String Theory

Using and Abusing the Taylor Equation

Rick Kemper, Sligo Harps


Brook Taylor 1685-1731

Remembered primarily for his work in Mathematics and Physics, Brook Taylor was also an accomplished artist and musician. Brook Taylor’s work propelled the budding science of differential calculus. Physicists admired his papers on capillary action, magnetism and thermometers. His experiments with strings helped unite the arts of music and luthiery with the practical applied mathematics and physics. His equation gave physicists and instrument builders a way to accurately predict the fundamental behavior of a plucked string;

 

T= M (2L F)2

Where

T is the tension of the string

M is the linear mass

F is the Frequency

L is the Length of the string

 

Let us take a fairly simple case, the 440 Hz A just above middle C. On most nylon strung harps, this will be about .045” in diameter, about 16 inches long. Nylon has a density of .0385 lbs/cubic inch. The Taylor formula can help us estimate the tension on this string;

 

First calculate the linear mass,

 

M.045 = Cross Sectional Area x = (.045in/2)2 = 1.599 x 10-7

 

Then use that to calculate the tension on the string

 

T = M (2L F)2 = 1.599 x 10-7 (2 x 16 inches x 440 Hz)2= 31.4 lbs

 

Now string makers use this equation to calculate the total tension on the harp, and it works well for that. The Taylor formula can also be used to set some likely limits on the length of a string given specific material and target pitch.

 

To illustrate this, let us take the example an Italian Lute builder, Giuseppe. Our Guiseppe, he wants to make an instrument with a 523 Hz C string. Can Taylor’s formula tell him what material or gauge to use? It does not. It can be used to set some limits on the length. Experience may tell our Giuseppe that a .080 nylon string sounds way too thunky for a lute at this pitch or that the fine .011 string he uses on his soprano lutes will be far too quiet. Giuseppe, he has been building the fabulous instruments for many, many years and he decides that this string should be .040 nylon.

 

We can now rearrange the terms of Taylor’s formula and calculate the longest length possible based on the breaking strength of 44,600 lbs/square inch for Tynex Nylon strings. First we need to calculate the linear mass and breaking tension for .040 nylon;

 

M.040 = (.040in/2)2 = 1.253 x 10-7

 

T.040max= Breaking Strength x Area = 44,600 lbs /in2 x (.040in/2)2 = 56 lbs

 

Now we can use Taylor’s formula to find what the breaking length for the .040 A string will be;

 

L.040max = = = 20.2 inches

 

An interesting thing happens when we decide to calculate the maximum breaking length for .025 nylon;

 

L.025max = = = 20.2 inches

 

In fact, for any monofilament 523 Hz C string

 

L.025max = L.028max = L.032max = L.036max = L.040max = L.045max =20.2 inches = Lmax for any nylon string!

 

The breaking length is independent of he string’s gauge! Taylor’s formula cannot be used to dictate a “right” string gauge (thickness) for a particular note. A harp designer can use heavier gauge strings to give different strings more power (the string has more energy to shake the sound board). He has limits because the increased tension will break the instrument at some point. A good string designer will also strive to give the harp an even “feel” across the harps range (more about that T/L ratio and “feel” later).

 

Orchestral Pedal harps place a premium on volume – the harp needs to be heard over a hundred other instruments. A designer makes these harps louder by using large soundboards, heavy gauge strings, and string lengths that are near the maximum. Playing these harps at volume requires a forceful technique and it is no surprise that many professional players develop stress injuries at some point in their playing career.

 

Now, few harp builders are willing to put up with string lengths that are right at the design limit – the harp will just break too many strings. Most designers will use a margin of safety and set an arbitrary limit, usually 65-95% of the string’s breaking strength. Using the Taylor equation, and that 95% limit, a designer can produce a table calculating the maximum lengths for each note, for nylon string:

 

Note

Pitch (Hz)

Max Length (inches)

C7

2093

4.9

B6

1976

5.2

A6

1760

5.9

G6

1568

6.6

F6

1397

7.4

E6

1319

7.8

D6

1175

8.8

C6

1047

9.8

B5

988

10.4

etc.

 

 

 

The lower limit for string length is not as distinct. Most designers will set an arbitrary limit where the string “feels to floppy” to sound effectively. Some place this limit at 40% of the breaking strength; others are willing to bend the rules, especially at the bass end to 15% of the breaking strength. If we arbitrarily set that lower limit at 30% we can generate a useful table that calculates the theoretical upper and lower limits for monofilament string lengths:

 

Note

Pitch

30% Min

95% Max

 

Note

Pitch

30% Min

95% Max

C7

2093

2.8

4.9

 

F4

349

16.6

29.5

B6

1976

2.9

5.2

 

E4

330

17.6

31.3

A6

1760

3.3

5.9

 

D4

294

19.7

35.1

G6

1568

3.7

6.6

 

C-m

262

22.1

39.4

F6

1397

4.1

7.4

 

B3

247

23.5

41.7

E6

1319

4.4

7.8

 

A3

220

26.3

46.8

D6

1175

4.9

8.8

 

G3

196

29.5

52.6

C6

1047

5.5

9.8

 

F3

175

33.2

59.0

B5

988

5.9

10.4

 

E3

165

35.1

62.5

A5

880

6.6

11.7

 

D3

147

39.4

70.2

G5

784

7.4

13.1

 

C3

131

44.3

78.8

F5

698

8.3

14.8

 

B2

123

46.9

83.5

E5

659

8.8

15.6

 

A2

110

52.6

93.7

D5

587

9.9

17.5

 

G2

98

59.1

105.2

C5

523

11.1

19.7

 

F2

87

66.3

118.0

B4

494

11.7

20.9

 

E2

82

70.3

125.1

A4

440

13.2

23.4

 

D2

73

78.9

140.4

G4

392

14.8

26.3

 

C2

65

88.5

157.6

Graphed out, they look like this:

 

Some observations – Bass string lengths greater than 60 or 70 inches are problematic for two reasons. The harp is unreasonably LARGE (to transport) and the amplitude of the plucked string will require greater spacing to keep it from hitting the adjacent strings or the harpists fingers as she reaches in to pluck an adjacent string. Spacing greater than 5” per octave or so reduces the player’s ability to play useful intervals like octaves and tenths with the left hand. Because of these considerations, most designs greater that three or four octaves will employ wound bass strings. Wound bass strings can be made in a variety of configurations. The bottom notes of five and six octave harps employ a steel core over-wrapped with bronze or copper wire. Most strings with a metal core use fiber bedding between the core and wrap. The core carries the tension, and the over wrap can dramatically increase the string’s linear mass, allowing a relatively short string to achieve big fat bass notes.

 

The trick is to come up with a string band that transitions gracefully from monofilament to wound strings without any abrupt changes in string’s tone, length or feel.

 

The Taylor formula is still used for wound strings, but calculating the linear mass and becomes more complex. Joseph Jourdain, Bolles and others have debated some variations on these formulas. In practice, most harp builders/designers turn to string makers who use computerized programs created by Bolles, Jourdain, Cady or others to do the tiresome calculations for wound strings. The programs produce tables that list the string’s pitch, length and tension along with two other parameters, %TS and T/L that are useful in string band design.

 

%TS or TEN% is short for Percentage of the tensile strength. This parameter gives the string’s tension as a percentage of its breaking strength. Let us calculate the tension on the 523hz C on a L&H Troubadour. The vibrating length is 14.4 inches. It is usually .040 or .045 gauge nylon. Let say it is .040 nylon. We can use the Taylor formula to estimate the tension on the string:

 

T= M (2L F)2 = 1.253E-07 (2 (14.4in)523Hz)2 = 28.4 lbs Tension

 

The breaking strength of .040 nylon was calculated earlier at 56.0 lbs, so

 

%TEN = 28.4/56.0 =.507 or 50.7%

 

It is important to note that the Taylor equation does not “prove” anything about this string or its suitability on the Troubadour or any other harp. Designers merely use the equation along with rules of thumb that have been developed over the years (specifically “the TEN% should be between 30 and 80”) to figure out if the string is going to break a lot or be really floppy.

 

T/L is called the Tension to length ratio and is the strings Tension divided by the length. Using the same string referenced above,

 

T/L = 28.4 lbs/14.4inches = 1.97

 

By itself, for an individual string, T/L is not a very useful number. The idea is that longer strings should have more tension. If they did not, they would feel floppy relative to shorter strings with the same tension. Harpists seem to feel most comfortable with string bands that start with a T/L ratio around 3-6 (Using the English units measurement of lb/in), typically descending to about .8 – 1.0 in the fifth octave. When they develop a string table, makers will usually scan down the T/L column looking for any sudden changes.

 

There are three good reasons for bass strings to have greater tension than the treble strings. Most music requires longer ringing chords in the bass end. To get that sustain, you need to take the initial energy (the pluck) and distribute it over time. Second, Higher tension limits the amplitude of the plucked strings which allows them to be spaced reasonably close. Lastly, bass notes need a significant amount of energy to be heard as well as higher notes. The speakers that make the bass notes are big and their cones are heavy (relative to a tweeter).

 

Silvia Woods with HarpNow, let’s take a look at some harp string bands to see how the TEN% and T/L vary among different designs. I will start with a vintage Witcher design, a 32 string harp similar to the one shown on the Sylvia Woods book, Teach Yourself to Play the Folk Harp. With a ruler and micrometer, I can measure the harps’ string lengths and gauges. The owner readily admitted that these are not the original gauges, and that the harp had been restrung several times. Using the physical properties of the nylon and Taylor’s equation, I can use a spreadsheet to calculate the tension on each string and the resulting T/L and TEN%.


 

Tension; The thinner gauge strings at the treble end only have 12 lbs on tension on them. The tension climbs fairly evenly up to 43 lbs at the bass end. All together there is about 840 lbs of tension on the harp.

 

Percent Breaking Strength; %TEN starts near 60% of the breaking strength, and descends in to about 30% at the last monofilament string, #22. It jumps up back up to 54% for the first Nylon/Nylon wound string, and then descends again to about 40%. All but one of the strings fall in between that 30-60% range. It is a pretty good medium tension design.

 

Tension to Length Ratio; The T/L ratio starts at 3.3 and descends fairly evenly in to 1.1 at the bass end. There are no big jumps in the T/L that would result in an uneven feel.

 

Now, let us take a look at three other models. What can these charts do to help us understand string band design?

 

The Music Maker’s Gothic 31 (Monofillament Strings):

 

Looking at the tension chart, one notes that the bass strings do not have as a high a tension as the Witcher. All together, they total about 640 lbs. This makes sense for kit harp, where inexperienced builders may not be able to produce a super strong frame with limited tools and clampage. The long bass strings give it a very low %TS (descending down to 15%) and T/L, (0.4). The bass strings that are quite loose. Music Makers offers an alternate set of wound strings that bring up the %TS and T/L considerably in that last octave.

 

Triplett Axline

 

Like the Witcher, the string set I found on this well-used Axline was probably not factory. I believe Triplett and Dusty both use bronze core strings in the bass, and these were steel core. There is a small jump in the total percent breaking strength (%TS) at the transition to wound nylon core nylon wrap strings, but it is not as dramatic as it was in the Witcher. The tension to length ratio drops evenly staying just above 1.0 at the bass end.

 

Fisher Eireann

The Eireann has been a real hit with hard core Celtic Traditionalists. The finely gauged incremental sizes of the Savarez monofilament strings yield some very smooth lines. Without any the nylon/nylon transition strings, the Tension, %TS and T/L all show a significant “jump” at that transition from monofilament to Steel core bronze wrap bass strings. When I play a descending scale on the harp, I can certainly hear that change, but this transition does not seem to bother many players.

 

Despite many these differences, there are some interesting similarities among the four harps,

 

Tension starts at 10-20 lbs, and tends to climb higher, reaching a maximum of 35-50 lbs in all four designs

%TS starts high and tends to go lower staying between an upper bound of 70% at the treble end and 15-20% at the bass end

T/L Starts around 3 and drops to about 1. The Music Makers harp is an exception, falling to 0.4.

 

These are “rules of thumb”. There are some very good sounding designs that push or break these limits. Some harp designers take these rules of thumb and attempt to translate them into equations that they claim will allow them to calculate the “optimal” string gauge or length for the “perfect” instrument. I think this is folly. After carefully measuring the strings on 60 harps and noting the diversity among the best sounding ones, I can confidently state there is not a single optimal solution. Selecting the best string lengths and gauges will depend on several other parameters, including but not limited to:

        The soundboard materials used

        The strength and technique of the player

        How loud the harp needs to be

        The builder’s ability to fabricate sound boxes, boards and neck/pillar assemblies

 

I would be remiss if I did not to not provide a table outlining the properties of various stringing materials. You can use these with the Taylor’s formula above to analyze monofilament strings. If you want to analyze composite (wound) strings, you can use the equations from Bolles, Jourdain, or Cady.

 

Physical Properties of String Materials

 

Density (lb/in3)

Density (kg/m3)

Tensile Strength

(lb/in2)

Tensile Strength

(kg/cm2)

Tynex

.0385

1,067

44,600

3,136

Synthetic Silk

.0412

1,140

52,000

3,656

Gut

.0469

1,300

52,000

3,656

Steel music wire

.2829

7,830

325,000

2,2850

Bronze

.3204

8,870

125,000

8,788

Brass

.3048

8,437

110,000

7,734

Copper

.3226

8,930

6,1000

4,289

Sourced from; Joseph Jourdain’s The Folk Harp Stringband, fortepiano.com

 

Occasionally, some well meaning nut will want to change the string material with which the harp is strung. Nylon, Gut and Fluorocarbon have densities and strengths that are close enough to be interchangeable provided one selects gauges that do not increase the overall tension of the harp. What if a player wants to substitute metal strings for nylon?

 

Cautious technicians will sternly warn that substituting different string materials will void any warranty for the harp. Often the levers do not work. If the harp owner persists in their wishes, can we use Mr. Brook Taylor’s equation to discourage them farther?

 

Using the Physical Properties of String Materials, let’s generate a table of monofilament string lengths for likely pitches so that the string will be at 30% and 70% of its theoretical breaking strength for brass, nylon and steel strings.

 

 

 

Brass

Nylon

Steel

 

 

30%

70%

30%

70%

30%

70%

Note

F (Hz)

BL30

BL70

NL30

NL70

SL30

SL70

C7

2093

1.5

2.4

2.8

4.2

2.8

4.2

B6

1976

1.6

2.5

2.9

4.5

2.9

4.5

A6

1760

1.8

2.8

3.3

5.0

3.3

5.0

G6

1568

2.1

3.1

3.7

5.6

3.7

5.6

F6

1397

2.3

3.5

4.1

6.3

4.1

6.3

E6

1319

2.5

3.7

4.4

6.7

4.4

6.7

D6

1175

2.8

4.2

4.9

7.5

4.9

7.5

C6

1047

3.1

4.7

5.5

8.5

5.5

8.4

B5

988

3.3

5.0

5.9

9.0

5.8

8.9

A5

880

3.7

5.6

6.6

10.1

6.6

10.0

G5

784

4.1

6.3

7.4

11.3

7.4

11.2

F5

698

4.6

7.1

8.3

12.7

8.3

12.6

E5

659

4.9

7.5

8.8

13.4

8.7

13.4

D5

587

5.5

8.4

9.9

15.1

9.8

15.0

C5

523

6.2

9.4

11.1

16.9

11.0

16.8

B4

494

6.5

10.0

11.7

17.9

11.7

17.8

A4

440

7.3

11.2

13.2

20.1

13.1

20.0

G4

392

8.2

12.6

14.8

22.6

14.7

22.5

F4

349

9.3

14.1

16.6

25.3

16.5

25.2

E4

330

9.8

15.0

17.6

26.8

17.5

26.7

D4

294

11.0

16.8

19.7

30.1

19.6

30.0

C-m

262

12.4

18.9

22.1

33.8

22.0

33.7

B3

247

13.1

20.0

23.5

35.8

23.4

35.7

A3

220

14.7

22.4

26.3

40.2

26.2

40.0

G3

196

16.5

25.2

29.5

45.1

29.4

44.9

F3

175

18.5

28.3

33.2

50.7

33.0

50.5

E3

165

19.6

30.0

35.1

53.7

35.0

53.5

D3

147

22.0

33.6

39.4

60.2

39.3

60.0

C3

131

24.7

37.7

44.3

67.6

44.1

67.3

B2

123

26.2

40.0

46.9

71.6

46.7

71.3

A2

110

29.4

44.9

52.6

80.4

52.4

80.1

G2

98

33.0

50.4

59.1

90.3

58.8

89.9

F2

87

37.0

56.6

66.3

101.3

66.1

100.9

E2

82

39.2

59.9

70.3

107.3

70.0

106.9

D2

73

44.0

67.3

78.9

120.5

78.6

120.0

C2

65

49.4

75.5

88.5

135.2

88.2

134.7

In this table,

BL30 = Length of a Brass string at 30% Tension BL70 = Length of a Brass string at 70% Tension

NL30 = Length of a Nylon string at 30% Tension NL70 = Length of a Nylon string at 70% Tension

SL30 = Length of a Steel string at 30% Tension SL70 = Length of a Steel string at 70% Tension

 

It is very interesting to note that limiting lengths for steel music wire and nylon are identical (when rounded to the nearest tenth of an inch). Remember these limits are set by two “rules of thumb”, that most successful string sets start with the strings at about 70% of their breaking strength in the treble end and descend to about 30% of their breaking strength in the fourth octave where many designs start using wound strings.

 

Are they really interchangeable? Well, I have restrung a L&H folk harp (designed and initially strung with Gut) with steel strings. To keep the tension in the same range as the gut string harp, the steel strings had to significantly finer. The second incarnation as a steel strung harp worked well. It did not break strings, the harp did not implode, and the client (an ardent fan of Patrick Ball) loved the new sound.

 

Dubious readers will say, “Well, OK maybe it is possible to replace nylon and gut with finer steel strings, but this table proves that you cannot replace nylon or gut strung harps with brass.” But does the table really “prove” anything like that? What if we shifted the range down six notes?

Nylon Length Ranges

 

Brass Length Ranges

 

 

30%

70%

 

 

 

30%

70%

Note

F (Hz)

NL30

NL80

 

Note

F (Hz)

BL30

BL70

C7

2093

2.8

4.2

 

D6

1175

2.8

4.2

B6

1976

2.9

4.5

 

C6

1047

3.1

4.7

A6

1760

3.3

5.0

 

B5

988

3.3

5.0

G6

1568

3.7

5.6

 

A5

880

3.7

5.6

F6

1397

4.1

6.3

 

G5

784

4.1

6.3

E6

1319

4.4

6.7

 

F5

698

4.6

7.1

D6

1175

4.9

7.5

 

E5

659

4.9

7.5

C6

1047

5.5

8.5

 

D5

587

5.5

8.4

B5

988

5.9

9.0

 

C5

523

6.2

9.4

A5

880

6.6

10.1

 

B4

494

6.5

10.0

G5

784

7.4

11.3

 

A4

440

7.3

11.2

F5

698

8.3

12.7

 

G4

392

8.2

12.6

E5

659

8.8

13.4

 

F4

349

9.3

14.1

D5

587

9.9

15.1

 

E4

330

9.8

15.0

C5

523

11.1

16.9

 

D4

294

11.0

16.8

B4

494

11.7

17.9

 

C-m

262

12.4

18.9

A4

440

13.2

20.1

 

B3

247

13.1

20.0

G4

392

14.8

22.6

 

A3

220

14.7

22.4

F4

349

16.6

25.3

 

G3

196

16.5

25.2

E4

330

17.6

26.8

 

F3

175

18.5

28.3

D4

294

19.7

30.1

 

E3

165

19.6

30.0

C-m

262

22.1

33.8

 

D3

147

22.0

33.6

B3

247

23.5

35.8

 

C3

131

24.7

37.7

A3

220

26.3

40.2

 

B2

123

26.2

40.0

G3

196

29.5

45.1

 

A2

110

29.4

44.9

F3

175

33.2

50.7

 

G2

98

33.0

50.4

E3

165

35.1

53.7

 

F2

87

37.0

56.6

D3

147

39.4

60.2

 

E2

82

39.2

59.9

C3

131

44.3

67.6

 

D2

73

44.0

67.3

B2

123

46.9

71.6

 

C2

65

49.4

75.5

 

If you scan across the rows to the NL and BL columns you will find the length ranges are quite close, if not identical. The discrepancies have to do with the half steps and whole steps used in common harp tuning scales. If one is willing to shift the range of the instrument and make the appropriate adjustment in string gauges, it would appear that many harps could be strung with a broad variety of different string materials.

 

I will emphasize again, before substituting different strings materials, the harp owner needs to understand that the levers may not work. This kind of experimentation will void any warranty for the harp. The harp owner must be willing to bear those risks.

 

Conclusions

On its own, the Taylor’s formula does not predict a perfect or optimal string length or tension. Taylor’s Equation is a useful tool that can help harp designers limit string breakage or string lengths that are so short, floppy strings will not sound well. Derivative measures like the tension to length ratio (T/L) can help the designer develop string bands with an even feel.

 

Adventurous experimenters will find Taylor’s Equation can be used as a tool to predict substitute strings, pitch ranges and gauges that will approximate the tension of the factory strings.

 

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