# Jan Arts Guitars

A scientific approach to guitar building

## Methodology

It is impossible to say what makes a perfect guitar but there are a number of  specific terms which make it possible to describe the sound and character of a particular guitar reasonably well.  Tone, volume, presence, projection, balance, separation, sustain, dynamic range, sensitivity, clarity, ease of playability are some of the terms to describe the sound and character of an acoustic guitar.

See http://www.acousticfingerstyle.com/gfaqs/GtarFAQFrameGsound.htm for a brief explanation of some of the terminology.

General design, construction, choice of wood, bracing, setup, strings are some of the many parameters which determine the character of an instrument and what it sounds like.

Parameters which can be quantified and measured include:

• Physical properties like mass, density, elasticity modulus, internal damping, resonance frequencies
• Geometry and dimensions like thicknesses, bridge height, air volume, diameter sound hole, scale length.

These parameters can be used to provide a quantitative approach during the building process and a base for understanding of the specific terms mentioned before. Having said this, at the end, player and listener are the final judges about the quality and tone of an instrument.

### Examples of simple calculation

The formulae which follow are briefly explained by simple examples of applications. I follow briefly the formulation and terminology used in "Left-Brain Lutherie" by David Hurd. For derivation of the equations you may have to look in handbooks about classical mechanics. All equations are in metric units.

#### String tension:

$T=4L^2f^2m$    (1)

Where T = force in N/m2, L = length (m), f = frequency (Hz), m mass/length (kg/m)
The equation tells that a 10% longer scale increases the tension by approximately 20% and the same can be said about a 10% increase in diameter of a steel string.

#### Resonance frequency air volume (Helmholtz frequency):

The Helmholtz frequency for a guitar which has a very stiff back and top can be estimated by the folowing equation.  The value will be lower in reality depending on the stiffness of the top and back of the instrument.

$f=c/(2\pi)\sqrt{(\pi r^2/(1.7Vr))}$

where f=Helmholtz frequency (Hz), c=speed of sound in air(approximately 340m/s), r = radius sound hole (m), V = volume guitar body $(m^3)$

For a classical guitar you will find a calculated value around 130 Hz. The measured value is likely to be in the range 100 to 110 Hz or even lower. Reducing thicknesses of top, back and bracing reduce the Helmholtz frequency and the peak frequencies of top and back as well.

#### Mechanics bridge (static and dynamic)

Simple mechanics can be applied to calculate the forces acting on the top of a guitar. Dynamic calculation are much more complicated. It shows however that a bridge acts as a filter depending on mass, stiffness and damping properties and it should be possible to model the transfer of energy from strings to top. A higher bridge may increase volume but reduce sustain at the same time.

#### Density and mass

These are important parameters. To accelerate mass takes energy and more energy for higher frequencies. A light top is more responsive than a heavy top and is better in reproducing the higher tones. See under tonewoods for more explanation.

#### Dynamic elasticity modulus and resonance frequency of a flat plate

$f=C \sqrt{(h^2E/(dL^4))}$

where L= length plate(m), h = thickness(m), d=density($kg/m^3)$, E=elasticity modulus, C = 1.02 for metric units

It's pretty easy to measure resonance frequencies and to calculate the modulus of elasticity. The values are dependent on grain direction. This equation is very useful when changing dimensions of an instrument or making adjustments for different tone-woods. The equation is very helpful in making quantitative judgements about  tonewoods.

#### Brace dimensions and stiffness

A first estimate is provided by the equation for deflection of a beam under static load

$y=KWL^3/Elh^3$

where K=0.25 for rectangular cross section, E elasticity modulus, L=length(m), l =width beam(m) h=thickness(m)

The equation quantifies sensitivity of deflection for length and thickness parameters. High narrow braces make sense to get more stiffness from light bracing.

#### Properties of composite braces

Reduced weight while maintaining stiffness can be achieved by combining light and stiff materials.

An interesting example is a surprisingly good sounding violin made out of balsa wood and graphite (see YouTube)! Use of double tops and lattice bracing are other examples.