Pickup positioning and sensing width

[jcnc]

Hi. I’m new to this site, but figured I’d find some like minded people here. I’ve spent the last couple of years analyzing guitar tones with the end goal of reverse engineering tones from my favorite players. Along the way, I got seriously sidetracked when I realized there were basic things about the physics of vibrating strings on electric guitars that I did not fully understand.

I am currently trying to match the predictions that Tillman’s online app modeling the effects of pickup width and placement to real-life samples produced with my own guitars. The first thing I learned is that the location where the string is plucked has a huge effect on the relative strength of each of its overtones and this effect is not modeled in Tillman’s app but has as strong an effect on the shape of the overtone series as the pickup’s location. Another thing I learned is that plucked guitar strings are inharmonic and I’ve calculated inharmonicties for standard gauge strings and mapped how inharmonicity changes as you fret up the string.

I’m posting today because I have realized that pickup sensing widths are much wider than I had been led to
believe. As was noted on this board, sensing width is correlated with distance from the string, but I am seeing sensing widths of as much as two inches with single coil pickups, which is twice as wide as I expected. My approach to determine sensing width is to look at the overtone series and match it to the expected overtones predicted by the formula’s in Tillman’s descriptions. I’ve written fft code in Python that gives 1 Hz resolution.

If anyone is interested I can post some screenshots, but I am curious if anyone else has tried this approach and what people here have concluded regarding pickup sensing widths.

[JohnH]

hi jcnc, welcome to GN2 and this is definitely the best and maybe the only place to discuss this stuff! There is an ongoing interest in pickup testing and modelling over the last few years.

I got intro it too, in the form of a spreadsheet to model both the electrical and the physical response of a guitar, picking up string vibration harmonics, pickup placement and picking position. You might like to read and download it. It runs in Excel:

guitarnuts2.proboards.com/thread/3627/guitarfreak-guitar-frequency-response-calculator

Tillmans stuff is built into it, along with a few other key theories. The idea is to model the output as seen by the amp.

Id be interested to see your charts too.

[ms]

The sensing width is somewhat wider than a pole diameter.  I believe that Zollner shows this (https://www.gitec-forum-eng.de/the-book/).  I am not sure that link gets you all the pdf files, but you can find them.

[aquin43]

Zollner measures pickup apertures in a way that sets an upper limit. They turn out to be quite narrow.

www.gitec-forum-eng.de/wp-content/uploads/2019/03/poteg-5-4-4-aperture.pdf

There are some papers on the web that discuss estimating the plucking point from the sound.

dafx17.eca.ed.ac.uk/papers/DAFx17_paper_43.pdf

www.eecs.qmul.ac.uk/~simond/pub/2017/MohamadDixonHarte-JASA2017.pdf

Modelling including multiple pickups:

www.researchgate.net/publication/234034228_Acoustics_and_Modeling_of_Pickups

Probably many more since there is a lot of commercial work on modelling guitar sounds.

[jcnc]

Thanks for the replies and the links; I look forward to sifting through it. Thus far when I analyze telecaster single coil bridge and neck samples, I can definitively identify the effects of inharmonicity, plectrum position, pickup position and pickup sensing width, and these all match expected behavior. But I’m having trouble with my humbucker samples, to the point where I wonder if different formulas are required for humbucker sensing width than for single coils.

[jcnc]

Jan 27, 2022 7:07:32 GMT -5 aquin43 said:

Zollner measures pickup apertures in a way that sets an upper limit. They turn out to be quite narrow.

www.gitec-forum-eng.de/wp-content/uploads/2019/03/poteg-5-4-4-aperture.pdf

There are some papers on the web that discuss estimating the plucking point from the sound.

dafx17.eca.ed.ac.uk/papers/DAFx17_paper_43.pdf

www.eecs.qmul.ac.uk/~simond/pub/2017/MohamadDixonHarte-JASA2017.pdf

Modelling including multiple pickups:

www.researchgate.net/publication/234034228_Acoustics_and_Modeling_of_Pickups

Probably many more since there is a lot of commercial work on modelling guitar sounds.

Thanks. A quick skim of Zollner’s paper makes me hopeful that it explainable why I am having trouble detecting humbucker sensing width effects.

I’ve read the MohamadDixonHarte paper before and downloaded the related source code from Github. I can execute the code and it generates numbers, presumably it’s predictions of pickup location and plectrum position, but the results have no units and I can’t relate them to the actual known plectrum and pickup positions used in my samples.

[timtam]

Jan 26, 2022 21:56:35 GMT -5 jcnc said:

Another thing I learned is that plucked guitar strings are inharmonic and I’ve calculated inharmonicties for standard gauge strings and mapped how inharmonicity changes as you fret up the string.

There's a small literature on inharmonicity in guitars - the relative sharpening of upper harmonics in comparison to the expected exact frequency multiples - but still some uncertainties. It would be good to see your results.

It think generally it's been thought that inharmonicity is not that significant for guitars, because of their operating parameters, in comparison to the things that make inharmonicity worse - bigger, stiffer, shorter, lower tension strings. It's obviously worse on (wider operating range) pianos, so much so that they are stretch-tuned to minimize its negative effects. It's also evident in bass guitars (Kemp 2020).

But French (2009) has argued that when adjusting a guitar's intonation, inharmonicity (from the string's bending stiffness) may have a slightly greater effect on the frequency errors than the usually-implicated fretting-induced string stretching/sharpening. But of course that error is compensated, within limits, when adjusting intonation at the saddle. In French's new book on acoustic guitars (2022) there is an FFT chart (fig 5.5 - see below) that shows the (remaining) measured sharpened harmonics for a Taylor acoustic guitar. It looks minimal for the high E, but maybe not for the low E - the red curve's upper harmonic peaks no longer line up with exact frequency multiples (vertical chart divisions at each normalised frequency).


Pertinent questions are can we typically hear/identify guitar inharmonicity (when not played alongside the same sounds without it), do we actually prefer our sounds that way, and does it cause any other problems, for example distorting the process of electronic or manual (heard) tuning ?


Fastl, H., & Völk, F. (2008). Inharmonicity of sounds from electric guitars: Physical flaw or musical asset? Proc. 10th Intern. Conf. on Music Perception and Cognition ICMPC10.
French, Mark. (2009). Engineering the Guitar. Springer US. doi.org/10.1007/978-0-387-74369-1
French, Mark. (2022). Acoustic Guitar Design. Springer. link.springer.com/book/9783030893804
Järveläinen, H., & Karjalainen, M. (2006). Perceptibility of inharmonicity in the acoustic guitar. Acta Acustica United with Acustica, 92, 842–847.
Kemp, J. A. (2020). On inharmonicity in bass guitar strings with application to tapered and lumped constructions. SN Applied Sciences, 2(4), 636.

[jcnc]

I don’t have an opinion as to whether the inharmonicity is audible, but it makes a huge difference when modeling frequency response of things like pickup positioning. I measured inharmonicity on all the strings on a couple of my guitars and also up the fretboard on a single string to get a feel for how inharmonicity varies with fretting position and strong tension. I’ll post the Beta values later, but the obvious takeaway was that you can’t accurately predict plectrum position nulls and pickup position nulls without taking inharmonicity into account. The effect increases greatly as you look at higher order harmonics. Also inharmonicity increases greatly as you fret the string because the length of the string is one of the variables that contributes to inharmonicity. I found a relatively easy way to measure inharmonicity by plotting the fft and also plotting calculated harmonic (integer) multiples of f0 and seeing how they line up. Then by trial and error, I increase beta until the calculated harmonics match the measured harmonics. It’s difficult to see any effect in the first dozen or so harmonics on a low pitched string but gets obvious as you look at higher overtones. For notes at the upper end of the fretboard, the effects of inharmonicity are quite visible in the even the low order harmonics.

I am curious about something related to inharmonicity that I haven’t figured out yet. When I try to plot a curve to project pickup position and pickup width effects, I do it by calculating the expected output across the harmonic series, then I take the frequency of each overtone and stretch it sharp according to the inharmonicity of the string (which I know from having measured it using the technique given above). For example, if the pickup is centered one inch from the bridge on a 24.75 inch string, I expect the first bull to appear at f25. However, I’ve noticed that for bridge pickups, the actual notch that shows up on the fft is sometimes a harmonic or two higher than expected, like at f26 or f27. I am guessing this could be due to inharmonicity effecting the physical location of the nodes in the standing wave near the terminating ends of the string. Like maybe when a string is vibrating in relatively small sections, the standing wave isn’t actually 25 equally sized divisions of the strings but instead the vibrating sections in a higher order harmonic are uneven in size as you get closer to the end of the string. I plan a test on a Tele with a slanted bridge pickup to see if the pickup nulls move progressively out of their expected position depending on how close the pickup polepiece is to the bridge. To rule out the effects of different f0 and string tension, I plan to use the same string at the same pitch in each string slot for this test.

Sorry my posts are sporadic but I am really interested in sharing what I have learned and getting feedback from other people who’ve conducted their own investigations.

[jcnc]

I was looking at the previous post on my phone and didn't see the graph of the Taylor until now. The effects of inharmonicity increase as you go up the harmonic series and are minimal in the first 10 harmonics.

Here's what I measured for beta values on two of my guitars:

B=0.00015 # SG low E

B=0.00007 # SG A

B=0.000055 # SG D

B=0.00012 # SG G

B=0.00004 # SG B

B=0.00001 # SG high E



B=0.000128 # Tele low E

B=0.00007 # Tele A

B=0.00006 # Tele D

B=0.000108 # Tele G

B=0.000035 # Tele B

B=0.00001 # Tele high E



The values are similar (probably because I strung them with similar strings...I don't remember the exact gauges but probably 10-46 on both. Note that the low E and G strings are the most inharmonic whereas the high E strings are the least inharmonic. 

Here is an example of how inharmonicity increases as you go up the fret board. Note that on the same string, notes fretted at the 12th fret are nearly 4 times as inharmonic as the open string and the 17th fret is nearly 8 times as inharmonic as the open string. 

tele G string: measured beta values
Fret0, f0=196.3, B=0.000108 
Fret3, f0=234, B=0.00014 
Fret5, f0=262, B=0.00018 
Fret 7, f0=294, B=0.000235
Fret 10, f0=350, B=0.00032
Fret 12, f0=392, B=0.0004 
Fret 15, f0=467, B=0.00058 
Fret 17, f0=524, B=0.00074

John

[jcnc]

Here are some graphs of plectrum position nulls on a Tele low E string:

[jcnc]

And lastly, here is what I am currently working on...trying to determine the pickup sensing width by looking at the strength of each harmonic. In the example below of a Tele low E string on the bridge pickup, I use a pickup width of 1" to generate the expected curve (labelled tillman in my graph) but I don't think 1" is the correct value. In my code, I can try out different values for pickup width and then visually compare the expected curve with the actual fft output.  For bridge pickups, I've had a hard time getting them to match.
The blue lines are the expected pickup position nulls (adjusted for the string's inharmonicity) and the red line is the expected plectrum position null (1 cm from bridge). The tillman curve is generated from his discussion of pickup position and pickup width (also adjusted to account for the string's inharmonicity). 

[ms]

The physics tells you that the sensing width is something like a pole diameter.  Why are you using 1 inch?  A humbucker has two sensing regions, and that matters for measurements and for the sound on the lower three strings.

[jcnc]

I used 1 inch because that was the value used in Tillman's description of sensing width:

"This sensing area is called the "aperture" of the pickup and is about an inch wide on a thin single coil pickup and about 2.5 inches wide on a wider pickup such as the Gibson humbucker."
www.till.com/articles/PickupResponse/

When I use the pole piece width instead, it takes pickup sensing width pretty much out of the equation as a factor in the pickup's frequency response. The pole pieces on my tele bridge pickup are about 5 millimeters in diameter, which yields a first null due to sensing width at over 20k. Even as a low pass filter, the sensing width only decreases high frequency response by a couple of dB by 10k. This isn't necessarily inconsistent with what I see in my data, but detecting sensing width visually in fft graphs is the main thing I am having trouble with. 

I ran some tests with a string length of 4" (using a dobro capo) and tested it with a tele pickup at 2" and the results implied that the sensing width was 1.14". I did a similar test at the 21st fret and got similar results.

[ms]

Feb 13, 2022 10:17:33 GMT -5 jcnc said:

I used 1 inch because that was the value used in Tillman's description of sensing width:

"This sensing area is called the "aperture" of the pickup and is about an inch wide on a thin single coil pickup and about 2.5 inches wide on a wider pickup such as the Gibson humbucker."
www.till.com/articles/PickupResponse/

When I use the pole piece width instead, it takes pickup sensing width pretty much out of the equation as a factor in the pickup's frequency response. The pole pieces on my tele bridge pickup are about 5 millimeters in diameter, which yields a first null due to sensing width at over 20k. Even as a low pass filter, the sensing width only decreases high frequency response by a couple of dB by 10k. This isn't necessarily inconsistent with what I see in my data, but detecting sensing width visually in fft graphs is the main thing I am having trouble with. 

I ran some tests with a string length of 4" (using a dobro capo) and tested it with a tele pickup at 2" and the results implied that the sensing width was 1.14". I did a similar test at the 21st fret and got similar results.

I think for the higher harmonics you have to account for damping from bending as well as filtering from spatial sampling.

[aquin43]

Zollner obtains a roughly Gaussian shape for the window of a pickup pole, one for a single coil, two for a humbucker. The shape of his sampling pulse on the string is not infinitesimally narrow so the actual window shapes may well be slightly sharper than this. Are there any other direct measurements of the aperture? Tillman's seem to be chosen arbitrarily for the purpose of illustration.

The frequency response due to the aperture will depend on the shape of the aperture as well as its "width", however that is defined. As ms points out, it will also be necessary to take into account the effect of the bending stiffness generally reducing higher harmonic amplitudes. In trying to measure the rather narrow pole sampling width via the frequency response or wave shape you are, in effect, assuming that the string can carry a sufficiently sharp curvature. It appears that an ordinary guitar string can not.

It seems to be generally true that the pole piece window width is small enough to have a negligible effect within the pickup bandwidth. Humbuckers, as we know, have a secondary mechanism that produces its own comb filtering.

[jcnc]

Yes, I also worry that I am confusing visual evidence for high frequency damping with evidence of pickup sensing width effects.
Here are two examples that appear to yield evidence of single coil pickup sensing width of a little over an inch on a  tele bridge pickup but I don't fully trust that pickup width is the cause of the missing harmonics.








In both graphics, the green vertical lines indicate expected the location of expected harmonics (f0 * n, adjusted for the string's inharmonicity). The blue curve at the top is the frequency response projected from Tillman's formulas.

[antigua]

I have a hard time following this, but I want to mention anyway that when it comes to higher harmonics and string rigidity, the question isn't just whether the string can support the tiny harmonic movement, but for how long. When the string is first plucked, there's a burst of higher harmonics, but many of them are gone before the first full cycle even completes. If harmonics are only heard in that transient, then it's perceived as being a percussive attack sound rather than being a tonal quality. That's the underlying reason why guitar picks sound different than fingertips.