Question for mathematical/analytical minds

[Dave Mudgett]

OK, here are some basic, non-prettied up graphs of slant angle in degrees vs. fret number for my 30's Ric 6-strings (one is a ca. 1939-40 Model 59, the other is a ca. 1935 B6), and my 2001 Zum 12-string universal.

The relevant parameters are as follows (the Rics were close enough that I amalgamated/averaged them into one) - first three numbers are full string spacing at indicated positions, 4th is average single-string spacing s, 5th is scale length L, 6th is s/L ratio - all measurements in inches, measured to about 1/32" and rounded to two decimal places):

Code: Select all

Guitars   Nut  Bridge  12th Fret   s       L     s/L
Rics     1.81   2.20    1.95      0.38   22.50  0.017
Zum      3.09   3.78    3.44      0.29   24.25  0.012

And here are the plots - sorry, the graphs run off the page, but they look terrible at lower resolution. If you look in this folder, you can find pdfs that view better - http://acs.ist.psu.edu/mudgett/files/slant-graphs/pdf/

Plot 1: Ric adjacent-string slant-angle vs fret #



Plot 2: Zum adjacent-string slant-angle vs fret #


Plot 3: Ric skipped-string slant-angle vs. fret #


Plot 4: Zum skipped-string slant-angle vs. fret #

[basilh]

What about "Split Slants?"
What's the optimum bar head diameter to maintain pitch on adjacent strings whilst still being able to slant for other strings (be they 1, 2 or whatever)
And reverse slants on non adjacent strings ?
I need to know..

[Doug Beaumier]

I need to know..

No, you don't Basil. You already know!

[Dave Mudgett]

All that other stuff - split-slants and other 2nd-order corrections for different nut/bridge string spacing, 3rd-order corrections for JI vs ET - fuggedaboudit. Seriously, that's not the point here. My point is to have a tool for mental imagery of what the slant ought to look like, geometrically, at various points along the fretboard for different steels, such as a fairly wide-spaced, short-scale lap steel and a tightly-spaced, longer-scale pedal steel.

One last graph - all of those earlier plots on one graph (image below, there's also a pdf in the same folder given earlier - http://acs.ist.psu.edu/mudgett/files/slant-graphs/pdf/) Red and Yellow are adjacent-string slants for Zum and Ric, respectively. Blue and Green are skipped-string slants for Zum and Ric, respectively. As you can see, the differences are significant, as one would expect.

[Mike Neer]

Dave, that is just fantastic. Thanks so much. This will be very helpful in a number of ways.

[Dave Mudgett]

Curmudgeonly? Cynical? On the Steel Guitar Forum? Surely you're joking (Mr. Feynmann!*)



*btw - if you've never read this book by Richard Feynmann, I highly recommend it - http://www.chem.fsu.edu/chemlab/isc3523 ... surely.pdf

[Mike Neer]

Edited: Having a bad day.

[Bob Hickish]

Curmudgeonly? Cynical? On the Steel Guitar Forum? Surely you're joking (Mr. Feynmann!*)



*btw - if you've never read this book by Richard Feynmann, I highly recommend it - http://www.chem.fsu.edu/chemlab/isc3523 ... surely.pdf

Rumor has it that Mr. Feynmann loved his Orange juice .

I like his simple explanation of an anti gravity machine . & steel guitar bar slants

[Earnest Bovine]

the increasing angles of the bar slants as one progresses up the neck.

Wouldn't you say that as you progress "up" the neck, the angle of slant decreases rather than increases?

[basilh]

Sorry if my interjection is/was a little off topic, I was only asking out of interest because Split Slants, three string slants and reverse slants would be of interest to most players as they are the most difficult to perform.

Forward slants on (adjacent strings or not) are quite easy to play for most of us, whereas the split slant using the bullet nose for adjacent strings and then the next string a fret lower (or more) is frequently used but not studied enough IMHO, as is the three string slant which I believe is not possible to accurately execute but has to be a compromise.

So, which notes in a three string slant are of the least importance (Pitchwise) and how does one figure this out.
i.e. The faux 9th played al la Sol in the C#m7 ?

[Earnest Bovine]

Many of us grew up way back when frequecies were measured in Cycles Per Second (cps). In the 1960s the world changed to a new unit called Hertz (Hz).
To help you convert between the 2 types of measurement, I am re-posting this graph:

[Earnest Bovine]

Rumor has it that Mr. Feynmann loved his Orange juice .

I like his simple explanation of an anti gravity machine . & steel guitar bar slants

Unlike Einstein, Teller, and a great many mathematicians & physicists, Feynman did not "get" music. He thought it was dopey and a waste of time. IMO he didn't even "get" drumming. Other than that he was my idol.

[basilh]

Many of us grew up way back when frequencies were measured in Cycles Per Second (cps). In the 1960s the world changed to a new unit called Hertz (Hz).
To help you convert between the 2 types of measurement, I am re-posting this graph:

Thanks Ernest, so basically Hz = CPS, it helps, and no longer hurts !

[Dave Mudgett]

the increasing angles of the bar slants as one progresses up the neck.

Wouldn't you say that as you progress "up" the neck, the angle of slant decreases rather than increases?

The angle with the string increases, the angle from the straight-bar position decreases. I distinguished between these in my first post. The former is the arctan, the latter is the arccot.

Earnest - that graph is wicked. I am also a disciple of Feynmann, but I agree, it wasn't clear what he 'got' about music.

[Earnest Bovine]

Thanks Ernest, so basically Hz = CPS, it helps, and no longer hurts !

To help with the conversion, here is the source page. It describes also how to convert Megacycles to Megahertz, which involves multiplication and extraction of square root. This helped me a lot back in the 60s when they made the change.
http://www.brainerdham.org/Tips/CPS_to_ ... rsion.html

If you still have magazines and books describing frequency in Cycles per Second or Megacycles, you will want to convert those numbers to Hz (Hertz) or MHz (Megahertz). To do that, use the chart above that was graciously provided by long time ham, Jim Barry WØIRE. To do it mathematically, multiply the frequency in cycles per second by cycles per second and then take the square root of the result. If you have additional questions, just ask at the next club meeting.

[Alan Brookes]

So how does one apply these calculations when actually playing slants on the steel guitar?...

I can supply slide rules that you can fix to your guitar with Velcro.

[Mike Neer]

the increasing angles of the bar slants as one progresses up the neck.

Wouldn't you say that as you progress "up" the neck, the angle of slant decreases rather than increases?

My perspective is different. I was measuring it against the horizontal plane of the strings--the back of the bar is the vortex. At straight position, the bar is 90°. A slant on the lower strings is much more acute against the plane of the string. As I move up the neck, the tip moves back toward straight position, increasing the angle.

[Alan Brookes]

Many of us grew up way back when frequecies were measured in Cycles Per Second (cps). In the 1960s the world changed to a new unit called Hertz (Hz).
To help you convert between the 2 types of measurement, I am re-posting this graph:

You're pulling our legs, Earnest.
1 Hz = 1 cps. The only reason the graph isn't a straight line is because the axes are different.

[Mike Neer]

Forward slants on (adjacent strings or not) are quite easy to play for most of us, whereas the split slant using the bullet nose for adjacent strings and then the next string a fret lower (or more) is frequently used but not studied enough IMHO, as is the three string slant which I believe is not possible to accurately execute but has to be a compromise.

So, which notes in a three string slant are of the least importance (Pitchwise) and how does one figure this out.
i.e. The faux 9th played al la Sol in the C#m7 ?

The bar essentially becomes two different line segments when doing a split slant, because the points from the inside to the tip note will be in a slightly different direction. Point A to B on the lower notes, and points B to C for the tip.

I'd better hit those calculus books.

[Bob Hickish]

There is a simple way to come to an angular description of a bar slant -- if the inquiring mind want to know -- there is a simple tool called a p-2w plotter used in navigation - below is a photo using a simple angle from string 5 @ fret 5 to fret 8 @ string 2 on a S-10 E9th, it = 20 degrees -- as the frets change spacing you would have to divide the odd angle -- say 22.5 degrees Or half a 45 , etc.

for what its worth , it simplifies all the high math equations to determine an angle of a bar slant --

as a personal note, we have convinced the world that a steel is a hard instrument to learn as it is , so all this should add to the mix - Just MO - no reflection on Mikes wanting for knowledge .

Earnest Bovine
Wrote
Other than that he was my idol.

Ditto that ! Mr. Bovine

[Dave Mudgett]

Forward slants on (adjacent strings or not) are quite easy to play for most of us...

Um ... not everybody, and it's certainly not as easy on a narrow-spacing, long scale-length pedal steel. I think a lot of players find low-adjacent-string slants of 70+ degrees quite challenging. I think a lot of pedal steel players never even think seriously about slanting, much less doing challenging slants like this.

... whereas the split slant using the bullet nose for adjacent strings and then the next string a fret lower (or more) is frequently used but not studied enough IMHO, as is the three string slant which I believe is not possible to accurately execute but has to be a compromise.

Yeah, split slants are just flat-out hard. My biggest issues with split slants are 1) getting the tip of the bullet bar to cleanly sound the two notes intended and 2) getting that nose lined up exactly along the fret marker of both split notes. I agree that this is closely related to the particulars of the tip of the bar, a whole 'nuther but interesting issue.

But back to the topic of this thread - the basic nominal split bar-slant angle isn't a big deal. By nominal, I'm talking about autonomically getting the bar close to the correct angle - I think we all agree that one needs to hone in on the correct pitches by ear, but this is sure gonna be a lot easier if you land in the ballpark to start. For example, if you're trying to split a slant to the next two adjacent strings - e.g., a split slant from string 6 to strings 5 and 4, then to the approximations I made, one would average the bar-slant angles for the next two adjacent strings - in that example, the slant angles from 6 to 5 and 56 to 4. Or if you wanted to split-slant two strings with a string skip - e.g., a split slant from string 6 to strings 4 and 3 - then the bar angle would be approximately the average of the angles to slant to string 4 and the angle to slant to string 3. This isn't exact - the angle relationship is not perfectly linear. But you can see from the plots that it's not far from linear.

for what its worth , it simplifies all the high math equations to determine an angle of a bar slant --

Absolutely - I'm all for experimental investigation, you can do all this with a straightedge and a protractor. But seriously, this is not 'higher math' - everything here is strictly sophomore to junior level high school algebra and trigonometry. Which, unfortunately, only a very small percentage of the US population deems as remotely important anymore.

BTW - I measured my Zum string 5 fret 5 to string 2 fret 8. I got string length of 2-11/16" and string spacing 15/16", which gives a computed - arctan((15/16)/(2+11/16)) - angle with the string (horizontal) of about 19.2 degrees, or 70.8 with the straight bar position (vertical), so I assume the Zum's string spacing is a bit tighter. Or measuring from fret 5 to fret 6 and over one string, I got string length almost exactly 1" and string spacing 9/32" (slightly smaller than the average I used), to give a computed angle with the string of about 15.7 degrees, or an angle with the straight bar (vertical) of about 74.3 degrees - nonetheless even with the approximations I made, this corresponds very closely with the red curve in the graph for n=5.

[David Mason]

I did notice that in trying to play ten-string backward slants, you need a bar that is curved convexly away from you, and in trying to play ten-string forward slants you need a bar that is curved towards you. And unless you want to sound all sloppy-like, you'll need a different bar for each set of frets. But I'm pretty sure that if you're slanting from, like, fret 2 to fret 11, you can just flip the same bar over - for forward vs. backward slants - this will help keep expenses down. Now, I just have to find a song that needs ten-string slants. Send money.

[Ulrich Sinn]

isn't the y-axis just plotted against a logarithmic scale?

reads like 1:1 to me (as it should if I remember the definition of 1 Hz correctly)

[Wally Taylor]

[Spats Davenport]

I too find the maths behind music fascinating, although I never apply it while playing. And while some will say "just play the thing" I can't see it hurting in anyway. And, I agree with Mike's idea of using the angles as a ball park from which to fine tune the slant, rather than as an absolute.

But, if the angles are being visually verified, there will be a parallax error introduced, and increasingly so towards the lower frets. So, visually, the slant angle would seem less down there than it actually is.

I hope I haven't said anything too controversial, it's only my second post.

Spats.