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I'm working on this piece of music on a sequencer, and I'm trying to emulate the effect of playing the same note on different strings of a guitar, say, an E3 on the 5th fret of the B string and an open E3 on the E (first) string.

Tab 1

Obviously, only repeating the exact same note doesn't cut it, so I figured I could achieve this by introducing a small difference in frequency between the notes played. But this is not a general solution, since the higher you go on the fingerboard the higher is the difference between frequencies.

enter image description here

The question is: what is the difference between any two notes, with the same pitch, played in strings of different thickness?

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What features do you have control over in synthesizing these sounds? –  Dave Aug 26 at 12:42

6 Answers 6

Technically speaking two notes with the same pitch have the same frequency as the fundamental.

However this does not explain why two notes of the same frequency also called unisons, sound different on strings of different diameters or lengths or both. The guitar and the entire orchestra string family as you may know have numerous unisons (unlike the piano).

For instance on the guitar, E4 can be played on 5 different strings.

As shown in the diagram below.

enter image description here

So the key question is why and how do these sound different although they are the same pitch?

Largely because each one of these unisons differs in the amplitudes of each harmonic. This is due to the change of thickness of the strings and the length of the string as in where the string is 'stopped' (fretted).

Conversely if you had two identical gauge strings tuned exactly the same as in the two upper E strings on a standard 12 string guitar, then the only difference would be phase related as to when the strings were struck. If the these two strings were engaged at the same time they would sound the same.

To understand why two unisons may sound different (thickness and/or length), you need to make a study of how a thick string playing say E4 vs. a thinner string playing E4 has differing strengths for the harmonics. One way would be to look at each waveform using an oscilloscope or an audio spectrum analyzer or software that would take your audio inputs and render a Fourier transform.

Wiki on Harmonics:

http://en.wikipedia.org/wiki/Harmonic_series_%28music%29

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+1 for Fourier transform! –  Ben Kushigian Aug 26 at 0:22
    
On guitars with 24 frets, there's an E4 on bottom E, fret 24. Also, as harmonics are in the equation, don't forget 5th and 24th fret bottom string, and the 'false ' harmonics found on 4th 5th and 6th strings, all making E4. How a harmonic of a harmonic affects the sound, I don't know. Although harmonics themselves seem to sound very similar, unlike the phenomenon in the question. –  Tim Aug 26 at 6:21
    
@Tim I think they're talking about the broader mathematical definition of a harmonic. Though they're definitely related to the guitar-playing kind. –  Matthew Phipps Aug 26 at 19:48
    
Just to be clear, I am referring to the harmonic series of a vibrating string (fundamental plus overtones) as per Pythagoras. Here's a nice description and more from NUSW in Sydney: phys.unsw.edu.au/jw/strings.html –  filzilla Aug 26 at 22:01

All else being equal, a thicker string will damp out transverse vibrations more rapidly because it experiences more drag (inter-molecular deformation) per unit length. (See section 4.6 of [not my work] http://www.people.fas.harvard.edu/~djmorin/waves/transverse.pdf.) (If we consider strings made of different materials or under different tensions, this rule may or may not apply.)

As noted in some other answers (for various correct and incorrect reasons), the spectrum of harmonics will also be different between the two strings. However, while this is true, my own experience is that the difference in tone produced by this effect is usually smaller than what can be achieved with other performance techniques. For example, by moving the bow closer to the bridge when playing on the heavier string, the difference between the overtone spectra may become negligible or even inverted, or acquire totally different characteristics like the noise of attacking each string from an angle. Thus, getting your sequencer to sound like a human musician may be somewhat counterintuitive, and you should play around with different ideas.

In addition to adjusting the rate of decay of one of your enharmonic notes, I also suggest you try adding vibrato (finger-speed frequency fluctuations) to one of them, since that is what a real player's finger would be doing on the fingered string, but obviously not on the open string.

As far as shifting the center of each pitch, which is also a realistic idea, you would simply need to scale the pitch difference logarithmically with frequency, as @BenKushigian suggests. I also like @smiley's idea of adding a very slight phasor over the entire waveform, to enhance the effect of beating between the strings.

Please let us know what you find sounds best!

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That's a nice reference you quoted there, but it doesn't really discuss dampening as it happens in stringed instruments, only dampening by fluid viscosity friction (unless you're playing underwater guitar, that won't be a relevant factor. And if it was relevant, your conclusion would be wrong: a heavier string is actually attenuated less by this effect, for much the same reason a ball of lead drops faster than a feather). In reality, attenuation on a steel string happens mostly at the bridge and fretboard and on dirt particles on the string. For nylon etc. strings it's more complicated. –  leftaroundabout Aug 26 at 10:53
    
You sound like you know a lot about string physics. One thing I've been wondering about is whether it would be practical to design guitars (especially smaller scale ones) in such a way that string tension would vary less with displacement. The biggest problem I've observed with low string tension is that loud notes are noticeably sharp compared with quieter notes; I would think that having a more constant string tension would help fix that. Do you know of any efforts to achieve such a result? –  supercat Aug 26 at 15:59
    
@leftroundabout, I'm not sure why fluid damping would be different from string thickness. According to the reference, to leading order the damping is directly proportional to mass density per unit length. This makes sense to me because a thicker string has to undergo more (inelastic) deformation which causes it to dissipate energy, the same as in the fluid case. So I disagree that a heavier string is attenuated less, and also that dirt or air resistance has anything to do with it. What are your sources? –  ninemileskid Aug 26 at 19:57
    
@supercat, that's an interesting idea. I suppose you could just crank up the tension so that all displacements are "small", but then it would be harder to play at all. –  ninemileskid Aug 26 at 20:08
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Where did you take "to leading order the damping is directly proportional to mass density per unit length" from anyway? That's not in your source. Again, it's wrong; for constant friction coefficient the viscous damping goes ∝ 1/ρ as per a simple energy argument (doubling the density reduces the average velocity (and hence force) by 1/√2, so each oscillation by a given amplitude dissipates that much less energy. It also lengthens each cycle to √2 T, so the time needed to dissipate a given amount of energy is doubled). –  leftaroundabout Aug 26 at 22:48

The other two answers are true however it seems your question is about sounding a unison as opposed to replicating one sound on a different string, even though you pointed out string thickness as a possible reason for the sound you are noticing. What happens when a string vibrates is that it actually stretches from side to side or up and down depending on which way it was plucked or snapped. This causes a slight warble in pitch which is almost unnoticeable. Depending on how hard the string is attacked this warble will become noticeable and will confuse your electronic tuner. When unison strings are sounded at the same time each one has its own warble pattern and the effect is somewhat akin to chorusing. I have a twelve string guitar that I keep strung with same octave strings so I can get this effect. It sounds amazing but don't try bending the strings. The minute difference in the amount each string gets bent sends the notes out of sink with each other. It makes a ring modulator sound harmonic!

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Some additional details.

There is a very small change in pitch due to the change in tension that occurs when the string is fretted. This change in tension varies along the neck, generally larger changes further up (away from the nut) the neck. This change is small enough that it is usually imperceptible in single note playing; however this difference does contribute to the chorusing type of sound that you hear when playing unisons.

Changing the length of the vibrating string also has effects on the inharmonicities in the overtones of the note; in general, the shorter the string the greater the change in frequency of the higher overtones away from their ideal values (this has to do with the fact that real strings have non-zero transverse stiffness). In addition, thicker strings (of a given material) will have greater degree of inharmonicities in the overtones. These effects will shift the locations of the harmonics in frequency.

I suspect that the relative strengths of the harmonics, as indicated in Filzilla's answer is the dominant effect, and would be what I'd look at trying to modify first, possibly by just applying different EQ curves to the differently fretted notes; however, if you are going into excruciating detail, there is a physical basis for incorporating effects that shift the frequencies of the harmonics, including the fundamental.

One point which is implicit in some of the other answers but not spelled out in detail is that when modifying pitches you usually want to shift them multiplicatively, something like F' = (1+df)*F where df is a fractional change in the pitch. My read of this question is that the OP tried doing the pitch shifts by F'=F+dF where dF is a frequency shift in Hertz, which does not produce uniform changes in perceived pitch as the original pitch is varied.

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So the difference between the quality of two notes that are the same pitch (two different strings, or even with two different instruments) is not in the frequency necessarily (though my guitar is always a bit out of tune...) but rather the overtones each string produces. I don't know exactly how to mimic that but read into the overtone series - maybe bassier strings should have more emphasis on the lower end of the overtone series (just a hair, too much and it would be too drastic!)

Also, as far as changing the frequencies are concerned, look into cents ( a logarithmic division of semitones in equal-temperment) instead of constant numerical changes. If I had an A440 and added 2Hz to it and wanted to mimic this at A880 then I would want to add 4Hz to it. The best way to do this is to automate the process (if you have any programing skills). I did up a little Python script, I can send it along if you want it - you input your first frequency, your second frequency (adjusted a few Hz) and then a third frequency. The script outputs a fourth frequency that is logarithmically proportional to the third frequency as the second is to the 1st.

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:) and I'd be happy to explain it a bit better too if you were so inclined –  Ben Kushigian Aug 25 at 23:55
    
Sure, please elaborate! Also, would you explain why the frequencies would be logarithmically proportional and what this means? –  Jovito Aug 26 at 0:02
    
Essentially if you have 4 frequencies w, x, y and z they are proportional if w is to x as y is to z (or some version of that). This won't work with cents in different octaves because the freq doubles every octave. There IS a pattern though, involving the logarithm: if lg(w/x) = lg(y/z) then I'm calling these logarithmically proportional. By the way, lg is notation for base-2 logarithm. If you're not up on logarithms don't worry, but they are very useful when analyzing frequencies! For example: lg(440/220) = lg(2) = 1. Also lg(880/440) = lg(2) = 1. Out of room, hold on... –  Ben Kushigian Aug 26 at 0:14
    
That means that whatever the distance is from 440Hz to 220Hz (as far as intervals or cents), that is the same distance from 880Hz to 440Hz. In this case both distances are 1 octave or 1200 cents. So what the program i wrote does is take two frequencies that are a certain distance from each other that you would like to emulate. It then takes a third frequency and outputs a fourth frequency that is the same 'logarithmic' distance from the third frequency as the second was from the first. From our previous example: myprogram (220, 440, 110) >>> outputs 220 It says '1 octave between 220 and 440 –  Ben Kushigian Aug 26 at 0:18
    
an octave from 110 is 220' It actually does this with cents but those are a bit harder to visualize (really the same difference as between inches and yards - we just don't usually use inches to measure football fields) OK, that was a bit mathy - if you want to learn more about this there are some excellent resources out there. Wikipedia is one but it tends to get a little technical. Just look around and read up on it - it's cool stuff –  Ben Kushigian Aug 26 at 0:20

There's a key difference in string gauges. Under the same tension, a thinner gauge has less mass and it can move more easily when it receives an impulse, as opposite to a thicker gauge. In thinner strings, more harmonics are generated because, aside from the fundamental vibration, smaller harmonic vibrations occur on the string more easily. That makes for a "brighter" sound to our minds, as opposite from the "darker" sound on thicker strings. There's also other things involved, as noted in Dave's answer.

You won't be able to generate this on a computer so easily, by just changing frequencies, it's not a matter of frecuencies, but physics. If you still want to try, you need to have a variable amount of harmonics (hint: can be achieved with a low-pass filter).

Those subtle differences are what make electronic generated music sound so artificial.

https://en.wikipedia.org/wiki/Harmonic#Harmonics_and_overtones

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