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Relative fret spacing all same no matter scale?

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Postby corbind » August 6th, 2011, 10:26 am

Let me see if I can ask this properly....

Standard guitars come in 25/5" scale (Strats) and 24.75" (Les Paul). Other guitars can be 24" like a 24" Jag-Stang but much less common. So you have a 1.5" difference in total scale length but tuned to the same pitch. I could not grasp how that is even possible. Seems the 24" scale would make strings really floppy and loose. Or to combat that you'd have to use thicker gauge strings.

Now say we go down to a barritone which is tuned lower to accomodate a similar "feel" or tension on the string since it's a longer guitar.

This made me start to think that the RELATIVE position of the frets on guitars and basses must be the same. And on banjos and mandolins? Just like

2 x 2 = 4...
2 x 2 x 2 = 8,
2 x 2 x 2 x 2 = 16 ....

Does anyone know if the RELATIVE position of frets is fixed by a formula of sorts? :?:
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Postby NoteBoat » August 6th, 2011, 1:33 pm

Yes, it is.

Doubling the frequency raises the pitch by one octave, and requires cutting the string length in half. So the 12th fret on any fretted instrument* will be 1/2 way between the nut and the saddle.

Now the real trick is finding out where to put the other frets. There are 12 slices in an octave, and each one has to be spaced a little closer than the last. If you have 24 frets, that will have to be 3/4 of the way from the nut to the bridge (because it'll have to be exactly halfway between the 12th fret and the bridge.

The mathematical formula for fret spacing is this: d = S - (s / ( 2 ^ (n/12)))
where d = the distance from the nut, s = the scale length, and n = the fret number

that 12th root of 2 is the real trick here - every 12th fret you'll get the string length / 2

*this applies only to instruments intonated to 12TET - 12 tone equal temperament. Other temperaments are possible, but rare, and they'll have a different formula for EACH of the frets.
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Postby Blue Jay » August 6th, 2011, 4:20 pm

Good question corbind, and awesome scientific/technical answer NoteBoat! :D
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Postby Nuno » August 7th, 2011, 6:35 am

Sorry, and "S"?
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Postby NoteBoat » August 7th, 2011, 8:48 am

It's in there - s is the scale length.

For example, let's say you've got a 20" scale length on an instrument, and you want to know where to put the 7th fret.

d = 20 - (20 / (2^(7/12)))

7/12 is .583 (this is the ratio of the desired fret to the octave)
2^.583 = 1.497 (this is the ratio of the desired fret to the midpoint of the string, measured from the bridge)
20 / 1.497 = 13.36 (this is the distance of the fret from the bridge)
20 - 13.36 = 6.64 (this is the fret location in inches from the nut)

We have to first find the distance from the saddle, because that's the "speaking length" of the ideal string - a small adjustment may be needed for intonation. But then we've got to subtract that from the total scale length, because the fretboard doesn't go all the way to the bridge.... to cut the slot in the right place, it's a lot easier to measure from the nut end.

You can double check the spacing at certain points (the places where you'll get harmonics). The 12th fret should end up being 1/2 the scale length; the 5th fret should be 1/4 the scale length, and the 7th fret should be 1/3 the scale length - which is why I chose it for this example. 6.64 x 3 = 19.92. The 4/10ths of 1% error is due to rounding, as I didn't want to go too far out here and make your eyes glaze over. In practice I'd carry the calculations to six or seven places, and the actual measurement for the fret slot should be as precise as your measuring tools will allow. You don't want to just use a tape measure... precision rulers for odd scale length luthier work will be very expensive. But if you stick to a common scale length, you should be able to get a precision machined fret spacing measure for between about $10-50 (depending on how common the length is).
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Postby corbind » August 7th, 2011, 9:40 am

This goes to show you are not one-dimensional and have a good grasp of so many things in the music world. As a math man I'm so happy to see that beautiful formula!

Last night I was thinking about your response and I pictured a piano. How it goes from very low notes (far left) to very high (far right). Piano can carry the bass notes, all guitar notes, and even higher. It covers it all.

If you were to lay a bass guitar, 6-string guitar, banjo, and a mandolin next to one another then line them up to where the fret spacing matches the other, It would be somewhat similar to a piano covering many frequencies. Based on what you said, the fret spacing is dictated by a formula so, setting all those instruments next to one another, at some point parts of each instrument would match the spacing of another.
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Postby NoteBoat » August 7th, 2011, 11:21 am

Not quite, Dennis. The frequency a string produces depends on the length, tension, and mass of the string. Since bass strings are heavier - and mandolin strings lighter - if the frets all lined up and the tension was the same, they'd each produce different pitches.

Pianos are kind of interesting in design. As the pitch goes up, they control the frequency in two ways: the strings get thinner, and they also get shorter. That helps keep the stresses on the harp (the brass support for the strings) within design tolerances. But as strings get thinner, they also lose dynamic range - they can't play as loud. So a bass key on the piano makes the hammer strike just one string, but as you go up in range the hammers will strike two strings (tuned identically), and finally three strings. That way you can play a nice forte across the spectrum.
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Postby Nuno » August 8th, 2011, 2:10 am

NoteBoat wrote:It's in there - s is the scale length.

You used both "S" and "s" in your original formula. They are different to me. Usually, when the same uppercase and lowercase letter are used in an expression they represent a kind of related term or a maximum or greater value. That confused to me. Thanks!
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Postby SeamanTickles » November 13th, 2012, 8:28 am

So... What you're saying is that a bolt on neck from a 24.75 fret guitar will have a different fret spacing than a 25.5? And what about this: if you have a 21 fret 25.5 scale guitar, wouldn't putting a 24 fret neck, even of a 24.75 scale increase the scale beyond 25.5 overall, making it impossible to intonate, without completely moving the bridge to compensate for the added tension? This is assuming it bolts on at the 21st fret for one neck, and the 24th for the other, rather than having the final three frets overlap the body, where the original neck did not. In essence, my question is when ordering a new neck, do you have to get it to a certain scale, to avoid off putting the formula, and ruining your intonation?
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Postby NoteBoat » December 27th, 2012, 8:00 am

For some reason I didn't see this until now...

Yes, a 24.75" scale length guitar will have a different spacing than a 25.5" scale length.

For the rest, I sense some confusion...

If you have a 21 fret 25.5" scale length guitar, you can put a 24 fret neck on it IF the 24th fret lands 19.125 inches from the nut, and 6.375" from the saddle. If it's anyplace else, your frets will be in the wrong place.

The length of the neck doesn't determine the scale length - the distance from the nut to the saddle does. So if you take a 24.75" scale length neck, and you put it on a guitar in such a way that the nut is anyplace other than 24.75" from the saddle, your frets will be in the wrong place.

You don't move the bridge to compensate for tension - you move it to get the scale length to match the fret positions. If you have too much tension on a guitar, your scale length is fixed - so the only way to reduce the tension is to get lighter strings.

If you decide to swap one neck for another with a different number of frets, you open up a whole can of worms... if the scale length doesn't match fret position, you will never be in tune (and this is different from intonation - no saddle piece adjustments will help). If the scale length matches fret position, you need to consider the placement of the pickups. For example, on a Strat (which has 21-22 frets depending on origin) the neck pickup is where the 24th fret would be - it was placed there for a reason. If you replace the neck with something custom engineered so the 24th fret falls right where the 21st would have been - giving you a 26.324" scale length if my math is right - the pickups will now be located closer to the bridge than they were. This will make your tone "twangier".
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