|"All things being equal, a guitar with a longer scale length will sound a bit brighter, with a quicker attack and more defined bass notes. The sound will be more chime-like. The short scaled guitar will be sweeter and warmer sounding. The basses may be rounder (some might say ‘muddier’) but the highs will be more lyrical and resonant sounding."|
SCALE LENGTH & INTONATION
by Chris Herrod
What is ‘scale length’? How does the scale length affect the guitar’s tone? Why is a guitar’s saddle slanted? Why do some guitars play more in tune than others?
Intonating a guitar (that is, building or repairing a guitar to play in tune) is a crucial skill for a luthier or repair person to learn. Still many beginning luthiers avoid discussion about scale length and intonation because inevitably, they are asked to wander into a description of the physics of string vibration, harmonic temperament and compensation -with all the icky math that goes along with these topics. We will attempt to cut a path through this material that will be accessible to the beginning luthier or to those who are beginning to work on (or just understand) their instruments. Those who would like to more deeply understand the science behind intonation are referred to Mike Doolin’s excellent article in ‘American Lutherie’ (Issue 92. Guild of American Luthiers, www.luth.org and Greg Byer’s article at www.byersguitars.com.
These articles help explain how the design of stringed fretted instruments presents inherent “problems” that the luthier (and to some degree, the musician) must work around and show that no instrument made with straight frets will play 100% in tune (according to a sensitive electronic tuner).
It’s a series of compromises and adjustments that allow the builder or repair person to intonate an instrument as well as is possible. The purpose of this article is to map out the reasons why these compromises must be made and to show how a guitar’s set-up and construction achieves the best result, with an emphasis on the real-world issues of nuts, saddles, strings, frets and scale lengths (that is to say, with not so much math and science!).
What Do You Mean This Guitar is Not in Tune?
As mentioned above, even the most expertly intonated guitar will not play perfectly in tune. Luthier Kevin Ryan says this is “inherent inharmonicity” and explains the concept concisely in this passage from his website (www.ryanguitars.com):
“…the fundamental of a string produces a tone along with, say, the 1st partial of that string (harmonic at the 12th fret). Let us assume the note is A (oscillating at 440 cycles). In theory, the 1st partial will be oscillating at 880 cycles per second. But it is not! It oscillates slightly faster-- i.e., sharper. Why? The answer lies in the physical properties of a vibrating string. The proportion of stiffness to diameter in a half-section of a string is greater than the proportion in the whole string. And the shorter the section, the greater the difference in the proportion. In any string length (whether fretted or open), the partials are all progressively sharper than the fundamental. This is Inherent Inharmonicity.”
We’ve veered terribly close to some arcane science by including that quote, but worry not! The point is that even a simple single vibrating string contains the seeds of dissonance.
It’s a similar situation if you double the length of a vibrating column of air (i.e. a flute). You might assume the sound will drop an octave, but in the real world, the pipe holding the column of air (or the metal in a length of string) has characteristics such as mass and thickness which do not change in proportion to the change in length of the vibrating object.
In essence, we are struggling with the limitations inherent in the material world! It would seem that we are cursed from the start. Luckily, all we need to do is get somewhere close to perfect intonation, and to balance our compromises equally, to make a nice sounding guitar. As we will see later on, there are some fairly simple rules and techniques to follow to make sure that happens.
What we hope to do is isolate and minimize some common intonation problems (the effects of inherent inharmonicity) that are familiar to most guitar players. For example, most of us are familiar with the experience of tuning a guitar up so that an open chord sounds good - just to find that a barre chord up the neck sounds “off”. Or, we tune up to an open G chord and the open D chord sounds off (most likely it’s the third, or F# note that will be the main offender. The 6th of a scale is also a problem tone).
A handful of builders have experimented with what is called a “true-tempered” fingerboard to help deal with these harmonic issues. Essentially, each note on this kind of fingerboard gets it’s own, individual little fret. This means string bending is forbidden; the luthier spends all week fretting a fingerboard and refret jobs on the instrument cost a small fortune! Obviously, it is an inelegant solution.
Luckily, a small degree of “inherent inharmonicity” is something we have all grown accustomed to. Says Doolin, “The fact is, intonation is part of the expressive palette of most instruments. Wind, string and vocal musicians adjust the intonation of each note, and not necessarily to achieve perfect consonance. Guitarists in particular often bend strings for pitch inflection. This is part of the essential tension and release of musical flow. All musicians, regardless of what instrument they play or sing through, must play their instrument in tune, and use intonation as an expressive tool.”
In order to understand intonation, we need to get an idea about why frets (and by extension, the guitar’s saddle) are positioned where they are. The position of a guitar’s frets, nut and saddle are all determined by what is called the scale length.
There is one common myth about ‘scale length’ that beginners sometimes hold and that we should dispel immediately; that scale length is equal to ‘string length’, it aint! It is best to think of scale length as the mathematical factor that determines the spacing of the frets. The position of the saddle is related to the position of the frets, but as we will see in the section on compensation below, we will want to alter the position of the saddle slightly. The same is also true for the nut.
Because compensation at the saddle is more common than compensation at the nut, the best way to find the scale length on an existing instrument is to measure from the nut to the twelfth fret and multiply by two, making sure to measure along the center of the fingerboard (most fingerboards are tapered, so it is best to measure between the 3rd and 4th strings on a 6 string guitar). But plenty of instruments do have a compensated nut, so this method is not 100% precise.
Theoretically, you can find the scale length by getting the precise measurement between the crown, or peak, of any two frets. This is because the fret’s spacings are related by the 12th root of 2.
Here’s the math (fast-forward if it gets too graphic!):
If the scale length is factor X, then the distance between the first and second fret is the 6th root of 2 times X, minus the 12th root of 2 times X. If Y is the distance between frets 1 and 2, then the scale length is Y divided by (the 6th root of 2, minus the 12th root of two).
To sum up then, the scale length is a numerical figure used to determine the placement of the frets. Luckily, modern luthiers can rely on simple computer generated spreadsheets to provide these distances –or can use precision-machined fret templates to make their fretboards. Both (fretting templates, fret scale spread sheets) are available from LMI. Essentially, once you have settled on a scale length, the placement of the frets is pre-determined and does not need to be altered or fussed with.
So why choose one scale length over another? Scale length affects two facets of the instrument that are of utmost importance to the guitar player; tone and playability.
All things being equal, a guitar with a longer scale length will sound a bit brighter, with a quicker attack and more defined bass notes. The sound will be more chime-like. The short scaled guitar will be sweeter and warmer sounding. The basses may be rounder (some might say ‘muddier’) but the highs will be more lyrical and resonant sounding.
It’s important to recognize that tone begins with the string! Everything else is a modifier. This includes the woods, body shape, pickups (on electrics)…the list goes on. When designing a guitar from scratch, start with the scale length and proceed from there!
Two basic facts about strings bear on playability: One, the longer a string is (i.e. longer scale length) the more tension is required to bring it to pitch. Two, the thicker a string is, the stiffer it is (more tension).
When designing an instrument, we want to choose a scale length that will allow the player to play in a chosen register (i.e. a bass guitar for low notes, mandolin for high notes) and to choose strings appropriate to that register. The strings should not be too slack (this causes rattles and buzzes) nor too stiff to play on.
Excess tension limits the string’s ability to vibrate freely. A stiff string is not only difficult to play on, it is also going to sound one-dimensional. This is because it can not divide up into enough harmonic nodes to make a pleasing sound. Furthermore, it will be hard to keep an overly stiff string in tune as it causes the tuning machines to work harder than necessary.
Keep in mind also that the angle of the string over the nut, and the degree to which a string moves at an angle from the nut to the tuning post, will increase tension in the string.
Players will often want a thicker string because the greater mass will generate more sound, but it’s a balancing act –we want to get the tension of the string in the sweet spot between too floppy and too stiff. One might like to improve the playability of a long scaled guitar by using lighter gauged strings, but doing this will not necessarily preserve the long-scaled instrument’s tonal envelope (i.e. the quick attack). Nor will it magically endow the instrument with the “sweetness” that we find in shorter scaled guitars. All of these factors come to play independently and their various combinations yield individual results.
Intonation and Compensation
So, assuming we have selected the appropriate scale length for the instrument we are building, we now have to evaluate a few other elements to determine how to add compensation to the saddle and thus properly intonate the instrument (we’ll cover nut compensation separately). Compensation here is defined as lengthening the individual string length beyond the scale length. Compensation therefore flattens the note.
Here are some factors that influence how and why we will compensate an instrument:
- The pressure of fretting the string, sharpens the note because the pressure adds tension.
- The higher the action, the more compensation is needed.
- The shorter the scale length the more compensation is needed.
- A lower or “slack” tuned guitar string will require greater compensation. For example, a guitar tuned with a dropped D tuning, or the popular DADGAD tuning, does not usually require extra compensation, but if a player tunes the 1st string down to C or B, more compensation is needed.
- The stiffer the string (do not confuse stiffness with tension here) the more compensation is needed. Steel is stiffer than nylon, for example, and requires more compensation. Similarly, thicker strings will require more compensation. So the bass strings are compensated more than the trebles and a guitar made with heavier gauged strings will require greater compensation.
A few of these elements intersect. For instance, the bass strings vibrate more and so require higher action to prevent the vibrating string from buzzing against the frets at the center of the fingerboard where the string vibration is the widest. So the higher action and thicker string gauge combine their effects and cause us to move the saddle back correspondingly.
At first glance we might see a compensated saddle as a straight, slanted line, but the need to compensate the strings does not necessarily graduate evenly from the lowest to the highest string. Some luthiers are now installing two separate saddle pieces, one for the lower four strings and a separate piece for the two unwound treble strings (acoustics typically have two unwound strings, electrics have three). Strings are made with windings to help increase their mass without adding stiffness. When we “jump” from the wound G string to the unwound B string, the relative mass of the B string is greater and therefore the string requires a little extra compensation (the jump goes between the wound D and unwound G on most electrics).
Another more popular approach, is to install an extra wide saddle piece so that the breakaway point for each string can be individually positioned (compensated) along the length of the saddle. Of course, electric guitar makers have it made when it comes to individually compensating the strings as many electrics have an individual, adjustable saddle for each string.
It should be mentioned that a good number of luthiers and repair people make their instruments as many of the factories do –without individually compensated strings. For these folks, there are standard ‘starting points’ for saddle position.
Here is a formula that is applied to many standard steel string acoustics:
Find the point on the bridge that is double the length of the nut to the 12th fret and add 1/16" where the 1st string will cross the saddle and add 5/32" where the 6th string will cross. Mark a pinpoint on those two points then draw a line between the points. This will be the forward edge (not the center) of a ramped 3/32" saddle slot.
Nylon string acoustics:
On many nylon stringed guitars you will often see a straight saddle as opposed to a ramped one. The saddle will (usually) be compensated (that is, set back) about 2mm. A minority of nylon string guitar builders slant the saddle a little bit with the high E string about 1mm back and the low E about 2.25mm. It is not uncommon for classical builders to include some extra compensation for the G string.
But, for many luthiers (especially those who are custom making instruments to their customer’s specifications for string gauge, action etc.) it is necessary to eschew these formulas and fine tune the compensation for each string individually. If you are using the wide saddle approach (mentioned above) here is the technique for intonating each string individually:
- After the saddle has been adjusted for action, height and the radius of the fingerboard, put on a new set of strings.
- Slide a small wire (a piece of .010 steel string will work) between the tuned string and the saddle.
- Play the note at the 12th fret. Then play the harmonic at the 12th fret.
- If the note is flat or sharp, move the wire toward or away from the nut until the note and the harmonic are perfectly in tune
- With a sharp pencil, mark the saddle on both sides of the wire.
- Repeat this procedure with each string and then remove the strings and file the saddle until the usual rounded edge is formed at the center point of each pair of pencil marks.
- Replace the strings and check the intonation to make sure it’s “perfect”.
It is now a common practice to compensate the nut by moving the breakaway point of the nut (which should be at the very edge) closer to the saddle a short distance. This will help minimize the intonation differences between fretted and open strings. This is considered necessary because fretted notes are sharpened by the tension added to the string when fretting. Because open strings do not ‘receive’ tension in this way, they are slightly flat in comparison).
Nut compensation also helps minimize the difference between notes played near the nut (which tend to be a bit sharp) and those between the 4th and 12th fret, which can run a little flat. As we mentioned before, higher action will necessitate greater saddle compensation because more tension is added to the string as it is pressed down on to the fret. Similarly, the action at the nut (that is, between the nut and the first fret) will have a bearing on the compensation required at the nut. Generally, the lower the action can go at the nut, without generating string buzz, the easier it will be to intonate the guitar and the less nut compensation will be required.
There are a number of complex theories about nut compensation, such as those put forth by John Gilbert and Greg Byers -and the patented method by Buzz Feiten which is licensed to manufacturers and repair persons. There are some products, such as the Earvana Nut, that incorporate compensation into their design. However, for the great majority of luthiers, it is simply a matter of moving the nut .012 to .014 closer to the saddle.
Luthier Lance McCollum has noted that for those who order pre-slotted fingerboards (such as the ones sold by LMI); there is usually a fret slot to mark the nut position. The slot width is .023 or .025 wide. If you cut the board, as most do, right at the slot and position your nut there, then you have automatically introduced this compensation, as the prescribed amount (.012) is incidentally half the width of the slot!
A note about 12 string guitars (also courtesy of Lance McCollum): On most guitars the slots in the nut are positioned so that the tops of the strings are level. On a twelve string it is important that the bottom of the strings be level for the various courses (pairs of strings in octaves) to intonate correctly. Also note that you will need to take the staggered string gauges on a 12 string into account when compensating the saddle.
As we mentioned above, there can be subtle differences in intonation at different positions on the neck. There can also be differences between open, fretted and harmonic tones. Many people have put forth different methods for tuning a guitar to balance out these differences. One method that is considered to be among the more thoughtfully constructed was formulated by Kevin Ryan and is described on his website (www.ryanguitars.com).
A) When you tune the following fretted notes to the harmonics, tune them "beatless"-- i.e., without any hint of "rolling" or pulsating as the two notes synchronize. When two notes get closer, their "beating" slows down until it disappears altogether when they are perfectly in tune. This is very important! This is the skill to be gained!
B) In each step below, pluck the harmonic first. Then fret and pluck the designated string. This allows you to hear both notes simultaneously. Then tune the appropriate string.
1. Tune the D string to a known source.
2. Pluck the 12th fret harmonic of the D, then tune the G (fretted at the 7th fret) to this harmonic.
3. Pluck the same 12th fret harmonic of the D then tune the B (fretted at the 3rd fret) to this harmonic.
4. Pluck the 12th fret harmonic of the G and tune the High E fretted at the 3rd fret to this harmonic.
5. Tune the 12th fret harmonic of the A to the G fretted at the 2nd fret (pluck the harmonic first!)
6. Tune the 5th fret harmonic of the Low E to the High E open (pluck the harmonic first!)
Note: To apply the tuning method to alternate tunings, all you have to do is find the proper fretted note on the string you are tuning and tune it beatless to a 12th fret harmonic on a string below it. Easy as pie.
Final advice: take note that old strings are more difficult to tune than new strings. This is because of uneven stretching of the string and the subsequent erratic vibration patterns. In some instances, old strings are impossible to tune correctly. If you have difficulty achieving good intonation, change strings.
A guitar is a complex and imperfect amalgam of elements and it is the job of the builder and/or repair person to balance, adjust and combine these elements so that the guitar performs as well as possible. Furthermore, this must be done with the musician’s specific requirements in mind. Not an easy task!
In the end, there is no substitute for experience. Listen to what guitarists have to say. Play a variety of instruments. Do research. And, for those who are not yet experienced in instrument set-up and intonation, it is important to respect the fact that a few thousandths of an inch difference here or there on the guitar can make a huge difference in how well a guitar sounds and plays.
Once again it becomes apparent that lutherie is a unique combination of art and science. However you understand this combination, it is important to know that the area within which your creativity as a luthier can move is restricted tremendously by the physics of the instrument -and by the limited tolerance musicians have for the variance from “perfect” intonation that we must, at least to some small degree, accept.