Intonation: How Much Change in Scale Length Per Cent of Pitch Change at 12th?

[Todd Stock]

Got a note from a semi-local novice repair person/builder that is trying to understand a discussion on intonation on another site. Her question to me was – reference the assertion of the old gray/balding head of the repair people that posted responses - “Does the suggested 0.007" per cent of change in pitch account for the change in scale length due to the correction?"

My response was as follows:

I use .014” adjustment at the saddle per cent of pitch change at the 12th, based on the notion that – while physically moving the 12th fret by 0.007 while maintaining a fixed scale length (i.e., no saddle adjustment) will indeed change the fretted pitch by one cent, the adjustment at the saddle to change scale length must be doubled, or ~0.014” scale length change at saddle to see one cent change in pitch at the 12th when fretted.

Confession time: years ago, I would have insisted that 0.007” per cent change was correct, but Jeff Highland pointed out that I was not accounting for scale length change at the saddle with that value. I hemmed and hawed and finally pulled out a Tele to verify his assertion with actual measurements. I adjusted E4 at the saddle to 20 cents flat at the 12th, measured the distance from saddle contact point to fixed location (OK - actually front of barrel and rear of bridge, then zeroed micrometer). I then adjusted the saddle to 10 cents flat and took another measurement at the 12th, then adjusted to 0 cents flat or sharp, and make a final measurement at 12th. I got ~ 0.140” for a 10 cent change each time, and have seen the same physical situation when I’ve bothered to measure.

So… please… feel free to tell me I am wrong and have steered this novice in the wrong direction, but both calculation and experimentation have led me to the conclusion that the common wisdom - 0.007” per cent change in pitch at 12th – only works when the scale is fixed length and the fret is moved, and that 0.014” is the correct value for saddle adjustment per cent of pitch change at the 12th where the scale length is varying.

[Michael Imbler]

Frank Ford has a very easy to understand explanation that agrees with your numbers:
(I edited this to remove the explanation, because it is copyrighted, and I don't know if this forum would exceed fair use requirements. It is easy to find on Frank's site, though for anyone interested.

A simple formula for any instrument
Calculating Intonation Correction
© Frank Ford, 2/28/96

[Todd Stock]

Another good quick approximation from Frank Ford! Good call. He is using average distance for a cent between nut and 1st fret, which is a good approximation for change at the saddle, so for my example, it works out about the same at the saddle:

25.0" scale distance to first fret: 1.403"
Error in Cents: 20 (this is a 20% or 0.2 error for Frank's formula)

So: Distance to First Fret X percent error in semitones = 1.403" * 20 cents/100 = 1.403" * 0.2 = .280"

Exactly twice the correction that the .007"/cent would suggest, or .014"/cent error at 12th.

[Trevor Gore]

Frank's method is correct (of course!), as is Jeff's explanation as to why.

[Todd Stock]

Jeff Highland and I had this discussion a few years back, and where my confusion stemmed from was remembering that the correction affects the scale length - my arithmetic was good, but my physical model was not.

Trevor - I am guessing you must have posted something on the subject that clarified things for my friend, as I just got a note from her to the effect of "...never mind - Trevor Gore posted something - got it now'. Funny how often that happens, eh?

[Trevor Gore]

Trevor - I am guessing you must have posted something on the subject that clarified things for my friend, as I just got a note from her to the effect of "...never mind - Trevor Gore posted something - got it now'.

Yep. The argument goes something like this:

Tune the string
Measure the cents error at fret 12. Assume for the moment it is sharp.
Then, if you just compensated for that error by adding extra length to correct the fretted note, the open string would now be flat because of the extra length. So the string is re-tuned, which leaves the fretted note sharp again. To get it right, both the open string and the fretted note need to be in tune which means adding twice the compensation that was first thought of.

Didn't even have to use any equations!

[David Collins]

Thanks Todd. If you re-read, you'll find that I thanked the person there who caught my mistake - guess I've been relying too much on using spreadsheets on my bench computer, and goofed on recalling the formulas I wrote in to it many years ago. It happens, and although I thank you for the nod toward my maturity, I'm not really graying yet beyond a few speckles in the beard.

[Todd Stock]

Generic term - I'd say graybeards, but that's almost everyone these days much over 40.

Jeff Highland straightened me out on a bad model a few years back... I think my apology was in the form of 'Crap - you're right!' I'm sure yours was a lot more gracious.

[David Collins]

No, no offense at all. When I was looking through the spreadsheet at the formulas I used, it was hard to decipher because I had to work in all sorts of corrections for the very mistake I'm having to back away with once again. My original versions did indeed prescribe the incorrect .007" correction!

Of course I'm terrible with arranging spreadsheets efficiently, so these corrections end up constituting a ridiculous number of lines for every time they are used. One of these days I'm going to have to learn how to use Excel the right way...

[David Collins]

I also have to admit that I had go back and write the formulas out longhand to be sure it was right. So many different embodiments and effects with nut compensation vs fret location vs saddle compensation, as much as it makes sense in the end, sometimes there are just too many numbers bouncing around in my head to keep track of.

[Todd Stock]

Numbers are the easier part of it for me - it's looking at all the moving pieces and seeing the physical model of the problem with correct relationships that is more difficult. Both craft and engineering are exercises in heuristics, but getting to those simple, usually usable rules involves either understanding or brute force iteration...and I hate iteration.

Charles (TGW) calls or sends me the occasional FB note on the stuff he thinks I'll get a chuckle out of (mostly entertaining melt-downs or 'Where's Waldo?" posts), but I got the snipped version of this thread with a bunch of questions from She Who (Mostly) Lurks. Shooting a 3 gun this weekend, so will try to find time to read the whole thing later. Re: Mike's post above and a snip I got of someone's (maybe his?) post, it does illustrate that Frank Ford has forgotten more about this stuff than most of us will ever learn.

Gotta jet...my turn to drive.