This harmonic series calculator lists the overtones sitting above a fundamental and measures each one against the equal-tempered note nearest to it. That measurement, in cents, is the reason the page exists. The harmonic series is generated by physics; equal temperament is a compromise designed by people; and the gap between them explains a long list of things musicians notice but rarely see written down as numbers.
It is deliberately a different job from the note frequency converter already on this site, which takes one pitch and converts it in either direction with its cents deviation. That page is about a single note. This one owns the series — the whole ladder of partials above a fundamental, and the pattern of how far each rung drifts from the piano. Arb Digital keeps the two apart rather than letting them compete.
What This Calculator Does
Given a fundamental, it computes the frequency of every harmonic up to the number you ask for, finds the nearest equal-tempered note to each, and reports the deviation in cents. It identifies which partial sits closest to the tempered scale and which sits furthest, ignoring the octave harmonics because those are exact by construction and would win every time. It also tells you how many octaves the series spans, which is a useful reminder of how quickly the partials crowd together as you go up.
A cent is one hundredth of an equal-tempered semitone, so 1,200 cents to the octave. Around five cents is the region where a trained ear starts to detect a difference on a sustained tone; twenty cents is unmistakable; fifty cents is a quarter tone and sounds like a different note. Having those anchors in mind makes the list far more readable.
How to Use It
- Set the fundamental by note name and octave, or type a frequency directly if you are working from a measurement.
- Adjust the A4 reference if you tune to something other than 440, because it changes every note name and cents figure.
- Choose how many partials to list. Sixteen covers four octaves; going higher shows how tightly the upper partials pack together.
- Read the cents column rather than the note names. The names tell you roughly where a partial lands; the cents tell you how badly it misses.
- Compare the odd harmonics with the even ones, because that split is the whole structure of the deviations.
The Formulas and Their Sources
The series itself is the simplest relation in acoustics. For an ideal string or air column, the nth mode of vibration has frequency n times the fundamental — as the University of New South Wales physics notes on strings, standing waves and harmonics put it, the nth harmonic has frequency fn = n f1. There is no approximation in that step for an idealised vibrator.
Converting a frequency into a note name and a cents deviation uses the equal-tempered relation set out in the same group’s note on note names, MIDI numbers and frequencies: each semitone is a ratio of the twelfth root of two, and the interval between two frequencies in cents is 1,200 times the base-two logarithm of their ratio. This tool converts each partial to a real-valued MIDI number, rounds to the nearest integer to get the note, and reports the difference in cents.
Worked example with the defaults, all of which you can verify on paper. A2 at a 440 Hz reference is 110 Hz exactly. The third harmonic is 330 Hz. Equal-tempered E4 is 329.6276 Hz, so the deviation is 1,200 × log₂(330 ÷ 329.6276) = +1.95 cents. The fifth harmonic is 550 Hz against a tempered C#5 of 554.365 Hz, which is −13.69 cents. The seventh is 770 Hz against G5 at 783.991 Hz, a full −31.17 cents. The eleventh lands almost exactly between two tempered notes at −48.68 cents, and the thirteenth is +40.53 cents the other way. Those five figures are constants of the harmonic series and do not depend on the fundamental you choose.
Why the Deviations Are Always the Same
Because the harmonic series is multiplicative and cents are logarithmic, the deviation of the nth partial depends only on n, not on the fundamental. The third harmonic is always +1.955 cents from the tempered fifth, whether you start on A2 or on E4. This is genuinely useful: it means the pattern is worth memorising, and it means a tool that recomputes it for every fundamental is confirming a constant rather than discovering one.
The reason the third harmonic misses by so little is that the tempered fifth is a very good approximation of the pure 3:2 ratio — 700 cents against the true 701.955. That two-cent error is why fifths sound clean on a piano. The tempered major third, at 400 cents, misses the pure 5:4 ratio of 386.31 cents by nearly fourteen, which is why thirds on a piano beat noticeably and why barbershop and brass ensembles pull them flat by ear.
The seventh harmonic is the famous one. At −31 cents it is a long way below the tempered minor seventh, close to a third of a semitone. It is the interval a dominant seventh chord approximates and never reaches, it is what makes a well-tuned barbershop seventh ring, and it is why the seventh partial is often described as a note that does not exist on a keyboard at all.
Where the Series Shows Up in Practice
On a brass instrument without valves, the harmonic series is the instrument’s entire range. A bugle plays partials 2 through 6 and nothing else, which is why bugle calls use the notes they do and no others. The natural horn repertoire is written around exactly this constraint, and the out-of-tune eleventh and thirteenth partials are the reason players hand-stop notes to bend them into place.
On a guitar, touching the string lightly at a node lets a specific partial sound while damping the fundamental, which is what a natural harmonic is. Touching at the twelfth fret gives the second partial, the seventh fret the third, the fifth fret the fourth. The fret spacing calculator covers where those positions fall along the string, and the guitar string tension calculator covers what tension puts the fundamental where you want it.
On any real instrument, the amplitudes of these partials are what timbre actually is. Two instruments playing the same fundamental sound different because they emphasise different partials. A clarinet, closed at one end, produces mainly odd harmonics, which is why it has its characteristic hollow tone in the low register. The wavelength calculator and the frequency converter are handy neighbours when you are working between the pitch and the physical dimension producing it.
Where the Ideal Model Stops Being True
Real strings are stiff, and stiffness makes the upper partials sharper than n times the fundamental. This is called inharmonicity, it is more pronounced on short thick strings, and it is why piano tuners stretch the tuning — deliberately tuning the top of the instrument sharp and the bottom flat so that the octaves match the partials rather than the theory. This tool computes the ideal series and does not model stiffness, so a real piano’s upper partials will read slightly sharper than the numbers here.
Percussion is further away still. A drumhead or a bell vibrates in modes whose frequencies are not integer multiples of anything, which is why a cymbal has no clear pitch and why tuned percussion is shaped specifically to force a few modes into harmonic relationships. Applying this page to a bell will produce numbers, and they will not describe the bell.
Arb Digital builds calculators that name the source, the assumptions and the cases they do not cover. Browse the library, or tell us what your readers keep asking for.
Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Confusing harmonic number with overtone number. The first overtone is the second harmonic. Off-by-one errors here account for most of the confusion in guitar and brass discussions.
- Assuming the partials are in tune. Only the octaves are exact against equal temperament. The seventh, eleventh and thirteenth are wildly out, and that is the point rather than a bug.
- Applying the ideal series to a piano. String stiffness sharpens the upper partials, which is why tuners stretch octaves. This tool computes the ideal case.
- Using it on drums or bells. Their modes are not integer multiples of a fundamental, so the whole model does not apply.
- Forgetting the reference pitch. Note names and cents are measured against equal temperament built on your A4 reference, so changing it changes the comparison.
Related Free Tools From Arb Digital
Convert a single pitch either way with the note frequency converter, place the frets that sound these partials with the fret spacing calculator, work out what tension puts a string at pitch with the guitar string tension calculator, relate frequency to physical size with the wavelength calculator, switch units with the frequency converter, and handle loudness ratios with the decibel calculator. Everything else is in the free online tools hub.
Frequently Asked Questions
It is the set of frequencies an ideal vibrating string or air column produces above its fundamental, where the nth harmonic has a frequency of exactly n times the fundamental. The first harmonic is the fundamental itself, the second is an octave above it, the third a perfect fifth above that, and so on.
Because equal temperament has no note near it. The seventh partial sits about 31 cents below the tempered minor seventh, close to a third of a semitone. It is the interval a dominant seventh chord approximates without reaching, and it is why a barbershop seventh sung by ear sounds different from one played on a piano.
No. Because the series is multiplicative and cents are logarithmic, the deviation of each partial depends only on its harmonic number. The third is always about +1.96 cents and the seventh always about −31.17, whatever note you start on.
The numbering. The fundamental is the first harmonic but has no overtone number; the second harmonic is the first overtone. Sources that mix the two conventions are the single biggest cause of confusion when people compare figures.
That tool converts one pitch in either direction and reports its cents deviation and MIDI number. This one takes a fundamental and lists the whole overtone series above it, naming and measuring every partial. Single note there, whole series here.
Twelve-tone equal temperament, built on the A4 reference you set. Every note name and every cents figure is a comparison against that scale, so changing the reference changes the comparison even though the harmonic frequencies themselves scale with it.
Because real strings have stiffness, which makes the upper partials sharper than the ideal integer multiples. That inharmonicity is why piano tuners stretch octaves. This calculator computes the ideal series and does not model stiffness.
Not meaningfully. Drumheads, bells and most percussion vibrate in modes that are not integer multiples of a fundamental, so there is no harmonic series to list. The tool will produce numbers, but they will not describe the instrument.
This page computes the ideal harmonic series for a perfectly flexible vibrator and compares it with twelve-tone equal temperament. It does not model string stiffness, inharmonicity, or the mode structure of percussion instruments, and the figures should be read as the theoretical series rather than as a measurement of any real instrument.
