A Simple Mathematical Approach To Choosing Strings...

[MWBass]

Hello All,

So, I am posting tonight to discuss string choice for those of us who like to venture into the custom realm of things. This discussion will get into the old balanced tension / progressive tension debate. I just wanted to say this up front so that I could save time for anyone uninterested in this topic.

Let me start with some background. I am a luthier that started building about a year and a half ago. After getting comfortable with the build process, I got into design and developing the designs that me and my building partner were going to manufacture. Once that was done, I was ready to start producing some product. Unfortunetly, at that time I was at a financial low, so starting was not possible. I figured with this time, I wanted to accomplish two things... Exhaust any remaining experimentation that needed to be done to appease my brain, and to develop an instrument that was the pinnacle of what I thought was possible.

So, this instrument unlike the rest of my instruments, started at the fretboard. What scale should I use? How many frets? How many strings? What tuning? Should I go with a multiscale fanned fret set up? What strings? What gauges? What tension? This is what got me into string tension research, and where this whole approach began...

[MWBass]

So, as most mathematically minded people like myself would do, I immediately seeked out the most useful equation for finding string tension. I quickly found the main tension equation...

T = tension (lbs.)
UW = string weight (linear ft.)
S = scale length (in.)
F = note frequency (hz.)

T = (UW X ( 2 X S X F )^2) / 386.4

So, I spent countless hours plugging into this equation, utilizing string gauge / weight / tension tables from D'Addario, LaBella, and Circle K. I worked with different scales lengths, as well as straight and multiscale fanned fret set ups. Not only that, but mixing bass and guitar strings as well. Now, most of these scales were short scales, because that was what was chosen for this big bass build project. I don't want to get into why in this thread. That is more for a luthier build thread. However, for all intensive purposes, the length of the scale will not effect the string to string tension too much, as long as the multiscale designs have a similar relationship between the bass and treble sides.

So, I will not share all of the charts and analysis, that would take days, but I will share the best results I got, for anyone who may want to use them...

And, as you'll be able to tell, this is for a balanced tension set up...

[MWBass]

All of the listed sets are standard 4 string sets, just to keep it consistent. I did tests for standard 5 strings, standard 6 strings, tenor 5 strings, tenor 6 strings, and 6 strings strung E, A, D, G, B, E in standard guitar tuning. It would just take ages to list everything. I feel with the 4 string sets, you get the idea...

Ok, so here they are...


1. Gauges: 105, 80, 60, 46

Strings: D'Add XLB105
D'Add NW080
D'Add NW060
D'Add NW046

This is a mix of bass and guitar strings, so it will only work with shorter scales, and just barely where I have it.

Scale: 31" on bass side, 29.7" on treble side. (fanned frets)

Tensions: E - 33.50, A - 33.65, D - 33.70, G - 33.51

So, the tolerance is within .2 lbs. for the whole set.


2. Gauges: 110, 85, 65, 50

Strings: D'Add XLB110
D'Add XLB085
D'Add XLB065
D'Add XLB050

Scale: 30" on bass side, 27.9" on treble side. (fanned frets)

Tensions: E - 35.97, A - 35.94, D - 36.30, G - 35.96

So, the tolerance is within .36 lbs. for the whole set.


3. Gauges: 106, 80, 61, 46

Strings: Circle K .106
Circle K .080
Circle K .061
Circle K .046

Scale: 30.0" on bass side, 28.2" on treble side. (fanned frets)

Tensions: E - 33.6, A - 33.1, D - 33.3, G - 33.6

So, the tolerance is within .5 lbs. for the whole set.


4. Gauges: 106, 80, 60, 45

Strings: LaBella SN-B106
LaBella SN-B080
LaBella SN-B060
LaBella SN-B045

*This is an extrapolation from D'Addarios Charts.
* I did this due to inconsistences in the La Bella charts.

Scale: 31.0" on bass side, 30.4" on treble side. (fanned frets)

Tensions: E - 34.17, A - 34.47, D - 34.75, G - 34.22

So, the tolerance is within .58 lbs. for the whole set.


5. Gauges: 105, 80, 60, 45

Strings: D'Add XLB105
D'Add XLB080
D'Add XLB060
D'Add XLB045

Scale: 30.9" on bass side, 30.0" on treble side. (fanned frets)

Tensions: E - 33.3, A - 34.0, D - 34.1, G - 33.3

So, the tolerance is within .8 lbs. for the whole set.


6. Gauges: 130, 95, 70, 56

Strings: D'Add XLB130
D'Add XLB095
D'Add XLB070
D'Add XLB056

*I know these gauges are kind of ridiculous, but it goes
to show that it can even be done for regular bass tension
on a Fende Strat Scale. (25.5")

Scale: 25.5" straight

Tensions: E - 34.50, A - 34.48, D - 33.81, G - 34.77

So, the tolerance is within .96 lbs. for the whole set.


7. Gauges: 95, 70, 56, 40

Strings: D'Add XLB095
D'Add XLB070
D'Add NWB056
D'Add XLB040

This uses one extra long scale guitar string. (.056)

Scale: 34" straight

Tensions: E - 34.40, A - 33.74, D - 34.68, G - 33.70

So, the tolerance is within .98 lbs. for the whole set.

[Thunderthumbs73]

Really interesting thread.

[MWBass]

So, as you could imagine, when I stumbled upon the #1 set above, I was elated. I found a set of strings and a multiscale fretboard that allowed for a balanced set of strings within .2 lb. tolerence. Basically, the perfect set.

But, then I started thinking. 3 of these strings are guitar strings. First of all, this really condenses string scale. It will work for me in my particular design, but not for others who want to change the scale. Well, fine I thought. This is my build I'll try it.

But, then I thought further. Guitar strings usually have thinner cores. Guitar strings are made for tensions in generally the 10 lb. to 20 lb. range. If I tried to maintain bass type tensions in the 30 lb. to 40 lb. range, the strings would most likely not be able to handle the tension and break much more often. Then I thought, I would hate to build an instrument based on the ideal of an almost perfect balanced tension, then have the strings break so often that I constantly question why i did't build the bass a different way...

I then began to question everything... Higher or lower tension? The feel of string thickness? Multiscale verses straight scale? And most importantly, what if I or any other prospective buyer does not want to just use D'Addario nickel strings? As soon as I switch to steel, or another company the perfection is lost...

I got so fed up. After all of this work, I felt I was back where I started. It made me think. I got into this because there was an equation and a mathematical approach. I am a high school math teacher, and I will admit, in a dorky manner, I enjoy figuring things like this out.

So, in the end, I said maybe I didn't analyze the math enough. Let me go back to the beginning and think a bit. Maybe there is an approach that can yield great results, satisfy everyone, and be easy enough for, unlike myself, the non-mathematically driven public to utilize...

[MWBass]

SO, WHEN DO WE GET TO THE POINT?

SO, WHERE'S THE SIMPLE MATHEMATICAL APPROACH?

Ok, let's get to it.

I started thinking about string weight. I immediately discounted thinking about the things that change between different strings and string manufacturers. Things like thin cores or thick cores... round cores or hex cores... plain strings or wound strings... single wraps, double wraps, or triple wraps...

I figured there is no way to analyze all of these.

So, for the sake of my math, I made some basic assumptions. I figured, lets assume that the strings are made of one solid material in order to ease the comparison of weight compaired to string thickness. I also assumed that 99.9% of players would not mix strings, like nickel and steel, hex and round, etc. I figured they would use the same type of string all the way across for consistency purposes.

So, the first thing I figured out quickly was that string weight will not increase linearly with thickness.

Each string is like a really long cylinder. Weight would be defined by the material used, its mass, the volume of the cylinder, and gravity, which I will not get into. So the main thing is that the volume of a cylinder is defined by...

V = 2(pi)(r^2)h

So, since the height can be the same for all strings, and since pi and 2 are constants, the radius squared is the key.

So, if arbitrarily one string has a radius of 2 units, and the twice as thick string has a radius of 4 units, by squaring, the volumes will be in a ratio of 2^2 to 4^2... which is 4 to 16. Thus, the weight will increase in this exponential ratio as well.

So, the assumption that a guage (which is the thickness of a string in thousandths) will double in weight as it doubles in size, is wrong! As guage gets larger weight increases exponentially.

So, I wanted to utilize the Tension Equation from before, but break down the Unit Weight to the above equation in order to pull out even more constants...

So, with balanced tension, what will remain constant...

Let's Assume...

T - tension is the same.
S - scale is the same. (just so we can get to the basics of string choice.)

So, let's look at the equation again...

T = ( UW X ( 2 X S X F )^2) / 386.4

Let's move some stuff around...

(T X 386.4) / UW = ( 2 X S X F )^2

More...

(Sq. Rt. ( (T X 386.4) / UW)) / ( 2 X S ) = F

More...

(Sq. Rt. ( T X 386.4)) / (Sq. Rt. (UW) X 2 X S ) = F

More...

(Sq. Rt. ( T X 386.4)) / (2 X S ) = F X (Sq. Rt. (UW))

And Finally...

((Sq. Rt. ( T X 386.4)) / (2 X S ))^2 = (F ^2) X UW

So, if you look at the left side of the equation now, the only variables are a T and an S. We previously stated that these were going to stay the same, so they are an arbitrary constant. So, in that case there is nothing but numbers on the left side. So, let's let all of these numbers absorb into one constant we will call K.

So the equation now...

K = (F ^2) X UW

Now, lets replace UW with what we defined it as above. We will make W stand for the cubic weight of whatever material is used in the string...

K = (F^2) X W X 2 X Pi X (R^2)

Well, we know that the W is going to be the same across all strings, and 2 and Pi are constant as well, so we can move them to the left side of the equation...

K / ( W X 2 X Pi ) = (F^2) X (R^2)

Let W, 2, and Pi get absorbed into the constant...

K = (F^2) X (R^2)

Finally, square root both sides and...

Sq. Rt. (K) = F X R

So...

K = F X R

So, we got down to a really basic equation. We know now that Note Frequency and the Radius of each string have a strong relationship. This means that since radius is only half of diameter, and diameter is the Gauge of a string, and since that 2 times the radius can be absorbed into the constant, we know that gauge has a huge relationship with frequency.

So, we can now say the final equation...

Frequency X Gauge = Constant

So, logically, as Frequency goes up, Gauge must go down to stay at the Constant.

So, now we know Frequency and Gauge have an inverse relationship. So, all we have to do is find out how Frequency changes from string to string, and do the opposite to string Gauge.

Frequency Of Notes:

E - 41.20 hz
A - 55.00 hz
D - 73.42 hz
G - 98.00 hz

So, the hertz don't change linearly. So, al you have to do is take a number and divide it by the following number...

41.20 / 55.00 = 0.75

55.00 / 73.42 = 0.75

73.42 / 98.00 = 0.75

So, every Frequency is 75% of the following Frequency.

Well, if the Frequency's go up in this perfect exponential fashion, the Gauges must go down (inversely) in the same exponential manner.

So, the only thing left to do is test this...

Let's say we choose a 100 gauge to use on the E string...
Using the rule of going down to 75% of the previous gauge, we get the following...

100
100 X .75 = 75
75 X .75 = 56.25
56.25 X .75 = 42.1875

So, feasibly these gauges (and I know the last two don't exist) should give us a perfect balanced tension...

So, we have to check it using the equation we made...

Frequency X Gauge = Constant

41.20 X 100 = 4120
55.00 X 75 = 4125
73.42 X 56.25 = 4130
98.00 X 42.1875 = 4134


So, as you can see the answers are extremely close for numbers in the 4000 range. For anyone who wants to know, why it isn't perfect, it is because the Frequencies are rounded and the Percent Change was rounded to 75%.



So, in the end, those who like simple...

**Use the "Rule of 75%" to choose gauges.**



For those of you, who want to be exact...
If you do some research you can find out that the note to note
change of frequency is based off of taking the previous note,
and multiplying by 2^(1/12). 1/12 for one twelth of the full octave.

So, 2^(1/12) = 1.059463, and from string to string we go up
by fifths, so, 1.059463^5 = 1.334839. Now, we just need the
inverse since string Gauges get thinner as Frequency goes
up... 1 / 1.334839 = 0.749154.



So, in the end, for those of you who like to be exact...

**Use the "Rule of 74.9154%" to choose string gauges.**



And, since my example of a 100 gauge on the E string was kind of ugly, let's do something better.

LaBella offers a 106 Gauge let's use that...

106
106.0 X .749154 = 79.4 closest = 80
79.4 X .749154 = 59.5 closest = 60
59.5 X .749154 = 44.6 closest = 45

So, a 106, 80, 60, 45 will give you a excellent balance in tension.

Coincidently, a 105, 80, 60, 45, which is attainable from most companies is an excellent choice.

Ok, I'm gonna stop for now. I'll add more soon.

[esa372]

Great info, MWBass - thank you!


~esa

[ixlramp]

Originally Posted by MWBass

K = F X R

MWBass, excellent thread! I've worked through the same mathematics and arrived at the same result. Assuming strings of constant density, tension and scale; gauge is inversely proportional to frequency. So if frequency is multiplied by M, gauge is multiplied by 1/M.

Since the frequency multipliers for the fourth and fifth intervals are so close to 4/3 and 3/2 I have been using 4/3 and 3/2 to design fourths tunings and fifths tunings.

Of course string density is constant for plain strings, but is lower for wound strings which have space in their structures. Also as wound strings get larger, the ratio of core gauge to full gauge changes, so the density may fall a little (knuckle_head could help us with this one i'm sure).

Originally Posted by MWBass

But, then I thought further. Guitar strings usually have thinner cores. Guitar strings are made for tensions in generally the 10 lb. to 20 lb. range. If I tried to maintain bass type tensions in the 30 lb. to 40 lb. range, the strings would most likely not be able to handle the tension and break much more often.

I'm so glad you mentioned this. I've spent years of designing sets for extended range basses, sets with tensions that allow all techniques to be used like tapping, chording, Ebow, string bending. I have tried to find the ideal tension gradient across the strings for an all-purpose, all-technique, full range fretboard.

While traditional bass sets are badly designed and have higher tension on the high strings, Circle K Strings are the first to correct this and have an even tension across the strings for most of their sets.

My ideal is the opposite of tradition, a gradual and steady fall in tension from the bass B string toward the highest string, which is of course essential for instruments with thin guitar strings, but there are many other advantages. The deviation from my ideal is to keep the bass B string and all strings below it at the same tension, since these strings get very large and it seems impractical to keep upping the tension and making these subcontra strings even fatter.

What else have you come up with ...?

[SBassman]

Originally Posted by MWBass

6. Gauges: 130, 95, 70, 56

Strings: D'Add XLB130
D'Add XLB095
D'Add XLB070
D'Add XLB056

*I know these gauges are kind of ridiculous, but it goes
to show that it can even be done for regular bass tension
on a Fende Strat Scale. (25.5")

Scale: 25.5" straight

Tensions: E - 34.50, A - 34.48, D - 33.81, G - 34.77

So, the tolerance is within .96 lbs. for the whole set.

Great thread.

And - the gauges for that 25.5 are not necessarily that ridiculous. There are some of us over in the sub-short scale thread trying to figure out what custom string gauges to use for our 25.5 or 26 inch scale [can't remember] Corsair MCR1 minis.

Wanna do the math for:
120 and 110?

[SBassman]

Also, doesn't the core type affect the tension?

[Rodger Bryan]

I like the fact that you are wanting to distill this problem mathematically . Very interesting! My left brain is numb after looking at the numbers.

Today I was contemplating mixing a D'Addario set for lower tension on the D string, but they didn't have an SL .042.

String manufacturers do their best to balance tension within the set, and some make that their top priority. The other balancing issue that has to be addressed is that of string to string volume balance, in terms of the effect on the magnetic field of the pickup. Sometimes this accounts for some of the tension variation within the set. peace,
R

[ixlramp]

In my opinion most string manufacturers do not do their best to balance tension. They follow the traditional sets of gauges, because that's what sells, because that's what most bassists want, because that's what most bassists are used too and familiar with, which is mostly because those are the only sets available. It's a cyclic thing.

Many bassists don't question the traditional choice of gauges, and don't know what the tensions are. I was one of them, I was a 40 60 80 100 125 guy. D'Addario print the tension of the individual strings on their packs, which is excellent because we can see how the tension varies and start to question it. They also have the excellent D'Addario tension chart pdf, so we can learn how to calculate tension and design our own sets. We can use their Unit Weight values as approximations of other manufacturers' strings.

The vast majority of sets are of the form 45 65 80 100 130 or 45 65 85 105 135.

D'Addario's own tension data ...

XLB045 G 0.045 42.8 lbf
XLB065 D 0.065 51.3
XLB080 A 0.080 42.0
XLB100 E 0.100 36.5
XLB130 B 0.130 34.5

XLB045 G 0.045 42.8 lbf
XLB065 D 0.065 51.3
XLB085 A 0.085 48.4
XLB105 E 0.105 40.3
XLB135 B 0.135 36.1

[ixlramp]

It seems to me that tension is what keeps the vibrating mass of a string in check, it's what keeps the vibration tight, with good tone and many harmonics. So as the mass of a string increases, you need more tension to keep that mass in check.

When going down a fourth, assuming equal tension, the mass of a string is multiplied by roughly (4/3)^2 = 16/9, it almost doubles.

It seems to me that the minimum practical tension for a string increases as gauge increases. This is why really fat strings like a B string are prone to dull, flabby tone and flopping about. While guitar plain strings work fine at low tensions like 10-15 pounds.

A guitar string at 20 pounds is tight and has great tone, but a bass B string at 20 pounds is an unusable, dull, loose floppy mess.

It follows that to achieve an even 'tightness' of vibration and even tone, that tension should actually fall towards the highest string, especially for extended range guitars and basses. Also to avoid breaking any thin plain strings on the instrument.

I think that even 4 string bass sets could benefit from a slight reduction in tension towards the highest string. To tighten and brighten the tone of the E and mellow the sound of the thinner D and G.

[calvinjones]

Hello MWBass! I have acoustic guitar. The tension you are talking about, are they valid for acoustic guitar strings also..thanks for this great mathematics information.

=================
Interest Savings Account Canada

[Andrew Jones]

Contact Chance at fearless Guitars in Portland Or.

He had passions for similar string sets and has a lot of ideas about this.And even a set or to to sell you.


Aj

[MWBass]

Great stuff guys, I love conversations like this. I am sorry it has been a few days since I have posted. As I said before, I am a teacher, and progress reports (remmember those) were due this week, so it has been hectic.

To respond...

Originally Posted by SBassman

Great thread.

And - the gauges for that 25.5 are not necessarily that ridiculous. There are some of us over in the sub-short scale thread trying to figure out what custom string gauges to use for our 25.5 or 26 inch scale [can't remember] Corsair MCR1 minis.

Wanna do the math for:
120 and 110?

For 120... I would say use 120, 90, 67.5(70 or 65), 50...

For 110... I would say use 110, 82.5(85 or 80), 60, 45

When using certain starting points, you run into some bumps, but on gauges like the 110, labella offers an 83 and a 61 which can make the balance even more even. Likewise, Circle K offers 84, 61 as well.

If you want a really nice 110 set, If your bass happens to have a multiscale with a treble side scale 2 inches shorter than the bass, a 110, 85, 65, 50 works great for balanced tension.

[MWBass]

Originally Posted by ixlramp

...They follow the traditional sets of gauges, because that's what sells, because that's what most bassists want, because that's what most bassists are used too and familiar with, which is mostly because those are the only sets available. It's a cyclic thing.

It's a cyclic thing. I love it. It is so true. I'll get back to this when I do my big post in a bit. But great stuff...

Originally Posted by ixlramp

Many bassists don't question the traditional choice of gauges, and don't know what the tensions are. I was one of them, I was a 40 60 80 100 125 guy.

Ha, I have been playing DR Hi Beam LR40's - 100, 80, 60, 40 for years... I will also get back to this in a bit...

[MWBass]

This post is going to be long and drawn out, but if you read the whole thing, you should have a good idea of my argument...

I started this thread talking about balanced tension and progressive tension, and I have not touched on the progressive tension thing at all. So, I will do that now.

I would like to start with some background info on the start of this project. I know what I am about to say is more of a luthier forum thing, but it will help in laying out my argument...

I started of by deciding to build an instrument that to me would be the pinnacle build for me as a luthier. In my thinking, I decided there were two highly important factors to me in this build.

First, is stability around the board and playing surface. Ultimatly, it would be amazing to have a 12" long neck made out of titanium. The stability of the neck would be virtually perfect. you could have incredibly easy set up with incredibly low action. The problem is that every note would sound like a tuning fork being stabbed into your ear drum... obviously unpratical, but the main concepts are there...

Second, is my mild jealousy of the electric guitar player. I played regular electric guitar for about a year when I was in middle school. It never felt that comfortable. I then became a jock and did sports through high school, scholarship to college for track and did that. When I was 22 I discovered the bass which became my true love in the instrument world. I never lost my envy over the electric guitarist. I watched a lot of friends play and a lot of professionals. The just seemed to look so cool. bending there notes, pouring out there emotions, makin there stupid O faces while they wailed a solo. Then I would watch players like Jon Petrucci, with there stable structured style. It was amazing to see how fast and regimented playing could be done. I watched how he used different inversions and the shorter scale of the guitar to his advantage. Short structured movements that allowed for incredible speeds. It really blew my mind.

Now, I know what your thinking, I'm sucking guitar player a$$ right now. Don't get me wrong, I am a bass player through and through. I am in love with this instrument. I do however feel that the sheer structure of guitar provides for a much more forgiving platform for ease of use and simplicity.

So, i started thinking by building a bass that has more of a closer relationship to a guitar, I can gain the benefits of both of the points I mentoned above. By shortening the scale, the neck will have less length and less of a chance of waning or torquing. This will also allow the fret reach to be shorter allowing for shorter quicker movements, as well as opening the door to new positions that may not have been attainable before. So, we would have a more stable neck / platform and a more forgiving playing field.

I then thought, how can I further tighten the gap between bass and guitar. What about doing a six string tuned E, A, D, G, B, E. I have seen this before. but I figured this could really tighten communication between guitarist and bassist, especially when noting on the B and E strings. Bassically both players would be on the same field and could visually share the same. This may be unimportant to people who really know their theory well, but to the layman, it could be very helpful. Now a lot of people say bass tuning is better because fourth tuning makes so much more sense, but guitars have been tuned E,A,D,G,B,E forever and will not change in the near future.

So, short scale 6 string bass with guitar tuning will close the communication gap between guitarist and bassist, give a shorter neck that has more stability, and closer frets that ease playability.

Sounds great. The only thing I wanted to do was make sure that the tensions were right for bass. A lot of short scale players kept the same strings, and the tensions got super loose which change the way the bass is played. Loose could be good, but too loose could be bad.

As I told you, my main goal is to gain the greatest neck stability I could get. I figured, by shortening the neck, I give the neck less chance to move. Also, a shorter neck reduces weight in the neck. If I use an ultra thin body and a thin neck profile I can greatly reduce weight. This may be great for those who like light instruments. But then I though further. Instead of reducing weight, I can use denser woods in the neck, this would provide even more stability. I could use neck and body combinations that people steered away from for fear of too much weight. This gets closer to the whole 12 inch titanium neck I spoke of. Then if I keep my tensions in the bass tension realm, but on the lighter side, the light tension will place less stress on the neck, but if its still in the bass realm, will still play like a bass and not be flabby.

So, it all boils down to getting those tensions right. And this is where all of the experimentation began...

[MWBass]

So, now your probably wondering, where is the Balanced Tension / Progressive Tension stuff... Well, here we go.

My main veiw on the whole thing is this. My main concern is stability of the neck and intonation. My thought was that a balanced tension would be great because it would put equal strain on the neck in all directons, and improve intonation iin the sense that it would close the gap on vibration differences between different gauged strings.

So, I began there and went through all of the testing I spoke of above, and this is where I arose at the string gauges and scale lengths I came up with.

So, I then started to read up on progressive tension, and having the tension get greater as the strings get thicker. All of the arguments I read were great.

The first one I read was about feel. Not necessarily having equal tensions, but string tensions that make each string FEEL equal in tension.

Then ixlramp made some great points above.

1. You said, higher tension is necessary to keep higher mass in check... and that is great.

2. The comparison of a plain steel guitar sized string at 20 lbs. of tension, and the thick low B string at 20 lbs, of tension... also makes clear sense.

3. The brightness of tone when the Lower strings have tighter tension, to more closely match the thinner strings...

This is all great logical stuff...

So naturally I started testing progrssive gauges. I found a great one using a 110 bass and then a bunch of guitar strings. It ran from 36 down to 32 across an E, A, D, G, B, E tuning, and at a straight scale of 30". Great great stuff...

After all of the testing was over, I started to think about balanced verses progressive tension...

[mongo2]

What's the end game here?