Behind The Bridge Sympathetic Resonance Calculator Thing

[rdnie]

Hi all,
(long time lurker, rare poster here)

I’ve been thinking about the JM tremolo/vibrato placement for a project guitar and would like to share something I came up with (in hopes that it could be useful or at least entertaining to the community).

TL;DR:
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I built a Google Sheet to try to ‘predict’ and highlight which harmonics are going to sympathetically resonate between the ‘regular’ and the behind-the-bridge sections of a string (both open and fretted notes) based on the length of string in front of and behind the bridge.

Here’s the spreadsheet as a web page (read-only):
https://docs.google.com/spreadsheets/d/ ... jL/pubhtml#

And here’s the link to the Google Sheet itself, which you can copy (File -> Make a copy) and try your own values:
https://docs.google.com/spreadsheets/d/ ... sp=sharing


———

So I had this idea of building a more compact JM vibrato-equipped guitar and it got me thinking into what happens if we move the vibrato closer to the bridge. Now there’s the tried and true ‘vintage’ spacing and the ‘modern’ spacing (e.g. Squier JMJM, Fender CP). The latter gets mixed opinions - one of the cons often mentioned is having less BTB resonance and feeling less ‘alive’. (Perhaps there’s also a bunch of other variations in other builders’ guitars, but it’s hard for me to quantify other than from pictures online, which isn’t accurate enough.)

Thinking that maybe it’s not just the shorter string length and the resulting higher BTB pitch, but how the BTB pitch relates to the notes on the rest of the string, I built a little Google Sheet that:
- Maps out the harmonics of open and fretted notes ‘in front’ of the bridge as well as the string section behind the bridge.
- Highlights the harmonics that are close to each other using 5 gradations - the closer the frequencies, the darker the cell color.
- Calculates the musical note and variance (in cents) for the BTB section of the string.

The idea is that darker color should indicate which fretted notes would be more responsive to picking behind the bridge, and vice versa, which fretted notes would induce more resonance behind the bridge.
Now I only have a JMJM that I can measure, so the ‘vintage’ JM and Jaguar figures are only guesstimates at this point.

Some caveats:
- Measuring the ’regular’ and BTB strings sections may not be the best way to calculate the ‘BTB ratio’. If it’s a real physical guitar, measuring the BTB pitch (e.g. using a spectral analyzer plugin) is going to be more accurate. I guess this is because the string section behind the bridge is far from the idealized model in that a) it’s thickness is not negligible compared to its length; and b) it’s thickness is not uniform because of the core wire wrapped around itself at the end.
- For the same reasons, assuming harmonicity for the BTB section of the string is a major simplification, because in reality I think it’s highly inharmonic (i.e. the harmonics are not integer multiples of the fundamental).
- Only first 4 harmonics of the BTB section are considered (displayed as red, orange, green and blue) - though higher order harmonics aren’t going to have as much effect anyway.

Some intermediate conclusions:
- The BTB harmonics on the ‘modern’-spaced JMJM tend to fall somewhere ‘between the notes’, and that seems one of the reasons why not as much resonance is happening compared to ‘vintage’ spacing.
- (Obviously) the whole thing shifts a semitone down on the Jaguar because of the shorter scale length.
- I wish I knew if it was a conscious decision for ‘vintage’ spacing to enhance the resonances, and for ‘modern’ to tone them down a bit.
- I imagine it would make sense to place the vibrato somewhere between the ‘vintage’ and ‘modern’ locations so that the BTB sections is 6 times shorter than the ‘regular’ section (see 'Custom 6:1' tab in the spreadsheet). The BTB section pitch would be 2 octaves + 5th higher above the open string. (To compare, I believe with ‘vintage’ spacing it’s somewhere around 2 octaves + 4th, and with ‘modern’ - somewhere in the minor to major 6th range (+ 2 octaves).

I hope this can be of some interest and I would appreciate any feedback and corrections, or if you'd like to provide measurements for some of your guitars. The JMJM table is consistent with what I'm hearing with my guitar, i.e. it does favor F#, A and C# indeed. Do the tables for the regular JM and Jaguar look plausible?

[timtam]

Nice work ! I am not sure that analysis of the residual strings before now has extended beyond the simple notion that they must resonate at their own fundamental frequency by virtue of their fixed length.

That residual string length is indeed short enough for inharmonic partials to be present and probably significant sonically (extending down into low harmonic numbers), particularly on the heavier strings (unlike on the normal-scale strings of 6-string guitars). So you might need a stiff string model in order to be more accurate than integer multiples of the fundamental in predicting their frequencies (or direct frequency measurement as you suggest).

Re "I wish I knew if it was a conscious decision for ‘vintage’ spacing to enhance the resonances, and for ‘modern’ to tone them down a bit", I think it's unlikely that Leo Fender would have been thinking that first way at all. Unless there was another desirable musical instrument around at that time that had residual strings that were sonically important ... and I don't know what instrument that would have been. But it seems to me most likely that any sonic effects of residual strings were just an unintended side effect of the trem design.

As for contemporary Fender, their motivation on the models with relocated trems was likely simply "let's move the trem closer to the bridge so it doesn't need a buzztop". Suggesting a complex understanding of offsets at Fender nowadays would be an inference without any supporting evidence.

The residual strings can of course be excited to vibrate either by directly strumming them, or (less so) by vibrations from the main strings passing over the bridge to excite their vibrations. The 'downside' of that second mechanism - which happens without the player doing anything - is that energy is lost from the speaking length of the main string. So that may affect the sonic profile of those speaking-length strings too, for example their sustain (which will be reduced). Where you mention residual string resonant frequencies lying between fretted note frequencies rather than nearer to them, that issue would be less apparent (kind of akin to how acoustic guitars can now be engineered to avoid wolf notes by shifting their frequencies). The notion of a guitar seeming more 'alive' when those residual string resonances are more obvious is in that sense maybe misleading - if the residual strings are more alive, then the main strings must be less alive (as long as the laws of physics apply).

[rdnie]

Thanks for your interest! Really appreciate it.

That residual string length is indeed short enough for inharmonic partials to be present and probably significant sonically (extending down into low harmonic numbers), particularly on the heavier strings (unlike on the normal-scale strings of 6-string guitars). So you might need a stiff string model in order to be more accurate than integer multiples of the fundamental in predicting their frequencies (or direct frequency measurement as you suggest).

Yes, I see now that it's quite evident on the frequency analyzer. And one could guess it from the way it sounds (I mean the bell-like quality).
I've recently found some relevant theory in Chapter 1 of the GITEC Book, so maybe I'll add some of that to the calculations. The partials' frequencies would change, however if I got the theory right, the relationships between the partials for the same string would stay the same ("for a given string and the same absolute frequency, the inharmonicity is always of the same strength irrespective of the fretted note"), so maybe there isn't much practical sense to it.

As for contemporary Fender, their motivation on the models with relocated trems was likely simply "let's move the trem closer to the bridge so it doesn't need a buzztop". Suggesting a complex understanding of offsets at Fender nowadays would be an inference without any supporting evidence.

Right, but maybe I'll give them the benefit of the doubt in that they could try several locations and think "yeah, let's call it a day"

The residual strings can of course be excited to vibrate either by directly strumming them, or (less so) by vibrations from the main strings passing over the bridge to excite their vibrations. The 'downside' of that second mechanism - which happens without the player doing anything - is that energy is lost from the speaking length of the main string. So that may affect the sonic profile of those speaking-length strings too, for example their sustain (which will be reduced). Where you mention residual string resonant frequencies lying between fretted note frequencies rather than nearer to them, that issue would be less apparent (kind of akin to how acoustic guitars can now be engineered to avoid wolf notes by shifting their frequencies). The notion of a guitar seeming more 'alive' when those residual string resonances are more obvious is in that sense maybe misleading - if the residual strings are more alive, then the main strings must be less alive (as long as the laws of physics apply).

Yes, absolutely, I should have articulated this better
Maybe I hear some of that effect with the F# on my JMJM - like it looses something a bit faster than neighboring notes, but I'm not sure. I always thought this whole BTB thing should reduce sustain, but now that I look at the tables I'm thinking the residual string should 'steal' energy only from symphathetic partials, whereas sustain would be mostly defined by the first few (or, really, the fundamental). Does this make sense? Is this how it works?

Also, in the (hypothetical) severe case of 17th fret A on a 'proper' JM it theoretically should make the note sound hollow-er (triangle wave-ish), like when picking right in the middle of a string, because the residual string vibrations would 'rob' the note of even order harmonics.

[B.T.]

Interesting topic, thanks for posting