Relationship between tension, gauge, and unit weight ?

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Musiicy
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Joined: 06/07/2016
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Hi there,

I'm looking for a 6-strings set that would allow me to tune my Stratocaster to drop Ab tuning (Ab Eb Ab Db F Bb).
Until now I've been using these strings meant for standard tuning. Gauges are 10-13-17-26-36-46.

I just read about this tension chart.
It left me clueless and dealing with a painful headache.

I'm lost.
What do I have to look for in order to achieve my tuning ?
Gauge ? Tension ? Unit Weight ?
Also, Tension and U.W. seem to be mathematically related, but how does Gauge relate to them ?

Thanks A LOT in advance if anyone helps me with understanding this all.

Isotropy
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Joined: 23/06/2014
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TL;DR:
the good news is, you can keep using those strings (if you ask me) just ditch the 10 and then get a 64 - 70 for the low Ab.

Honestly I think 68 is just right. 64 for more bwow, 70 for more tightness and tuning stability (though tuning should be fine with a 68 too). Also it's totally personal preference.

Here's an awesome tension calculator:

http://www.stringulator.com/

Now for the science.

Tension, gauge, and unit weight - so let's quickly clean up those variables, I'm assuming by unit weight you're referring to just what unit of measurement is being used (typically lbs for American string manufacturers)

The formula is actually really simple - bear in mind I'm leaving out the derivation of this formula from my post (derived here):

T=4mLf^2
Where:
T = tension in Newtons (kg m/s^2)
m = mass of the string (in kg)
L = the length of the string (in meters)
f = the frequency (in HZ or s^2)

That formula there is the relationship between length, tension, frequency (note), and gauge.

The higher the gauge, the higher the tension at the same note. The shorter the scale length of your guitar, the lower the tension of the same note. The higher you tune the note, the higher the tension, etc.

So let's do cool maths now for your guitar. You're using a 25.5" scale length guitar tuned to E standard (I'm assuming, since you didn't say that specifically). You've got a .046 on the low E.

The mass per unit length of D'addario .046 = ~0.0068 kg/m and your length is 25.5" which is 0.6477m making the mass of string at that length = .00442kg. and E2 vibrates at 82.41Hz.

This gives us: T = 4(.00442)(0.6477)(82.41^2) = 77.75 Newtons which, when converted to lbs, gives us 17.5lbs

I apologize because I did some of the unit conversions outside of this post. If you look at D'addario's tension shart you'll see 17.5 lbs is exactly what's listed for .046 gauge string tuned to E2 on a 25.5" scale guitar.

Now that we've done that, let's see which string gauge would equal that tension for Ab1.

This is a little trickier because we know what tension we want but we need to find gauge which relates to mass.

T = 77.75N (important this is in Newtons)
L = 25.5 in = .6477m (scale length)
f = 51.91Hz (A flat octave 1)

T = 4mLf^2 solve for m ==> T/(4Lf^2) = m

77.75/(4 * .6477m * 51.91Hz^2) = .01114kg

Now bear in mind, that is the mass of a 25.5" (.6477m) length of string, not the mass/unit length. To determine that we have a little bit of work left, since we want our answer in lbs/inch. So convert kg to lbs .01114kg = .02456lbs

Now divide that by 25.5in = .02456/25.5 = .00096314 lbs/in. Which isn't very precise because I rounded most of these and we're talking about really small numbers so my answer is only somewhat accurate but it proves the point.

If you look for ~.0009 under the unit weight column on D'addario's string tension chart you'll see you get that between a .070 and a .072, so that's how you know right around those gauges will feel the same as your .046 tuned to E.

In practice, you may not actually like that feel on such a heavy gauge. It is something I recommend experimenting with. I also can say from personal experience I have not had favorable results trying to tune a 6 string that low, especially given the 25.5" length. I have a 7 string that I tune to drop Ab and it's a 26.5" and that is perfect if you ask me.

References:

D'addario String Tension Chart