# Middle c

February 5, 2018, 3:20 PM · Could someone please play a middle c on their violin and tell me if there is any sympathetic resonance on the e string?
Thank you

## Replies (25)

February 5, 2018, 3:30 PM · I don't think the E string rings while playing a middle C.
February 5, 2018, 3:43 PM · Not necessarily a resonance as such, but my violin likes to give me a shrill afterglow which retreats when I mute the e
Edited: February 6, 2018, 6:24 AM · Warning! some maths here!

Assuming the violin is tuned to A=440Hz and the strings are tuned in perfect 5ths, then the frequency of the E string will be 660Hz. Its first harmonic is the octave E at 1320Hz. If a C at the frequency of 264Hz is played then one of its higher harmonics will be that octave E at 1320Hz. I would therefore expect some sort of resonant response from the E string, the magnitude of the response depending on how loud the C is played and the general response of the instrument. Whether it will be audible is another matter.

Unfortunately, the middle C of the piano, whether played on the piano or violin, is 261.6Hz, and its relevant harmonic is 1308Hz, which is too low to resonate the E harmonic 1320Hz. Similarly, if you were to play the C that is a perfect 4th above the violin G string you would be playing a frequency of 260.7Hz, which is again too low to resonate that E harmonic.

The only way to get that E string to resonate from a C is to play that C sharp at 264Hz. Whether it works in practice depends on the response of the violin, as I mentioned above. Some professional string quartets get their violist and cellist to tune their C strings a little sharp so as to get a resonance with the violin Es. The viola C string would have to be tuned from 130Hz up to 132Hz to get the effect, and the cello C up from 65Hz to 66Hz.

The violin E is also affected by the open A at 440Hz, because one of the A's important harmonics is - wait for it! - the E harmonic 1320Hz. This is sometimes used, not for tuning as such, but as a quick check.

[Edit Feb 6, 2018: a typo wherein I inadvertently did the calculation for the cello low F instead of the C has been corrected. Many thanks to those who drew it to my attention.]

February 5, 2018, 6:57 PM · Very interesting, Trevor!
February 5, 2018, 9:06 PM · There is a characteristic resonance whenever a perfect 4th above an open string is played, including the C. While it doesn't ring with an octave below, it still has it's own ring. I don't know what the partials involved are, but they're there.
February 5, 2018, 10:47 PM · Trevor: "The viola C string would have to be tuned from 130Hz up to 132Hz to get the effect, and the cello C up from 87Hz to 88Hz."

I think you meant the cello F, not C.

February 6, 2018, 1:35 AM · Trevor - I'm not sure the reason for "close 5th's" tuning by string quartets is to achieve extra resonance. I'm no expert but I understand that if you tune by ear in perfect fifths starting with the top two strings, the bottom string and its harmonics will simply sound flat as compared with the top. Even in orchestral playing is it only me who often finds, when required to to play open G for the first time, that it has "gone flat"? So maybe everyone should do it as a matter of course, although it only seems to be taught in the context of string quartets.
February 6, 2018, 5:08 AM · I get a harmonic one octave above the open e on the e-string. That's two octaves and a third above middle c which is a natural overtone. @Trevor: This also works on a piano. If you silently press e' on a piano and play the middle c, the e will ring sympathetically. I guess, even though the e is not exactly on the overtone, it's still close enough. I can also move the c up and down a bit on the violin and still get the harmonic on the e-string, so there's some slack there too.
February 6, 2018, 5:56 AM · Cello C string is 65 Hz (give or take depending on the tuning system).
February 6, 2018, 6:23 AM · @Han, @Andrew. Many thanks for drawing my inadvertent wrong calculation (cello F instead of C) to my attention. I've amended my original post accordingly.
February 6, 2018, 7:26 AM · There is a major resonance around middle C on a violin which shakes the top and the bridge rather severely, so all of the other strings can be excited to some extent.
Edited: February 6, 2018, 8:10 AM · I dimly recall from string quartet courses that after tuning our instruments in close fifths, the check was to make sure the cellist's open C makes a sweet major 17th (!) with the violinists' open E. Does that answer the question?
February 6, 2018, 8:57 AM · All unmuted open strings will vibrate at one or more harmonics of the note being played. How loudly they vibrate depends on how close one of the harmonics of the played note is to one of the harmonics the open string is tuned to.

The term "resonance" basically means that the frequency of the played note, or one of its harmonics, can be expressed as an integer ratio of one or more of the harmonics of the unmuted open string. In this case, the open string can vibrate quit loudly, but at one of the harmonics of the PLAYED note.

For example, if you have a very responsive E string on your violin, you can hear a high pitched shimmer or ring when playing many notes on the other strings. The E string is vibrating at some harmonic of the played note.

It can be especially noticeable when you stop playing the note but leave the bow on the string. The open string will continue to ring but can decay to one of the open string's natural resonances. You might perceive a note "shift" when you cease bowing.

February 6, 2018, 11:23 AM · Trevor, I think your theory isn't correct. I have to admit it got me thinking, and digging....I found this chart which plots the "Reaction". There is a rollooff as the input frequency increases and nothing occur when the input frequency is 2X or 3x, etc....

https://en.wikipedia.org/wiki/Resonance#/media/File:Resonance.PNG

February 6, 2018, 11:42 AM · Try https://en.wikipedia.org/wiki/String_vibration instead.
February 6, 2018, 2:26 PM · Han,

The link you suggested describe the math behind at what frequency a string will vibrate at. It relates the density, tension, etc to indicate the vibration (resonance).
Although its an educational article, it doesn't address the OP's comment.

February 6, 2018, 2:51 PM · I don't get what relation you see between the frequency response of a single, strongly damped oscillator (single mass-spring-damper system) and that of a string, which behaves as a series of weakly coupled, weakly damped oscillators.
February 6, 2018, 5:17 PM ·
February 6, 2018, 5:26 PM · It has probably been obvious to everyone but me (until the last few months) that when one string is excited by the bow, that the frequency is transmitted through the bridge into the other three strings.

With this excitation, the other three strings find appropriate harmonics; if there are any. But they are still being moved.

ABL

February 7, 2018, 11:05 AM · Just to muddy the waters a little, middle C on my Rippen piano is 256Hz.
February 7, 2018, 11:56 AM · Han,
From what I'm interpreting on the chart I sent, and my general knowledge, the type of system (or amount of damping) is a don't care. Those items would impact how much (amplitude) of any resonance that may occur due to some input. The chart I posted simply indicates that the input to cause an excitation of any given element needs to line up with the resonance of that given element. So in order for the e string to vibrate as a result of vibrations input via the bridge, nut, tailpiece, or air, there must be a frequency at/nears 1318 Hz. A frequency that has its fundamental at 2636 hz, would NOT cause the string to vibrate. This is what I'm interpreting. And actually I can test this at home. So I would amplify a E5 tone, ensure I can hear the E string vibrate, then change the tone to E6, and see if I can detect an E string vibration. I'm relying on my senses here so false negative is possible.....
February 7, 2018, 3:23 PM · Thanks Trevor! In my weekly tuning of about sixty violins in the student orchestras that I assist, sometimes I get a "C" registering on my chromatic tuner (no, I don't have perfect pitch and tuning all those instruments in a noisy room...).

I was assuming that somehow there was an harmonic at work. I appreciate the science behind it all. Perhaps I'll even teach some of the more interested young musicians about music, math and harmonics.

February 7, 2018, 4:33 PM · Arnie Cohen write:

"So in order for the e string to vibrate as a result of vibrations input via the bridge, nut, tailpiece, or air, there must be a frequency at/nears 1318 Hz."

In the specific case of the bridge driving the open E string while another string is being bowed, what you posted is not what happens. See my previous post for a description of what happens.

Edited: February 7, 2018, 5:47 PM · If you tap the side of a bridge, all the strings will sound. It can be a single tap, and doesn't need to be at any particular frequency.

Similarly, playing about a middle C (and also around an octave above) induces a rather extreme motion of the instrument top and bridge, and that motion of the bridge will "play" the strings. The bow isn't the only thing that plays the strings. They are also played by the vibration of the violin, fed through the bridge, shaking the strings.

February 7, 2018, 10:21 PM · Thanks all for the informative discussion. I have been reading attentively. Thank you

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