Piano vs Guitar Strings? Tension vs length?

Question

Why do pianos have different length strings, while guitars have the same length strings? According to the guitar, it would seem that the pitch of a string is determined based solely off of tension, because all the strings are the same length. Yet a piano has different length strings, so which is it? Length or tension that creates a certain pitch?

Essentially, my question is: does it matter how long a string is when creating a certain pitch or can it be any length and its the tension that creates the pitch?

Answer 1

The best-written summary I could find of this was on Wikipedia.

Technical preliminaries (you can skip this if you don't care)

All chordophones (musical instruments based on vibrating strings) can be analyzed using the same physics model of a string under tension that is fixed on both ends. The model is slightly simplified and differs from reality in two ways: in the model, the string has zero thickness, and the string has zero stiffness. When comparing results calculated based on the model with real-world behavior of such strings, it turns out those two simplifictions do not make a significant difference in most areas. One exception to that is that the need for stretch tuning on some instruments like the piano is not predicted by the model because of the omission of stiffness.

The following discussion will be based on the model of string motion described above. When the string is at rest, no sound is produced. When the position of the string is disturbed, by plucking, striking, etc., the tension on the string acts to restore the string to its rest position. The mass of the string gives the string inertia which causes the string to overshoot its rest position and be displaced again in the opposite direction, and again the tension acts to restore the string to rest position, etc. The tension and mass interact in this way to cause the string to keep moving back and forth until something stops the motion of the string. A string that is not intentionally damped will eventually stop due to friction that converts the kinetic energy of the string into heat.

The formula (this is the math answer to your question)

In both our model and real-world strings, the back and forth motion of the disturbed string quickly settles down to a resonant frequency. That frequency is determined by the properties of the string, and based on the model can be calculated (with reasonable precision) by the following equation:

Where:

• f is the resonant frequency (the note being played)
• v is the speed of propagation of a disturbance (wave) along the length of the string
• L is the sounding length of the string (the length between the two fixed points on either end of the disturbed portion of the string)
• T is the tension of the string (along the sounding length, if the tension is not everywhere the same)
• mu is the linear density, or mass per unit length

Notice that the resonant frequency does not depend on the total mass of the string along the sounding length. In the model and formula, the length component that would be multiplied by mu to get the total mass "cancels out", so it is only the linear density that matters.

The musical consequences (read at least this)

The model and formula we have created for our vibrating strings suggest the following things about changing the resonant frequency (and therefore the note we are playing) of our strings:

• To play a higher note, we can reduce the length, reduce the linear density (in practice, the thickness), or increase the tension of a string.
• To play a lower note, we can increase the length, increase the thickness, or decrease the tension on the string.

As mentioned above, once our model hits the real world we quickly encounter a problem: the thickness of the strings (which is ignored in the model) is the easiest way to control the linear density, but has some effects on the final tone and tuning of the musical instrument. The thinner the string, the closer to the model the real string is (since the model assumes zero thickness), which makes the math of building the instrument easier. On the other hand, thicker strings tend to have a richer tone (citation needed - subjective). On the other other hand, thicker strings also have greater stiffness, which further deviates from the model and causes other problems (Aside: This is one reason why thicker strings are made from wire wrapped around wire, rather than just using a really thick piece of solid wire. Wound strings are much less stiff than solid strings of the same linear density.) There are also complicated concerns about mechanical impedance matching between strings and bridges or fixed points, etc., which I won't dive deeply into, here.

With a little experimentation, we can come up with an ideal string thickness and composition for any given instrument that uses strings. But now we start to have other problems. If we fix the thickness (linear density) of all the strings on our instrument, then the only way to change the notes those strings are playing is by changing the length and/or the tension. Tension can be a problem because the materials we use to stretch the strings have to be strong enough to withstand that tension. This is why cast iron piano frames were significant when they were introduced. On the other hand, strings that are too loose will not have as good tone or intonation. Length is a problem because again there are instrument construction concerns as well as playability. For an instrument like the guitar, it's not practical to make the strings have different unfretted sounding lengths. For an instrument like the piano, the ideal length of the lowest strings would be extremely long. Even a 9 foot grand piano is a compromise on length!

Piano construction realities

With the piano, there is an excellent opportunity for balancing the thickness, tension, and length of strings, because every note gets its own dedicated string(s)! More or less, modern pianos are built to try to have the optimum length, tension, and thickness for each note, but there are many challenges and compromises made.

Thinner strings with shorter sounding lengths may play the right note, but they also are quieter and have a shorter sustain. That is why if you look at high notes on the piano, you will see three strings used for each note. That doesn't help very much with the sustain, (notice dampers are omitted entirely for the highest notes on many pianos) but it does help even out the volume.

Thicker strings help with lowering the note a string produces, but then we have stiffness issues. We also need some tension to keep the string from being "too floppy" (technical term). So let's make the bass notes on the piano really long! Well, that causes problems with fitting the piano into a living room or on a crowded stage and making the piano very expensive. Symphony orchestras can afford to take up a lot of room and money on the best pianos, so you can see 9 foot long concert grands being played for concerti. At home, you might have a spinet with a maximum sounding length of only about three feet and a less rich bass tone to go along with that.

So, finding the ideal balance of length, tension, thickness, tone, cost, and size are all factors in determining how long each string is on a given piano.

Guitar construction realities

On a guitar, the strings all have to be approximately the same length. If they weren't, it wouldn't be a guitar and it couldn't be played the way a guitar is played. Each string also needs to be about the same tension or else the mis-matched tension on the strings can eventually deform the guitar and destroy its tone and intonation. That means that thickness is our main tool for creating the open string notes on a guitar.

That brings us to a fundamental difference between the guitar and the piano. Each note on the piano has its own string(s), whereas each string on the guitar is used to play multiple notes. It's not practical to change the thickness or tension of a string in real time to play music, so on the guitar (and all the instruments in the strings family), different notes are played on the same string by changing the sounding length of the string.

So even though the basic "open" sounding lengths of all the strings on a guitar (or violin or cello, etc.) are the same, while it's being played the sounding lengths are all changing and potentially very different. In case it's not obvious, the string lengths on the guitar are changed by fretting, i.e., pressing the string down against a metal bar (which is placed in a particular location) which acts to shorten the sounding length of the string.

Summary

So that's way too much information to simply say: It's length, tension, and thickness combined that determine the note(s) made by a string. On a piano, those are all pre-chosen and tuned before a single key is struck. On a guitar, the thickness and tension are pre-chosen, along with six basic lengths, and then the lengths are dynamically changed during performance.

Answer 2

String diameter and scale length and tension are all factors, but you are overlooking an entirely different dimension to your question.

Frets.

A guitar string has a fixed length, but have you noticed the frets? When you stop a string against a fret, you are then temporarily creating a shorter speaking string length. So one string on a guitar can be stopped in 19 different fret positions, which is the equivalent of having 20 different string lengths formed from a single string. So in that practical sense of how the notes are played on the instrument, it is not correct to say that there is only one string length on a guitar.

The piano, on the other hand, is a keyboard instrument and it does not use frets. Each different pitch is triggered independently by a different key striking open strings, so each key has to have its own set of strings tuned to a fixed pitch. So to play those same 19 pitches from the guitar string on a piano, you need 19 separate groups of piano strings.

Can you picture all this? More to the point, please get a guitar and a piano and play them and observe how they work. Talking about them in the abstract isn't very helpful, but putting your hands on them and playing them is.

Actually there was a historical keyboard called the clavichord (in the 1600s) which had frets. (actually they are called tangents, but they function like frets on a guitar.) It was a kind of a hybrid between the piano and the guitar. Some clavichords would have, for example, 36 keys but only 18 pairs of strings, because one key would strike a pair of strings at one length, while the adjacent key would strike the same string but stop it at a different speaking length (or fret, as it were). The result of this is that some chords that you could play on a piano could not be played on a clavichord, because there weren't that many available strings to play all the notes in the chord. This is analogous to the situation on the guitar, which only has six strings from which to form any given chord.

Answer 3

The pitch that a string produces is determined by the frequency of the vibration of the string. In other words how fast is it vibrating. The rate of vibration of a string when it is plucked or struck is dependant on several factors.

The tension of the string is only one of the things that will affect the frequency. A string placed under higher tension will vibrate faster and the pitch that you hear will be higher than the same string with less tension. On both a piano and guitar, the pitch of the strings is fined tuned by tightening the strings until the desired frequency is emitted when the string is excited.

The length of a string at a given tension will affect the frequency as well. The shorter a given string at a given tension, the faster it will vibrate when played. On an instrument such as the guitar, pressing the string against the fretboard behind various frets will raise the pitch of the note the string plays by shortening the string and causing it to vibrate at a higher frequency.

The diameter of the string will also affect the frequency that it will vibrate at a given tension. Larger diameter strings will vibrate more slowly and produce a lower frequency sound than thinner strings.

Finally the density of the material the string is made of will affect how fast it vibrates at a given tension. So a steel string on a guitar will vibrate at a different speed than a nylon string under the same amount of tension.

Using a six string guitar to illustrate, yes we start with six strings of relatively equal length. But each string has a different diameter which affects the frequency that the string will vibrate under a given tension. A set of guitar strings will be sized so that the tension is relatively uniform in standard tuning across all six strings. But the low E string will be much thicker than the high e-string and also will be made of a material with a different density which will also vibrate slower due to greater mass.

When you shorten the strings by fretting notes on the fretboard, you cause them to vibrate at a faster frequency and play higher notes. You have 6 strings which each play a different note. But each string on a standard acoustic guitar can play 20 different notes by shortening the string with your fretting fingers.

On a piano you need 88 different frequencies. And you can't shorten the strings with your fingers like you can on a guitar. So each string must be tuned to vibrate at the appropriate frequency that corresponds to each certain key on the piano. Different string lengths in combination with different string diameters and materials are used on the piano to optimally tune each key to the proper note without needing too much tension. Just like the guitar, the bass strings on a piano are thicker - so they don't have to be as long to vibrate at the slow frequency required.

So you can alter the pitch of the note created by a vibrating string by increasing or decreasing the tension and/or changing the length. But to keep the size of our instruments manageable, the best approach is to use a combination of tension, length, diameter, and material to maintain optimal balance without excess tension that could damage the instrument.

Answer 4

Both length and tension work together to create pitch. Note that the strings on a guitar are all approximately the same length and tension,but the bottom is about 4/5 times the diameter of the top string. On any string instrument, it's important that each string is about the same tension as the others, so that becomes static to a degree. So the two variables are gauge (basically thickness) and length. The bigger the gauge, the shorter a given string needs to be to produce a particular note.

On guitar, string length is pretty much governed by how long the guitar is, to be playable. A bass will have both longer and thicker strings. On piano, there's more leeway - strings can be vertical or overstrung, as there's more room. So to an extent it's easier to accommodate longer/shorter strings on piano than guitar.

I hope someone will expand this with scientific figures, but for now, this is a layman's explanation.

Answer 5

One point has been left out so far. The design of a guitar leaves us no option but to have equal-length strings with different thickness (with the frets used to control the length for all notes except E-A-D-G-B-E for a conventionally tuned guitar). A piano changes both length and thickness: that seems overkill until you realize that there is the issue of energy and perceived intensity of the sound - how loud is it, and how long does it ring? For a piano to sound good, you would like the same intensity of sound when you hit the keys in the same way, and you would like (if possible) the strings to ring down in the same time. You would also like the strings to sound the same. That is not really possible, but a piano, having an additional degree of freedom in the design, gets much closer than a guitar.

What follows borrows heavily from this excellent paper on the acoustics of pianos - I highly recommend that you read the original rather than this poor summary. That goes both for the figures, the equations, and the theory described.

In chapter 5, the paper discusses the relationship between length l, tension T, mass per unit length λ, and frequency f - giving

If you have two strings with frequencies one octave apart, you can in principle achieve this different by changing just one of the factors - length, tension, or mass. In practice, it is desirable to change all of them. The change in frequency is the product of the change in length, change in diameter, and square root of change in tension:

Now if you want all notes to have roughly the same tension, you would want the strings to have the same diameter - but this would mean, over the range of a piano, a massive change in length or diameter to compensate. So instead, strings get a bit longer, and a bit thicker, and have a slightly lower tension, as we go down in frequency.

But there's an important additional factor: the stiffness of the string. Where a "perfect" string would bend easily, so that only the tension determines the frequency and wave propagation, in a steel string there is a certain mode of sound propagation due simply to the stiffness of the string. This stiffness affects the higher harmonics more - and has a bigger impact on the short, high-tension strings at the top of the range.

In principle, the lower strings, being longer, suffer less of this problem; but for practical reasons, the lowest notes have shorter strings to keep the piano size "reasonable".

But here's a funny thing. The human ear can't hear the pitch of the lowest notes on a piano very well - instead, we use the harmonic content to "guess" the fundamental. This makes the lowest notes on a vertical (with shorter strings at the bottom end) sound less pleasant than the same notes on a grand - which has longer strings and therefore better-behaved harmonics. In fact, the length of the strings that one would like at the lower end exceeds even what can fit inside a grand - so the lowest strings are shorter than one might otherwise like (and at a lower tension):

Finally - the question of loudness and decay. The following graph shows how sound intensity decays across the keyboard, with the lower notes being sustained for longer:

Making the lower strings heavier has the effect of making them 'ring' longer; but it also makes it harder to give them a large kick - so that they will give a more similar intensity for the same touch on the keyboard.

All in all a piano is an extremely complex instrument. By contrast, a guitar is much simpler.

Answer 6

The formula is

frequency = sqrt(tension/mass per unit length)/(2*length)

The factor of 2 comes from the fact that an in-tune fundamental vibration has to travel all the way down the string and back up again in order to be in phase.

You can play around with all the parameters. A guitar has 6 strings of about the same tension, but differing thickness. This enables a difference of 2 octaves between the heaviest and lightest strings. Fretting the string at the 12th fret halves the string length, doubles the frequency and raises the pitch by an octave. Some electric guitars have 24 Frets, which means they can play a 2 octave range on a single string by reducing the speaking length to a quarter. The total range is therefore 2+2=4 octaves. It's also possible (but very unusual) to have a bass guitar combined into the same instrument, which would make a total of 5 octaves.

This however does create problems. To have the same scale as a regular guitar, bass guitar strings would have to be either very loose or very thick, which would make the instrument difficult to play. Therefore bass guitars have a longer scale than regular guitars.

The piano on the other hand uses much greater variations of length and thickness, and achieves a range of 7 octaves.

For wind instruments, the speed of sound in air is constant. This means there are less variables to play with and it's rare to get more than an octave and a half from fingering positions. The clarinet manages to squeeze over 2 octaves more by use of harmonics, giving it a range of nearly 4 octaves, which is impressive, but still less than most electric guitars.

Answer 7

Both length and tension factor into the frequency of a vibrating string, along with the mass of the string. The reasons for the differences arise from how one plays the instruments, and physics.

In the case of the guitar, chords are played by using the fingers to push the strings down onto the frets, decreasing the length of the string. This shows that the guitar strings do "change length" to make chords, but doesn't account for why the strings are all the same length when open. This is a matter of physiology. If the high frequency strings were shorter, you wouldn't be able to reach your fingers wide enough to manipulate both the high and low strings at the same time. It would become impossible to play chords.

A piano is constructed differently with one or more strings for every note that can be played. This permits all sorts of sounds that a piano can create which cannot be created by a guitar, such as chords with many notes. Nothing in a piano's structure prevents a Cmaj chord from being played simultaniously over 4 or 5 octaves! Pianos gain this ability by having enough strings to play every note on the piano simultaneously. However, this also means there are many more strings pulling on the soundboard. The forces on a piano are so great that there are thick cast iron plates to take the strain; the piano would crack or bend (and thus lose tune) if it wasn't so strong!¹ Trying to make all strings the same length would be an exercise in futility. You'd want to make the high strings ultra-tight. Piano strings are already near the tensile strength of high quality steel! The alternative would be to make thinner wires (to decrease their mass), but making wires thick enough on the low end and thin enough on the high end would be a materials nightmare! They already have to do this for the low notes by wrapping the strings. It would be 10x worse if they didn't vary the length as well.

And of course, having a piano player's bias, I think it's prettier that way ;-)

¹ I am not a piano tuner, but from what I've heard, the forces on a piano are actually enough to flex the cast iron plate the strings are attached to. Apparently they flex enough that as you tune up some strings, it bends the plate enough to take others out of tune. Piano tuning is a real art!

Answer 8

First to clarify some misconceptions. All strings on a guitar are actually a different length but they look the same. The bridge on the body is angled slightly so technically the strings all have a different length:

The main reason why the strings can be around the same length but have different pitches has to do with the thickness of the string. On a piano you want each string to have its own unique sound for the note so you have to vary the thickness and length and not just one or the other like on guitar.

Answer 9

All strings in the guitar (or other necked string instruments) are different: they have partially different materials (the lower strings tend to be wound with wire) and they certainly have different thickness. They sound best at a particular tension.

You usually buy them in sets even though they age differently.

Piano strings are very durable and rarely changed: there are no actual "sets". If a piano tuner would have to carry separate replacement strings for all different 88 notes, it would be messy. So the piano gets by with a lot fewer different strings, instead varying their length along with varying pitches and only occassionally moving to the next string type/thickness. That way, a tuner needs to carry only a limited number of replacement string sizes (the lower bass strings have winding stopping short of the string ends so they need to be individually sized after all and can't be cut to size on-site, but it's much less usual for them to require change than with the higher ones).