```

Stringed instruments

L    =  String length
V    =  Speed of a wave on a string
F    =  String vibration frequency
=  .5 V / L
D    =  String diameter
A    =  String cross-sectional area
Q    =  String density
T    =  String tension (Newtons)
P    =  Tensile stress
=  T / A
Pmax =  Maximum string tensile stress before breaking
=  Tensile strength
Z    =  Material strength-to-weight ratio
=  Pmax / Q
Pi   =  3.14159
K    =  Pi L^2 F^2

The speed of a wave on a string is

V^2  =  T / (Q*A)

Hence

T = Q D^2 Pi L^2 F^2
= Q D^2 K

Tension = Density * Diameter^2 * K

where K = Pi L^2 F^2.

For example, for a viola C string, you can vary tension, density, and diameter,
but you can't vary K.  K is fixed by the frequency and length of the string.

Transform to dimensionless units.
T  =  t  *   100 Newtons
Q  =  q  * 10000 kg/meter^3
D  =  d  *  .001 meters
K  =  k  *  .0001

t = q d^2 k

String  Freq  Length  Characterisic    k
(Hz)   (mm)    tension (N)
Violin      G     195.6   320       50         1.231
Viola       C     130.4   388       60          .804
Cello       C      65.2   690      140          .636
Bass        E      41.2  1060      160          .599
Guitar      E      82.5   650      140          .903
Bass guitar E      41.2   860      220          .394

Maximum frequency of a string

T  =  Q D^2 F^2 Pi L^2             -->         F^2  =  4 L^-2 T / (Q*A)

The maximum frequency of a string is determined by the strength-to-weight ratio Z.

Z = Tensile strength / Density

MaximumFrequency^2  =  .25  L^-2  Z

Tensile   Density   Z/10^6
strength  g/cm^3
GPa
Nylon              .045    1.15      .04
Gut                .2      1.5       .13
Magnesium alloy    .4      1.8       .22
Titanium alloy     .94     4.5       .21
Steel alloy     ~ 2        7.9       .25
Tungsten           .55    19.25      .029
Kevlar            3.6      1.44     2.5
Zylon             5.8      1.5      3.9
Carbon nanotube   7         .116   60.3      Technology not yet developed

A space elevator requires a material with Z > 100

Maximum frequency (Hertz)

Gut   Steel   Zylon   Carbon
nanotube
Violin      563    781    2960    12100
Viola       465    644    2440     3160
Cello       261    362    1370     5620
Bass        170    236     895     3660
Guitar      277    385    1519     5973

Density

Density   Price
g/cm^3    \$/g
Water           1.0
Gut           ~ 1.5
Synthetic     ~ 2.5
Aluminum        2.8      <.01
Titanium        4.5       .01
Steel           7.9      <.01
Nickel          8.9       .01
Silver         10.5       .6
Tungsten       19.2       .05
Gold           19.3     24
Rhenium        21.0      6
Platinum       21.4     88
Iridium        22.4     13
Osmium         22.6     12        Densest element

Tungsten is the only dense metal that is not expensive.

Tungsten string diameters

Density = 19.25 g/cm^3
Diameters in mm.

Freq   Violin  Viola
(Hz)    (mm)   (mm)
E   660     .14     .12
A   440     .20     .18
D   293     .31     .28
G   196     .46     .42
C   130     .69     .62
F    86.9   1.03    .93
Bb   57.9   1.55   1.40
Eb   38.6   2.33   2.10
Ab   25.8   3.49   3.15

Freq   Cello
(Hz)   (mm)
A   220     .32
D   147     .48
G    97.8   .71
C    65.2  1.07
F    43.5  1.60
Bb   29.0  2.47
Eb   19.3  3.61

Freq    String   Bass
(Hz)     Bass   Guitar
(mm)    (mm)
G    97.8    .50     .72
D    73.3    .66     .96
A    55.0    .88    1.28
E    41.2   1.18    1.70
B    30.9   1.57    2.27
F#   23.2   2.09    3.02
C#   17.4   2.79    4.03
G#   13.1   3.72    5.37

Freq   Guitar
(Hz)    (mm)
E   330     .22
B   248     .30
G   196     .38
D   147     .50
A   110     .67
E    82.5   .90
B    61.9  1.20
F#   46.4  1.60
C#   34.8  2.13
G#   26.1  2.84

A string can be made an octave lower by doubling the diameter.
Diameter * Frequency = Constant

Guitar

Tension = 140 Newtons

String  Freq    Material  Density  Diameter
(Hz)              (g/cm^3)   (mm)
G     782.2   Zylon       1.5      .34
D     586.7   Zylon       1.5      .45
A     440.0   Zylon       1.5      .60
E     330.0
B     247.5
G     195.6
D     146.7
A     110.0   Tungsten   19.3      .67
E      82.5   Tungsten   19.3      .90
B      55.0   Tungsten   19.3     1.34
F#     36.7   Tungsten   19.3     2.01
C#     24.4   Tungsten   19.3     3.03

Characteristic tension

Height of   Height of
String    Tension (N)       top string  bottom string
(mm)    E   A   D   G   C    (mm)       (mm)
Violin      320    80  50  45  45         3.2        5.2
Viola       388        65  55  55  55     4.8        6.2
Cello       690       160 130 130 130     5.2        8.2
Bass       1060       160 160 160 160

```