Guitar Playing and Standing Waves

[fog37]

Hello,

The guitar is a stringed instrument with six strings of equal length but different linear mass density. Fretting is about shortening the length of a string which causes the fundamental mode and higher modes to have higher resonant frequencies.

• When a guitar string is plucked (by hand or with a pick), the initial string form is triangular. The plucking is done not in the middle of the string but near the sound hole. Why? Is that the case for all stringed instruments? What do we gain by plucking far from the middle point?

• When a guitar player plays guitar by plucking and fretting strings, do the generated sounds correspond to the fundamental frequency of that string (higher order modes are always less dominant)? Music is a sequence of sounds having a specific frequency content but the fundamental mode in each sound is dominant and determines how the note sounds. All the added overtones are simply shaping the fundamental mode in a slightly different manner. Is that correct?

Thank you!

 

[DrClaude]

• When a guitar string is plucked (by hand or with a pick), the initial string form is triangular. The plucking is done not in the middle of the string but near the sound hole. Why? Is that the case for all stringed instruments? What do we gain by plucking far from the middle point?

The sound is different depending on where the string is plucked. In particular, the proportion of the fundamental changes, and the sound is "fuller" when the string is plucked ~2/3 of the way. Most stringed instruments are played off center (check for instance the violin).

Note that the guitar can be played other ways for effect.

• When a guitar player plays guitar by plucking and fretting strings, do the generated sounds correspond to the fundamental frequency of that string (higher order modes are always less dominant)? Music is a sequence of sounds having a specific frequency content but the fundamental mode in each sound is dominant and determines how the note sounds. All the added overtones are simply shaping the fundamental mode in a slightly different manner. Is that correct?

Overall, yes.

 

[Timothy P White]

The plucking point on a guitar string determines the relative mix of low and high harmonics in the vibrational signal sent to the body of the guitar. Plucking near the middle causes the fundamental to predominate. Plucking near the end causes the higher harmonics to predominate. Further, the softness of the pick determines the harmonic mix as well. The side of your thumb slips off the plucked string slowly, preventing high harmonics to be transferred to the string's motion, while a hard pick enables a rich mixture of harmonics to enter the string's motion. The best way to view the various transfers of vibrational energy from finger-to-string, string-to-saddle/nut, saddle/nut-to-wood, wood-to-air, air-to ear and finally ear to brain is to view each transfer as being characterized as a modulation transfer function (MTF), where the energy transferred at each interface exhibits an amplitude as a function of frequency. Extraordinary second- and third-order complexities arise in the detailed analysis of such things as the relationships of string harmonics, for example how "old" strings differ from "new" strings, how the vibrating strings interact with the three-dimensional shape of body and soundboard resonance modes, the physical properties of different wood types, and the acoustical environment in which the instrument is played (bathrooms sound best). Finally, what happens between the ear and the mind, ie. psychophysics, has an extraordinary effect on tonal perception. As an example, if you play a group of higher harmonics with frequencies that are integer multiples of a fundamental frequency which is not part of the mix, your ear/mind analyzes the high frequency harmonic mixture in real time to determine the least common denominator frequency shared by all of the high frequency harmonics, ie. the "missing fundamental", and that missing fundamental is what you hear, even though it is quite literally "not there". The study of psycho-acoustics is fascinating, and reveals how so much of what we think we are hearing is actually hallucinated from clues, analogous to how our visual field is created from small optical "hints".

 

[sophiecentaur]

All the added overtones are simply shaping the fundamental mode in a slightly different manner. Is that correct?

How nice to read someone using the correct word - "overtone" for the modes of a plucked string. The overtones may differ quite significantly from harmonics of the fundamental and those frequency differences will produce more interesting sounds than if they were just harmonics as they will 'run through' the harmonic frequencies of the fundamental frequency. End effects due to the profile of both the bridge and frets and the placing and pressure of the fretting fingers will also account for subtleties in the sound - in addition to the place and how the string is plucked.

 

[fog37]

Lots of good information. Thank you. I am processing it.

As far as overtones go: my understanding is that overtones are called both higher order modes having a frequency that is an integer multiple of the fundamental (the harmonics) and higher order modes whose frequency is not an integer multiple of the fundamental (called anharmonics).

• I have always thought that in the case of a string under tension the possible modes would be only the harmonics and that any general string vibration would be a weighted superposition of those harmonics. From Fourier theory, a periodic waveform is a superposition of harmonics with frequencies that are multiples of the fundamental but the waveform on a string is not a periodic signal (unless we look at it as a portion of a periodic signal)...
• "Plucking near the end causes the higher harmonics to predominate...". As time goes by, do the higher order modes lose energy more quickly than the fundamental and eventually we primarily end up hearing the fundamental?

Thanks!

 

[sophiecentaur]

a superposition of harmonics with frequencies that are multiples of the fundamental but the waveform on a string is not a periodic signal (unless we look at it as a portion of a periodic signal)...

This is the crux of the matter. The modes of oscillation refer to 'distances', when the release shape is analysed, rather than frequencies. Take any random shape of object and there is no reason to assume any harmonic relationship between the lowest frequency mode and the others. A drum or cymbal may have no identifiable fundamental frequency and the main 'sound' is more noise like than musical. A string is a fairly ideal system but the ends are not clamped very tightly and there is a radius to the curve a vibrating string forms at the ends. So, as you say, the signal is not exactly periodic (even ignoring the decay factor). The individual frequency components of the note will not necessarily be harmonically related at all. I have looked at traces of musical notes, including guitar notes and one of the things that strikes me is that there are fine structures which march along the basic fundamental sinusoid to the right or to the left. IF they were harmonics, they would not move from side to side but stay locked in position.
The difference between harmonics and overtones accounts for how badly early synthesisers managed to mimic real instruments. Hammond is a great sound but you cannot assign a particular set of drawbars (Harmonic content) to any particular instrument.
Further justification of my view is that Quartz Crystals (in many ways they are ideal oscillators) are designed and specified to operate at a particular overtone mode. Put them in the wrong (or just mis-tuned) circuit and you will get a Harmonic of the fundamental and not the specified overtone frequency. End effect is at work.

 

[fog37]

Thank you. I am still processing.

But, in general, when a certain note is played (like do, mi, sol, la, ti) the sound associated to each note mostly corresponds to a specific dominant frequency which should be lowest frequency in the mix (aka the fundamental). For example, the middle C note is said to be 264 Hz in frequency. This must be the frequency of the fundamental standing wave (other overtones are surely present).

 

[sophiecentaur]

Thank you. I am still processing.

But, in general, when a certain note is played (like do, mi, sol, la, ti) the sound associated to each note mostly corresponds to a specific dominant frequency which should be lowest frequency in the mix (aka the fundamental). For example, the middle C note is said to be 264 Hz in frequency. This must be the frequency of the fundamental standing wave (other overtones are surely present).

Absolutely. Our music system is based on that. But the sound colour or timbre of instruments is what give extra interest and enjoyment. People wouldn't go to a concert in which the notes played were all sinusoids produced electronically - except perhaps once for a novel experience.
Instruments have a range of different 'purity' but mostly there is one clear fundamental note. See this link for some pictures (note the scale on the y axis). The spectrum for the flute is confusing, though and reflects its idiosynchratic sound. Mostly, overtones are at a significantly lower level so a 'scope trace will show a readily identifiable fundamental. A Google search would probably get you some interesting sounds and pictures.

 

[fog37]

Thanks sophiecentaur. I am clearly not versed in music and the physics of music. But I would like to learn more about it and learn to play the guitar soon.

I don't see the link you mentioned posted in your reply.

For stringed, wind and percussion instruments (for all instruments essentially), are there always 7 basic notes (and notes in between, i.e. the sharp and the bass) called A, B, C, D, E, F, and G? Are these the same as the DO RE MI FA SO LA TI? Beside those 7 notes, there are other sets of notes having double frequency. Each set of 7 notes and their half step notes is called an octave. The "middle" octave is the 4th octave. The octave below in frequency would be the 3rd and the octave above in frequency would be the 5th.

Is that correct? Do musician "talk" music in terms of DO RE MI etc or A B C D...?

 

[DrClaude]

For stringed, wind and percussion instruments (for all instruments essentially), are there always 7 basic notes (and notes in between, i.e. the sharp and the bass)

In western music, mostly. This is not the case in other musical traditions (e.g., arabic or asian).

called A, B, C, D, E, F, and G? Are these the same as the DO RE MI FA SO LA TI?

In the English-speaking world, A-G are almost exclusively used. The German notation is A, H, C, D, E, F, G.

Beside those 7 notes, there are other sets of notes having double frequency.

Not necessarily. It depends on temperament.

 

[Dr. Courtney]

http://cds.cern.ch/record/951607/files/0605154.pdf

The above paper may help the less experienced understand the discussion.

 

[BWV]

In western music, mostly. This is not the case in other musical traditions (e.g., arabic or asian).

Most all the Arabic and Indian scales are 7 notes, with 5 note scales being common in East Asian traditions. Seven note scales seems to factor common to music systems of the Mediterranean and near-east with references to diatonic scales going back to ancient Mesopotamia

In the English-speaking world, A-G are almost exclusively used. The German notation is A, H, C, D, E, F, G.

Classical musicians tend to use solfege (do re mi etc) as do refers to the tonic pitch, which depending on the key, can be any of the 12 notes of the chromatic scale. So for example, in a piece in Ab major, Ab is do, Bb is re, C is mi etc.

For an acoustic guitar the other key factor is a downward pressure in plucking the string to fully engage the soundboard to vibrate, otherwise the sound will be thin and will not project.

 

[sophiecentaur]

I don't see the link you mentioned posted in your reply.

Sorry, it dropped off somewhere. Here it is.

 

[sophiecentaur]

For example, the middle C note is said to be 264 Hz in frequency.

It's all relative. The 'official' frequency associated with notes was steadily climbing, a bit at a time, over the centuries. I guess it's been located more forcefully recently. But it is still true to say that many soloists will play 'a bit sharp' to take the frequency of their instrument above that of the accompaniment. This makes them stand out in the ensemble. The brightness obtained that way must be a bit similar to the brightness that some instruments have, due to the relationships between some of their overtones and actual harmonics of the fundamental.
It's worth mentioning that overtones for many wind instruments are very far away from harmonics. If you take a length of cylindrical pipe with a brass mouthpiece on it and try to play all the available 'straight' notes on it, you get fundamental, octave, fifth, next octave, third, fifth, seventh and so on but the tuning is absolutely hopeless. 'Roman' trumpets are used in historical films to set the scene and they sound horrible. The Bell on brass instruments helps and the player has to help things along by varying the embouchure of lips and air pressure. The overtone content gives the timbre so a trumpet (with a conical bore) sounds significantly different from a cornet (with a cylindrical bore) and that's not just due to the different frequency response (relative overtone levels; it's down to the overtone frequencies too.
I would love to think that people would take those things on board and start using the right terms for things they are describing.

 

[cosmik debris]

Each set of 7 notes and their half step notes is called an octave. The "middle" octave is the 4th octave. The octave below in frequency would be the 3rd and the octave above in frequency would be the 5th.

Is that correct? Do musician "talk" music in terms of DO RE MI etc or A B C D...?

You are correct about the 7 (8) notes that form an octave but then you go a little astray. The middle of an octave is the flat 5th, also called the tritone. The 3rd is not an octave below and the 5th is not an octave above as these notes lie within an octave and as you correctly said an octave is 7(8) notes. An octave above and below are twice and half the frequency respectively. The 3rd and the 5th are intervals. A quick look at simple music theory will reveal a lot to you and would only take a short time to understand.

Cheers

 

[sophiecentaur]

I think we are going a bit off topic. We started off talking about overtone mix of a not on a plucked string. The frequency is not too relevant and the 'musicality' is probably going a bit too far too fast for only 15 posts.

 

[fog37]

Ok. Let me rewind for a minute.

Basic physics books talk about how a general oscillation of string of length $L$ and under tension $T$, fixed at both ends, can be expressed as a superposition of modes each having their own amplitude and a frequency $f_n$ that is an integer multiple of the fundamental frequency $f_0= \frac{v}{2L}$ of the fundamental mode ($v$ is the speed of the waves). What is wrong or what assumptions are underneath that concept?

Does that mean that a mode having frequency that is not an integer of $f_0$ cannot oscillate on the string?

 

[BWV]

Don’t know the physics but the soundboard makes the tone of a guitar - and the soundboard vibrates with irregular modes like a drumhead

 

[sophiecentaur]

Basic physics books

These books discuss strings because a string is nearer ideal than most instruments. The modes that are assumed in a string assume that the equation of motion of the string at the ends is the same as it is in the middle. Imagine you had a string which was not uniform thickness all the way along it. Say it had a thin section one end and a thick section at the other. You wouldn't expect the modes to be harmonics, even if all modes had their end Nodes in the same place. Any real string has a non-ideal termination at each end and that will disturb the modes. End effects are different for different modes.
The general principle of all instruments is that the frequencies of the modes are not harmonically related but strings are probably the least worst in that respect. They are still not perfect, though - imperfect enough to sound the (good) way they do. If you don't have access to a guitar and oscilloscope then browse around and find scope traces of instrumental notes. The higher components are not stationary on the fundamental. That means they are not harmonics.

 

[fog37]

Thank you. I didn't know any of that. I suspected that the string described in introductory physics book was something highly ideal...

 

[BWV]

The general principle of all instruments is that the frequencies of the modes are not harmonically related but strings are probably the least worst in that respect. They are still not perfect, though - imperfect enough to sound the (good) way they do. If you don't have access to a guitar and oscilloscope then browse around and find scope traces of instrumental notes. The higher components are not stationary on the fundamental. That means they are not harmonics.

Question - on a guitar should the individual partials on a plucked string be the same as the pitch of the corresponding plucked harmonic? I.e. if I pluck the open 6th (e) string the intonation of B, the second partial is the same as if I pluck the harmonic near the 7th fret - any imperfections in the string that would cause the intonation of the 2nd partial to be off in the spectral analysis of the fundamental would be the same imperfection on the plucked harmonic?

 

[sophiecentaur]

Question - on a guitar should the individual partials on a plucked string be the same as the pitch of the corresponding plucked harmonic? I.e. if I pluck the open 6th (e) string the intonation of B, the second partial is the same as if I pluck the harmonic near the 7th fret - any imperfections in the string that would cause the intonation of the 2nd partial to be off in the spectral analysis of the fundamental would be the same imperfection on the plucked harmonic?

Guitars are 'not bad' in this respect and many guitarists tune their guitars by tuning one string to the harmonic of another. But stopping a string on a fret can change the tension and I believe that a least worst solution is used. Imperfections with modern strings are a lot less than in the past so these differences are more significant than when you had to roll and stretch a new string to get it to behave over a range of frets.
I would doubt whether we would ever be having this sort of conversation if we were all horn players.