Tried the Math and Physics majors and no help...

[robj144]

Originally Posted by Jackknifegypsy

This is a mystery to me. I have no idea how I can use this equation without resorting to a calculator:


ArcTan(6.5/420) = .89 degrees


6.5 Divided by 420 = 0.0154

Then hit The ArcTan or tan^(-1) on your calculator or type or into google... there's an online calculator.

[robj144]

Originally Posted by Jackknifegypsy

This is a mystery to me. I have no idea how I can use this equation without resorting to a calculator:


ArcTan(6.5/420) = .89 degrees


6.5 Divided by 420 = 0.0154

Something is missing in that equation since I do not understand the 'function'-- ArcTan.

I'm not getting into the details, but this also works for this case:

(6.5/420)*(180/3.14159) = 0.89

Extra points go to anyone following the thread that can explain why the above calculation gives the sane answer as using the full trigonometry functions. :-)

[charles Tauber]

Originally Posted by robj144

Extra points go to anyone following the thread

You should have stopped there.

[charles Tauber]

This isn't aimed at anyone nor is it a criticism of anyone; it's just a story.

In the 30+ years since I started making instruments, I've trained three apprentices. The first one, early on, was a young man in his 20's who had not completed high school. One day we were doing something that required employment of the Pythagorean Theorem, of right-angled triangles, to which he had not been exposed. That was the first of an increasing number of challenges and impediment to training him. That he was unfamiliar with very basic, "common" knowledge that underlies much of what goes on around us in the world - things that we encounter often on a daily basis - was what eventually led to us parting company.

There are still many things that I know nothing about, but, with each passing day, I attempt to reduce a little more of what I don't know and am thankful for those along the way who help me do so.

[John Arnold]

(6.5/420)*(180/3.14159) = 0.89

Extra points go to anyone following the thread that can explain why the above calculation gives the same answer as using the full trigonometry functions. :-)

http://en.wikipedia.org/wiki/Radian

[robj144]

Originally Posted by John Arnold

http://en.wikipedia.org/wiki/Radian

That has something to do with it, but for instance,

ArcTan(400/420) = 43.60 degrees

but,

(400/420)*(180/3.1459) = 54.57 degrees.

In other words, it doesn't work all the time. I was kind of asking why it works for the previous case.

Sorry, I'm a teacher... I like to teach.

[Jackknifegypsy]

"There are still far many more things that I know nothing about than I do (and those are suspect), but, with each passing day, I attempt to reduce a little more of what I don't know and am thankful for those along the way who help me do so". ( P'phrased from Charles)

That's what I was doing with the Trig thing as it applies to the fretboard and so many other techniques I have learned (and still fighting with) on this board.

Thx to you all.

[charles Tauber]

Originally Posted by Jackknifegypsy

That's what I was doing with the Trig thing as it applies to the fretboard and so many other techniques I have learned (and still fighting with) on this board.

And, I applaud that. But, John has given you good advice regarding using your chosen method. Trying to measure angles accurate to two or so decimal places isn't very practical (i.e. difficult to achieve) in a typical woodworking shop. Being even .1 degrees off on a 20ish inch long board isn't going to give you the results you want.

I use the same method John described: cut the taper on a bandsaw, clean-up and to the line with a hand plane or jointer.

As an aside, the ancient Egyptians, who were master architects and builders, did not know of irrational numbers - numbers with non-repeating, non-ending numbers of decimal places. However, they indirectly worked with them all of the time. They did so using the method John mentioned of "triangulation". They measured a desired rise and run that produced the desired angle.

Similarly, if you "triangulate" the taper (angle) you want so that you measure a simple-to-measure rise and run - such as a whole number or easily-read fraction - you can eliminate the need to accurately measure and produce angles, while still obtaining the same result. Roofers and stair-makers use the same basic method.

The Theorem of Thales can be used in a similar way to layout the fret positions in any fret scale, without the need to calculate anything or perform individual measurements. That, however, is definitely, a subject for another time.

[Jackknifegypsy]

Having tried both John's ---and I assume many other luthiers' method-- and then using the 'degree' method:

John's (and others') method is far superior. The Angle (Theta) is better "the result of" rather than "a factor used to determine" the correct taper on a fretboard.

[Howard Klepper]

Originally Posted by robj144

I'm not getting into the details, but this also works for this case:

(6.5/420)*(180/3.14159) = 0.89

Extra points go to anyone following the thread that can explain why the above calculation gives the sane answer as using the full trigonometry functions. :-)

You want an explanation of why you get the sane answer?

Because for very small angles, the side opposite approximates the arc of the circle subtended by the angle, and the side adjacent approximates the radius of that circle. So the tangent is roughly equal to the angle in radians. You can get the angle in degrees by converting radians to degrees, i.e., multiplying the tangent by 180/pi.

As a practical matter, I think Louie nailed it.

[robj144]

Originally Posted by Howard Klepper

You want an explanation of why you get the sane answer?

Because for very small angles, the side opposite approximates the arc of the circle subtended by the angle, and the side adjacent approximates the radius of that circle. So the tangent is roughly equal to the angle in radians. You can get the angle in degrees by converting radians to degrees, i.e., multiplying the tangent by 180/pi.

As a practical matter, I think Louie nailed it.

That's correct Howard. You get a gold star!

I was typing on my tablet and I don't always see the auto correct mistakes, but that was one sane answer. :-)

[Jackknifegypsy]

....but this thread has led to the place on the map where there be Dragons.

Among other things I have learned in it is the word, 'subtended'.

[LouieAtienza]

It would be interesting to note, extending Howard's explanation, that with any right triangle, the hypothenuse could be the diameter of a circle, and the (right) angle opposite the hypothenuse would always lay exactly on the circle's circumference, regardless of the other two angles.

[robj144]

Originally Posted by LouieAtienza

It would be interesting to note, extending Howard's explanation, that with any right triangle, the hypothenuse could be the diameter of a circle, and the (right) angle opposite the hypothenuse would always lay exactly on the circle's circumference, regardless of the other two angles.

You mean the hypotenuse is the radius (not the diameter) and the side (not angle) opposite to the angle (adjacent to the right angle) is the perimeter, or arc, of the circle. The circumference is defined is defined to be the length of the perimeter of the entire circle. Sorry, I teach this stuff, so I feel compelled to correct.

[tadol]

I think he's saying that if you lay a right angle in a circle, with the right angle touching the circumference, the hypotenuse of that triangle is the diameter. It's the old carpenters trick of using a square to find the center point of an arc or circle -



To give credit -