \"But that A3 is too flat!\" You don't believe, so you play the second measure again in itself:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2039,"value":"melody(\"3/4\", \"(A3 D4 F#4)/h.\", \"D3/h.\", [d3, a3FromF, d4, fis4])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2045,"value":"md`Indeed, you hear a slight beating, which indicates there must be a problem... But what exactly? Did the A3 go out of tune in the meantime?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2032,"value":"interval(\"f/3\", \"a/3\", true, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2036,"value":"tuner(f3, a3FromF, a3FromF, false, true, Mode.readonly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2069,"value":"md`That's clearly in tune; you don't understand what went wrong. Then a discovery shocks you: you tuned A3 based on F3, while it is now part of a V chord in G major, so you should have tuned it based on D3!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1996,"value":"interval(\"d/3\", \"a/3\", true, \"bass\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2003,"value":"tuner(d3, a3FromD, a3FromD, false, true, Mode.readonly)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2084,"value":"md`These two A3s differ by a significant amount (more than two Hertzes)! That means you have to tweak the tuning of A3 *every time* you switch between C major and G major, even in case of such a simple song as *Happy Birthday*!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2102,"value":"md`### Resolving the tension`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2104,"value":"md`What strategies could you employ to overcome this problem?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2114,"value":"md`* Tune that A3 to the average frequency (201.25 Hz), which means a compromise: neither Tim's nor Alice's accompaniment will be perfect, but hopefully they won't notice the ~1 Hz difference.\n\n

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2116,"value":"md`* Change Tim's accompaniment: replace IV with ii, so D will be the base note when tuning this chord.\n\n

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2118,"value":"md`* Observe the usage of this note: it is used only for one measure in the C major version, but more than two measures in the G major version, so tune it based on D and convince Tim that IV is only a transient chord, and hearing it for just one measure will not hurt anyone, on the contrary: it will make the resolution to the pure I chord even more satisfying.\n\nInstead of this:\n

\n...this is preferrable:\n

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1158,"value":"md`## To be continued...`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":936,"value":"md`## Code`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":827,"value":"md`### Intervals`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":855,"value":"function multiplyInterval(interval, times) {\n return Math.pow(interval, times)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":841,"value":"octave = 2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1083,"value":"detune = 1.02","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":830,"value":"equalMinorSecond = Math.pow(2, 1/12)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":843,"value":"pureMajorThird = 5/4","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":848,"value":"purePerfectFifth = 3/2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1834,"value":"f3Direction = Generators.input(f3Tuning.direction)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1931,"value":"f3Interval = Generators.input(f3Tuning.interval)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":833,"value":"md`### Notes`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":766,"value":"function addInterval(baseFrequency, interval) {\n return baseFrequency * interval;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":770,"value":"function subtractInterval(baseFrequency, interval) {\n return baseFrequency / interval;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1702,"value":"b2 = subtractInterval(b3, octave)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":858,"value":"c3 = 120","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2006,"value":"d3 = subtractInterval(d4, octave)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1767,"value":"e3 = addInterval(c3, pureMajorThird)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1842,"value":"f3 = subtractInterval(c4, purePerfectFifth)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":860,"value":"g3 = addInterval(c3, purePerfectFifth)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1080,"value":"g3Impure = subtractInterval(g3, detune)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1617,"value":"b3 = addInterval(g3, pureMajorThird)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":871,"value":"c4 = addInterval(c3, octave)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":873,"value":"c4Impure = subtractInterval(c4, detune)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1464,"value":"d4 = addInterval(g3, purePerfectFifth)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":926,"value":"e4 = addInterval(c4, pureMajorThird)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":928,"value":"e4Impure = addInterval(e4, detune)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1955,"value":"fis4 = addInterval(d4, pureMajorThird)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1957,"value":"fis4Impure = addInterval(fis4, detune)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":945,"value":"g4 = addInterval(c4, purePerfectFifth)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":947,"value":"g4Impure = subtractInterval(g4, detune)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1944,"value":"a3FromF = addInterval(f3, pureMajorThird)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1941,"value":"a3FromD = subtractInterval(a4, octave)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1903,"value":"a4 = addInterval(d4, purePerfectFifth)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1831,"value":"f3Base = Generators.input(f3Tuning.base)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":869,"value":"md`### Tuner`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":889,"value":"Pizzicato = require(\"pizzicato\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":893,"value":"Vex = require(\"vexflow\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1276,"value":"tunedColor = \"blue\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1279,"value":"baseColor = \"black\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1201,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":361,"value":"function control(input, note) {\n input.addEventListener(\"mousedown\", () => note.play());\n input.addEventListener(\"mouseup\", () => note.stop());\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1039,"value":"function updateResult(result, tuningAid) {\n const frequency = result.value.tuned;\n const pureFrequency = result.value.pure;\n const range = addInterval(pureFrequency, equalMinorSecond) - subtractInterval(pureFrequency, equalMinorSecond);\n const distance = Math.min(Math.abs(pureFrequency - frequency) / range, 1);\n const threshold = 0.05;\n const red = distance * 255;\n var green = 0;\n const blue = 0;\n var message;\n const outOfTune = \"✗ Out of tune!\"\n if (frequency < pureFrequency - threshold) {\n message = tuningAid ? \"▲ Flat!\" : outOfTune;\n } else if (frequency > pureFrequency + threshold) {\n message = tuningAid ? \"▼ Sharp!\" : outOfTune;\n } else {\n green = 255;\n message = \"✓ In tune!\";\n }\n result.result.innerHTML = `${message}`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1483,"value":"Mode = {\n return {\n interactive: 0,\n numeric: 1,\n readonly: 2\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":74,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2052,"value":"function playButton() {\n return html``;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":334,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":804,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1094,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1848,"value":"function noteName(frequency) {\n return notes.find(note => note.frequency == frequency).name;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1855,"value":"notes = [\n { name: \"B2\", frequency: b2 },\n { name: \"C3\", frequency: c3 },\n { name: \"D3\", frequency: d3 },\n { name: \"E3\", frequency: e3 },\n { name: \"F3\", frequency: f3 },\n { name: \"G3\", frequency: g3 },\n { name: \"A3\", frequency: a3FromD },\n { name: \"A3\", frequency: a3FromF },\n { name: \"B3\", frequency: b3 },\n { name: \"C4\", frequency: c4 },\n { name: \"D4\", frequency: d4 },\n { name: \"E4\", frequency: e4 },\n { name: \"F#4\", frequency: fis4 },\n { name: \"G4\", frequency: g4 },\n] ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1970,"value":"notesBeforeF3 = [\n { name: \"B2\", frequency: b2 },\n { name: \"C3\", frequency: c3 },\n { name: \"E3\", frequency: e3 },\n { name: \"G3\", frequency: g3 },\n { name: \"B3\", frequency: b3 },\n { name: \"C4\", frequency: c4 },\n { name: \"D4\", frequency: d4 },\n { name: \"E4\", frequency: e4 },\n { name: \"G4\", frequency: g4 },\n] ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W39","stdlib":"1","titling":"auto"}} 33:"$32:body"