Why is the highest frequency on a piano 4186 Hertz?

Question

The human hearing range is between 20 Hz and 20,000 Hz. The lowest frequency on a piano is 27.5 Hz, which is almost at the start of the human hearing range. However, the highest frequency on a piano is only 4186.01 Hz. Why is this the highest frequency on a piano? In theory there could be a couple more octaves at higher frequencies. Why aren't there?

My own theory is this: The human ear's ability to distinguish between these higher frequencies is less effective than at lower frequencies. So higher frequencies all kind of sound the same, and having these frequencies doesn't add much to a pianist's toolset. Is that right?

Answer 1

Wheat stated in his comment that

"Theoretical physics or mathematics are largely irrelevant to musical instruments, or music performance and practice"

Let's see if we can answer this question using science adding psycho physics and engineering to the mix. I'm not trying to dispute what was already said, I'm just offering a different view point and approach to answer the same question.

1. Physics says that any periodic motions (such as the movement of a string) can be expressed as the sum of harmonic motions (motions of a single frequency). The frequencies are all integer multiples of the lowest or "fundamental" frequency
2. Physics says that the spectral difference between different instruments is primarily a function of the relative balance of the different frequency components.
3. Physics says that a piano note consists of a percussive onset (hammer hitting the string) and then a periodic motion of the string. The onset has a fairly wide (non-tonal) spectrum and the ringing string has a harmonic spectrum.
4. Psycho-physics says that human can perceive frequencies up to about 20 kHz. This is not a "black & white" cut off point but the sensitivity of the ear drops fairly quickly as the frequency gets higher. This effect is impacted by age and exposure.
5. Psycho physics says that perception of tonality and timbre is simply a function of the short term frequency spectrum of the sound at the ear drum
6. Physics says that the frequency at which a string vibrates is a function of mass, length & tension. In order to produce very high frequencies the string must be light, short and have a lot of tension.
7. Physics says that there are multiple ways that a piano radiates sound energy. The volume velocity of the string itself (how much air does it move), and mechanical energy transferred into the resonance structure of the pianos body and radiated through the surfaces of the piano
8. Physics says that mechanical loss in a piano surfaces gets higher with frequency

Taking all this we can draw some conclusions:

1. The maximum number of keys you could add would be 2 octaves. Above that nearly no one would be able to hear the fundamental any more
2. Even below that you would only get the fundamental and the first harmonic at max. At these high frequencies all instruments would sound the same as any difference in the harmonic spectrum are outside the human hearing range
3. As the frequency increases the harmonic motion of the string has less energy as compared the percussive onset. So you hear more and more "boink" and less and less "note". It turns into an a-tonal percussion instrument.
4. Getting any harmonic energy radiated is difficult. Since the string is very short the mechanical energy in the string is small. Loss in the piano's frame and wood is high so not a lot gets out. At the same time the radiated volume velocity is small as well (again, short string), so that doesn't radiate well either. In order to get enough sound radiated, you would have to add A LOT of strings.

So the scientists answer would be: "don't bother adding more keys: it would sound more like a percussion instrument that a piano, the tonal quality would be poor, and it's really difficult to get it loud enough"

Answer 2

Nobody can hear fundamental musical pitches up to anywhere near 20,000 Hz, not even new-born babies with perfect ears. The upper range of human hearing is only useful for hearing overtones and harmonics that are basically "felt" rather than "heard".

There are a few notes above the range of the piano that can be heard, but they are not musically useful. A very skilled violinist can produce these using harmonics. You could produce these pitches with some analog electronic synthesizers, but not with an acoustic instrument. If you try this yourself, it will occur to you that you've never heard these pitches used in any actual music because they are not musically useful and because few people can hear them, and those that can find them unpleasant to listen to.

Neil, piano design aside, there's a fundamental misunderstanding in your line of questioning. You read in a book somewhere that the range of human hearing goes up to 20,000 Hz. Well, so what? That is only a theoretical construct. Different individuals have different ranges of hearing. Most people can hear the pitches in the middle of the theoretical range of human hearing more or less the same, but there is a tremendous variation among people in how they hear pitches at the extreme low or high end. For virtually all people, the range of hearing decreases with age. Men as a group lose high-end hearing sooner in age than women. I am a man who has always been regarded as having excellent hearing, but at the age of 34 I discovered in a recording studio that I could no longer perceive a loud sine wave at 11,000 Hz, while younger women in the studio could hear that pitch (and clapped their hands over their ears in pain).

Furthermore, the frequency response of the human ear is not at all linear. It doesn't even follow a mathematically describable curve, because, again, human hearing is not based on mathematical laws; it's based on the physiology of the human ear and the brain. The loudness with which people perceive extremely high pitches is sharply attenuated.

A statement like "the human hearing range goes up to 20,000 Hz" is like saying "the range of human height goes up to 7-foot 2-inches (218cm)." Does the clothing industry only mass-produce clothes that fit people who are 7-foot 2? No, they make clothes that average people can wear. That way they sell more clothes.

Answer 3

The highest fundamental pitch on the piano is such that it is, and no higher, because it is not feasible to build any more higher-pitched strings into the harp of the piano. It's down to the limits of mechanical engineering and the properties of steel strings: string length, tensile strength, tension. What makes you think that it's possible to build an acoustic string percussion instrument that can play pitches up to the limits of human hearing? That would defy the laws of metallurgy and physics.

Please reference this question: Why piano keys are not integer factors of octave notes?

Musical instruments are built based on what is practical and possible to construct in the real world, not based on theoretical mathematics.

Answer 4

One factor not covered in the other answers is the physical width of the piano keyboard.

The size of a key is optimised for the typical human hand; let's assume it's not amenable to change.

Pianos have to fit in people's homes, classrooms, theatres, places of worship. Extra width has to prove its value.

The deepest note and the highest note should be reachable by a player seated in the middle.

I suspect market forces would have shaped the range of keyboard instruments. Perhaps once an instrument maker produced an extended-range piano. He probably didn't sell all that many, and hence it didn't catch on.

There is a market for keyboards with a reduced range - for reasons of cost and size. But enough people are willing to pay for, and make room for, 88 keys, that they continue to be the norm.

Bösendorfer make two grand pianos with extended ranges (92 and 97 keys), but clearly they do not sell well enough that their competitors are flocking to imitate them.

Answer 5

I think Josh's comment has the right idea. Since musical tones comprise many partial tones, as you approach the highest few octaves below 20,000Hz the upper partials of the notes fall off the top; and the tones lose their character. The notes lack richness and coherence.

Observe that 5,000Hz is only 2 octaves below 20,000Hz. So the 3rd partial and above have passed over the threshold and are inaudible.

I suspect Wheat's answer about the limits of frame and strings is probably the reason pianos stop where they do (the physical materials dictated a stopping point). My answer goes to explain why nobody found this to be a problem (no extra octave of tinkerbells were desired enough to become popular). You ("Joe Western Culture") really don't need notes any higher.

Inspired by Wheat, here's some anecdotal evidence:

Now, I'm probably considered hyperacute to pitches. Even in this seemingly quite room, I can hear the television monitor squeal, the laptop's teensy little squeek, and a 60cycle hum from the bathroom and the refrigerator (and cars and birds outside). But I've got a cousin who's much worse off. He can't go in to certain stores because of their shrieking broken-beam doorbell systems. When he pointed it out and stepped back outside, I became aware of sound he described, but I'm able largely to filter it out of my attention. I suspect my cousin may have artificially increased his sensitivity by his work with restoring old radios, as I have with trying to read-along with the Rite of Spring.

Answer 6

The oft-quoted 20Hz-20Khz range does not mean that people can perceive pitches in the range 20Hz-20KHz range but not 19Hz nor 20.1KHz. Rather, there is a range of frequencies people can perceive as pitches, with people's ability to perceive things as pitches falling off near the ends of that range. Beyond that, there is a range of frequencies which people cannot hear as pitches in and of themselves, but which will alter people's perception of how other frequencies sound. It is the latter range which extends from 20Hz-20Khz; as with the former, perception falls off near the end of the range (adding a 10Khz signal to a 2Khz signal will make it sound different; adding a 15KHz signal may do likewise with a 3kHz signal, but the 15Khz signal might have to be louder to achieve the same effect).

Pipe organs often have some pipes which play pitches higher than the highest keys on a piano, but such pipes in isolation produce an annoying sound without a really discernible pitch. Playing such pipes in combination with pipes that some octaves down, however, will yield a "brighter" sound than if those upper pipes were omitted. The organist doesn't have to do any extra work to play those pipes beyond "turning them on". Once they are enabled, anything the organist plays will be echoed some octaves up on the smaller pipes.

If a piano were to include an extra octave at the top, playing the notes in that octave in combination with notes an octave or two down could probably add a pleasant brilliance to the sound produced. Unfortunately, playing everything in doubled octaves would limit what else the pianist could do. Reed organs, in addition to having two sets of reeds an octave apart which could be individually enabled or disabled, and often had an octave coupler which, when enabled, would automatically operate the keys an octave above the ones the player was pressing. I am unaware of any pianos ever having included such mechanisms, however.

Answer 7

Another obvious factor for not having that many scales in a piano, of course, would be the practical aspects of cost and space.

In particular, as to space, meaning the effort required to reach such high keys vs. the sound contribution to most music would not justify the cost nor space occupied by the instrument.

Not all instruments are suitable for sounding high frequencies. For high frequencies, in relation to hearing range, for instance, violins would have a preference.

High frequencies do come out of a piano, likely close to hearing limits, but they do that as harmonics, would say.

Answer 8

From my experience as an audio engineer, I can tell you that higher frequencies are distorted by the space you are in. It is impossible to maintain the ordinary ratios used in music (Just tuning's 1:2:3:4:5:6) because high frequency sound causes too much interference with its own reflections. As a result, high frequency noise isn't used for music, but it does tell you what sort of a room you are in. This is also why bats use higher pitches to navigate, instead of lower ones.

Some friends of mine on the sound crew and I found this out when trying to hear the difference between cd quality sample rates (44100 Hz) and the same files at lower sample rates (usually 11025 Hz). With most of the recordings, we could reliably pick which was which, and the way we did it was by trying to hear what sort of space the recording was made in. At lower sample rates that information wasn't there.

Answer 9

Because it would sound shrill and generally awful, the frequencies would also start to bunch up more causing the actual note being played to disappear, that's just from a frequency based outlook, like science is why is basically what I'm saying. From a musical standpoint they'd rarely be used though. If you wanted more keys you could tune the notes to half or quarter tones and have a massive keyboard though.