Question for mathematical/analytical minds

Mike Neer · From: NJ

I would like to be able to calculate, for each different scale length, the increasing angles of the bar slants as one progresses up the neck. Obviously, there would be many variables, such as string spacing, the type of slant, etc., but someone must have a clear idea on how to get some sort of formula to figure this out.

I tried to use a protractor to measure the angles, but that didn't work too well.

I know there are a lot of engineering types here, so I, an engineering school drop-out, turn to you. Laughing

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Dave Grafe · From: Hudson River Valley NY

If you are speaking to the logarithmic change in the distance between the frets, it's all based upon the square root of 2, if that helps any...

Ulrich Sinn · From: California, USA

http://www.hawaiiansteel.com/instruments/slant_angles.php
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Dylan Keating · From: Montreal QC

It seems like it's just a trigonometry problem. If you know the string spacing between the two strings you're slanting the bar between, and the distance between the "pivot" fret (the one closest to you) and the slanted fret, divide the string spacing by the fret distance and take the inverse tan of that ratio to get the angle between bar and pivot string.

Or:
angle = arctan( string spacing / fret distance)

You can use the scale length to compute the distance between frets (I think?) so you can combine the two formulas to find the angles of slants up and down the neck, and between strings.
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Dave Mudgett · From: Central Pennsylvania and Gallatin, Tennessee

Just a quick calculation with string spacing the same at nut and bridge - I also haven't carefully checked the algebra. If someone sees an error, please chime in.

The distance of the nth fret, d(n), from the nut is given by the classic fret formula:

d(n)/L = 1 - (1/(2^(n/12)))

where L is the scale length. So the distance between the nth fret and the (n+1)th fret is

Î”d(n)/L = (d(n+1)-d(n))/s = (1 - (1/(2^((n+1)/12))) - (1 - (1/(2^(n/12)))

= (1/(2^(n/12)) - (1/(2^((n+1)/12))

= (2^((n+1)/12)-2^(n/12))/(2^(n/12)*2^((n+1)/12))

= (2^((n+1)/12)-2^(n/12))/(2^(2n+1)/12)

= (2^((n+1)/12))*(1-2^((n-(n+1))/12))/(2^(2n+1)/12)

= (2^((n+1)/12))*(1-2^(-1/12))/(2^(2n+1)/12)

= (1-(1/(2^(1/12))))/(2^(n/12))

Since (1-(1/(2^(1/12)))) â‰ˆ 0.05612568732, then

Î”d(n) â‰ˆ 0.05612568732*L/(2^(n/12))

Now consider the string spacing between adjacent strings, s. For, let's say, a forward slant (the bar makes a positive angle, Î¸, between 0 and 90 degrees with the strings, or a 90deg - Î¸ with the straight bar angle), then tan(Î¸) is the ratio of the string spacing (the vertical distance of the slant) divided by the fret distance (the horizontal distance of the slant (tan(Î¸) = opposite/adjacent), so (recall, arctan is the inverse tangent function):

Î¸ â‰ˆ arctan(s*(2^(n/12)/0.05612568732*L)

â‰ˆ arctan(17.81715374*(2^(n/12)*(s/L))

This generally makes sense - the bar angle from the strings (horizontal) increases as you move up the neck {note that 2^(n/12) increases as you move up the neck}, and also increases as the string-spacing/scale-length ratio increases.

But we normally think about the slant angle as the angle away from the straight bar position, so the angle of interest is 90 degrees (Ï€/2 radians) - Î¸, which is the same as the inverse cotangent (arccot), i.e.,

Ï† = Ï€/2 - Î¸ â‰ˆ Ï€/2 - arctan(17.81715374*(2^(n/12)*(s/L))

= arccot(17.81715374*(2^(n/12)*(s/L))

One could quickly plot these relationships in, let's say, MATLAB, Mathematica, or Mupad for various scale lengths and string spacings.

Erv Niehaus · From: Litchfield, MN, USA

Just use your ears. Why does everything have to be SO complicated!!! Whoa!

Mike Neer · From: NJ

Erv, let boys be boys. Besides, it's math that makes it all possible, from Pythagoras to Edison. Laughing

Professor Mudgett, that is incredible. Thank you for sharing it. I wish I'd paid a little more attention in calc.

Ulrich, that is a cool tool on John's site, never saw it before. Thanks.

Dylan, respect. You casually ripped that out. I like it.
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Dave Mudgett · From: Central Pennsylvania and Gallatin, Tennessee

Of course, we use our ears to play slants. But sometimes, I find it useful to also think about and try to understand the physical relationships that I'm dealing with as I play. Sometimes a strictly qualitative understanding is fine, sometimes a quantitative understanding gives additional insight.

I made it complex by starting with the fretting formula, which measures the fret position from the nut. For actual fretting, this is better than dealing with distances between frets because errors can accumulate between frets. But I guess I could have started with the distance-between-frets formula.

But the final formula is simple:

angle-from-straight-bar = Ï€/2 - arctan(17.81715374*(2^(n/12)*(s/L))

For a given guitar, s/L is a constant string-spacing to scale-length ratio, so you just plug in n for all the frets from 1 to however high you want to go, and just calculate and plot (I'd convert from radians to degrees, personally).

If you want to take into account the full complexity, consider dissimilar nut and bridge spacings. But I think this is good enough to get a basic quantitative and qualitative feel for the relationship as one moves up the neck.

Erv Niehaus · From: Litchfield, MN, USA

The little boy came home from school and his dad asked him what he was studying in math.
The boy said that they were studying pi r square. His dad said: "Son, you've got that all wrong, pie are round. Laughing

Doug Beaumier · From: Northampton, MA

So how does one apply these calculations when actually playing slants on the steel guitar, short of using a ruler and memorizing all the numbers for slants on all the frets? In the end, the player has to execute the slants by ear on the fly.

edit: I just noticed Dave's reply about playing slants primarily by ear.
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Erv Niehaus · From: Litchfield, MN, USA

Doug,
Yes, I agree.
Just think, they could be spending all this time practicing! Very Happy

Dave Mudgett · From: Central Pennsylvania and Gallatin, Tennessee

Doug - sure, in the end, one needs to get notes on a steel, however one is playing them, in-tune by ear. But to me, having some basic understanding, qualitatively and quantitatively, how much to angle the bar as one moves up the neck is sort of analogous to having fret markers to guide nominal placement of the bar.

Of course, one can, in principle, play steel guitars without any fret markers, as do violin, viola, cello, and bass players. And I'd guess that as players get more and more years of playing, they need the fret markers less and less. But perhaps a chart of typical bar slant angles as one moves up the neck might help some players as they push into this territory. I guess that's why this seemed interesting enough to respond.

Mike Neer · From: NJ

Dylan Keating · From: Montreal QC

I've plotted Dave's (much better than mine) formula for the slant angle from straight bar position (for arbitrary string spacing and scale length):

Which is cool, because it shows what you'd think intuitively- as you go higher up the fretboard, the slant angle gets smaller.
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John Sluszny · From: Brussels, Belgium

Barry Blackwood · Posted 24 Jul 2014 3:12 pm

Mike Neer · From: NJ

Dave Mudgett · From: Central Pennsylvania and Gallatin, Tennessee

Dylan, you had the relationship correct.

For me, the utility of this is to develop some visual images in my head about how much slant angle is required at several important fret positions up the neck (e.g, 5th, 9th, 12th, 15th and interpolate/extrapolate the rest) for several different guitars I use with different scale lengths and string spacings. Playing my 24.25" scale length 10-12-string pedal steels with pretty narrow string spacing is quite a lot different than my 30's 22" (or thereabouts) scale length 6-string Rick B6 and Model 59 with pretty wide string spacing, or my 8-string Fender Dual Pros for that matter. Then there are string-skipping slants, which multiply the string spacing.

There are different ways of getting this kind of insight. The old-school music-teacher way is to just practice and practice for decades until it's second nature by feel. That's cool and I highly advocate the sheddin' time. But if some mathematics and basic graphs can help me or anybody else develop some mental imagery that can speed the process up, then I'm all in favor of that. I didn't start playing steel at birth and I ain't a kid, to say the least. Wink

So I'll use any trick I can to speed up the process. I already am an engineer/scientist/mathematician kinda' guy, so this kind of geometric/mathematical imagery is useful to me. YMMV, no problem.

I'll put some graphs up at some point using specific data from my guitars, but I gotta be somewhere tonight.

Mike Neer · From: NJ

I'm with you, Dave. Science plays an extremely powerful role in making music. If you have any doubt, read George Kochevitsky's book on the Art Of Piano Playing.

Hence my curiosity with all of this. It's been on my mind for a while. I can play music, but I also like to see all of the hidden beauty in it. I take music very seriously, not casually. Let's say it is my religion.
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David Mason · From: Cambridge, MD, USA

did you include the adjustment to the calculations for just intonation yet?

WOO HOO HOO HAR HAR HAR
WOO HOO HOO HAR HAR HAR
WOO HOO HOO HAR HAR HAR

WAAAAAAAAAAAAAAAIT a minute - it's ALL just intonation, right? Confused

Mike Neer · From: NJ

You rascal, you!
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Bob Hoffnar · From: Austin, Tx

I am looking forward to seeing the graph when you get to it Dave. I'm thinking it may reveal some interesting insights.
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Mike Neer · From: NJ

Dave, thanks for updating your post. I went to the library last night and took out some calculus books.

Looking forward to the graphs, if you ever get around to it.

And if anyone with app development skills wants to hook up, I have really good ideas and a strong concept for this. I doubt if it would ever be a moneymaker, but that's not the point.
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Bob Hickish · From: Port Ludlow, Washington, USA, R.I.P.

it all depends on tuning -- straight 440 or JI -- I have always used sight - to estimate bar angle /slant

basilh · From: United Kingdom

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