How to generate the series
Enter a fundamental frequency in hertz and the highest harmonic number. That number determines the row count: 8 produces harmonics 1 through 8, including the fundamental. The initial example is 110 Hz and 8 harmonics.
Read the frequency column first. The note column is the nearest note in 12-tone equal temperament, using A4 = 440 Hz. Reference frequency is the exact equal-tempered frequency of that note before display rounding. The cents column measures the harmonic’s offset from that reference. Copy result exports all rows and the tuning reference as plain text.
The form starts with a valid example. Press Calculate immediately to run it. Focusing or first editing a numeric field clears all example numbers and any confirmed result once, while preserving new typing and unit selections. Later focus changes keep your own entries. The diagram updates with valid draft values; empty or invalid drafts retain a frame marked —. Calculate confirms the separate result and enables Copy result. Clear leaves inputs empty and never restores the example.
Harmonics, overtones and the chart
Harmonic n has frequency fₙ = n × f₁. Harmonic 1 is the fundamental, harmonic 2 is the first overtone, and harmonic 3 is the second overtone. This page uses harmonic numbering throughout. The same numbering can describe ideal harmonic partials, but real sounds may contain inharmonic partials that are not exact integer multiples.
Bar lengths use one common linear frequency scale. A second-harmonic bar is twice as long as the fundamental bar. Bars describe frequency only, not amplitude, energy, audibility or perceived volume. Adding more rows changes the chart’s maximum scale but does not change any existing harmonic frequency. The table remains the exact place to read each displayed value.
How to interpret cents
For frequency f, the nearest equal-tempered note is found by rounding 69 + 12 × log₂(f/440) to an integer note number. Its reference frequency is 440 × 2^((m−69)/12). The offset is 1200 × log₂(f/reference). A positive offset is sharper than the reference; a negative offset is flatter. At an exact halfway tie, the higher note is chosen.
One octave is 1,200 cents and an equal-tempered semitone is 100 cents. Integer harmonics often fall between equal-tempered notes: a nearby note name is a comparison, not a claim that the frequencies match. Octave numbering uses middle C = C4 and international letter names; B is B natural. Sharps provide one consistent spelling rather than a key-dependent musical analysis.
Two worked examples
Enter 110 Hz and 8. Harmonics 1, 2, 3 and 4 are 110, 220, 330 and 440 Hz. The third is close to E4 at 329.63 Hz, with an offset of about +1.96 cents. The seventh is 770 Hz, close to G5 at 783.99 Hz, about −31.17 cents. The eighth is 880 Hz, exactly A5 in the reference tuning.
Enter 200 Hz and 5. The frequencies are 200, 400, 600, 800 and 1,000 Hz. These integer multiples are exact even though their note comparisons are generally not zero cents. The fifth is nearest B5 at 987.77 Hz, about +21.31 cents. Changing the fundamental scales every frequency proportionally; changing only the row count extends or shortens the same series.
Input limits and using the output
The fundamental must be 1–20,000 Hz and the highest harmonic an integer from 1 to 32. A decimal point or decimal comma is accepted without thousands separators. Empty, nonnumeric and out-of-range inputs show a field error. Frequencies and cents display at most two decimals; calculations keep internal precision, and negligible signed zero offsets display as zero.
Calculate again to confirm the new series. Clear empties the inputs, results and copy status. This tool is useful for theory study, comparing natural harmonic ratios with equal temperament, and checking arithmetic. It does not record a microphone, synthesize sound, diagnose hearing, infer timbre or promise that high harmonics can be heard. Output may exceed 20,000 Hz; that is a mathematical result, not an audibility judgment.
Frequently asked questions
Is the first harmonic the first overtone?
No. The first harmonic is the fundamental. The first overtone is harmonic 2.
Why is the seventh harmonic below the nearest note?
Its integer ratio is 7:1, whereas equal temperament divides octaves into equal frequency ratios. These systems do not match at every harmonic.
Can I change the reference tuning?
This page fixes A4 at 440 Hz for a consistent comparison. The harmonic frequencies themselves depend only on your fundamental.
Does a tall bar mean a louder sound?
No. The horizontal bar length represents frequency. No amplitude or audio data is calculated.