Arithmetic vs Geometric String Tension - Which Do You Prefer & Why?

[Federighi]

Hey TB'ers!

What's your experience on mixing gauges to achieve arithmetic vs geometric balance in string to string tension?

IME I've had relative success with a given instrument (cello, upright, classical guitar, etc) when mixing light/med/heavy gauges as needed with the intention of equilibrium between strings.

There was a just a thread on most-easily-intonated-gauges.1639043 where post #7 per @micguy states: "If you look into the Physics, one of the things that helps you get more consistent intonation is matching the tension of the strings - you want gauges that are a geometric progression of diameters, not an arithmetic one. 40-60-80-100 is arithmetic, 40-55-75-100 is close to a geometric progression (you want the diameter of the bigger string of two adjacent ones to be 1.33 x the diameter of the smaller one).

Thoughts? Comments?

 

[blr2]

String diameter and mass per unit length would be linearly dependent only if the composite density is constant across the strings. This, most probably is not the case. Within a set, different strings would have different number of windings above the core (E and A typically have two, D and G-one), and different core to diameter ratio. This would affect their density.
Don't care about the diameter, but about the string tension. This should be kept close to constant across strings.

 

[HaphAsSard]

String diameter and mass per unit length would be linearly dependent only if the composite density is constant across the strings. This, most probably is not the case. Within a set, different strings would have different number of windings above the core (E and A typically have two, D and G-one), and different core to diameter ratio. This would affect their density.
Don't care about the diameter, but about the string tension. This should be kept close to constant across strings.

It is true that (bar some exceptions, like plain steel strings, and possibly Hellborg signature sets - the old one by DR and the current Dogals) strings aren't proportionally scaled across available gauges in a product line, and discontinuities like those you describe occur. On the other hand, differences in expected tensions (due to the appearance of a different number of underwindings in the construction between adjacent gauges) aren't so dramatic as to invalidate the rule of thumb: applying gives a result closer to equal tension than, well, not.
As for your last couple sentences, the geometric progression is especially useful if tension data happen not to be disclosed by the manufacturer.

 

[blr2]

This can be easily measured. All you need is a precise scale (resolving to10 mg, calibration isn't important, since you are doing it relative), a ruler and a wire cutter. Cut exact same length (or known ones) and weigh them then scale to diameter. Mind that string gauges on the package are not always accurate. Rotosound, for example, have been known to measure thicker than stated. The best is to measure it with a good micrometer to a 0.01 mm. From these measurements, you can fairly accurately calculate composite density for each string in a set, and whether it scales linearly with diameter/2 (radius) squared.

 

[Prodigus Filius]

I messed about with the .75 ratio which gives a 055 075 100 132 as far as the closest Chromes go to that ratio.
I'm now doing
.720 132>095
.738 095>070
.714 070>050
These ratios give a progressive tension pretty much. It was demonstrably progressive low to high using the D'addario tension pdf, but apparently it is no longer accurate and the middle two strings have swapped around in the tension stakes by a smidge based on the D'addario website tension specs.

Chromes 050 070 095 132
C#1/.132/48.78lbf
F#1/.095/45.61lbf
B1/.070/46.01lbf
E2/.050/45.33lbf
TT = 185.73lbf
.





.

 

[Prodigus Filius]

You can see how the progressive tension uniformity is affected vis a vis the website tensions compared to the pdf listed tensions in these stats

Chromes 050 070 095 132
Using D'addario Tension pdf
C#1/.132/45.08lbf
F#/.095/44.36lbf
B1/.070/42.5lbf
E1/.050/38.9lbf
TT = TT = 169.84lbf
Over 16lb of difference

 

[Tim Skaggs]

Hey TB'ers!

What's your experience on mixing gauges to achieve arithmetic vs geometric balance in string to string tension?

IME I've had relative success with a given instrument (cello, upright, classical guitar, etc) when mixing light/med/heavy gauges as needed with the intention of equilibrium between strings.

There was a just a thread on most-easily-intonated-gauges.1639043 where post #7 per @micguy states: "If you look into the Physics, one of the things that helps you get more consistent intonation is matching the tension of the strings - you want gauges that are a geometric progression of diameters, not an arithmetic one. 40-60-80-100 is arithmetic, 40-55-75-100 is close to a geometric progression (you want the diameter of the bigger string of two adjacent ones to be 1.33 x the diameter of the smaller one).

Thoughts? Comments?

It will depend on the string construction, core shape, core diameter, winding wire size, core metal alloy, winding wire alloy, etc. The two characteristics you mention would only matter if the two gauges listed were identical in every other variable I listed (and I doubt I included all other variables).

The mention of the intonation thread; setting the intonation on a string is not affected by any other string. Matching the tension of strings won’t affect how any of the strings intonate unless to get the tension matched, you use an extrodinarily small or large core wire on one of the strings which makes it intonate poorly.

 

[Federighi]

It will depend on the string construction, core shape, core diameter, winding wire size, core metal alloy, winding wire alloy, etc. The two characteristics you mention would only matter if the two gauges listed were identical in every other variable I listed (and I doubt I included all other variables).

How would this concept apply to natural and / or synthetic gut strings? How about strings with a silk, rope, perlon, etc core? For the bass strings, outer winding material can be anything from silver, alum, steel, chrome, tungsten, etc.

I play a lot of 'period' music at historical pitch (A=415 hz) and strive to achieve even string to string tension, tone and projection.

 

[Derek Kay]

I approached the progression using a hypothetical 6 string set, based on various set's averages, to see what worked at the extremes eg the B string & the C string.
I then tried to space out the other strings by some progression and realised that the relationship could not be arithmetic.

Te best fit was geometric, and exponential:

30:######
+10
40:########
+15
55:###########
+20
75:###############
+25
100####################
+30
130##########################

The sizes increased by an increasing amount.
A regular exponential curve, depicted above.

This theoretical set worked on my 5 string (without C string) but not my 4 string,

However the same pattern did work when it was transposed to a heavier range ie 45-60-80-105.

An arithmetic progression does work eg 45-65-85-105
but the expanded range of a 6 string set suggests a curve is more likely a better progression.

 

[Riff Ranger]

While I don’t dispute any of the factual information above, I stopped painstakingly balancing tension between strings when I noticed an apparent imbalance in output, i.e. the thinner strings on my balanced tension set got buried in a live band mix; I wasn’t measuring decibels, but neither was my audience. Perhaps I could have addressed this some other way, but when I went to a not-terribly-balanced set where even arithmetic-ally the numbers looked weird, it solved my problem without changing my sound or introducing any challenges to my playing. Balanced tension sets do feel great, and when they also sound balanced, by all means let me at them. .125/.095/.075/.055 for BEAD is just perfect for me—balanced in every way but arithmetic-ally. The same approach did not get the results I wanted in GCGCF; I’m happier with .135/.105/.085/.070/.050 than I ever was with any combination of diameters I tried that remotely balanced tension (in ADGCF; GCGCF just ain’t a balanced tuning ).

I suppose if I had to choose, give me balanced tension rather than even intervals between diameters. But more than anything, I want a set that sounds balanced in my situations.

 

[Prodigus Filius]

I approached the progression using a hypothetical 6 string set, based on various set's averages, to see what worked at the extremes eg the B string & the C string.
I then tried to space out the other strings by some progression and realised that the relationship could not be arithmetic.

Te best fit was geometric, and exponential:

30:######
+10
40:########
+15
55:###########
+20
75:###############
+25
100####################
+30
130##########################

The sizes increased by an increasing amount.
A regular exponential curve, depicted above.

This theoretical set worked on my 5 string (without C string) but not my 4 string,

However the same pattern did work when it was transposed to a heavier range ie 45-60-80-105.

An arithmetic progression does work eg 45-65-85-105
but the expanded range of a 6 string set suggests a curve is more likely a better progression.

The proportional equation for this approach is
(Highest Gauge/Lowest gauge) ^(1÷ number of strings less1)
So with your hypothetical set the equation would
(.030÷.130)^(1÷5) = .7458
Dividing the highest to lowest by .7458 gives
.030 .040 053 .072 .096 .130
With a bit of rounding up or down, if you could find 30 40 55 70 95 130 it would be very balanced if that was desired outcome

 

[Prodigus Filius]

While I don’t dispute any of the factual information above, I stopped painstakingly balancing tension between strings when I noticed an apparent imbalance in output, i.e. the thinner strings on my balanced tension set got buried in a live band mix; I wasn’t measuring decibels, but neither was my audience. Perhaps I could have addressed this some other way, but when I went to a not-terribly-balanced set where even arithmetic-ally the numbers looked weird, it solved my problem without changing my sound or introducing any challenges to my playing. Balanced tension sets do feel great, and when they also sound balanced, by all means let me at them. .125/.095/.075/.055 for BEAD is just perfect for me—balanced in every way but arithmetic-ally. The same approach did not get the results I wanted in GCGCF; I’m happier with .135/.105/.085/.070/.050 than I ever was with any combination of diameters I tried that remotely balanced tension (in ADGCF; GCGCF just ain’t a balanced tuning ).

I suppose if I had to choose, give me balanced tension rather than even intervals between diameters. But more than anything, I want a set that sounds balanced in my situations.

Have you done any progressive tension from low to high? I'm not saying it leads to balanced sound per se .
For instance a set of XL nickel round singles just to use as an example for BEAD with gauges 052 070 095 130 would look like this
B0/.130/32.14lbf
E1/.095/31.66lbf
A1/.070/31.21lbf
D2/.052/29.80lbf

TT = 124.81lbf
Low total tension. Pretty close to La Bella LTF's 127lbf in BEAD, which isn't balanced or progressive

 

[Riff Ranger]

Have you done any progressive tension from low to high? I'm not saying it leads to balanced sound per se

Do you mean progressively higher tension on progressively higher-pitched strings? That’s sort of what I’m doing with my 5-strings now, but it’s not anything terribly thought out on my part. I had had a set of 5-string Power Slinkies—not the set for anyone seeking balanced tension between strings in any reasonable tuning!—sitting around for years after the show where I realized my big chords on my higher-pitched strings weren’t sounding big with the balanced tension set I had on. Switching to the Power Slinkies instantly solved my problem.

Assuming Ernie Ball’s numbers aren’t far off of Stringjoy’s, here’s what I had going on:

And here’s what they would look like if tuned in even intervals:

Strictly speaking, .140s would balance tension better but neither Stringjoy nor Ernie Ball currently make them. (I do have GHS .140s tuned to G0 on a pair of instruments right now and like them.)

I seem to remember Billy Sheehan favoring sets that get progressively heavier in the other direction (would that make them regressive? ), i.e. having higher tension on lower-pitched strings, his reasoning being that he wanted easier bending of the skinny strings and more power from the thicker ones. They seem to work for him. He’s definitely working with advantages I don’t have EDIT: Billy’s .110/.080/.065/.043 set tuned EADG isn’t exactly progressive. The E and A should be about the same for him, with a tighter D and looser G. Still, that’s a thicker E and thinner G than in typical sets.

 

[Prodigus Filius]

Do you mean progressively higher tension on progressively higher-pitched strings? That’s sort of what I’m doing with my 5-strings now, but it’s not anything terribly thought out on my part. I had had a set of 5-string Power Slinkies—not the set for anyone seeking balanced tension between strings in any reasonable tuning!—sitting around for years after the show where I realized my big chords on my higher-pitched strings weren’t sounding big with the balanced tension set I had on. Switching to the Power Slinkies instantly solved my problem.

Assuming Ernie Ball’s numbers aren’t far off of Stringjoy’s, here’s what I had going on:

View attachment 5346943 View attachment 5346944

And here’s what they would look like if tuned in even intervals:

View attachment 5346945 View attachment 5346946

Strictly speaking, .140s would balance tension better but neither Stringjoy nor Ernie Ball currently make them. (I do have GHS .140s tuned to G0 on a pair of instruments right now and like them.)

I seem to remember Billy Sheehan favoring sets that get progressively heavier in the other direction (would that make them regressive? ), i.e. having higher tension on lower-pitched strings, his reasoning being that he wanted easier bending of the skinny strings and more power from the thicker ones. They seem to work for him. He’s definitely working with advantages I don’t have EDIT: Billy’s .110/.080/.065/.043 set tuned EADG isn’t exactly progressive. The E and A should be about the same for him, with a tighter D and looser G. Still, that’s a thicker E and thinner G than in typical sets.

I meant progressive decreasing tension from low to high, as you mentioned in reference to Billy Sheehan. The BS custom set in the Rotosound Swing Bass 66 catalogue is as follows
E1/.110/42.25lbf
A1/.080/43.80lbf
D2/.065/51.30lbf
G2/.043/40.94lbf

TT = 178.29lbf

In my opininion the tension listed for Rotosound .065 in this set and all Roto Bass and the Nickel variant of Swing 66 is wrong. It jumps up 13lb from the listed spec for the 060 .If you compare to an XL Nickel round or a Pro Steel, the tension is more likely 47ish which would make Sheehan's set look like this
E1/.110/42.25lbf
A1/.080/43.80lbf
D2/.065/47.00lbf
G2/.043/40.94lbf

TT = 172.99lbf
Still no balance or progression downwards from low to high though.
A near progression with Swing 66 for BEAD for example could look like this

Gauges 050 070 095 130

B0/.130/32.56lbf
E1/.095/31.69lbf
A1/.070/31.71lbf
D2/.050/29.86lbf

TT = 125.82lbf

An example of progressive tension the other way, decreasing from high to low, is

BEAD with TI Jazz Rounds

Gauges 051 068 089 118

B0/.118/27.55lbf
E1/.089/27.99lbf
A1/.068/29.54lbf
D2/.051/30.20lbf
TT = 115.28lbf