Over Return, by the numbers

ed packard · From: Show Low AZ

Forum over return issue:
OK, the philosophy session is over….time to run the numbers.
We are analyzing the OVER RETURN issue for possible sources and solutions.

Let’s take a 25.0000” scale, a common 0.014000”diameter string tuned to an E4 note = 329.627557 Hz (fundamental = h1) at 0 cents shift.



cents off	Hz	Beats = Hz


-10	327.729042	1.90


-9	327.918400	1.71


-8	328.107868	1.52


-7	328.297445	1.33


-6	328.487132	1.14


-5	328.676929	0.95


-4	328.866835	0.76


-3	329.056851	0.57


-2	329.246976	0.38


-1	329.437212	0.19


E4 =	329.627557	0.00


+1	329.818012	0.19


+2	330.008578	0.38


+3	330.199253	0.57


+4	330.390039	0.76


+5	330.580935	0.95


+6	330.771941	1.14


+7	330.963057	1.34


+8	331.154284	1.53


+9	331.345622	1.72


+10	331.537070	1.91

Beats are in cycles per second = Hz. This is what you would hear if the changing string was heard against another string tuned to 329.627557 Hz.

For a scale length of 25.0000”:

The string stretch to get from Eb4 to E4 is about 0.020”…that is 100 cents.
If the cents were linear, and they are not (they are per a 2^(1/1200) scale), each cent would be the amount of string stretch divided by 100 = 0.020”/100 = 0.00020” or less than 1/10th the thickness of a human hair. For an E4 to F4 change it would be about 0.00030”.

The tension on the E4 note at 0 cents shift would be about 30 pounds.
The tension for an Eb4 note would be about 27 pounds.
The tension for an F4 note would be about 34 pounds.

What can cause the E4 note frequency to change by a cent? …a change in tension of about 0.035 pounds. To do this, the string would need to change length by about 0.00020”.

Activating the changes to b and # the E4 note will cause a temperature change centered about the points of friction, and bending. The faster the bending, either to b or # the note, the greater the temperature increase at these points. These temperature increase points are against a mass of metal with the ability to absorb this increase in temperature via thermal conductivity…true for both Roller nuts, non roller nuts, and changer fingers. Not room or time enough here to run the thermal transfer analysis.

By experiment, I find no measurable change in the amount of over return for slow, fast, single, multiple activations (using the BEAST and the Peterson VS-II), and no frequency drift back as would occur from cooling off…the conclusion is ….it ain’t a thermal thing folks.

My test bed for this is the BEAST, which, as many know has no roller nut, no string wrap in contact with the changer finger, and shallow bends of the string across the nut and the changer finger. The total string length is about 31.5 inches, and the scale length is 29.730”. String 4 is tuned to C4. The string diameter is 0.014”, so the tension and string stretch is in the ballpark of the numbers given above.

Even though the over return that I measure is less than a cent, it IS an over return phenomena.

If I use 1 cent for my over return, then I must find a source of 0.00020” or so of string length change, or the source of a 0.034 pound (0.002 oz) or so tension change to account for the phenomena. Let’s assume that it is not in or beyond the changer for the moment, so we need to find between the ends of the string.

The 31.5” string contracts about 0.020” when lowered to Eb, and stretches about 0.030” when raised to F. The total motion is caused at the changer end, and diminishes to zero at the anchored end beyond the nut. The amount of motion across the nut is 1/31.5th of the total motion at the changer end because the nut is 1 inch from the anchored end beyond the nut….so the amount of motion across the nut (assuming no friction) for the halftone activation is about 0.0008” (using 0.025” as the total motion at the changer), or less by some amount to the degree that there is some friction involved).

If friction were present, could it contribute to the over return problem? Yes…we only need a hang up of 0.0002” or so total for a one cent over return.

If we had used a string lock in place of the nut, the amount of motion at the nut would have been zero = infinite friction (for practical purposes), hence no source of over return at that location/device, and all the over return burden would be centered at the other end of the string = the ball/wrap/changer end. Same would be true if there was a roller nut that provided zero friction….zero?????

The best solution that I can see at the moment (from a design standpoint) is the String Lock instead of the roller nut, or of the non roller nut (on keyed or keyless), no ball and wrap/twist, anchor the string via a screw clamp at the back top of the changer finger (machine a little flat there).

Possible complaints:
1. This would change my sound!...Maybe, but you might even like it.
OK, now you add your complaint.

Please check this as it was done on the fly.

edited for some corrections...

[This message was edited by ed packard on 21 May 2006 at 07:44 AM.]

[This message was edited by ed packard on 21 May 2006 at 07:52 AM.]

Curt Langston · Posted 20 May 2006 4:18 pm

Good job Ed!
I appreciate your calculations. Now give us a compressed summary my good man!

ed packard · From: Show Low AZ

Curt....see the last paragraph befor the "complaint" section.

Stu Schulman · From: Ulster Park New Yawk (deceased)

Ed:Would a bearing like an abec7 with a groove instead of the standard roller nut help at all,or would that be overkill?Stu

ed packard · From: Show Low AZ

Stu...as I recall, bearings suffer from stiction problems, and detent problem at the not moving much or often levels.

Might be worth a try...I think that I would try a polished Zirconia rod to the degree that I suspect friction at the nut, and didn't want to use a string lock.

Webb Kline · From: Orangeville, PA

yikes

Colby Tipton · From: Crosby, Texas, USA

Keep up the good work ED.

richard burton · From: Britain

A string lock at the nut would necessitate fine tuners on the changer fingers.

Bobby Lee · From: Cloverdale, California, USA

quote:
Let’s take a 25.0000” scale, a common 0.014000”diameter string tuned to an E4 note = 329.627557 Hz (fundamental = h1) at 0 cents shift.
+1 cent = 332.925737 Hz, beats heard against the fundamental would be about 2.3.
+2 cents = 336.223917 Hz, beats heard against the fundamental would be about 6.4.
-1 cent = 326.329377 Hz, beats heard against the fundamental would be about 3.3.
-2 cents = 323.031197 Hz, beats heard against the fundamental would be about 6.6.

Check your math, Ed. I think the +1 cent figure is worng.

Henry Nagle · From: Santa Rosa, California

grumble, grumble, newfangled (!), grumble, grumble... "worng"?

[This message was edited by Henry Nagle on 21 May 2006 at 02:13 AM.]

ed packard · From: Show Low AZ

Richard...you are correct as I see it. On the BEAST, the tuner is part of the changer...It can be done.

b0b....I will check it...thought that I multiplied the E Hz by 2^(1/1200) which is a + one cent change (not percent). I would be surprised if that were to be the only miscalc. The point is not the exact numbers, but the magnitude (perhaps minitude would be a better word)of forces and motions involved. 0.034 pounds AIN'T much.

Howard Tate · From: Leesville, Louisiana, USA, R.I.P.

Wow, this makes me feel like I never went to school. For what it's worth I just checked my Mullen and didn't seem to have the problem. Del Mullen told me the hardest part was machining the rollers to such a tolerance that they roll freely without being sloppy, or words to that effect.

Charlie McDonald · From: out of the blue

I too wondered at the +1 and +2 cents values, but only intuitively, because if

quote:
The tension on the E4 note at 0 cents shift would be about 30 pounds.
The tension for an Eb4 note would be about 27 pounds.
The tension for an F4 note would be about 34 pounds.

then I would expect the beat rate for raised notes to be greater than the beat rate for lowered notes.

Be that as it may, Ed, I wonder if a static nut (similar to your string retainer) on the left end with the changer/tuning mechanism on the right end would work as well. Was always curious about this.

ed packard · From: Show Low AZ

Charlie...Check the new (corrected and extended) freq/beat chart above....thanks to old eagle eye b0b...My spastic finger entered an extra digit in my calcs.

A standard (changer on the right) was modified to have the tuner integrated into it, and the terminator on the players left to prove out the concept before the BEAST was built. It sits next to the BEAST at the moment...seems to behave the same.

[This message was edited by ed packard on 21 May 2006 at 08:13 AM.]

Charlie McDonald · From: out of the blue

Cool.

Bobby Lee · From: Cloverdale, California, USA

Okay, now that we have some real numbers...

It's enlightening to see that a difference of 10 cents on the E note produces less than 2 beats per second!

I submit that over-return on the order of 5 cents or less will be unheard by all but the most discerning player's ears. It would certainly be unnoticed by anyone listening without bar in hand.

Also consider that the pitch difference between the notes of a properly centered JI triad and their ET counterparts is about 6 cents - in the neighborhood of 1 beat per second on the E note. Is this hair's width really an issue in practical terms? Does 1 beat per second really make something sound out of unison?

Do you ever really hold the bar perfectly still for a full second? What about the widening effect that reverb and/or electronic delay has on a moving pitch?

------------------

Bobby Lee (a.k.a. b0b) - email: quasar@b0b.com - gigs - CDs, Open Hearts
Williams D-12 E9, C6add9, Sierra Olympic S-12 (F Diatonic)
Sierra Laptop S-8 (E6add9), Fender Stringmaster D-8 (E13, C6 or A6) My Blog

ed packard · From: Show Low AZ

bob....I suppose that someone will point out that the beats will double if the note is raised an octave, so 3.8 beats might annoy someone.

My personal take is that a little off pitch probably livens up the music.

It just seems like the thing to do to understand how the various concerns re off pitch, over return, cabinet drop, thermal excursion, etc. can be controlled. These are sort of like doing puzzles.

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