Scale length, fret positioning, tuning and intonation.

In my previous article I focused on the structure and materials used in building guitar necks. This time we’ll take a look at scale length, fret spacing, tuning, intonation and how the neck dimensions relate to tuning.

Scale length

Scale length is one of the basic measurements that determines the size or scale of a stringed instrument and therefore the length of the neck. On fretted instruments this is a design value, used as the starting point for calculating fret positions. Although it is often described as being the vibrating length of the strings this isn’t really true, since string lengths are always slightly longer than the scale length to allow for intonation compensation (see the section on intonation compensation). The most accurate way to determine the scale length of a (conventionally fretted) guitar is to measure from the inside face of the nut (or the middle of the zero fret if the instrument has one) to the middle of the twelfth fret and then double this measurement. As we will see later, even this isn’t accurate for those very few instruments that have nut compensation.

Scale length, instrument pitch, tone and string construction are closely related and early instrument design was constrained by the relatively crude string technology of the time. There is still no fixed standard scale length for steel strung acoustic guitar and it’s not unknown for manufacturers to make the same model of guitar in a choice of scale lengths. Typical scale lengths used today for the steel strung acoustic guitar range from 24.9 to 26.44 inches. Martin commonly uses 25.4 inches and Taylor 25.5 inches (approx. 645mm and 648mm, see Issue 04 for Doyle Dykes comments about his new Taylor 24 and 7/8 inch – 632mm approx., short scale guitar).

Classical guitar scale length has today settled on an almost standard scale of 650mm although classical guitars are made with scale lengths ranging from 640mm up to 665mm.

The ancestors of the modern guitar, the early four course guitar and the Vihuela, had a scale length of around 540mm for the guitar and somewhere around 720mm to 798mm for the Vihuela, so the modern guitar has wound up between the two.

The scale length of an instrument has a number of implications for the guitarist. Given the same strings, a guitar with a shorter scale length has noticeably less string tension than a guitar with a longer scale length, tuned to the same pitch. As scale length changes so does the fret spacing and this affects playability. Scale lengths may vary by perhaps as much as an inch (25.4mm) and this has a surprisingly large effect on tone. Longer, higher tensioned strings are louder with a well defined bass, while shorter slacker strings are quieter and not as strong in the bass, but can produce a warm attractive tone. However an experienced luthier, having chosen for example, a short scale length, can tune the soundboard and bracing to get the best from the shorter scale.

A player may feel a shorter scale length extends their technique by allowing a wider spread of intervals under the hand and may try compensating for tonal difference by using heavier strings. Guitars with longer scale lengths are a better choice for use in dropped tunings since their operating tension is higher.

To recap - Scale length, playability versus tone

Short scale guitars with their lower string tension and closer fret spacing are easier to play, but longer scale lengths provide richer tone and may sound more in tune. This effect of longer scale length on tone is quite evident when you examine a baritone guitar. These instruments usually have a huge, rich, sonorous sound and great sustain, compared to many standard guitars.

Longer scale instruments tend to have a longer sustain because the higher mass and higher tension of the strings stores more energy, while the shorter scale instruments can have a faster note attack with less sustain, so different scale instruments suite different styles of music.

Novax Fanned Fretting

One solution that attempts to provide the best of both worlds as far as scale length is concerned, is the Novax fanned fretboard pioneered by Ralph Novak who holds patents on the idea. Other stringed instruments like the piano and the harp, with individual strings for each note, are forced to address the inharmonicity problem (see the section on String Stiffness & Inharmonicity) by using both a mixture of heavy wound strings plus plain strings and a wide range of string lengths or scales. Novak has applied this principle to the guitar and makes his strings gradually longer from the treble to the bass. This means the fret spacing needs to get wider as the strings get longer. Although stepped frets could be used, fanned or slanted frets are more convenient and easier to play on.

With the Novax fanned fretboard, inharmonicity of the lower, thicker (therefore stiffer) strings is reduced and the whole instrument sounds harmonically rich and more in tune. Novak himself currently uses fanned fretting only on electric instruments and perhaps to greatest effect on the eight string instrument he developed for jazz player, Charlie Hunter. Other luthiers such as Ervin Somogyi have used it for acoustic instruments, for example on guitars he has built for the California Guitar Trio.

Novak also believes that using a variable scale length sounds better because it takes into account another mode of string vibration he refers to as the ‘clang tone’. It seems that piano designers may know things about strings and string vibration that guitar makers have forgotten. Clang tones have the peculiar property of being unaffected by string tension. Every particular string gauge and length has its respective clang tone and piano designers take these tones into account when choosing the length of each string in a piano. This is the reason for the sinuous curved shape of the string frame of the piano and harp.

www.novaxguitars.com

Novax electric guitar with fanned fretboard.

The Evolution of the Guitar

While there have been numerous opinions about the origins of the guitar, with some authorities claiming antecedents back to the ancient Greeks and before, the evidence amassed today indicates that the first guitar-like instruments actually called guitars appeared in Malaga during the 15th century. These instruments had four courses of two strings each, often tuned in a fourth, a third and a fourth to DD, GG, BB and EE, although other tunings were used. This was a high tuning with often a re-entrant tuning for the DD strings so they were in the same octave as the EE strings. Since it was a high pitched instrument it had a small body and short scale length. Another similar, but much larger instrument of the time, the Vihuela, had six courses of double strings and looks to the modern eye more like a guitar (a 12 string) than the four course guitar of the same period. Vihuela were made in different types intended to be picked with a quill, plucked with the fingers and even to be bowed. Today the term ‘Vihuela’ is used to refer to the bass-like guitar played in mariachi bands. All these early instruments had short necks with the bridge fitted, as it is on the lute, near the bottom edge of the soundboard, a rose filling the soundhole and long narrow bodies, with only a slight waist. There was little point in long necks with more frets, since the strings of the time were so poor that true notes couldn’t be obtained much past a twelfth fret. Also the bridge position dictates a short neck. This bridge position was probably used on lutes and other early instruments because it puts less strain on the soundboard.

String doubling or even tripling was a common technique used to try and obtain more volume, particularly with small bodied high pitched instruments. The mandolin is perhaps the most well known modern instrument still using this technique. From the 15th to the 18th centuries there was a lot of experimentation with the forms of instruments in general, driven by gradual improvements in the technology of, for example - strings and developments in compositional style and taste. Due to pressure from composers wanting to extend the range of the guitar, the guitar became larger, a fifth bass string was added and finally all the strings became single strings, along with the addition of the sixth string, towards the end of the 18th century, resulting in the E, A, D, G, B, E tuning we use today. Some experiments even extended beyond this with eight and ten string designs.

Another commonly held belief about the guitar is that it developed out of the Lute family, which in turn is based on the ancient Arabic Oud (the name ‘Lute’ being derived from al’Oud). The Oud is still extant with a large repertoire and many really excellent players. Modern Oud’s may have as many as six, double-string, courses but the early Oud had four, tuned in fourths to E, A, D and G. Like the Oud, the first Lutes had only four courses, but unlike the Oud, were fretted with gut frets. Over time the Lute gradually acquired more strings, becoming technically difficult to play. This also led to the rather peculiar situation of different types of lute being required to play different parts of the lute repertoire. Lutes today are available with six, seven, eight, nine, ten or even thirteen courses. Even the six course Lute requires a wide neck and with ten courses the neck gets very broad and difficult to finger. This is probably why the guitar wound up with six single strings as a good compromise between musical range and a comfortable neck width. Rather than being a direct development of the Lute, the Guitar, which would have been made in the same workshops and by the same craftsmen, simply supplanted the more complex instrument.

If you are interested in the early Vihuela and Baroque Guitar take a look at the website of Stephen Barber and Sandi Harris who make exquisite reproduction instruments based on their own meticulous research - www.lutesandguitars.co.uk.

Just intonation and equal temperament

Pythagoras and the monochord

The theory and structure of western music is supposedly based on the ideas of the Greek philosopher Pythagoras. However no original writings of Pythagoras have survived. Pythagoras was the leader of a mystical cult now referred to by the imaginative title of ‘the Pythagoreans’ who were extremely secretive about their knowledge and beliefs. Any information we now have about them is from later commentators. Pythagoras is credited with developing a system of simple ratios to describe the notes in a musical scale (now called Just Intonation) from his observations of the vibration of a single stretched string, with its length divided by a moveable bridge. This device is called a monochord, because it has only a single string, but can play two notes at once, one on either side of the moveable bridge, to form a chord. Pythagoras is also supposed to have shown that the entire scale can be derived from only one interval or ratio, the major fifth or the ratio 3/2. This tuning is now known as Pythagorean tuning and like all tuning systems produces errors when compared to natural harmonics.

Just Intonation

Just intonation is a tuning system based on natural harmonics, where the relationships between the notes in a scale are all ‘simple’ ratios. The harmonies produced by Just intonation are perfect, but unfortunately only in the key of the root note, so Just intonation does not allow music in different keys to be played on the same instrument without extensive re-tuning.

Most other tuning systems are based on, or at least use, Just Intonation as a reference because, for any specific key, it is harmonically perfect. The problem with JI is that it does not follow a regular mathematical progression. This makes key changes impossible because the note spacings are not equal. Other systems, such as Equal temperament are attempts to solve the key modulation problem by ‘tempering’ – by altering the tuning of the note intervals, while sacrificing some harmonic accuracy.

Equal temperament

While there are documents from the late 16th century that describe Equal temperament and there is an existing guitar that was made around 1800 and shortly afterwards converted from moveable gut frets to Equal tempered metal frets, Equal temperament only came into general use for European music around the middle of the 19th century. Although Equal temperament can be described as ‘Well tempered’, the term is more accurately used when referring to a number of different tuning schemes that allow instruments to be played in a variety of keys without re-tuning and without sounding too far out of tune.

Before the adoption of Equal temperament, lutes, early guitars and other fretted stringed instruments of the time used tied on frets, made of gut. This allowed the frets to be moved along the neck and the instrument could be re-tuned, or intonated, for any specific key or tuning system. Perhaps more importantly, the frets could be adjusted to make up for the tuning irregularity (see inharmonicity) of the crude gut strings of the time. Keyboard instruments of the time were either re-tuned or maintained in sets, with each instrument tuned for different keys. Some even had extra keys and strings fitted to allow some key modulations.

Equal temperament uses a simple mathematical formula to calculate the tuning intervals used and for fretted instruments, the fret positions. The formula is based on the twelfth root of two (since there are twelve intervals in the common western scale and the ratio of two notes an octave apart is 2 to 1). The ratio in frequency between any successive pair of notes in the equal tempered scale is equal to the twelfth root of 2 or 1.059463275. So it follows that, with all fretted instruments tuned to the tempered scale, because the pitch of a vibrating string is directly proportional to length, the ratio between the length of a string from the bridge to a fret, and the length of the string from the bridge to the next fret, is exactly the twelfth root of 2 (ignoring intonation compensation for the moment). Mathematically speaking Equal temperament is a geometric series and it is known as Equal temperament because it makes the interval between each semitone in the scale the same.
Although equal temperament allows for reasonably accurate tuning in all keys, all the notes in an Equal tempered scale, except for the octave, are slightly out of tune when compared to Just intonation for the same key or root. The biggest errors are for the major seventh, minor seventh and minor second.

It is this discrepancy between the ‘perfect’ sounding harmonic intervals and the equal temperament spacing of the frets that contributes to guitarists sometimes having tuning problems. Other contributing factors are string inharmonicity and inaccurate intonation compensation.

Another way to look at the differences between Just intonation and Equal temperament is to plot the note frequencies against their intervallic relation, or scale position, on a graph. The Equal temperament values can be plotted as a smooth continuous line, as they are shown here, because no matter where the starting pitch, all related intervals fall on the same curve. The Just intonation intervals for two starting pitches; middle C and for the Major Second (D) are shown as points because they don’t form a continuous progression. The differences between the Just scale values and the Equal temperament curve and between a C scale and a D scale in Just intonation are obvious (the scale interval labels on the horizontal axis are for the C scale).

The eighteen rule for calculating fret positions

There is another well known method of calculating fret positions called the 18 rule or the rule of 18. This method was in use in the 16th century and was probably arrived at by trial and error. Although it isn’t as precise as calculating fret positions using the pure form of equal temperament based on the twelfth root of two, in practice, the error it produces results in some degree of intonation compensation, because the higher frets wind up further away from the saddle than they do using the twelfth root of two. It’s possible to replace the 18 value with a constant derived from the twelfth root of two (17.81715375, usually rounded down to 17.817) which simply converts this method into Equal tempered fret spacing.

The simplest, although not very elegant method of using the 18 rule is to first find the distance of the first fret from the nut by dividing the Scale Length by 18 (or 17.817). Then, to find the spacing of the next fret from the first, the previously calculated value is subtracted from the Scale Length and the result is again divided by 18. This process is repeated to find the rest of the fret positions.

Other practical methods for fret positioning

Many builders don’t bother with the calculations and either use tables that have been calculated for different scale lengths, buy in pre-slotted fingerboards, use layout templates or fretting boxes.

String stiffness and inharmonicity

It is a common scientific approach to use simple models to explain the behaviour of physical systems, because it’s easier to understand basic behaviour when using simplified components. Factors that give rise to very complex behaviour, or only have a small effect, are often just ignored. For example vibrating strings are often assumed to be uniform, to be under constant tension and to have length and mass, but no stiffness. In diagrams, strings are often shown as vibrating freely, all the way up to the fixing points at either end and it’s the distance between these two points (in the case of the guitar the distance between the nut and the saddle) that is taken as the vibrating length and used to calculate the strings vibrating pitch.

Stiffness is quite a major factor that’s missing from the simplistic approach and is one of the major problems that string technology has had to minimise. If strings had no stiffness they would sustain for far longer than they actually do. Real vibrating strings have significant stiffness and don’t flex freely at the nut and saddle, but behave as though their vibrating length is slightly shorter than their physical length. To complicate things further, stiffness has a gradually increasing effect on the higher harmonic modes of a vibrating string. Also, as the higher notes are played and the vibrating length of a string is shortened, because the stiffness is a constant, it becomes gradually more significant. This effect is known as inharmonicity. A short and very thick string will sound discordant because even low order harmonics won’t be correctly related to the fundamental, but will be slightly sharp. All vibrating strings suffer from some degree of inharmonicity, instrument designers just have to choose a long enough scale for gauge of the strings they are using so that the inharmonicity is minimised. This also explains why it can be more difficult to tune a guitar with new strings. New strings sound brighter – the upper harmonics are more prominent. Because of inharmonicity, these harmonics are not perfect multiples of the fundamental note and the overall pitch is perceived as sharp. Low quality, stiff and non-uniform strings will sound even more out of tune.

A string can be tuned to a lower note by reducing its tension, but at some point a string may be so slack that it produces a note that audibly falls in pitch as it decays. All vibrating strings do this, since the tension is raised when the string is plucked and then gradually reduces as the string sounds and the note decays. At normal tensions the pitch change is small enough not to matter. Although this slight pitch drop is one factor that may distinguish real instrument sounds from synthesised sounds. Pitch in general (this is true for all non-synthesised instruments) also rises slightly as you play louder.

The other ways to get lower notes are to use longer or thicker/heavier strings. Unfortunately just making strings thicker tends to make them stiffer and their inharmonicity gets worse. This was a real problem for the early string makers in their attempts to make bass strings. Shortcomings in string manufacture had a big effect on the design of early instruments and on instruments like the Arch Lute, long auxiliary necks were added for extra bass strings.

String makers first had to develop the multi-strand twisted gut string and then the metal wire-wound string, to produce bass strings that had relatively high mass or weight, but remained reasonably flexible. Winding metal wire around a core makes a string heavier, while leaving the flexibility almost the same as the inner core.

Fret marker positions

Most guitarists probably just take fret markers for granted as a handy visual reference that helps them check where their fretting hand is on the neck. The early guitars usually had either plain or intricately inlaid fretboards, with no markers. This is probably because, with moveable frets, markers might have looked a bit strange and in any case they only had twelve frets to worry about.

The most straightforward and scientific way to explain why fret markers are where they are is to say that up to the twelfth fret they mark the first five harmonics of the natural harmonic series of the open strings (none of the higher harmonics occur over a fret position). As follows –

After the twelfth fret the marker pattern simply repeats.

Note – These fret positions only approximate the harmonic nodes due to the differences in Just intonation and Equal temperament.

This is all very well, but you may ask; why is this important to guitarists?
Starting at the twelfth fret it’s obvious that there is a pretty strong reason for a marker here since the twelfth fret is at the fretted octave which is also the strong second harmonic point.

The fifth and seventh fret markers mark harmonics of the same note for adjacent pairs of strings when the two strings are tuned a fourth apart. This is significant for standard guitar tuning as all the intervals between the strings except the third between the G and B, are fourths.

The practice of tuning stringed instruments in either fourths or fifths between the strings extends back to the very beginning of stringed instruments. This is because frets are a relatively recent invention and on an unfretted instrument, with no pitch references available, the practice was either to tune the highest string as high as it would go without snapping (which must have required great experience or possibly clairvoyance) or to tune the lowest string to the lowest comfortably sung bass note. Natural harmonics were then used to tune across the strings one to another. Without frets or markers these harmonic points are easily established by measuring, although skilled musicians would either have relied on experience, or probably would have made small marks on the necks of their instruments.
For example, the 4th harmonic of the bottom string and the 3rd harmonic of the fifth string on the guitar are the same note when the two strings are tuned to an interval of a fourth and so can easily be used to tune the two strings to this interval. The fifth fret and seventh fret markers mark these historically important points. The harmonics over the third and fifth frets can be used in the same way to tune to an interval of a fifth between strings. However on instruments like the modern guitar a certain degree of beating between these harmonics must be allowed to accommodate Equal temperament tuning.

Other than the convenience of the available harmonics for tuning, having the strings at an interval of a fourth apart makes a lot of sense because scales and chords can be played across the strings without having to stretch or move the left hand very much.
On shorter scale instruments like the violin and mandolin, tuning in fifths becomes more practical.

Many altered tunings for the guitar maintain intervals of either fourths or fifths between the strings, with usually a smaller interval for the second or sometimes third string in order to make the top and bottom strings the same note, usually a D or an E.
Dadgad tuning for example is a 5th, a 4th, a 4th, a 2nd and a 4th with top and bottom tuned to D. The ‘harmonics’ tuning technique here, or even for ‘standard’ tuning, is to tune the open top string to the open bottom and then use the harmonics to tune the second string to the top string.

Intonation & Intonation Compensation

Most guitars have some degree of intonation compensation applied to improve their tuning accuracy. Tuning inaccuracies are due to two factors – changes in string tension pulling notes sharp, which are introduced as each string is deflected on fretted notes and inharmonicity, due to string stiffness, that also makes notes sharp.

Notes of equal temperament are produced on the guitar when the fret positions are established using the methods already described. However there is a snag, the strings are stretched above the fretboard and when they are deflected as each note is fretted, the deflection increases the tension in the string and each note sounds slightly sharp. The degree of error depends on the string type and gauge and it gets worse for guitars with high actions. Simply positioning the frets according to the calculated position using the twelfth root of two does not allow for this error.

On an acoustic guitar the usual method of dealing with this is to compensate for the rise in pitch by positioning the bridge and saddle so that the open string length is slightly longer than the designed scale length, normally by around 1 to 2mm. The saddle on a steel strung guitar is also slightly angled to allow for a variation in the tensioning error across the six strings. Traditionally gut or nylon strung guitars have the saddle at right angles to the strings because the effect is not as marked with nylon strings. However, tuning on nylon strung guitars can certainly be improved with intonation compensation.

Saddle position intonation compensation is found on most guitars and it reduces the error caused by string tension changes when the strings are fretted. The correct saddle position is determined by moving the saddle until the fretted note on the twelfth fret is exactly one octave above the open string note. Any change in action height or in string gauge and type ideally requires the compensation to be re-adjusted and the precise compensation is different for each string. On most acoustic guitars the saddle and bridge are glued in place and the position chosen will be ideal for the manufacturers chosen action and string type. Compensation across the strings is usually a compromise reached by setting the saddle at an angle, although recently modern saddles have started to include offsets to suite each string. Sometimes a split saddle is used. This form of compensation is very common but is only an approximation because, even with the twelfth fret note sounding the perfect octave, the notes on the lower frets will still play slightly sharp.

This picture of a Tusq composite saddle shows the slight saddle angle required to compensate the four wound strings and the step offset on the B string needed to compensate the plain strings]

The purpose of compensation is to make the guitars tuning conform more accurately to Equal temperament. Even then, as has been described earlier in this article, perfect Equal temperament tuning will not result in beat free ‘perfect’ intervals.

Most electric guitars have some form of adjustable bridge saddles that allow the length of each string to be adjusted. So far no similar solutions have been adopted for the acoustic guitar and saddle compensation takes the form of fixed offsets, carved into the saddle. Some makers even split the saddle into two lengths set in two slots in the bridge.

Although in general the term ‘intonation’ refers to the tuning scheme that’s being used, guitarists and luthieres often use it in a more limited fashion to refer to the accuracy with which a guitar conforms to equal temperament. In practice the meaning is even more limited, since the accuracy is often set or checked by how closely the fretted octave at the twelfth fret on each string matches the harmonic at the twelfth fret. This doesn’t take account of the fact that even when the compensation is corrected for the twelfth, the notes at the lower frets will still not conform perfectly to the intended Equal temperament values.

Even greater compensation accuracy can be obtained by moving the nut towards the bridge a millimetre or so. This in effect moves all the frets away from the bridge, flattening the notes and compensates for the inharmonicity caused by string stiffness. A further refinement is to introduce individual offsets at the nut and this nut movement/compensation has to be balanced with an appropriate adjustment at the saddle. This dual compensation system is at present very rarely seen. This is partly because the required degree of compensation will vary depending on the string gauge, string brand and height of the action.

In recent years there have been a number of commercial products marketed that aim to improve guitar intonation. One of the most notorious is sold as a system that requires installers to gain a training certificate and the precise details of the system are shrouded in secrecy, despite the fact that they are on public record with the US patent office. Examination of the patent indicates that the details of the system were probably arrived at empirically and later supported, for the purposes of the patent document, by some very poorly understood and inaccurate theory.

There are other products, for example the Earvana nuts and saddles that support individual compensation for string, based on the more rigorous theoretical approach of Greg Byers, see – www.earvana.com and www.byersguitars.com.

The Earvana adjustable compensated nut and compensated acoustic saddle for use with the nut. Note how the compensation differs from the compensation on the conventional compensated saddle.

Nut compensation on a Greg Byres guitar

New developments

Do you fancy having an acoustic guitar that you can shift to any tuning you like just by touching a button? Thanks to the efforts of Steve Klein such a thing is now possible. The Transperformance/Klein design incorporates the Performer automatic tuning system, a computer controlled, servo driven, tuning mechanism, into a Klein designed acoustic guitar. With this system the guitarist has a programmed choice of 144 preset tunings and 240 user defined tunings.

http://www.kleinguitars.com/transperformance.htm

http://transperformance.com (NO www)

Other tuning systems

Although equal temperament is the accepted norm, there are a huge number of tuning systems designed to overcome or at least minimise, the key change problem with Just or harmonic intonation. One group of solutions involves providing alternate or extra notes, through dividing the octave up into more than twelve steps. One example of this is the Lucy system developed by Charles Lucy of Lucy Scale Developments, based on the work of John Harrison, the same carpenter and clock maker who dedicated his life to solving the Longitude problem of navigation. Harrison developed a meantone tuning system based on PI. His system uses two ‘intervals’ derived from PI, the Larger note (a ratio of 1 to 1.116633) and the Lesser note (a ratio of 1 to 1.073344). His scales are then constructed by adding different numbers of Larger and Lesser notes together. Harrison also specified some compensation at the nut and bridge. Lucy guitars have between nineteen and twenty five frets to the octave; the appropriate frets are fingered according to the key.

www.lucytune.com

Next month

In next months article I’ll be dealing with the body, bridge and soundboard of the guitar.

By – Terry Relph-Knight

My thanks for their help in writing this article to – Greg Byers, Deneen Patti at Earvana & Hugh Burns for his help with musical theory.