Frequency Calculator

Frequency Calculator converts a period, a wavelength, a cycle count, a shaft speed with poles or a music note into hertz, then adds the period, rad/s, wavelength and six harmonics.

What the Frequency Calculator Solves

The Frequency Calculator turns a period, a wavelength, a cycle count, a rotation rate or a music note into a frequency in hertz. Two more modes run the other way. One gives the period and wavelength for a known frequency, and the other gives the wave speed.

Frequency counts how many full cycles happen in one second. One cycle per second is one hertz (Hz), and the SI Brochure writes that unit as s⁻¹, a reciprocal second.

Every answer arrives with four cards. They carry cycles per minute, angular frequency in rad/s, the wavelength in the chosen medium, and the nearest musical note with its spectrum band and photon energy.

Time to Frequency With the Period Formula

Period is the time one cycle takes. Frequency is its reciprocal, so a time to frequency calculator needs only that one number.

$$f = \frac{1}{T}$$

Here f is the frequency in hertz and T is the period in seconds. A period entered in milliseconds, microseconds, nanoseconds, minutes, hours, days or years is converted to seconds before the division, with a year counted as 365.25 days.

The default entry is 20 ms. That is 0.02 s, and 1 ÷ 0.02 gives 50 Hz, the mains frequency used across Europe. A period of 1 s is exactly 1 Hz, so 1 Hz means one full cycle every second.

On a handheld calculator, change the time to seconds, type it in, and press the reciprocal key marked x⁻¹ or 1/x. The same key turns a frequency back into a period, because T = 1 ÷ f.

Each row was run through the Frequency Calculator in its “Period and Wavelength from f” mode. Sound wavelengths use air at 20 °C, and the radio or light column uses the speed of light in a vacuum.

Above 10 MHz the calculator stops giving a wavelength in air. With an air medium selected, the wavelength results switch to light in a vacuum, and the message under the cards says so.

Frequency Calculator With Wavelength

A wave moves forward one wavelength in one period. Divide the wave speed by the wavelength and the result is the frequency. OpenStax College Physics gives the same relationship for all waves as v = fλ.

$$f = \frac{v}{\lambda}$$

In this formula v is the wave speed in m/s and λ is the wavelength in meters. Wavelengths can be typed in meters, centimeters, millimeters, micrometers, nanometers, kilometers, feet or inches.

Choose “Frequency from Wavelength” in the Frequency Calculator and the Wave Medium list moves up next to the wavelength field. The default is a 0.78 m wave traveling as sound in air at 20 °C. The speed there is 343.2 m/s, and 343.2 ÷ 0.78 comes to 440 Hz.

The same 0.78 m as light in a vacuum is a very different wave. Light covers 299,792,458 m every second, which puts the frequency at 384.3 MHz. The medium matters as much as the wavelength does.

The light options divide c by the refractive index of air, fresh water and crown glass, the values in the OpenStax refraction table. Light keeps its frequency when it enters water or glass, so only the wavelength shrinks.

Solids carry sound at more than one speed. The copper and aluminum entries are bulk speeds for rolled metal, and a thin rod is slower. HyperPhysics lists 5,100 m/s for aluminum and 3,560 m/s for copper.

Rubber varies even more. The Sound in Vulcanized Rubber entry is 54 m/s, the OpenStax figure, while HyperPhysics gives 1,600 m/s for rubber. The closed Wave Medium bar prints the speed in use, and Custom Wave Speed takes any other figure.

Sound in Air, Set Temp opens an Air Temperature field in °C or °F. The speed of sound then follows a square-root rule, so warm air carries sound a little faster.

$$v = 331.3 \sqrt{1 + \frac{T}{273.15}}$$

T is the air temperature in °C and v is in m/s. At 25 °C the speed is 346.1 m/s, against 343.2 m/s at 20 °C. OpenStax states the same rule with the rounded constants 331 m/s and 273 K.

Green light makes a good check. Enter 550 nm with Light in a Vacuum and the Frequency Calculator returns 545.1 THz. The fourth card places it in visible light with a photon energy of 2.254 eV.

Going from frequency to speed uses the same formula turned around, v = f × λ. The “Wave Speed from f and λ” mode does that product. Its default of 440 Hz and 0.78 m returns 343.2 m/s, or 1,126 ft/s with US result units.

Counting Cycles Over a Time

Some periods are too short to time, yet the cycles are easy to count. Frequency is then the count divided by the time the count took.

$$f = \frac{N}{t}$$

N is the number of whole cycles and t is the elapsed time in seconds. The default of 150 cycles in 30 s works out to 5 Hz, with a period of 200 ms.

This is the mode for finding your own frequency, whether that is a pulse, a walking pace or a breathing rate. A pulse of 72 beats in 60 s is 1.2 Hz. A longer count shrinks the error from one missed or half-finished cycle.

An oscilloscope trace is read in a similar way. Count the horizontal divisions that one cycle covers and multiply by the time per division. One cycle across 4 divisions at 5 ms per division lasts 20 ms, which is 50 Hz.

Real oscillations also fade as they repeat. The Damping Ratio Calculator takes the peaks of a decaying trace and returns how quickly that swing dies away.

RPM, rad/s and the Frequency Calculator in Hz

A spinning shaft has a frequency too. One revolution is one cycle, so the Frequency Calculator divides revolutions per minute by 60 to get hertz.

$$f = \frac{\text{RPM}}{60}$$

Switch the unit to rad/s and the field is relabeled Angular Frequency (ω). The calculator then divides by 2π. Degrees per second are accepted as well and are divided by 360.

$$f = \frac{\omega}{2\pi}$$

The default of 1,800 RPM gives 30 Hz, a period of 33.33 ms and 188.5 rad/s. The Poles field beside it starts at 2, which leaves the plain shaft rate. Going the other way, 1 Hz is equal to 60 RPM, 6.283 rad/s and 360 degrees per second.

The “Frequency in Every Unit” table under the cards lists one answer in Hz, mHz, kHz, MHz, GHz and THz. It also repeats the value as RPM, rad/s and degrees per second, which covers most unit conversions in one place.

For ω from a spring, a pendulum or a rim speed, the Angular Frequency Calculator has those inputs. The Angular Velocity Calculator handles a turning shaft with a radius, and the Centrifuge Calculator turns rotor RPM into g-force.

An object moving around a circle has a period as well. The Circular Motion Calculator finds that period from speed and radius, and its reciprocal is the orbit frequency.

Frequency Calculator for Sound and Music

OpenStax puts the audible range at 20 to 20,000 Hz. The fourth card tests every result against that range. Inside it, the card names the nearest note. Outside it, the card reports the frequency as below or above hearing.

Notes follow equal temperament with A4 fixed at 440 Hz. The distance from A4, counted in semitones, comes from a base-2 logarithm of the frequency ratio.

$$n = 12 \log_2\left(\frac{f}{440}\right)$$

The value n is rounded to the nearest whole semitone, and the leftover is shown in cents. One semitone is 100 cents. A 50 Hz hum lands on G1 and sits 35 cents sharp, while 440 Hz is A4 exactly.

The “Harmonics of This Frequency” table lists the first six whole-number multiples. For 440 Hz those are 880, 1,320, 1,760, 2,200 and 2,640 Hz. The third harmonic falls on E6 and the fifth on C♯7, 14 cents flat.

Each harmonic row carries its own period and wavelength. In air the 440 Hz series runs from 78 cm down to 13 cm, which is handy when sizing a pipe or placing a microphone.

The “Frequency of a Music Note” mode works in reverse. Pick a Note Name and an Octave Number from −1 to 10, and the calculator counts the semitones from A4.

$$f = 440 \times 2^{n/12}$$

Middle C is C4, 9 semitones below A4, which gives 261.6 Hz. The fourth card then shows the MIDI number, which is 60 for C4 and 69 for A4.

Frequency by sound here means entering a number or a note. The tool does not listen through a microphone, so a tuner app is still needed to read the pitch of a live sound.

Frequency in Electricity: 50 Hz and 60 Hz

Mains frequency is the number of full voltage swings each second. Enter 60 Hz in the Frequency Calculator under “Period and Wavelength from f” and the period is 16.67 ms. At 50 Hz it is 20 ms. The Half Cycle row gives the gap between zero crossings, 8.333 ms and 10 ms.

A 50 Hz supply cannot be converted to 60 Hz with arithmetic. The ratio is 1.2, so anything locked to line frequency runs 20% faster on 60 Hz. Changing the supply itself takes hardware such as a variable frequency drive.

Generators and synchronous motors tie frequency to shaft speed through the pole count. Electrical frequency equals poles × RPM ÷ 120, and the Frequency Calculator applies it through the Poles field.

$$f = \frac{P \times \text{RPM}}{120}$$

P is the number of poles. Set Poles to 4 in the “Frequency from RPM or ω” mode and 1,800 RPM returns 60 Hz. A 2-pole machine needs 3,600 RPM for 60 Hz, and a 4-pole machine at 1,500 RPM gives 50 Hz.

Poles must be an even whole number, because magnetic poles come in north and south pairs. Leaving the field at 2 returns the shaft rate itself, 30 Hz at 1,800 RPM. The resonant frequency of an inductor and capacitor follows a different formula and is outside this tool.

Spectrum Band and Photon Energy

Radio bands are named by decade. Under ITU-R Recommendation V.431, band number N runs from 0.3 × 10^N to 3 × 10^N Hz. That makes 3 to 30 kHz VLF, 30 to 300 MHz VHF and 300 to 3,000 MHz UHF.

The fourth card shows the band for any result. Past the radio bands it switches to infrared, visible light, ultraviolet, X-rays and gamma rays. A 2.45 GHz entry reads UHF radio, and 100 MHz reads VHF radio. Results past the terahertz range are written in PHz and EHz, so an X-ray at 3 × 10¹⁹ Hz reads 30 EHz.

The same card gives the energy of one photon at that frequency. Energy is the Planck constant times the frequency, so it rises in step with f.

$$E = h f$$

The NIST CODATA value of h is exact at 6.62607015 × 10⁻³⁴ J per hertz, or 4.135667696 × 10⁻¹⁵ eV per hertz. A 50 Hz wave carries 2.068 × 10⁻¹³ eV per photon, against 2.254 eV for green light.

Frequency in Statistics and Math

In statistics a frequency is a count, not a rate. OpenStax Introductory Statistics defines it as the number of times a value of the data occurs. The Frequency Calculator covers waves and rotation only, so these are worked by hand.

The total frequency is the sum of every count, and it equals the sample size. Relative frequency is one count divided by that total. A value that appears 5 times in a sample of 20 has a relative frequency of 0.25.

Four kinds are usually taught together. They are the simple frequency, relative frequency, cumulative frequency and cumulative relative frequency. The cumulative versions are running totals down the table, and the last cumulative relative entry is 1.

Frequency density belongs to histograms with unequal class widths. It is the class frequency divided by the class width. A class holding 12 values across a width of 5 has a frequency density of 2.4.

Common Frequency Questions

These answers use figures from the Frequency Calculator at its default settings, with sound in air at 20 °C and SI result units.

What does 300 Hz mean?

It means 300 complete cycles every second. Each cycle lasts 3.333 ms, and as a sound in air the wave is 1.144 m long. The nearest note is D4, 37 cents sharp, which is just above middle C at 261.6 Hz.

How fast is 2 Hz?

Two hertz is two cycles per second, with half a second for each one. That is 120 cycles a minute and 12.57 rad/s. A march at 120 steps a minute is a 2 Hz rhythm.

Can humans hear 1 Hz?

No. Human hearing starts near 20 Hz, and 1 Hz is twenty times lower than that. A 1 Hz pressure wave in air is 343.2 m long. The card reports it as “Below the Hearing Range” and shows no note.

What frequency is 20,000 Hz?

It is 20 kHz, the upper edge of human hearing. One cycle takes 50 µs and the wavelength in air is 1.716 cm. The same number on the radio spectrum sits in the VLF band, with a wavelength of 14.99 km.

What is short for frequency?

The symbol is a lowercase f, and its unit is the hertz, written Hz. Light and photon formulas often use the Greek letter ν instead. Angular frequency has its own symbol, ω, and is measured in rad/s.

What is a good frequency?

A good frequency is the one the equipment was built for. US appliances expect 60 Hz from the wall, and orchestras tune A4 to 440 Hz. A sound has to fall between 20 Hz and 20,000 Hz to be heard at all.

The Frequency Calculator sits with the other tools for spinning and repeating motion on the Rotational Motion Calculators hub. Angular speed, circular motion and centrifuge tools are grouped there.