Some Findings

on the Tension vs Pitch of a 27-5/8" String

Here is where we look at what all these numbers mean, and how we can use this info to choose strings more carefully than we could before. The discussion is divided into five topics:

**I. General remarks on the findings. **Here we talk about what we have learned about all strings in general (but especially about wound strings).

**II. Exploring the data. **We find out whether the "engineering assumption" that wound strings behave more or less like plain music wire is warranted. And we find out something about an artificial chemical element.

**III. Using the data. **Discuss how to choose strings well, using Arto Wikla's String Calculator as a tool.

**IV. Wrapping it up.** A short summary of what we've learned about choosing and using strings.

**V.Calculating your own string set.** A short example of how to use my data tables to compute the optimum diameter and tension for strings of different lengths (including the length on your own dulcimer!).

These are the reference materials I'll be citing from time to time in this discussion:

1. My data and chart and string setup table

2. Steven K. Smith's article, "String Choice for Dulcimer Technoids"

3. Arto Wikla's String Calculator

**I. General remarks on the findings**

First, note that these data are on a 27-5/8" scale length. If your string is shorter, your tension at a given pitch will be lower.

As Tom Smothers has already observed in a posting on the Dulcimer-List, the data for many of the strings fall into three regions, defined by sharp changes in the slope of the tension vs pitch line:

1. Low tension region. In this region, increasing tension does not result in as much achange in pitch as in region 2. This suggests that the wire has not been sufficiently tensioned to perform as expected.

2. Elastic region. In this region there is a long series of tension changes which produce linear changes in pitch. The string is operating elastically, and, presumably thesis the region of best musical performance.

3. Inelastic region. In this region, the string has become over-stressed and is no longerable to return fully to its un-stressed condition. The tension beyond which an elasticbody cannot be stretched without permanent change in shape is called Young's Modulus.(We're all familiar with the concept if not the name, because you cannot get through a whole lifetime in this culture without popping at least one rubber band! You've all exceeded Young's Modulus of rubber at least once.) So in this region, it takes more and more tension to get the string to vibrate at a higher pitch, as it is beginning to just stretch out under the load. The string is starting to break.

Notice also that wound strings of equal diameter behave very similarly to tension. Though their different materials may indeed result in their having differences of timbre, you can substitute one string for another of given diameter, as far as tension is concerned.

Thirdly, notice the effect of changing the string between two pitches. Moving a .012 string 27-5/8" long from A=220 to D=294, for instance, changes the tension from 4.8 to 9.0 Kg, almost doubling the tension! Notice that at about Db, the string begins to enter Region 3, and is beginning to stretch (a precursor to breaking).

As Steve notes in "String Choice for Dulcimer Technoids," the density of steel is .283 lb/in**3 (read in**3 as "cubic inch"). That corresponds to 7842 kg/m**3, very close to the 7800 Kg/m**3 quoted by Arto Wikla at his website for string calculations. Perhaps we can learn something from this data about how wound strings approximate plain music wire. If we go to Arto's calculator and plug in the tension required to raise a 27-5/8" (72 mm) .024" (.6096 mm) wire to A=110, we find 5.854 Kg is required. Comparing that to the figures for the DR Strings DR024 (4.8 Kg) and the GHSS Steel Wound LW24 (4.1 Kg), we immediately see the effect of the windings: the mass of the windings allows the string to vibrate at an lower pitch than could a plain wire of equal diameter at a given tension. In other words, you would need to put a plain .024" wire under about 20% more tension than .024" wound wire. Can we generalize on that? Here's a little table of tension vs tension for a plain wire and for a wound wire, taken at pitch A=110:

Dia |
Wound |
Plain |
% Diff |

020 |
4.06 |
2.8-3.0 |
45-35 |

024 |
5.85 |
4.1-4.8 |
42-21 |

029 |
8.55 |
6.4 |
34 |

030 |
9.15 |
7.6-8.0 |
20-14 |

032 |
10.4 |
8.2-8.7 |
26-19 |

Looks like it's all over the map. We'd be better off making the measurements of tension vs pitch, it would seem, rather than making our standard "engineering assumption" that a wound string will behave like a plain wire.

Is there anything to be learned about the effective mass of a wound string? If we plug in our .024" wire into the equation and adjust the mass until the tension at A=220 is 4.8 Kg,the density turns out to be 6430 Kg/m**3, or 81% of the mass of mass of music wire, or.232 lb/in**3. That's about the density of Tellurium (.2257), an artificial element made in atomic accelerators, or Antimony (.2390). See what fascinating stuff you learn on the Dulcimer List??!

1. One general lesson to be drawn here is that if you break strings often and can associate that to lots of re-tuning, try getting lighter strings so you will be operating at lower tension.

2. So, what tension should I use? Often, musicians will wonder what the effect of heavier or lighter string will be on their instruments. Here is what Arto Wikla says about varying the diameter (and thus the tension) of strings on his lutes:

- "My first thought for a lute string tension is about 3 Kg.
- "But the matter is quite complicated: For top strings I use more (for example my archlute's 67 cm top g has 4.0 Kg, but that is quite much). If the instrument has single strings the tension can be more than with double courses. The octave doubles should have less tension than their counterparts. It might be good idea to lessen the tension slightly towards bass. When there is difference of length between neighboring courses (lutes with bass extension), the lowest short basses could have more tension than the highest long basses,to lessen the difference of sound quality.
- "Experimenting is the only way of finding well sounding tensions for each lute and string material. And the sound of all courses should be balanced..."

My experience is that 6-9 Kg is right for a 27-5/8" dulcimer string, depending on the kind of action you want. See my article on the subject of "Choosing Strings" for more details on what I mean by that at strings but I think I'll be revising my baritone and Medium Tension setup in light of my findings. (For one thing, I'm going to rename the "Medium Tension Setup" to "High Tension Setup", in view of how close the strings are operating to Region 3!) Arto's advice to slightly decrease the tension of the bass string is interesting and suggestive of getting a little more "interesting" voice from the low end.

3. Using my Tension vs Pitch tables. If you want to pursue some experiments with your strings, try choose new sets that will keep them all at the same tension.

So for instance, if you have a yen to see what replacing your D=294 .012" with .014", change your A=110 and D=147 so they have more or less the same tension. First, go to Arto Wikla's String Calculator and find the tension of your .014" string at your scale length at D=294. (Hints: multiply .014" times 25.4 mm/inch to convert. And use 7842 Kg/m**3 as the density of music wire.)

Next, plug in the pitch and the same tension for A=110 and solve for diameter of your A string.

Then go to my Tension vs Pitch table to find a wound string that is at D=147 for that same (or a little lower) tension. You've solved the problem for a DAd tuning. Using similar logic, you can solve for any other tuning also.

If you find that you need a string that doesn't appear on my table, send me a string that you think you need, and I'll measure it and add it to the table for you. That could be especially helpful for a bass dulcimer.

1. Use Steven K. Smith's "String Choice for Dulcimer Technoids" for basic string physics

2. Use Arto Wikla's String Calculator for a fast check on pitch vs diameter vs tension for wire strings

3. Use my Tension-vs Pitch Chart for getting the right wound string to match the tension of your wire strings

4. If you are breaking strings for no seeming reason except for re-tuning, you may be operating in Region 3. Get a lighter string.

5. Don't over-stress or under-stress your strings or your dulcimer. The music will sound much more "interesting" when you choose strings that will have the correct tension for your instrument. At 27-5/8", that's between 6-9 Kg. Play around with different string diameters and see if your tone improves.

**V.Calculating your own string set**

You can convert my data for a 27-5/8" string to any other string with a simple formula found on Steven K. Smith's site. Steven observes that the tension in a string can be expressed as a function of length, diameter, material density r(rho), frequency, and gravity (actually g, the gravitational constant is not a variable as long as all dulcimers are played on earth):

So the tension in a string of diameter (d) at the same pitch (f) of two different lengths, L (my string length) and L' (your string length), can be expressed as a ratio of the squares of their lengths:

, where F is the tension of the string at my length, and F' is the tension at your length.

The square of 27-5/8" is 763.14, so

So, to find the tension your string is operating under, at the pitch you have it tuned, find the tension and pitch for its diameter on my chart, multiply the length of the string by itself, multiply that result by the tension you read from my chart, then divide by 763.14 The result is the tension your string of length L is at.

As an example, say your fretboard is 26", and you have a .024 string for your low D string. What tension is the string at? My chart gives a 27-5/8" long .024" string tuned to D as 8 Kg. (Ignore the DR 24 string, as it is very unusual).

First, square the length of your string's length: 26x26=676

Multiply by the tension of a 27-5/8" string:

676x8=5408

Then divide by 763.14:

5408/763.14=7.08

Your string is at 7.1 Kg of tension.

And conversely, if you want to find what string would be at a specific pitch at a given tension, you can solve the equation for F instead of F':

Go to Arto Wikla's String Calculator and plug in the tension you want the string to be (F') and the length of your string squared () and find F. Then look in the table for that value of tension for the string you want at the pitch you want, and the head of the column will tell you the diameter of the string you want.