Harmonic Series Frequency Calculator

This calculator computes the frequencies of the harmonic series for a given fundamental frequency. You enter the fundamental frequency (in Hz) and the number of harmonics you want, and it returns a list of harmonic frequencies (in Hz) for harmonic numbers 1 through N. The first harmonic is the fundamental itself, and each subsequent harmonic is an integer multiple of that fundamental.

Formula

The harmonic series frequency formula is straightforward:

f_n = n × f₀

Where:

• f_n = frequency of the n-th harmonic (in Hz)
• n = harmonic number (1, 2, 3, …)
• f₀ = fundamental frequency (in Hz)

This formula applies to any periodic waveform—sine, square, sawtooth, or a real instrument tone. The harmonic series is the set of frequencies that are integer multiples of the fundamental. For example, if the fundamental is 100 Hz, the harmonics are 100 Hz, 200 Hz, 300 Hz, 400 Hz, and so on.

Worked Example

Using the default values from the calculator above (fundamental = 100 Hz, number of harmonics = 8):

1. 1st harmonic: 1 × 100 = 100 Hz
2. 2nd harmonic: 2 × 100 = 200 Hz
3. 3rd harmonic: 3 × 100 = 300 Hz
4. 4th harmonic: 4 × 100 = 400 Hz
5. 5th harmonic: 5 × 100 = 500 Hz
6. 6th harmonic: 6 × 100 = 600 Hz
7. 7th harmonic: 7 × 100 = 700 Hz
8. 8th harmonic: 8 × 100 = 800 Hz

The calculator shows these exact values. Notice that the first harmonic equals the fundamental—this is a common point of confusion. Some traditions call the fundamental the “1st harmonic,” while others call the first overtone the “2nd harmonic.” This calculator follows the physics convention where the harmonic number equals the integer multiplier.

What Is the Harmonic Series?

In acoustics and music, the harmonic series is a sequence of frequencies where each frequency is an integer multiple of a base frequency. When a string vibrates, a column of air oscillates, or a membrane moves, it doesn’t just produce a single frequency—it produces a fundamental plus a series of higher frequencies called harmonics or overtones. The relative strengths of these harmonics determine the timbre (tone color) of the sound. For example, a flute has few strong harmonics, while a violin or a brass instrument has many.

The harmonic series is not just a musical curiosity; it appears in physics (standing waves on strings and in pipes), electronics (harmonics in power systems), and even in the analysis of periodic signals via Fourier series. Any periodic waveform can be decomposed into a sum of sine waves at harmonic frequencies.

How to Use the Calculator

To use this harmonic series calculator:

• Enter the fundamental frequency in hertz (Hz). This could be the pitch of a guitar string (e.g., 110 Hz for A2), a tuning fork (440 Hz for A4), or any other base tone.
• Enter the number of harmonics you want to generate. The calculator will list the first N harmonics, starting from the fundamental.
• Read the output list. Each line shows the harmonic number and its frequency in Hz.

If you need frequencies for a specific harmonic beyond the 8th, you can either increase the “number of harmonics” input (up to 50) or simply multiply the fundamental by the harmonic number yourself. For instance, the 12th harmonic of 100 Hz is 1200 Hz.

Common Mistakes and Misconceptions

One frequent mistake is confusing harmonics with overtones. In music, the first overtone is the second harmonic (for most instruments). So the “1st overtone” is not the same as the “1st harmonic.” Always clarify which convention you’re using.

Another error is using the wrong fundamental. If you’re analyzing a guitar string, the fundamental is the open string frequency, not the fret frequency. For a pipe, the fundamental depends on whether the pipe is open at both ends (fundamental = speed of sound / (2 × length)) or closed at one end (fundamental = speed of sound / (4 × length)).

Also, remember that the harmonic series is exact only for ideal, perfectly periodic waveforms. Real instruments have slight inharmonicity—for example, piano strings have stiffness that makes higher harmonics slightly sharp. But for most purposes, the integer-multiple rule is a very good approximation.

Why the Harmonic Series Matters

Understanding the harmonic series helps in many practical areas:

• Instrument design: The placement of frets, the length of organ pipes, and the shape of bells all depend on harmonic relationships.
• Audio engineering: Equalizers and synthesizers shape sound by boosting or cutting specific harmonics.
• Music theory: Chords and intervals are based on simple frequency ratios, which are ratios of harmonic numbers (e.g., 2:1 is an octave, 3:2 is a perfect fifth).
• Physics education: The harmonic series demonstrates wave superposition and resonance.

For example, if you have a 440 Hz tuning fork, the second harmonic is 880 Hz (one octave higher), the third is 1320 Hz (a perfect fifth above that), and the fourth is 1760 Hz (two octaves above). These relationships are why the same note played on different instruments sounds different—each instrument emphasizes different harmonics.

Limitations and Additional Considerations

This calculator assumes a perfectly linear system with no damping or nonlinearity. In real life, harmonics may be slightly detuned, and the amplitude of each harmonic varies. The calculator only gives frequencies, not amplitudes or phases. If you need to know how loud each harmonic is, you’d need a spectrum analyzer or a Fourier transform of the waveform.

Also, the human ear perceives frequency logarithmically. The difference between 100 Hz and 200 Hz sounds much larger than the difference between 1000 Hz and 1100 Hz, even though both are 100 Hz apart. When using harmonic frequencies for musical intervals, it’s often helpful to think in terms of ratios rather than absolute differences.

For very high harmonics, the frequencies may exceed the audible range (roughly 20 Hz to 20 kHz). For example, the 50th harmonic of 1000 Hz is 50,000 Hz, which is ultrasonic. The calculator will still compute it, but you won’t hear it.

Frequently Asked Questions

What is the difference between a harmonic and an overtone?

A harmonic is any integer multiple of the fundamental frequency, including the fundamental itself (the 1st harmonic). An overtone is any frequency higher than the fundamental. So the 1st overtone is the 2nd harmonic, the 2nd overtone is the 3rd harmonic, and so on. The terms are often used interchangeably, but the numbering differs.

How do I find the frequency of the 10th harmonic?

Simply multiply the fundamental frequency by 10. For example, if the fundamental is 50 Hz, the 10th harmonic is 500 Hz. You can use this calculator by setting the number of harmonics to at least 10, or just do the multiplication manually.

Does the harmonic series apply to all instruments?

Most instruments (strings, brass, woodwinds, and the human voice) produce harmonics that are close to integer multiples of the fundamental. However, some instruments, like bells and cymbals, produce inharmonic partials that are not integer multiples. For those, the harmonic series formula does not apply accurately.

Why does the fundamental frequency matter in tuning?

The fundamental frequency determines the perceived pitch of a note. When you tune an instrument, you set the fundamental frequencies of its strings or tubes. The harmonics then fall into place automatically, but their relative strengths affect the tone quality. A well-tuned instrument has harmonics that align with the harmonic series, producing a consonant, pleasant sound.

Can I use this calculator for electrical signals?

Yes. The harmonic series applies to any periodic electrical signal, such as a square wave or a power line waveform. For example, a 60 Hz power line has harmonics at 120 Hz, 180 Hz, 240 Hz, and so on. This calculator works for those frequencies just as well as for audio.

What is the fundamental frequency of a guitar string?

It depends on the string’s length, mass, and tension. The formula is f = (1/2L) × sqrt(T/μ), where L is the length, T is the tension, and μ is the linear mass density. For a standard tuned guitar, the low E string is about 82.4 Hz, A is 110 Hz, D is 146.8 Hz, G is 196 Hz, B is 246.9 Hz, and high E is 329.6 Hz. You can enter any of these into the calculator to see their harmonics.

Related

• Cents to Hertz Calculator
• Guitar Fret Calculator (Fret Position from the Nut)

in f0 | Fundamental frequency | Hz | 100 | 1 | 20000
in n | Number of harmonics | - | 8 | 1 | 50
out f1 | 1st harmonic | Hz | f0 | 2
out f2 | 2nd harmonic | Hz | f0 * 2 | 2
out f3 | 3rd harmonic | Hz | f0 * 3 | 2
out f4 | 4th harmonic | Hz | f0 * 4 | 2
out f5 | 5th harmonic | Hz | f0 * 5 | 2
out f6 | 6th harmonic | Hz | f0 * 6 | 2
out f7 | 7th harmonic | Hz | f0 * 7 | 2
out f8 | 8th harmonic | Hz | f0 * 8 | 2