in PDF format. . This means that you have to know some music theory. It is a challenging task to write lessons on music theory for guitar players who mainly. Most of these free music theory lessons contain a free PDF download that contains the actual tutorial. The best thing to do is to simply print out the PDF files and. Just because something is "theory", doesn't mean it has to be boring and taught by aging professors with grey hair and glasses! The "theory" of something is just.

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guitar player to work with groups of musicians (“OK, everyone – this song is in the key of the practical aspects of music theory, not the near-infinite complexities. Crash Course in Music Theory for Guitarists by Andy Drudy. An in-depth knowledge of music theory is essential for any musician. Learning the ropes so-to -speak. working toward a doctorate in music theory at Northwestern, Phillips taught served as Music Editor of the magazines Guitar and Guitar One.

Welcome home. Somehow, being creative just comes easy for them. How do they do it? Scales and modes are one of the most difficult and controversial topic in music theory Armed with the core knowledge this system will give you, you will be able to learn and connect everything there is to know about the modes, and at the same time use everything you are going to learn to play real music.

Learn how to finally master completely scales and modes on guitar in a ways that is musically useful and will help you to write, improvise, and play the music YOU want to play!

It explains harmony on the guitar fretboard for both beginner an advanced players What if you already knew and were able to play any chord on the guitar, in any position and could freely create your own awesome music? It seems like there is soooo much stuff to learn, and it would take a lifetime to really learn it all!

It was so frustrating. You too can learn how to understand and apply chords and harmony on guitar and be able to write interesting chord progression, use complex chords without even thinking, play awesome rhythm parts on your guitar in any style YOU want! Are you thinking too much about "what to play next" rather than letting your emotions speak through your playing?

Do you struggle in playing the Blues in all keys and over all the fretboard as opposed in just one basic position? If you have answered "yes" to any of the questions above, then you need to learn how music theory applies to Blues, and how to implement this knowledge into your guitar playing in an effective way.

Read about how to finally master all the scales for Blues guitar , how to get rid of your frustration and lack of confidence once and for all, and how to finally play Blues from your heart!

You know that you have to hit the chord notes, but you struggle to do it in real time? Or maybe you have not idea at all of how to find the 'right' notes. In this course we will explore together how to find the right notes on the fretboard not just in theory and how to practice to do everything in real time so that you will be ready to have fun at your next jam.

The next big development was the solid body electric guitar. In , Leo Fender introduced the Broadcaster solid body electric guitar. After a trademark dispute, it was renamed the Telecaster in Figure 1. Because it had a clear tone and was essentially impervious to feedback, it quickly became popular with working musicians. While electromagnetic pickups had been in use for years, they were generally installed on acoustic instruments. The increase in volume that came with pickups and tube amplifiers caused feedback problems.

The body of an acoustic guitar is intentionally flexible so it can produce sound. However, this also makes it respond to the sound field around it — a high enough sound level will make the body vibrate. If there is a pickup on the guitar, this vibration is detected and fed to the History of the Guitar 5 amplifier, which then further amplifies the sound and outputs it to a speaker.

The speaker then causes the body to vibrate at a higher amplitude and the loop is repeated. The result is the familiar screech of acoustic feedback. Because kinetic energy in the strings is not radiated away as sound, solid body guitars also offer the possibility of long decay times long sustain.

Since the body of an electric guitar has only a secondary acoustical function, it can be made in any shape and of any suitable material. The result has been the emergence of the electric guitar as an art form in which the body is as much a palette as it is part of the instrument Figure 1.

It should be noted that differences in body materials do have an effect on the tonal quality of solid body electric guitars, but it tends to be subtle. Still, accomplished players often have very definite preferences in body materials — alder and swamp ash are favorites. Unlike violins in which the design of most instruments is rigidly dictated by tradition to be fair, there are some innovative electric violins, but they are still a small minority of the market , guitar designs are constantly evolving.

In particular, designers and manufacturers have adopted synthetic materials. Ovation was the first major manufacturer to make widely-used instruments from fiber composites. Other manufacturers have since adopted composite materials and their use is increasing as the supply of high quality tone woods dwindles.

The author has played a similar instrument and found to the tone to be warm and rich — comparable to a fine guitar made from traditional materials. Another interesting line of instruments from Martin uses high pressure laminates in the body and laminated wood in the neck. The tops, being synthetic, are sometimes used as palettes for elaborate graphics. The author has played several of these instruments and found the tonal quality to be good.

While they are relatively new at this writing, they should prove to be quite durable. One interesting experiment in guitar design is the construction of a family of guitars based on principles of structural dynamics [11].

Graham Caldersmith has constructed a group of four guitars consisting of a treble, a standard, a baritone and a bass. This effort is analogous to a project in which a group of eight instruments in the violin family were designed and built using dynamic scaling concepts [12]. These can take many forms, but are generally instruments with sophisticated pickups, pre-amplifiers and sometimes graphic equalizers mounted in a thin, semi-acoustic body.

The body is hollow and the soundboard is flexible so that the instrument radiates sound even when not amplified. The bodies are thin for comfort and the resulting un-amplified sound can be thin and unsatisfying. These instruments are intended to be amplified and, when plugged in, can sound very good indeed. They have the additional advantage that they can be produce both acoustic and electric sounds which allows a stage musician to use the same instrument for different pieces.

He divided a vibrating string into two parts so that each part could vibrate and produce its own frequency. He also found that, if the lengths of the two parts were related by simple ratios like 2: Modern terminology labels these combinations consonant.

Some of the most familiar features of the modern guitar are dictated by the structure of music and by the fact that the frequencies of strings vary according to length. The foundations of western music were in place certainly by the end of the Renaissance [14], so the ideas and the terminology are very old. They have an ancient and solid feel about them that contrasts with the cool, technical descriptions of the underlying concepts.

Music, at its most fundamental level, is just a progression of carefully structured sounds. So, along with an understanding of the music should also come an understanding of some basic ideas in acoustics. In particular, it is helpful to see how amplitudes are quantified and how the human ear responds to sounds. A pure tone of any frequency in the human hearing range can be made with a signal generator and a speaker.

Idealized notes in a musical scale are sounds consisting of pure tones sine waves combined with harmonic overtones. They have specific frequencies chosen very precisely from the continuum of possible frequencies in the human hearing range. We are all different, but, based on extensive measurements, the human hearing range is generally assumed to be 20 Hz — 20 kHz [15].

From the beginnings of music, people realized that certain combinations of notes sounded good together. These combinations became widely used and were given names to make them easy to work with. In the early s Galileo built on the work of Pythagoras and showed that the length of a string under a constant tension is inversely proportional to its lowest natural frequency [16].

From this discovery came the realization that these pleasing combinations of notes corresponded to frequencies that were the ratios of small integers like 2: In western music, the notes are related to each other by given ratios [17].

Thus, if we know the absolute frequency of a single note and the frequency ratios that define the relationships between that note and the others, we can find the frequency of every possible note. By international agreement, the frequency standard used for music is the A note, now defined to be Hz [18].

If you are like most guitar players, you want a place where you can get:

The A note with a frequency of Hz is more precisely called A4. The note one octave higher in pitch is called A5 and has a frequency of Hz. Similarly, the note one octave lower is called A3 and has a frequency of Hz. The octave is, in turn made of smaller intervals. In western music, the smallest possible interval between two notes is called variously a semi-tone, a half tone or a half step. In this book, this interval will be called a half step.

In western music, there are 12 half steps in an octave, but other cultures have used other numbers; there is no physical requirement to use 12 half steps. Predictably, an interval of two half steps is called a whole step or a whole tone.

For consistency, here it will be called a whole step. The various intervals between two notes have unique names to identify them. For each interval, there is a corresponding inversion and the sum of any interval and its inversion is an interval of one octave 12 half steps. Figure 2. The scale is the next level of musical structure and dictates which eight of the 12 notes are used.

A musical scale is just a fixed pattern of intervals [19]. There are several different scales, though even nonmusicians might be familiar with the names of the most widely used, the major scale and the minor scale. The structure of the major and minor scales are shown in Figure 2. The names of the notes themselves are simply the letters A through G with the names repeating every octave. There is a whole step between all the letter names except for B-C and E-F, which are separated by half steps.

A scale can start on any note and it helps to give the scale a name that compactly indicates which notes are in it.

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This name is called the key and it is simply the first note in the scale. For example, the name C major tells the musician everything needed about which notes will appear in a piece of music written in this key. Beginning musicians often play in the key of C major since it has no sharps or flats. The notes in a scale are often also identified with Roman numerals as shown. As another example, consider the key of G. This process works for all keys and all scales.

It is clear from the way the keyboard is arranged that playing in the key of C major means only playing the white keys. For simplicity, the discussion here is confined to major scales. However, other scales can be defined using the same kind of pattern used to define the major scale. A scale made from a group of seven notes consisting of five whole steps and two half steps, in which the half steps are separated by either two or three whole steps is called a Diatonic scale [20, 21].

Diatonic scales are the foundation of Western music and the piano keyboard is laid out in Diatonic intervals. By far, the most widely used are the major and the natural minor. However, guitarists, particularly those who play lead guitar in a band, are often students of the different musical scales. It is perhaps helpful to see that they are all logically related to one another; indeed, they are formed simply by choosing different starting points on the repeating pattern of intervals used to define the major scale.

There is one more piece of the scale puzzle — the numerical relationship between the notes. Music has evolved over thousands of years, and, for historical reasons there is more than one way to define the frequency ratios between the notes [22, 23]. While this might seem to be an arcane diversion, it is central to the development of the guitar. The most consonant ratio of frequencies is the octave, a 2: The next two are 3: Note from Figure 2. Thus, going up a fourth results in the same letter as going down a fifth [17].

This is true no matter what starting point is chosen. Furthermore, any note can be reached from any other note by moving up in steps of fifths only or by moving down in steps of fourths only.

A Pythagorean scale results when frequency ratios based on increasing fifths or decreasing fourths are used to determine the notes. For example, the interval between C and D is two half tones. D can also be reached by increasing pitch by two successive fifths. The remainder of the scale can be developed in this manner so that the frequency ratios for the Pythagorean scale are as shown in Figure 2. There are only two different intervals in the Pythagorean scale shown in Figure 2.

These are the whole step and the half step respectively. The result is that some combinations of notes sound out of tune. A modification to the Pythagorean scale is Just Intonation. The fundamental idea is that groups of three specific notes should all have the same frequency ratios. The major chords based on the I, IV and V notes of the scale are called the tonic, dominant and subdominant chords.

Just Intonation requires that the frequency ratios of the three notes called a triad that form these chords is 4: If the first note in the series is assigned a value of 1, the frequency ratio becomes 1: The frequency ratios in the just scale are shown in Figure 2. For example, not all intervals of 7 half steps are perfect fifths with a frequency ratio of 3: The frequency ratio of D: A in the key of C shown in Figure 2.

This effect is even more of a problem when one considers that not all instruments in an orchestra play in the same key. A really is 3: The practical solution is to slightly modify or temper the frequency ratios of the just scale. There are several common temperament schemes and the one used for guitars, called equal temperament, starts with a fixed half step frequency ratio — one that is identical for all half step intervals, no matter what the root note is.

All the other intervals are then derived from this grounding assumption. The equal tempered scale is not perfect either, and can produce some unpleasantly dissonant combinations of notes. In practice, different temperaments are used as necessary. The twelfth note in the series is a full octave above the original one. Since increasing pitch by an octave doubles the frequency, it is easy to find r.

To avoid confusion, each octave is traditionally given a subscript. The lowest note, C0, has a frequency of Using this notation, the human hearing range spans roughly E0 — E For comparison, the lowest note on a Piano is A0 with a frequency of Table 2. The guitar player is basically trading the ability to play a just scale for the ability to use a regularly spaced fret pattern and the ability to play easily up and down the neck.

Another solution to the problem is to dispense with frets as on instruments of the violin family. Thus, a skilled violinist can have some of the same flexibility of the guitar without being bound to equal temperament.

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This too comes at a price, though. With no frets, the accuracy of any note is completely dependent on the skill of the player. Any parent of a young strings violin, viola, cello or bass player can tell you that the early stages are as much a chore for anyone within earshot as for the child. In addition to standardized frequencies for the individual notes, the six strings of the guitar also have a standard tuning pattern.

However, a large majority of guitar music is written for standard tuning. Guitars are used to play both single notes and chords. In fact, rhythm guitarists are mostly known as chord players.

A chord is just a collection of at least three notes from a diatonic scale which are played together three note groups are called triads. Each different kind of chord uses a different group of notes. Consider a major chord, made using the 1, 3 and 5 notes in a major scale using A as the starting point; the notes in the triad are A, C , E. From a strictly mechanical point of view, a guitar is a device that connects strings under tension to a resonator with flexible walls and offers a convenient way to shorten the strings to raise their frequencies.

The frets on a guitar neck are there to stop the strings, effectively shortening them. They are arranged so that the space between two successive frets on the same string is always a half step. For instance, doubling the length of a string would lower the frequency by half or one octave. Knowing this, it is not hard to write down the expression for the position of the frets on the neck. An equal tempered scale allows regular fret spacing, though the resulting notes are not at the exact frequency as they would be with just intonation.

Having all the frets spaced the same way makes the guitar very versatile since there are often several ways of playing the same note on different strings. This allows interesting combinations of sounds that can give a skilled player added expressive freedom. Conversely, on a piano, there is one key for every note.

The combinations of notes that can be played are limited by what an individual player can reach. Indeed, some piano pieces are intended for two people playing at once on the same piano. There are many different scale lengths in use. However, two common ones are The fret spacings for those two scale lengths are presented in Table 2. Fret spacings for other scale lengths can be calculated readily using a spreadsheet program.

Acoustics and Musical Theory 23 Table 2. The lowest note on the neck is E on the 6th string — Doubling that frequency gives Doubling again gives If your guitar has 24 frets, you could raise string 1 by another two octaves. Thus, the instrument has a 4 octave range. This distinction is often mentioned in books on musical instruments, but not always explained.

A C instrument is one for which playing a note called C yields the standard frequency, also called also called concert pitch, for the C note as shown in Table 2. This would seem to be trivial except that not all instruments are C instruments. These are called transposing instruments; the guitar is a non-transposing instrument and guitar music can be written in concert pitch. A piece written for clarinet and guitar would then have the two instruments playing in different keys so that they would be in tune with one another.

Acoustics and Musical Theory 25 2. Sound is the name we give to time-varying air pressure within a frequency range of 20 Hz—20 kHz. The dB is different from most other familiar units because it is dimensionless and it is logarithmic. The RMS average is required for sound measurements because the mean of any sinusoidal signal that oscillates about zero is zero no matter what its amplitude might be.

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RMS captures the amplitude of a signal that oscillates about zero. The RMS pressure in Pa is called sound pressure. The level reported in dB is called sound pressure level SPL.

It is useful to develop some intuition about what a sound pressure level means in real life. A guitar has two types of radiation sources, the soundhole and the vibrating plates. The top plate is generally assumed to radiate much more sound energy than the back.

Acousticians often speak in terms of how many poles a radiator has. A monopole is a point source that radiates out in all directions.

A dipole is formed by two closely-space monopoles and so on [25, 26]. The center of the radial pattern is the location of the source and the waves radiating out represent positive and negative air pressure at an instant in time.

Again, this is a two-dimensional analogy since it is quite difficult to illustrate the actual three-dimensional pressure field on a sheet of paper. Note the line on which the amplitude of the wave is zero. These two patterns relate directly to the guitar since the different parts of the top can vibrate out of phase with one another.

The soundhole acts roughly like a monopole radiator for the first coupled airbody mode of a guitar [27]. Mathematically, it can be treated as a piston in the same manner as a speaker cone in a cabinet. In the region near a piston the near field , the sound is concentrated on the axis of motion. Far from the piston the far 5 4. For a guitar, the far field might be defined as the region more than a few body lengths from the top.

While a listener could not conveniently be within the near field, it is certainly possible to place a microphone there. Acoustic radiation from the body is strongly conditioned by the mode shapes of the top. In a manner analogous to the example shown in Figure 2. The math becomes much easier without changing the underlying concept. The mode shapes of a simply supported plate one whose edges can rotate, but not move up or down are a product of sine waves. This is called the 1,1 mode [29] as shown in Figure 2.

The 1,1 mode has no interior node line. The 1,2 mode has one interior node line and two anti-nodes and so on. Note that there is more than one method of naming plate modes in the literature.

Sometimes, they are named according to the number of internal node lines. Thus, the mode labeled here as 1,1 would be labeled 0,0. The underlying mechanical principles are unchanged by the naming convention. The 1,1 mode acts as a monopole radiator since the entire plate moves in phase and there are no internal node lines lines where there is no motion. The 1,2 mode acts as a dipole radiator. Because of the phase difference, the transient pressures generated at the surface of the upper and lower parts of plate are of opposite signs.

This basic idea extends to the higher modes in which there are more and smaller areas of the plate out of phase with each other. The definitions of the near field and far field are a little loose, since there is no step change in the sound field as one moves farther from the instrument.

In the near field, the contributions of the different parts of the vibrating plate are clear. The region that is significantly farther from the source than the source size is called the far field. In the far field, the dimensions of the source are small enough to be ignored, so calculations and measurements become easier.

In practice, the far field might be a meter or more from the plate. Because of the internal structure of a guitar, only the portion of the soundboard in the lower bout generally moves enough to be an effective radiator at low frequencies Figure 2.

The ideal microphone senses all frequencies between 20 Hz and 20 kHz with equal gain so it can perfectly reproduce the part of the sound field it hears.

High quality microphones used for acoustic testing come close to this ideal. Other types of microphones have flat response throughout the human hearing range. The outer ear called the pinna, this is the part you can see and the inner ear have their own frequency response characteristics and the result is that people are not equally sensitive to all frequencies [30]. While everyone is different, people typically have a peak in their acoustic sensitivity near Hz.

Additionally, human sensitivity to sound generally decreases significantly below Hz and above 10 kHz. Of course, hearing damage from exposure to loud noises and normal biological variations can change this. There have been many attempts to correct SPL to match human perceptions of sound levels as functions of both frequency and amplitude. The simplest, and probably most widely used, is the A-weight filter [31]. This is a simple weighting function that attenuates or amplifies a signal based only on its frequency.

When sound pressure level is calculated on a signal that has been modified with an Aweight filter, the result is expressed in dB A. This is why many microphone calibrators generate a Hz signal. Note also how insensitive human ears are to very low frequencies. The vertical axis is expressed as power spectral density PSD. PSD is the complex Fourier transform multiplied by its complex conjugate and divided by the sample time.

The result is a real function. The unweighted and A-weighted results are similar above about Hz remember this is a log scale plot. Only below this frequency do the two curves significantly diverge. This is reasonable since it is harder to isolate a chamber from low frequency noise than from high frequency noise. The unweighted SPL is Human perception of sound levels is a function not only of frequency, but also of level. The A-weight filter addresses the frequency dependence, but not the level dependence.

An additional measure of sound called loudness was developed to account for both the frequency and the amplitude dependence. Loudness is fundamentally different from SPL in that it is a subjective measure; the mathematical descriptions of loudness are based on the subjective responses of groups of listeners to test sounds.

The initial description of the loudness was proposed by Fletcher and Munson in [32]. They also defined the unit of loudness, the Phon. Acoustics and Musical Theory 33 Figure 2. One other interesting feature of human hearing is how we perceive frequency intervals. The part of the ear that determines the frequency of a sound is the cochlea. This is a tube coiled into a spiral that serves to convert mechanical vibrations into nerve signals that can be interpreted by the brain. A tapered membrane called the basilar membrane in the cochlea vibrates in response to sound transmitted through the eardrum and the tiny bones of the middle ear.

Since the basilar membrane is tapered, different frequencies excite different portions of it Figure 2. Research has showed that distance to the point of maximum displacement is proportional to the logarithm of the input frequency. That suggests people are hard wired to perceive frequency intervals on a log scale. Of course, this is exactly how frequency intervals are described on a musical scale.

By international standard, octave bands are described in terms of the center frequencies of the bands with one band centered on Hz [34]. The next level of sophistication is sound quality — quantifying the subtle psychoacoustic process by which humans determine whether they like sounds.

The two sounds have identical levels in each band in fact, one was filtered to match the other. One plot was calculated from a segment of orchestral music and the other was calculated from a recording of a hydraulic pump. In rough terms, psychoacoustics is the field concerned with explaining why one of the sounds represented in Figure 2. In musical acoustics, the word timbre is often used to describe differences in sounds. Unfortunately, there is no universal definition of timbre. This is essentially a subjective description in contrast to the objective definitions of loudness and pitch.

Reducing the problem to its most basic elements, a sound is good if it is what the listener expects. A good sounding race car engine obviously makes a very different noise than a good sounding acoustic guitar, yet both sounds can elicit an emotional response in a listener.

Think of a nice small block V-8, breathing through dual quad carburetors and exhausting through headers. Then think of Willie Nelson playing his legendary classical guitar, Trigger. While both sounds might represent transcendent achievements in their own area, they have nothing else in common. Acoustics and Musical Theory 37 Figure 2. Time domain amplitude vs. Frequency domain amplitude vs. Time-frequency domain shows not only what frequencies are present in the sound, but also when those frequencies are present [36].

The time domain plot at the top clearly shows the five notes along with their starting times, duration and decay.

However, frequency information is not discernable. The second plot shows which frequencies are present in the recording, but not when they occur. The time-frequency plot spectrogram at the bottom shows not only what frequencies are present, but when each is present. The first note increases in pitch quickly before settling at a stable frequency. This is a slide. The fourth note starts at a fixed frequency and then increases in pitch over about half a second.

This is a sting bend. The final note oscillates in pitch for about a second and a half. This is vibrato. Also discernable is that the lower frequencies of the 4th and 5th notes sustain much longer than do the higher frequencies and that decay time is approximately inversely related to frequency.

The spectrogram is the most widely used time-frequency method, mostly because it is a simple extension of the fast Fourier transform FFT and is computationally efficient. Improved methods are available [37, 38], though at the price of increased computational time.

Sounds from stringed instruments almost never consist of a single frequency component. Rather, they contain a fundamental frequency and harmonics — components whose frequencies are integer multiples of the fundamental frequency. These harmonics are clearly visible in the spectrogram of the five note phrase shown at the bottom of Figure 2.

The phrase consists of five single notes, but each note clearly has many harmonics of the fundamental frequency. Harmonics can be described using two equivalent ideas, one physical and one mathematical. The natural frequencies in Hertz of an ideal string are integer multiples of the fundamental frequency.

For reasons that will be described in more detail later, plucking a string excites many modes and the resulting sound is made of the fundamental and higher natural frequencies. These higher modes are clearly visible in both frequency and time-frequency plots. A more mathematical explanation is based on the Fourier series approximation that forms the basis of the Fourier transform.

The fundamental assumption is that any periodic signal like the sound made by a plucked string can be approximated by a summation of sinusoidal functions.

The two expressions in Equation 2. These coefficients define how much of each term must be added to the summation in order to reconstitute the original signal. Plotting the coefficients as a function of frequency gives the frequency domain form of the original signal.

After all, it should only take one sine wave to approximate a sine wave. However, if the original signal is not sinusoidal, additional terms are required. The frequencies of these additional terms are integer multiples of the fundamental and are, thus, harmonics.

Since there is undesirable noise at very low frequencies and these frequencies are below the first resonant frequency of the string, an A-weight filter was applied to the data. By transforming the data into frequency domain, one can clearly see that the signal is composed of many discrete frequencies and that they appear to be evenly spaced along the frequency axis as shown in Figure 2.

These amplitudes show the relative contribution of each frequency component, so they can be used as a tool for quantifying tonal quality. A note with lots of treble would have a larger proportion of the energy at higher frequencies.

By zooming in on the range from 0— Hz, one can identify the fundamental frequency and verify that successive peaks are indeed integer multiples of the fundamental Figure 2. They are, thus, harmonics as defined earlier. The amplitude of the peak is determined by the constants — A1 and B1 or C1 depending on the formulation. If someone wanted to change the tonal quality of a note, the way to approach the problem in mathematical terms is to change the amplitudes of the constants in the Fourier series representation of the note.

This is essentially what electronic filters like the ones in a graphic equalizer do. The strings must be brought to the correct tension for the instrument to be in tune, so the neck and body must resist the resulting compressive load. In addition, there are dynamic loads when the instrument is played. Finally, there are forces generated by temperature and humidity changes. While these might not be the most obvious sources of loading, they are a very common source of structural failure.

It makes sense, then, to explore the structure of guitars and how they are designed to be strong enough to withstand both playing and environmental loads while still being light enough to radiate sound.

These two components must resist the tensile forces in the strings as shown in Figure 3. Like all practical structures, guitars are a compromise between mutually opposing requirements.

Start first with the neck. It must be strong enough to withstand the string loads while still being light enough that the instrument will balance in a comfortable way. The neck also has to have a profile that makes it comfortable to play; it can not be so wide or thick that it is difficult to grasp, nor so narrow or shallow that it is difficult to play. Compressive Force Figure 3.

Assuming the portion of the string between the nut and saddle is essentially parallel to the fretboard, the free body diagram in Figure 3. The cross section of the neck cannot be accurately approximated by some simple shape like a triangle or rectangle, so it is necessary at this point to develop expressions for the centroid and the area moment of inertia of the neck.

One way to account for the variation in acceptable neck profiles is to define a generic profile in terms of variable parameters. One possibility is to use one equation that approximates a wide, flat neck profile and another that approximates more of a V-shaped profile. The two functions can then be summed in various proportions to make a wide range of neck profiles. In addition, a scale factor can be added so that the resulting neck profile can be adjusted to accommodate the neck taper.

Figure 3.

The unscaled profile is 2 in. If the two functions are to be combined in equal proportions and the neck is to be 0. The resulting profile is shown in Figure 3. The original 2 in. It is important to note that guitar necks are not generally intended to be straight.

Rather, they generally have a slight curvature that increases the distance between the strings and the neck. To put guitar design on a firmer analytical footing, it is useful to present a method for calculating the neck deflection.

In order to calculate the deformation of the neck as a result of string tension, it is necessary to know the vertical location of the centroid and the area moment of inertia of the neck cross section. The deformation of a beam undergoing small deflections is described by a simple expression relating moment and curvature [40]. The area moment of inertia for the example neck shape is 0.

Equations 3. This is not generally the case since most instruments use different materials for the neck and the fretboard. Additionally, most guitars with the exception of classical guitars also have truss rods made of metal or graphite.

In this case, 48 Engineering the Guitar the concepts of the centroid and the area moment of inertia must be modified to account for changes in the elastic moduli of the different materials. An easy way to account for the change in elastic modulus is to change the effective cross sectional areas of the different materials. Consider the case of the neck cross section shown in Figure 3. Assume the neck is made from mahogany and has a width of 2.

Finally, assume a rosewood fretboard has been added increasing the depth of the neck. If a set of calipers was used to measure the depth of the neck, the result would be 1. However, the portion of the neck that can actually carry a load is 0.

The resulting cross section is shown in Figure 3. The elastic modulus of mahogany is 1. The respective effective areas can be modified so that subsequent calculations can proceed as if the elastic modulus of all the materials was the same. For this example, the modulus values will be normalized to match mahogany. Effective areas are determined by multiplying the width of the element by the ratio of elastic moduli since the area moment of inertia in this case is proportional to width.

There are four areas to be calculated, three for the different elements and one for the rectangle of mahogany that is removed to so the truss rod can be installed. The material removed will be considered to have a negative area. Properties of the different components are given in Table 3. Note that the cross-sectional area of the fretboard in column two is 0.

The centroid location for the neck cross section with the fretboard and truss rod is —0. Ai is the effective area and I i is the effective area moment of inertia of element i. Table 3. Since the neck is tapered, the area moment of inertia tapers along the neck. While not strictly necessary, it is helpful to develop a simple expression for I x before calculating the deformation.

Assume the width at the nut is 2. An equivalent expression can certainly be derived directly, but this approach is a convenient shortcut that avoids lengthy symbolic manipulations.

To find the deflection along the neck due to the moment exerted by the strings, Equation 3. Then, all the terms on the right side of Equation 3. For this particular case, both constants of integration are zero. Assume also that the neck is made of mahogany with an elastic modulus of 1.

The moment is constant along the neck and the distance from the centroid to the top surface of the neck fretboard is 0. If the strings are assumed to be 0. Evaluating Equation 3.

Note that solution for the tapered neck is compared to that for the mathematically much simpler case of a constant cross section. While practical guitars are not actually made with untapered necks, this simplifying approximation gives a surprisingly accurate result. The first thing to note is that it is not generally desirable for the neck to be perfectly flat. Rather, a slight upward curvature known as relief is intentionally built into most guitars. There is no one correct value for relief since it can depend on the height of strings above the soundboard, the style of music being played and the type of guitar.

A simple way to measure neck relief is to place a capo a clamp that stops the strings wherever it is placed on the neck on the first fret and to fret a selected string where the neck joins the body.

For steel string acoustic guitars, this is usually the 14th fret and for classical guitars, this is usually the 12th fret. Since the string is straight between the two frets, relief is apparent as space between the strings and the frets between these two points as shown Figure 3. Measured this way, neck relief in the range of 0. Wood is sensitive to changes in both temperature and humidity. Furthermore, wood can creep under load — permanently deform even though the stresses are lower than the yield stress.

Thus, the neck relief of a guitar can change over time and may need to be adjusted periodically. With the exception of classical guitars, most guitars have a truss rod to stiffen the neck, allow some adjustment of the neck deformation or both. The different designs can be roughly grouped into three categories: Fixed non-adjustable , Tension and Double Acting.

Fixed truss rods are simply reinforcing elements in the neck that increase the bending stiffness. Current practice is to use unidirectional carbon fiber, though steel rods are still sometimes used. Even a relatively small fixed truss rod can have a major effect on the stiffness of the neck because of the differences in elastic modulus of the materials.

For example, the elastic modulus of wood is on the order of 1. First Fret Neck Relief Figure 3. Early tension rods were straight, but had to be placed below the elastic axis of the neck. Tightening a nut on one end places the rod in tension, thus, generating both a compressive force and a moment as shown in Figure 3. Straight rods were first used in acoustic guitars with fixed necks glued on and not easily removed , so adjustments were made at the head end.

That end was generally accessed though a slot in the top of the headstock, near the nut. Generally, tension rods are installed with a slight dip in the middle, increasing the resulting vertical force and making them more effective. These curved rods are routinely installed in both electric and acoustic guitars.

The end at which the rod is adjusted depends on how it is installed. Guitars with the adjustment at the head end often have a small, removable cover over the end of the truss rod. Whether straight or curved, though, a tension rod can only induce convex curvature along the fretboard. While designs differ, a representative one consists of two circular rods with threaded ends and mounted into end blocks as shown in Figure 3.

Turning the adjustment nut at the left forces the bottom of the blocks apart or together depending on the direction of rotation and, thus, causes the rod to arch either up or down.

Some double acting truss rods are wrapped with fiberglass tape to keep the two rods from separating in the center. Note, this design does not require a tape wrap since the end blocks apply a moment to the two rods, causing them both to bend. The soundboard is clearly a very important structural element in any acoustic guitar.

Unfortunately, it is very difficult to analyze. There is a plate equation that describes the static deformations of an isotropic plate with known boundary conditions in a compact form.

However, it is a fourth order partial differential equation; it has been solved analytically only for simple shapes like rectangles and circles and for simple boundary conditions. Thus, a mathematical representation of a guitar top simple enough to be solved analytically cannot capture the details of the actual structure. Analyses of guitar tops appearing in the literature often use one of two approaches.

The first is to make a simple analytical model and attempt to extend the results to an actual guitar top by analogy. The second is to make a discretized model of the structure, usually a finite element model [41].

A good finite element model can easily have thousands of degrees of freedom, particularly if it includes the air volume in and around the instrument. Mathematical modeling for soundboards will be discussed in more detail in Chapter 5. The body of an acoustic guitar is constrained by the same types of opposing requirements as the neck. The combined string forces are generally 60 lb — lb N — N and the top must be able to withstand this force while retaining enough flexibility to vibrate in response to the strings.

Because of this conflict between strength and light weight, the static string tension is large enough to noticeably deform the soundboard of an acoustic guitar. For this discussion, a model made using thousands of degrees of freedom is not necessary for a basic understanding of how string forces deform the top plate. Indeed, a useful qualitative description can result from treating the plate as a pinned beam as shown in Figure 3. The in-plane force does not cause out of plane deformation, so the beam equation can be solved using the moment as the only applied load.

The resulting loadshear-moment diagram is shown in Figure 3. The solid line shows the deformed shape assuming the stiffness is constant along the beam and the dotted line shows the effect of doubling the stiffness on the left side of the applied moment the left side of the bridge. It is typical for acoustic guitars to have heavy bracing near the sound hole and around the neck interface. Thus the stiffness of the top is significantly higher in the lower bout than the upper. When this change is made, the area to the right of the moment has positive deflection — the same behavior as generally seen in acoustic instruments.

One significant exception, of course, is the strings. The central feature of the modern classical guitar is the nylon strings while earlier ones were fitted with gut strings.

Because of the material properties of nylon, the strings need to be brought to about 80 lb N tension for the instrument to be at standard tuning [42]. The instrument consists basically of the body and the neck. Guitars are regularly made from a range of different woods, but classical guitars have traditionally used spruce for the top and rosewood for the back and sides. Initially, tops were made from European species of spruce; now, Engelmann and Sitka spruce are widely used.

The neck is often mahogany and the fretboard is ebony. There is a decorative rosette around the soundhole that is usually made from very small tiles of dyed wood. The bridge is usually made of ebony or rosewood. The load-carrying structure of the classical guitar is generally very simple, though it is quite refined. There is no single accepted design for classical guitars as there is for violins, so some representative design has to be used. The description here is of a traditional instrument established by Torres and others in the 19th century.

Structure of the Guitar 59 Figure 3. Quarter-sawn wood is cut from the log so that the grain is perpendicular to the face of the resulting boards. The name comes from the fact that the log is split into quarters before being sawn into planks. Note that there are several common sawing patterns for quarter sawing logs and this is just one of them. The vast majority of commercial lumber is slab sawn because it results in much less waste than does quarter sawing.

Quarter-Sawn Log Figure 3. From a structural standpoint slab-sawn boards are problematic because the grain lines tend to straighten out as the wood dries and the boards tend to cup.

Instruments made from quarter-sawn lumber tend to be more dimensionally stable [43]. There are aesthetic reasons as well; the grain of quarter-sawn wood tends to be even and sometimes displays subtle, attractive effects. Finally, centuries of tradition have conditioned musicians to associate quality instruments with quartersawn wood. Note that the center board from a slab sawn log has vertical grain just like a quarter sawn board.

Guitar tops and backs are generally book matched. A book matched board is one that is cut in half and joined to make a plank that is twice as wide and half as thick. Book matching results in a board whose grain is symmetric about the centerline and can be done for aesthetic or structural reasons. The guitar back in Figure 3. Indeed, the vast majority of classical guitars have proportions similar to these.

Note that these are just proportions and not absolute measurements. The length of the body is 1 and all other dimensions are to scale. If, for example, the body is to be 21 in mm long, simply multiply all the numbers by 21 in.

While the majority of classical guitars generally follow traditional designs, there is a rich tradition of experimentation and many luthiers go through a stage of making very unconventional instruments.

As an example, Figure 3. This is a very non-standard instrument made by the author after seeing a picture of a similar instrument made by Bob Benedetto [45]. Structure of the Guitar Figure 3. For example, dispensing with the braces and simply making the entire plate thicker might make it strong enough to bear the in-plane loads, but the result would likely be inferior sound. The bracing has to stiffen the top directly and distribute loads so that the strength of material making up the top is used efficiently.

The number of bracing patterns in the literature is far too large to summarize here. However, the traditional pattern is fan bracing popularized by Torres. While many bracing patterns have evolved empirically, some are the product of more analytical processes based on the underlying physics. One that has attracted attention is the Kasha-Schneider pattern.

Developed by Dr. Michael Kasha and implemented by luthier Richard Schneider, it was intended to improve both sound quality and volume of the resulting instrument [46] by tuning different portions of the soundboard to respond at different frequencies. The bracing pattern is extremely asymmetric and Kasha style instruments are generally fitted with asymmetric bridges that may also have a notch in the center.

This geometry is intended Figure 3. Kashabraced guitars often feature an offset soundhole, though it is important to note that offset soundholes have been used on more conventional designs as well in an effort to generate a larger radiating surface on the guitar top. This bracing pattern has not been widely accepted, though a number of successful luthiers use it and attest to its superior sound quality.

He has proposed a method for determining brace geometry that requires curved braces [47] and makes instruments using this system. The lattice tends to distribute the bracing stiffness more evenly across the top than does fan bracing. The elements of the lattice are sometimes made of wood, though graphite reinforced wood has been used successfully. Some luthiers are even using balsa for the wood portion of the lattice and gluing strips of unidirectional graphite to the top of the braces.

This approach can result in extremely light tops. If the luthier wishes to distribute bracing stiffness more evenly, the limiting case would be to make the top with a sandwich construction. Some luthiers have been successful in using thin wood plates separated by a thin layer of Nomex honeycomb [48].

This can be a particularly efficient structure since the parts of the structure that can withstand tensile and compressive stresses the wood plates are placed where those stresses are the highest — at the outer surfaces.

The core layer basically carries shear loads. The soundboard shown here will have a wood insert to transmit the bridge loads to the wood plates. The insert also will form the edge of the soundhole after it is cut and will reinforce the upper bout. Structure of the Guitar 67 Figure 3. This design is probably not as efficient as the one shown in Figure 3. A key structural element in any guitar is the joint between the neck and body. On almost all acoustic guitars, the end of the neck that joins the body has a curved extension called a heel that extends down the full depth of the body.

Classical necks are generally fixed to the body in one of two ways. One is to fit the bent sides into notches sawn into the heel block. When an instrument is made this way, the neck is attached before the body is complete. The three instruments shown in Figure 3. The other common way of fixing the neck to the body is a mortise and tenon joint as shown in Figure 3.

The tenon is an extension of the heel and the mortise is milled into the body. The neck is generally glued to the body after the body structure is complete, though often before the bridge is attached. On classical guitars, the sides of the tenon are traditionally parallel rather than tapered as in a dovetail joint; the body in Figure 3.

A variation on mortise and tenon joints for fixing necks to bodies is spline joints.Also, the working time is short, so the parts must be brought together quickly after the glue is applied. An experienced violin maker once told me his rule of thumb was that any finish that will stick to wood will stick to shellac.

This is true on all instruments.

You too can learn how to understand and apply chords and harmony on guitar and be able to write interesting chord progression, use complex chords without even thinking, play awesome rhythm parts on your guitar in any style YOU want! Guitar Triads Structure, shapes, how to extend them Learning guitar triads will expand greatly your fretboard knowledge.

All of the chord shapes above contain exactly the same notes C, E and G. In addition to the introduction of geared tuners, the flush mounted fretboard was replaced by one that covered the neck as well as part of the soundboard. Some solid guitar bodies use more than one type of wood. The first note increases in pitch quickly before settling at a stable frequency.

The bodies are generally thin to mimic the feel of a solid body electric guitar and are being made from both wood and composite materials.

ANTONETTA from Murrieta
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