# Can human ear hear 4 Hz frequency, if I tap my hand 4 times per second on table?

Frequency means the number of repetitions per second. Humans can hear between 20 Hz and 20 kHz, but I have a very basic question: if I tap my hand four times on a table per second, it means I am producing a 4 Hz wave, and it is audible to me. I am sorry if I sound stupid, but I searched for this online and didn't get a satisfactory answer.

• If you are saying it is audible to you who am I to disagree?
– lcv
Dec 6, 2023 at 10:54
• It's not sound frequency, but rather your "sound generator" frequency. Dec 6, 2023 at 12:00
• "Frequency means no of repetitions per second" That is one possible meaning, but if we're talking about what "frequencies" can be heard by human ears, then we're talking specifically about the frequencies of pure sine waves. You may be able to hear your hand tapping the desk four times per second, but you cannot hear a 4Hz pure sine wave. Dec 6, 2023 at 14:07
• If you tap only once, are you hearing a 0 Hz tone? Dec 6, 2023 at 19:11
• Analogously, you can see a light flashing four times a second, but you cannot see 4 Hz electromagnetic radiation. Dec 6, 2023 at 21:19

You are not producing a 4 Hz wave—you are producing four sets of sounds, each with a spread of audible frequencies. If you had a 4 Hz sinusoidal wave you would not hear it.

If you tap a drum, for example, several times, you get a waveform like this, which includes a wide spread of audible frequencies.

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– Buzz
Dec 6, 2023 at 22:14

Strictly considering whether the ear is hearing a 4 Hz wave, the answer is definitely no. Your ear doesn't transduce (pure) 4 Hz waves, and as the other answers mention, there isn't even much energy at 4 Hz in that signal.

If you are asking whether you "hear" 4 Hz in that signal, that's kind of subjective, but there is some cool psychoacoustic research into how we perceive repeated sounds at different repetition frequencies! At about 30 Hz and above, repetitive sounds are perceived to have a pitch of the repetition frequency, even if there isn't any acoustic energy at the perceived pitch. This “pitch of the missing fundamental” phenomenon has played a key role in the study of how we perceive sounds. Below a repetition rate of 30 Hz, you don't really perceive pitch, whether there is truly acoustic energy present at that frequency or not. The percept of repeated sounds below that range can be variously described as something like a motorboat or various other analogies.

• +1 for pointing out that perception involves more than just the Fourier spectrum! Dec 31, 2023 at 0:35

The sound that the clap itself creates has a higher frequecy than 20 Hz, since it is audible. The number of sounds in a given period can also be expressed as a frequency, but this should be seen as separate from the frequency itself.

For example, a lot of music references beats per minute (BPM) for this specific frequency.

In the discussion under Marco Ocram's answer there was a discussion about whether the frequency that a sound is repeated at is reflected as a component in the frequency spectra. I was asserting that it does while DonQuiKong was asserting that it doesn't. After running through some numerical software I am inclined to agree with him.

In the data, below, a 100Hz sine wave is burst for 200ms followed by 800ms of silence and then repeated. Essentially, a 100Hz sine wave is burst with a duty cycle of 20% at a rate of 1Hz. The simulation runs for 10 seconds with a resolution of 1ms. Shown here in the time domain:

Since I couldn't remember how to properly scale the frequency axis in an FFT, instead made a spectra of the pure, eternal sine wave without start or end to mark the fundamental of the tone being burst on the spectral plot:

Now here is the spectra of the burst tone:

Using the first plot to find the fundamental in the second plot, you can see there aren't significant components other than the fundamental of the tone.

This is a plot that marks the position of 1Hz in the spectra: I was expecting to see some more dominant 1Hz component responsible for the repeating burst in there but, as you can see from the spectra of the burst tone, nothing stands out.

So the answer to your question is that, although the human ear cannot hear 4Hz to begin with, tapping at 4Hz produces no 4Hz frequency component in the first place. I still find it strange that you can have a signal in it with something repeating, but not require that frequency to be present in the spectra.

1Hz is so close to the plot edge it is difficult to see so here is a second simulation where the 100Hz tone is burst for 100ms followed by 100ms of silence before repeating to result in a burst that repeats at 5Hz:

This spectra marks 5Hz, and again, no significant component is found there in the spectra for the burst:

• Although you may find it strange, your result makes sense from a theoretical perspective. What you do mathematically by repeating a sound is taking its amplitude $x(t)$ and convolving it with a train of Dirac impulses, known as a Dirac comb $III(t)$, which is its own Fourier transform (apart from some scaling in amplitude and frequency). The spectrum is hence $\mathscr{F}\{x(t) * III(t)\} = X(f) \cdot c\,III(c\,f)$ for some $c$. If you want a spike at $f_0$, add $\cos(2\pi f_0 t)$.
– Mew
Dec 12, 2023 at 20:02
• @Mew It also makes sense from summations and products of sine waves not introducing any new frequencies. Dec 12, 2023 at 20:10
• Although perhaps unintuitive, this result is well-known in the laser physics community. One often generates light with a pulsed laser, rather than a CW laser. In such a case, the resulting mode profile is an optical comb of pulses offset by a frequency spacing determined by the repetition rate. See, e.g., en.wikipedia.org/wiki/Frequency_comb Dec 31, 2023 at 0:49

As others have said, the tapping frequency in your example is not the same as a sound frequency (its pitch).

Now, musically, the transition from both types of rhythms does exist. It has notably been explored by Karlheinz Stockhausen in his Kontakte.

You can actually hear this transition, and have it explained, in this nice blog post: Rhythm / Pitch Duality: hear rhythm become pitch before your ears.

This video of Euler's disk provides a useful example. Listen to the sounds. Around 0:30, you can hear a noise in 4Hz or so. You will notice that it's clearly "pulsing" - you can tell the difference between different peaks. The tempo rises and rises, but one can still hear the beats up until 1:00 and even afterwards. It is only close to the end when the noise undergoes a weird transition, with its frequency rising higher and higher until it goes into the transitory frequency range between beams and continuous noise. During the last 2-3 seconds (1:25-1:28) I can only hear a continuous noise at some tens of Hertzs, and then it rests.