Scale length vs tension and harmonics

Gear
#1

I like my 1960s 22.5" “student scale” guitars for their smaller overall size. It’s not a fretboard reach issue— it’s just the overall size of the instrument in every dimension. A standard-sized Strat feels like a boat on me. To keep the tension in a playable range, I run 11s on these guitars. But it’s not a perfect solution.

The low strings are still a little floppy, and increasing the gauge doesn’t quite solve the problem. I used to run 12s, but the string thickness was so fat on the wound strings at that point that you’d get buzzing when fretting notes even with good action and good technique.

There simply comes a point where the scale length is too short for the pitch you’re trying to hit, and you can’t fix it by increasing the gauge. For example we have a five-string electric mandolin with a low C string that is essentially unplayable — it’s too floppy, barely holds a consistent pitch when fretted, and goes out of tune at the slightest provocation. If you really need such a wide spread of frequencies on a single instrument, multi-scale is clearly the way to go.

Second issue: There is a sonic change when you have a mismatch between scale length, string gauge, and tuning. A Fender Strat tuned to E at 25.5" is going to have that chimey sound that you simply can’t get on a Mustang tuned to E, even with beefier strings. Some of this is a matter of personal preference. I don’t mind the darker sound of shorter-scale guitars tuned to E. I’m just mentioning this to acknowledge that there are changes due to physics that you can’t dial out by string choice alone.

So… here’s my question for the more mathematically inclined:

With a 22.5" scale length, what string gauge would I need to run, and to what pitch would I need to tune, to get Strat-like string tension, playability, and chime?

That’s an experiment I’d like to try, just to see what it’s like.

#2

I’d try the same string gauges as the Strat, tuned a whole step higher. 2nd fret capo on 25.5" Strat scale = about 22.5")

#3

Interesting. So it’s that simple — find the spot on a Strat where the capo matches the scale length of the shorter guitar, then tune to that note? Does the fact that the Strat has extra string mass that extends past the capo play any kind of role?

For example, there are archtop guitars where the strings go way past the bridge and terminate in a tailpiece at the end of the body. In theory you’ve potentially got a lot more string mass there, albeit mass that is not technically vibrating to produce a pitch, since the extra mass is beyond the bridge and nut.

How does this extra mass affect the resulting tension and harmonics? If you had, let’s say, an extreme case with a foot of extra string past the bridge, what would the difference in sound and playability be compared to a guitar of the same exact scale length running a hardtail bridge where the strings terminate at the bridge itself?

#4

Formulae for playing forces required to fret and bend strings are given in this document.

poteg-7-4-1-playing-forces.pdf (154.2 KB)

I had made an Excel file calculating fretting forces for all standard acoustic and electric D’Addario strings based on their string tensions, with a range of actions and scale lengths. I can’t find that file right now, but I wouldn’t have deleted it. I’ll try to find it again tomorrow, it’s getting a little late here.

#5

I would consider each string individually and decide how much tension you want there. Most tunings have very different tension per string.

This calculator might work, but there are others:

https://tension.stringjoy.com/

1 Like
#6

Ha, that’s awesome. Someone got published for this.

Interesting, this also addresses my follow-up question which is the extra string beyond the nut and bridge. More bending force is required when there’s more “residual” string. So does the whole string have higher tension then for the same pitch? Or is it just the bending force that’s higher?

1 Like
#7

Tension: not at all, but more length of string that can actually stretch contributes to the elasticity and might feel like less tension. 22.5" to 22.5" is about the same change as the 4th string with a locking vs. non-locking nut (and string able to move freely) with a standard 6 inline headstock.

Harmonics: Too complex to give a specific answer, I think. Extra sections of string could ring sympathetically or not-quite-sympathetically enough and enhance/diminish different harmonics in different ways.

#8

Just bending forces are increased, the tension of an un-bent strings are exactly the same whatever the residuals.

EDIT: Found the excel file, but I can’t upload it to the forum. I’ve sent it via email. It will give you an idea of the mean playing fretting forces you’re comfortable with.

EDIT 2: 22.5" scale has (22.5/25.5) = 0.8823 times the fretting force of 25.5". Tuning up a half step brings the playing force to 0.8823*(2^(1/6)) = 0.99 the fretting force of 25.5" scale.

So, choose your string gauge based on the 25.5" tension from the chart and tune to F standard. It’s going to be significantly easier to bend strings, so you might want to go a gauge or half gauge heavier based on that.

#9

Ignoring bending, pick any three and physics gives you the fourth:

  • length between bridge and nut
  • weight of string per length
  • tension in the string
  • tuning of open string

The equation is here in very convenient units,

1 Like
#10

I once setup a guitar with nearly equal tension on each string, based on data from a string tension calculator. The result was not pleasing; chords sounded all out of whack due to higher strings over powering lower strings. I don’t assume that will be the case for every scale length/tuning/guitar but that was my experience.

#11

I tuned one of my guitars to baritone. It has a 25.5" scale length, and I used the stringjoy calculator that @kgk posted to choose a string set that resulted in the same tension on the strings as having the guitar tuned to E. It actually sounds pretty good, but definitely feels somehow different than playing an actual baritone with a longer scale, or the low strings of a seven string. Fretting notes feels the same as when it had higher notes, even with the heavier strings. I think the big difference is in the string vibration. I don’t know how to explain it, but for example the low B just feels like it moves a lot more than the low E. Maybe this is another use case scenario for the Magnet, film the strings on a guitar tuned to E and compare it the baritone tuned one!

1 Like
#12

Source for the physics document supplied above:

https://gitec-forum-eng.de/the-book/

I’ve also uploaded the excel worksheet for fretting forces to my google drive

1 Like
#13

That works good it’s what I use when trying to maintain a similar tension on different guitars with different scale lengths in various tunings.

1 Like