So.. where's my math error?

[Mike Kolb]

I expect to catch heat for being a mathematical idiot, but here goes:  

Refretted a customer's electric with s/s frets. Ordered the frets (4ft) and it all worked out fine. The end.

Before doing the work, and while standing-around waiting for something to dry on another job, my mind wandered to the frets. The guitar (an Ibanez RG) has 24 frets... so I measured the 1st fret (1.730") and then the 24th fret (2.270").  

My thinking was: by adding the 2 lengths together and dividing by 2 I'd get the average fret length ... which I did and it was exactly 2-inches. 

Then panic set-in. "If the average length is 2" and there's 24 frets, that's 48" and that's the length of the fretwire I ordered, with no room for error whatsoever".

As it turned-out, there was plenty of fretwire left-over... maybe 10" or so. The job's done, the customer's happy.... but where'd I go wrong here?  My better half thinks the supplier may have been generous, but I never measured the actual wire when it was received.  

There's bound to be someone here who'll get a chuckle before setting me straight?

[Mike Kolb]

and yet, Murray, when all was said & done, my original simple formula (average of the shortest + longest and multiplied by 24) gave an initial prediction of 48".  

In fact, the end result of measuring & adding all of the finished frets together yielded 48.740".  So how did that happen?  

Yes, it was a bit longer than anticipated... about 3/4 of an inch...but certainly not "significantly". 

[eliya]

It happened because it's a fluke. I can arrange the frets on the frets on the board in infinitely many ways that would yield a fret average of 2", or in other words, it'll add up to 48".

This is sort of a min/max problem. The shortest fret length average is 1.730", meaning all the frets are the length of that fret. The longest is 2.270", meaning all the frets are the length of the last one. Now what's the overall length in the first case? 41.52", and in the second case? 54.48". What's the average of these two lengths? You guessed it - exactly 48". So with this fretboard, we're always dancing around 48" total fret length, which is why when you actually went out and measured the frets, you got 48.740".

I wouldn't use the system of adding the first and last fret and dividing by two. Or at least add a couple feet to it, just in case.

[Andrew]

I have to retract my previous statements after measuring and adding a few times. I came to several different totals after repeating the same steps of measurement on the same guitar. Apparently I dont know what Im talking about :D.

Ill continue to just order eight feet of wire.

[Mike Kolb]

Yeah, the sum of this has taught me....

(a). I'm a mathematical idiot.... aways have been, always will be.

(b). order way more fretwire than I *think* is needed. (see above).

(c). there's a whole lot of nice & interesting folks here and it's always my pleasure to interact with them :)

So thanks for the input, one & all.

[Murray MacLeod]

Mike, let's try viusualising this geometrically instead of arithmetically or algebraically.

The fretboard is a symmetrical trapezium (or trapezoid, depending on which side of the pond you are on) bounded at either end by the first and last (24th) frets, (forget about the nut and the bit beyond the last fret).

Now, you have already measured the sum of the lengths of the 1st and the 24th fret to be exactly 4", and obviously the average of these two is 2". It also follows inexorably that 2" is the exact width of the fretboard at the midpoint between the first and last frets.

Now, if the average length of all the frets was in fact to be 2", that would then mean that there would be 12 matched pairs, each with a total length of 4". In other words, the second and the 23rd fret would total 4", the third and 22nd fret would total 4", the fourth and 21st frets would total 4" ... etc ...etc ...etc.

But, if you look at the fretboard, you will see immediately that that is impossible, because, for the second and the 23rd fret combined to total 4" , the second fret would have to be the same distance from the first fret as the 23rd fret is from the 24th. It is immediately apparent that if that were the case, then the second fret would be shorter than it actually is, therefore the sum of the lengths of the actual second fret and the 23rd fret is greater than 4". This is so for all the other pairs, and the amount in excess of 4" increases fractionally for each pair.

By the time you reach the 9th fret (which is paired with the 16th fret) you are already past the 2" wide midpoint, so even the shorter of the pair is wider than 2". Same goes for 10-15, 11-14, and 12-13.

As you already found out, the difference is not huge, but it is still significant, not in any practical way, since you only need another inch or so of fretwire, but mathematically ...yes, I would say it is significant.

[Mike Kolb]

Thanks, Murray, your explanation will surely make sense to somebody more adept at understanding it.... so, if it's all the same, let's not try visualizing anything :)

[Andrew]

I understand it now. Well enough that it doesnt bother me any more. Great explanation Murray. Very clear and succinct.

[Ned Knepp]

Eveready called...we're now officially facing a shortage of calculator battery.