Harmonic Series Explorer
Every harmonic is a whole-number multiple of the fundamental: harmonic n sounds at n × f0. Play up the ladder, or stack the partials into a drone, and watch how far each one drifts from equal temperament.
Move the fundamental and every frequency changes, but the cents column never does — a harmonic’s distance from equal temperament depends only on its number. B♭2 gives the bugle and natural-trumpet partials.
Tap any rung to sound that harmonic on its own.
Which harmonic?
You hear the fundamental, then one mystery partial above it. Name the partial — tap a number below or the matching rung on the ladder.
Press New question and listen. The fundamental comes first as your reference.
Get a miss and the widget plays your answer first, then the true partial, back to back — that pairing is what actually trains the ear. Octaves (2, 4, 8, 16) are the easiest anchors; 3 and 6 sit a fifth above them; 5 and 10 are the sweet major third; 7 is the flat, bluesy one. Nothing is stored between sessions, so the score resets when the page reloads.
Build a timbre
Switch partials in and out and set how loud each one is. Every musical tone colour is just a recipe of harmonics — build one and hear it.
Tap Hold the tone to start the composite, then switch partials in and out while it rings.
The whole tone is trimmed automatically as you add partials, so the sum of all the sine amplitudes never exceeds full scale — adding harmonic 16 will duck the rest very slightly rather than clip. Pluck it gives every partial its own decay, with the high ones dying away first, which is what makes a plucked string sound dull as it fades.
Bugle & natural horn
Before valves, a brass instrument could sound only its harmonic series. The player picked a partial with the lips — nothing else was available.
A bugle has no valves and no slide. Everything it can play is on these five buttons.
Brass players read written notes, and on any natural brass instrument partial 2 is written C. That is why the written set is always C, G, C, E, G, B♭, C, D, E, F♯, G … whatever key the instrument is actually in. The sounding column shows what comes out over the fundamental you have set: pick B♭2 for a real bugle and F2 for the classic natural horn. Written partial 11 is notated F♯ but sounds 49 cents flat of it — horn players lipped or hand-stopped it into place.
Pure harmonic vs equal temperament
The cents column tells you the gap. This makes you hear it: the same partial, first as nature tunes it, then as a piano would play it.
—
Both at once is the giveaway: two sine tones a few hertz apart beat against each other, and the beat rate in the readout is exactly the frequency difference. A pure octave or fifth beats slowly or not at all; harmonic 11 against its equal-tempered neighbour churns. With the fundamental underneath, the pure version locks into the drone and the tempered one grinds against it — that roughness is the whole argument between just intonation and the piano.
How many partials can you hear?
Partials slide in one at a time, each one quieter than the last. Press I hear it every time you notice the tone change, and stop when you honestly cannot tell any more.
Use headphones and a quiet room. The test forces the ladder to all 16 partials.
Each new partial enters about 2.9 dB below the one before it, so partial 16 arrives roughly 43 dB under the fundamental. Most listeners lose the thread somewhere between the 7th and the 11th on small speakers, and go several partials further on headphones. This is not a hearing test — it measures how well you can pick a faint new component out of a tone you are already hearing.
The six panels
- The row of tabs under the title switches the workbench below the ladder: Explore, Which harmonic?, Build a timbre, Bugle & horn, 7th & 11th A/B and Fade test. The fundamental, transport, ladder and volume stay put in every panel.
- What a ladder rung does changes with the panel, and the line just above the ladder always says which: in Explore it sounds the harmonic, in the game it is your answer, in Build a timbre it switches that partial in or out, in Bugle mode it tries to lip that partial, in A/B it loads that harmonic into the comparison.
- Switching panels stops every sound first, so nothing is ever left ringing behind you.
Mouse & touch
- Click or tap any ladder row to sound that harmonic on its own. The row lights up while it rings and stays outlined as the current selection, so you can compare two partials by alternating between them.
- Play up the series walks from harmonic 1 to the top of the ladder, one partial per step, lighting each row as it sounds. Every note is scheduled in advance against the audio clock, so the rhythm never drifts. Press it again (or Stop) to cancel.
- Stack partials holds harmonic 1 as a drone and then fades in harmonic 2, 3, 4 … one step at a time without dropping anything. The tone starts as a pure sine and thickens into a bright, reedy timbre — that is exactly how a complex waveform is built from partials. It keeps sounding until you press Stop.
- Fundamental: drag the slider (C2 to C4, one semitone per step), use the − / + buttons, or hit a preset chip. The big readout shows the note name, the frequency in hertz and the MIDI number. Changing the fundamental while the stack is running glides every partial to its new pitch.
- Harmonics shown switches the ladder between partials 1–8 and 1–16.
- Step speed sets how long each rung lasts in the walk and the stack: 0.75 s, 0.50 s or 0.32 s.
- Master volume starts at a safe 30%. Upper partials are trimmed automatically as the harmonic number rises, because high sine tones sound much shriller at the same amplitude.
- Reset stops all sound and restores C2, 16 partials, medium speed and 30% volume — and also clears the game score and per-harmonic table, empties the timbre back to partials 1–6, returns the brass panel to Bugle, the A/B panel to H7 with the fundamental on, and blanks the fade-test result.
- Stop kills every sound source in every panel at once: the walk, the stack, a held timbre, a bugle call, an A/B comparison or a fade test.
- Fullscreen expands the widget to fill the screen; press it again (or Esc) to come back.
Which harmonic? — the listening game
- New question plays the fundamental as a reference, then one random partial from the pool. Name it by tapping a number in the Your answer grid or the matching rung on the ladder.
- Pool chooses how hard it is: 2–6 (the bugle partials, wide apart and easy), 2–10 or 2–16, where the top partials sit less than a tone apart. Changing the pool clears the current question and widens the ladder if it needs to.
- Replay it plays the same question again, as many times as you like, without penalty. Give up · reveal names and sounds the answer but counts as a miss.
- Get it right and the score, streak and best streak all move, and the partial sounds once more so the correct answer is the last thing you hear. Get it wrong and the widget plays your answer first and the true partial straight after, lighting each rung as it goes — hearing the two side by side is what actually builds the skill.
- Per-harmonic accuracy shows right-out-of-asked for every partial in the pool, so you can see which ones you keep confusing. Cells you always get turn accent-coloured; cells you get wrong more often than not turn warning-coloured.
- Clear score wipes the counters and the per-harmonic table. Nothing is ever stored between sessions — reload the page and the score is gone.
Build a timbre
- The grid of sixteen cells is the tone you are building. Tap a cell to switch that partial in or out; it also becomes the cell the Level slider edits. Dragging the level to 0 switches the partial off, dragging it up switches it back on.
- Hold the tone starts the composite and keeps it ringing so you can switch partials in and out and hear the colour change live. Pluck it fires the same recipe as one decaying note, with the upper partials dying away first — the reason a plucked or struck note goes dull as it fades.
- All off empties the tone. All 16 on loads every partial at 1/n, the full sawtooth.
- The recipes are real partial sets: Flute is partials 1, 2 and 3 only, so it is nearly a pure sine. Clarinet is odd partials only (1, 3, 5, 7, 9, 11, 13) — a cylindrical bore stopped at the reed suppresses the evens, which is why a clarinet sounds hollow and why it overblows to the twelfth. String / saw is every partial at 1/n, the bright buzzy spectrum bowed strings and reed organs sit near. Brass keeps everything but pushes the energy up into partials 2 to 6 with 7 clearly present, which is the blare. Bell-ish is a sparse, gappy 2, 3, 4, 6, 8, 9, 12 with no fundamental at all — a real bell is inharmonic, so this only hints at it. Bluesy 7th is partials 1 to 7 with the seventh turned up loud.
- Compare Clarinet against String / saw for the classic lesson: odd-only sounds hollow and dark, all-partials sounds bright and rich. Then load Bluesy 7th and toggle H7 off and on.
- The whole tone is scaled by the sum of its partial amplitudes, so however many you switch on it can never clip — adding a partial ducks the rest very slightly instead.
Bugle & natural horn
- A brass tube with no valves can only sound its own harmonic series, so this panel takes everything else away. Bugle leaves partials 2 to 6; Natural horn leaves partials 2 to 16. Rungs the instrument cannot reach are greyed on the ladder, and tapping one tells you so.
- Each button shows the written note in large type and the partial number plus the sounding note underneath. Brass parts are written so that partial 2 is always C, whatever key the instrument is really in — which is why the written set reads C, G, C, E, G, B♭, C, D, E, F♯, G, A♭, B♭, B, C on every natural brass instrument ever built.
- Taps · opening and Charge! play real calls built only from partials 3, 4, 5 and 6. Run the whole set walks every note the instrument has, in order.
- Set the fundamental to B♭2 for a real bugle and F2 for the classic natural horn, then compare the written column against the sounding column.
- Written partial 11 is notated F♯ but sounds 49 cents flat of it, and written partial 13 is a neutral sixth: horn players lipped and hand-stopped those notes into tune, and composers mostly avoided them.
7th & 11th A/B
- Pick a harmonic with the number buttons (or tap a rung on the ladder, which loads it and immediately alternates). The readout gives the pure frequency, the nearest equal-tempered note and frequency, the cents gap and the beat rate the two would produce together.
- Pure harmonic plays the partial exactly n times the fundamental. Equal-tempered plays the note a piano would give you instead. A / B / A / B alternates them four times with the readout naming each one as it sounds. Both at once plays them together for nearly three seconds so you can count the beating.
- Fundamental underneath adds the fundamental to every example. With it on, the pure version locks into the drone and the tempered version grinds — that roughness is the entire argument between just intonation and the piano.
- Start with H7 (−31 cents) and H11 (−49 cents), then try H4 to hear what a genuine 0-cent match sounds like: no beating at all.
Fade test
- Start the fade test forces the ladder to all 16 partials and brings them in one at a time, each about 2.9 dB quieter than the one before, so partial 16 arrives roughly 43 dB under the fundamental.
- Press I hear it every time you notice a new partial arrive. The highest one you log is your score. Stop & score ends the run and reports how far you got and how far down that was.
- The three tiles show which partial is entering now, how far below the fundamental it is, and the highest one you have logged.
Keyboard
- Space start / stop the walk when the widget itself has focus. When a ladder row has focus, Space or Enter sounds that harmonic instead.
- ↑ / ↓ move between harmonics · Home / End jump to the first or last rung.
- ← / → move the fundamental down / up one semitone.
- 1 … 9 sound harmonics 1–9 · 0 sounds harmonic 10 · hold Shift with 1–6 for harmonics 11–16.
- P play up the series · S stack partials · X or Esc stop · R reset · H this panel · F fullscreen.
- Arrow keys are left alone while a slider has focus, so you can still nudge the fundamental or the volume one step at a time.
- V steps forward through the six panels, Shift + V steps back.
- The number keys follow the panel, exactly like a ladder tap: in the game they are your answer, in Build a timbre they switch partials in and out, in A/B they load a harmonic. In Bugle & horn they behave like lip positions instead — 1 is the lowest partial the instrument has, 2 the next, and so on, so a bugle really is five keys.
- Panel keys: game — N new question, A replay it, G give up. Timbre — T hold the tone, N pluck it. Bugle — B swap bugle / natural horn, N run the whole set. A/B — N pure, A alternate, B both at once, T fundamental on / off. Fade test — N start, then I hear it; T stop and score.
- X or Esc always stops everything, in every panel.
What the cents column means
A cent is one hundredth of an equal-tempered semitone, so 100 cents = 1 semitone and 1200 cents = 1 octave. For each harmonic the widget finds the nearest note of 12-tone equal temperament and reports the gap: cents = 1200 × log2(f / f nearest). A positive number means the harmonic is sharper than the piano note, negative means flatter. The bar draws the same number on a scale of ±50 cents, with the hairline in the middle marking a perfect match.
The deviations depend only on the harmonic number, never on the fundamental — move the slider and the cents column stays exactly the same. That is why they are worth memorising: octaves (1, 2, 4, 8, 16) are dead on 0; the fifths (3, 6, 12) are +2; the major thirds (5, 10) are −14; harmonic 9 is +4 and harmonic 15 is −12.
Why 7, 11 and 13 sound strange
Equal temperament splits the octave into twelve identical steps, and those steps simply have no slot for the 7th, 11th and 13th partials. Harmonic 7 lands 31 cents below the equal-tempered minor seventh — the “harmonic seventh”, noticeably flat but beautifully consonant in a barbershop or brass chord. Harmonic 11 is the extreme case at −49 cents: it sits almost exactly halfway between the perfect fourth and the augmented fourth, which is why the alphorn and the natural trumpet produce the famous quarter-tone “alphorn fa”. Harmonic 13 is +41 cents, a neutral sixth that is neither major nor minor. Harmonic 14 is just harmonic 7 an octave up, so it is flat by the same 31 cents. These four rungs are the ones marked in the second colour.
Tips
1. Start the stack on a low fundamental such as C2 at slow speed and listen for the moment harmonic 7 enters — the drone suddenly gains a bluesy, slightly sour edge. That single partial is the difference between a “clean” and a “brassy” timbre.
2. Harmonics 2 to 8 over a B♭2 fundamental are exactly the bugle notes: no valves, no slide, just the partials a player can lip. Set the fundamental to B♭2, show 1–8 and walk up the series to hear a bugle call’s entire vocabulary.
3. In Build a timbre, load Clarinet, then String / saw, then back again. The only difference is the even partials, and it is the whole difference between hollow and bright. Then load Bluesy 7th and switch H7 off and on while the tone is held.
4. In 7th & 11th A/B, turn the fundamental underneath on, pick H11 and press Both at once. That churn is 49 cents. Then pick H4 and press it again: dead silence where the beating was. Nothing in the cents column makes the point as fast.
5. In the game, start on the 2–6 pool until your streak is comfortable, then jump straight to 2–16. Watch the per-harmonic table: almost everyone confuses 5 with 6, and 9 with 8, long after the octaves have become obvious.
Explore the natural sequence of tones produced by a single vibrating tube; it’s called the harmonic series. The interface can be played from your keyboard or mouse. Depending on the mode you choose, the interface will change. What doesn’t change is the underlying concept: You’re hearing the pure intervals as they would have been before adjustment to equal tuning.
Begin in the Explore view (this is the default). Look at the ladder of buttons. Each button represent one partial from the series. From the bottom button up: First, there’s the fundamental tone. Second, it is an octave higher. Third, there is a fifth above that.
How to Use the Harmonic Series Tool
Tap on a button and you’ll hear it played as a brief pulse. Or tap Play to listen to the entire series in sequence. That way, you can hear the intervals approaching each other as you ascend the scale.
It’s very playable without a mouse which is partly due to the way that it maps to the keyboard. To move around in the selection highlight, simply press the up and down arrow keys. To play what is highlighted, press the space bar. Alternatively, you can skip straight to each individual partial by hitting its number key on the keyboard. For example, the top row of numbers will play partials 1 to 9. If you hold down shift at the same time as you hit a number then it will play partials 10 to 18. This lets you easily compare a fundamental to a higher partial. It also lets you see if you can hear how close two intervals are.
The whole series is changed in pitch using the transpose controls. You can move the fundamental frequency up and down by semitones using either the left or right arrow keys, or by clicking on the arrows that appear alongside the frequency display. That means if the melody sounds too high or too low it can be shifted immediately. Your selection update the fundamental note name and frequency accordingly. Nearby the transpose controls there is a volume slider that can be used to control the overall loudness of the instrument. It may be worth turning this down initially if using headphones in case they distort. Starting quietly can also help you avoid distortion, especially since higher partials can be bright and buzzy.
There is also an option to slow or speed up how quickly the stack or walk modes play through. For example, slowing it down allows you to have more time to listen to each interval, while speeding it up makes you hear the series as one evolving sound.
For creating complex sounds from scratch switch to the Timbre view. The grid of buttons enables turning on/off individual partials. Each one is represented by a button, and the slider lets you adjust its volume relative to the others. The slider allows for adjusting the volume level for each partial relative to the rest. When you hold down the Hold button it sustains whatever combination of partials are currently active. That way you can listen and compare the difference that adding/removing just one harmonic makes to the color of the sound. There’s also an option to load pre-made recipe presets so you can see what different instruments approximate their own tone.
If you click the pluck button it strikes all active partials, allowing them to naturaly decay. This demonstrates how low frequencies drop away slower then higher frequencies.
Your ear is put through its paces in the Game mode. Here, it presents the basic tone followed by a random part chosen from a set of options. Clicking the matching button/number indicates which partial has been played. By doing this you will be training yourself to distinguish between equal-tempered and pure intervals. Your score and current streak are tracked by the widget. From here you recieve feedback about the partials that are proving tricky for you to pick out. As your ear gets better, you can expand the pool to include higher partials.
This brass section shows what it would of sounded like on an instrument without valves, such as the natural horn or bugle. The bugle is restricted to partials 2-6 whereas the horn view has more. By clicking the button you play this note with a realistic envelope. It’s also possible to listen to recorded bugle calls and you’ll notice that all of these melodies use just the limited pitches available. This gives us some insight into why brass players often push up into higher parts of range. In that range, the partials are closer together, allowing for more step-wise movement.
In the A/B view, you can compare the equal-tempered note against a pure harmonic. Choose a partial and then select to play just the pure frequency, just the equal-tempered frequency, or both combined. If you play them together, you’ll hear beats if they are not the same. This shows how pure intervals sound stable. Depending on the context, equal-tempered intervals can sound slightly flat or sharp. Toggle the fundamental on or off to hear it as an isolated interval.
Lastly there is the Fade test which tests your hearing threshold. The Fade test presents the partials one after another, fading them in starting from higher values. When you think you’ve heard the next partial, you press the button. This will help you realize just how ‘loud’ the upper harmonics actually are in a natural sound. It only works effectively with headphone use and a quiet environment.
At any point you can hit the reset button to go back to default and start again. Spend some time getting to grips with each of the views so that you can truly understand what the harmonic series is all about.