String Article by Bill Foley

One of the never ending pleasures in life is the joy of playing your guitar when it is sounding really good. What makes a guitar sound really good is a complex issue, but it is a topic nonetheless that can be analyzed, broken down into its constituent components, and further analyzed.

One of the key constituent components of a good sounding guitar, or any string instrument, is the user's choice of strings. This is where things get interesting and fun.

The marketplace is a tangle of brands and constructions, so it is helpful to have a rudimentary understanding of string behavior to better interpret the manufacturers' accompanying claims.

Modern  instrument strings are the culmination of many millennia of intellectual advancements from a disparate group of innovators. Prehistory hunters, classical philosophers, mathematicians and physicists all have contributed to the contemporary body of knowledge that governs the manufacture of today's strings.

Our most elementary experiences with musical string behavior stretch back into the unwritten voids of prehistory. One would surmise that anyone who had ever used a hunting bow would have noticed a relationship between a tensioned string and musical pitch, but when and where this event may have happened is a matter of conjecture. The oldest archaeological evidence I have seen is a picture of a drawing made circa 15,500 BC in the Magdalenian cave of Les Trois Freres in southern France. This cave art shows

what may be interpreted to be a hunter using a musical bow. The musical bow uses the mouth as a resonant chamber much like a jaw harp. A hollow tortoise shell or a gourd could also be attached to the bow to increase volume. The musical bow shows use of one length of string with variable tensioning.

From 3000 BC onward the historic record (pottery, murals, written records, etc.) shows use of differing string lengths and tensionings in multi-string instruments. Harps, lyres, citharas and lutes strung with gut and horsehair strings appeared at this time.

The first major advancement in the study of string behavior occurred sometime during the lifetime of Pythagoras of Samos, who lived from c.569 BC to c. 475 BC. Pythagoras' primary contribution to civilization was the revolutionary concept of mathematical analysis. Aristotle wrote that "the Pythagorean thought that things are numbers, and that the whole cosmos is a scale and a number." In other words, Pythagoras pioneered the abstraction of objects into numbers.

Pythagoras' method of perpetuating his deeply held beliefs should also be noted. In his time of many gods and great legends, passed along by word of mouth, he chose to establish a religious school to further his philosophy. Inspired by his studies in Egypt at the temple of Diospolis (the only one who would accept him), he structured his school into a society of a secretive inner circle of "mathematikoi" and a not so exclusive outer circle of' �acousmatics". He moved the school from Samos to Croton in southern Italy c.518 BC. After his death, the schools expanded to other cities and split into several factions. As is often the case with any new religion or philosophy, the pre existing old schools can be less than welcoming to any perception of competition. In 460 BC the meeting houses were destroyed, between 50 and 60 Pythagoreans were killed in Croton, and the survivors scattered to parts unknown. Truths tend to endure fires and murders; Pythagoras' math and methods became entrenched in civilization. When it comes to the power of word of mouth, we who live now are not so unlike those who lived long ago.

One of Pythagoras' experiments involved the study of differing lengths of strings under differing tensions. He observed that harmonic tones are produced when the ratios of differing lengths of strings under equal tension are whole numbers. From this base of knowledge he organized the first science derived Western musical scale.

Pythagoras' main contribution to the understanding of string behavior was the ability to use numbers to describe relationships between string lengths, tensions, and pitches.

Though the art of mathematics and music progressed, fourteen centuries elapsed before the next significant development in string evolution occurred. Around 1100 AD, a Westphalian monk named Theophilus wrote one of the earliest known descriptions of wire drawing. The process involved hammering a metal bar into a long, thin rod, then pulling the rod through iron dies to create wire. By the early 1300's wire drawing had become an important industry, and within the next century iron and brass strings began to appear on instruments. Wire strings not only were used on guitars, mandoras, and bass lutes, but gave rise to a whole new family of string instruments, which included  zithers, citterns, the Irish harp, psaltery, clavichord and others during the 15th and 16th centuries.

In 1588, lutenist and music theorist Vincenzo Galilei performed experiments showing that the ratios of tensions of strings of equal lengths tuned an octave apart is 4:1, disproving the accepted notion that the ratio was 2:1. His methodology of using experiment to try to answer theoretical questions was emulated by his more renowned son, Galileo Galilei.

STRING CHOICE

In 1636 French mathematician Marin Mersenne (whose translations were responsible for the spread of Galileo's works outside of Italy) formulated three basic laws governing string motion:

1. When the string's density and tension remain constant, but the string's length is varied, the string's musical pitch (frequency) is proportional to its length. (This is Pythagoras' law restated.)

2. When the string's density and length remain constant, but its tension is varied, the string's musical pitch is proportional to the square root of its tension.

3. For different composition strings of constant length and tension, the strings' musical pitches are proportional to the square root of the weight (density) of the strings.

Mersenne proved simply that as tension is increased in the string, the forces tending to pull the string back to its original position are increased, and the motion of the string is proportionately increased.  Conversely, if the mass of the string is increased and the same tension is applied, the motion of the string is proportionately decreased.

In 1686 Isaac Newton described mathematically how sound travels in his Principia, and in 1747, Jean d'Alembert derived the general wave equation in his study of vibrating strings.

These studies describe the two fundamental types of wave motion: longitudinal and transverse.

Longitudinal waves propagate through a medium by a series of compressions and rarefactions of the particles of the medium. This spring like motion is the primary mechanism by which the resonant character of an instrument's parts (bridge, body, etc.) bounce back into the string.

Transverse waves propagate through a medium by disturbing the particles of the medium in a direction perpendicular to the direction of the wave flow. The transverse wave is the waveform of primary interest in the description of string behavior.

Idealized, each transverse wave starts from an equilibrium point of no motion, moves up to a peak and is pulled back down to the equilibrium point, overshoots the starting point by a distance equivalent to its peak, and returns to the equilibrium point. This is considered one complete cycle of the wave. The time required for one complete cycle is called the period of the wave and is represented by, t. The length covered from start to finish of one cycle of a transverse wave is called the wavelength and is represented by, λ.   The number of cycles a wave completes in one second is called frequency, and is represented by, f.  The velocity of a wave in a string is represented by, v, and can be defined as

                           (1.)

Combining Mersenne's second and third laws we can get a more string specific definition of wave velocity, thus

                                  (2.)

Equation ( 2.) shows wave velocity as the product of the square root of the string's tension, t,  divided by the string's average weight, or linear mass density, μ.

So, if we want to calculate any frequency a string will support, we can combine equations (1.) and (2.), producing

                        (3.)

If we simply want the fundamental, or first harmonic, of the open string, equation (3.) becomes.

                         (3.a.)

Since the length of vibrating string between the nut and saddle is one half wavelength, the λ product becomes twice the length,, of the open string.

Of special interest for us in equations (2.) and (3.) is the �, or mass density value of the string. We can see mathematically that as the size of this quantity changes, so changes the wave velocity and thus the frequencies at which the string can resonate.

Another step forward in string evolution occurred in 1834 when Webster and Horsfall's of England made steel wire commercially available. This year could be considered the inception date of all contemporary steel string instruments.

In 1843 Georg Simon Ohm published a description of the way combination tones are heard. His work needed some refinement, however, which was undertaken by Hermann von Helmholz in 1862. Helmholz proved that the tone of a musical pitch of a string is determined by the proportions of the harmonics constituting the note. Helmholz recommended the use of Joseph Fourier's ( 1768-1830 ) mathematical analysis of curves to show that a vibrating string's motion is the sum total of its component harmonic motions.

Their work can be used to illustrate the distribution of the energies of the harmonic series on a vibrating string. On an ideal plucked string ( other starting motions will produce different results ) , neglecting resonant contributions, where the fundamental frequency would have an arbitrary value of 1, the second harmonic's value would become 1/4 the third harmonic 1/9, the fourth harmonic 1/16, the fifth harmonic 1/25, etc. The point along the length of the string at which it is plucked will also determine which group of harmonics is excited or damped to some degree.

The 1900's were a time of great technological advancement, especially with the advent of the computer age at the end of the century. As computers became entwined into the fabric of civilization, string variety and availability reached unprecedented levels, which is good news for all of us instrument players.

In the contemporary marketplace there are steel strings, nickel plated steel strings, bronze plated steel strings, cryogenically treated steel strings, gold plated steel strings, anodized steel strings, plus nickel, bronze, phosphor bronze, copper, silver plated copper, nylon, silk core, and many other constructions. The  linear mass density of each of these materials is different.

From this impressive assortment of materials , an equally impressive variety of construction methods are in use, specifically in wound strings. Thinner wound guitar strings are constructed from a wire core with another length of wire wound around the core. To attain any diameter string, the thickness of both pieces of wire can be varied. A thinner core could be used with a thicker wrap and vice-versa to create the same diameter string. The core wire is also variable as to round or hex shaped. The wrap wire can be round, ground to a smooth surface, burnished, roller-pressed to a smooth surface, flat ribbon wound, plastic coated, etc. Thicker wound strings can be built with two outer wraps using the above methods. The termination of a thicker bass string can taper toward the ball end or even leave the core exposed. All these factors, of course, vary the μ value.

This variation in the linear mass density and the specific mechanical manifestation of this density sets the conditions for the harmonic grouping the string will be able to produce.

Enough analysis! Let's look at this same picture now through the perspective of experimentation.

The following tables show some real life examples of different types of strings tuned to musical pitch. The string types included in the tests are GHS manufactured plain steel, wound steel ( Super Steels� ), wound nickel ( Nickel Rockers� ), wound nickel plated steel ( Boomers� ), and wound alloy 52 ( B52� ). The tests show different gauges of strings tuned to standard pitches, using the frequencies of equal temperament, on a scale length of 25 inches. These tests were performed by assistant David Ferbrache and myself in the fall of 2003. Special thanks go to Elizabeth Randall and Ben Cole of GHS Strings for supplying the test samples.

String Data - Click on the link below and save to your computer.

Nickel Calculations 1

Steel Calculations 1

NPS Calculations 1

B52 Calculations 1

Note:Click on Tabs at bottom of each table to see different string results.

The experimental data shows that the tension, as well as the intonation points, will vary with different types of strings.

It can be concluded, then, that each type of string will have a tone unique to its construction. This tone, when activated by playing, will be the starting force for the inherent resonant character of the instrument. The string's interaction with the resonant character of the instrument will likewise be unique.

This brings us to the final part of string choice, and this is where I will leave you on your own. The aural examination is the test you will have to perform on your own instruments to discover which unique voicing appeals the most powerfully to your own tastes. My recommendation is try every construction and brand of string that interests you; one type will generally stand out from the crowd, but you won't know which one it is until you hear it. Keeping a string journal for each instrument may be helpful, too. You can record the type of string used, the playing conditions, your observations, and any other relevant factors.

The best string will ultimately be the one that you like the best on any particular instrument. It is not unusual to end up using different strings on different guitars after extensive application of the three E words: experiment, experiment, and experiment.

Have fun!

� Copyright 2004 GVM Publishing                 Columbus, Ohio USA

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