Perfect Intonation Long Version
William Compiano, the insightful author of the seminal lutherie book, Guitarmaking: Tradition and Technology, has astutely described the guitar as a cultural artifact and prescribed evidenced based processes to improve the quality of our instruments. We have a history and a relationship, and embrace a set of expectations surrounding this phenomenon we call a guitar. Reasonably, one of the things we anticipate is that the guitar should play in tune. Not surprisingly, it turns out that guitars sound better when they are in tune, but tragically few if any of us have ever actually heard such a guitar. We can use mathematics and a knowledge of physics to design a perfectly intonated guitar and to tell us where we should install the frets and saddles precisely and accurately… theoretically. However, the physics of a real world instrument will make our guitar play the actual notes somewhat inaccurately. The degree to which our real instrument approaches the ideal can be considered a measure of the quality of the instrument. Happily, this measurement is only tedious, and not too difficult. It would be to our benefit if we could adjust the intonation of our guitars once they are set up so that they do play in tune or as close to it as possible.
The following pages describe a method to make an objective and useful measurement of the intonation quality, a practical method to approach the ideal, and the assessment of a variety of guitars for comparison purposes. This comparison demonstrates that the described intonation process improves intonation by several factors.
On these following pages I reference a guitar intonation system designed for an acoustic guitar, diagramed below in Figure 1. The system employs both a movable nut and a movable bridge. The system in theory is well known and the mathematics has been fully described elsewhere (see Gore & Gilet).
In addition this system takes advantage of a User Adjustable Tilt Action Neck that allows the action of the strings to be changed from high to low. The feature is fully described below.
The Equal Tempered Scale (the goal, for good or bad)
Strings vibrate with a fundamental frequency and a set of overtones, or harmonics which we can use to compose a pleasing 12 note scale. Each key in this “just tempered” scale has a certain sound associated with it. That is, if we play a song in the key of C, it will sound very different than when we play the same song in the key of F. This is because the just tempered scale is made up of 12 unequal steps. Two separate intervals in the scale may or may not sound the same. This scale sounds interesting, natural and pleasing, but has the unfortunate property that a keyboard or fretboard that is tuned to this scale can’t transpose a song into another key without changing the sound and character of the song. As the story goes, to correct this problem, in the 17th and 18th century, about the time pianos became popular, our Western European culture (or the piano technicians) chose to split the octave into twelve equal parts, called an “equal tempered” scale. Each interval sounds the same as every other interval. This means that most of the notes don’t fall on the natural notes of the just tempered scale anymore, and the scale should sound wrong. It however does have the happy result that we only need one keyboard or fretboard to play in all of the keys. Luckily our brains are very accommodating and we have learned to like, expect, and anticipate the uniform intervals of our equal tempered scale. In fact, anything else now sounds wrong, exotic, or strange to us. Be careful though, this is a cultural bias, other cultures have different scales that sound just as good and natural to them as ours does to us. Go forth and forage…
Intonation and the Intonation Quality (IQ) Number
By definition, intonation is the accuracy in frequency of a set of notes compared to a standard. We have generally agreed that our standard is to tune concert pitch A to 440 hz and divide the octave into 12 equal parts. The precise frequency of each note is easily calculated and is derived elsewhere. To help enable the useful assessment of our instruments, I propose Intonation Quality (IQ) as an objective measure of this accuracy. In technical terms the quality or Q of a resonant system is defined as the frequency of resonance divided by the full with half max frequency spread of the resonance, i.e., Q= Frequency/(Delta Frequency). This means that a high Q system has a clear piercing tone, while a low Q system has an indistinct muffled tone. For example, a brass bell is a high Q system while a feather pillow is a low Q system. A high Q number means less variation and higher quality while a low Q number means more variation from the center. Remaining consistent with this definition I propose an Intonation Quality number as:
IQ = 440hz/(Average of Intonation Errors) = 100 *440/(Ave. %Error*440) = 100/Ave. %Error
The average is calculated by taking the average of the absolute values. This is because the errors are both positive and negative and we want a measure of how far they are from the zero line.
For example, a guitar with an average error of 1.8 cents would have an IQ number = 100/1.8 = 55.5. As long as the testing protocols are the same these IQ numbers can be used to objectively and directly compare the results of one guitar against another and more importantly they can help us get our guitar pretty well intonated.
The IQ number is the most important of a variety of statistical metrics that are useful in analyzing the quality of intonation. A list of some other metrics are included and explained below in the Analysis section.
A Method to Measure the Intonation Quality
The primary method of determining intonation quality is to measure the frequency of each note on the guitar and compare them to their ideal frequencies and then calculate the errors. The average value of the errors, the median, kurtosis, and the standard deviation are also calculated. Thankfully a spreadsheet makes short work of these calculations and helps keep everything organized. The data can be presented in a variety of ways for visualization and analysis purposes. The intonation errors of each individual string can be can be plotted, and the combined plots for each string can be graphed together to get an overall visual sense for the intonation quality of the instrument. Fig 2 is an example of an aggregate plot. A more thorough explanation is given in the analysis section. The horizontal axis is the note and the vertical axis is the number of cents error. To be in-tune we want the notes to play as close to the zero line as possible. The farther away the note is from the zero line, the more out of tune it is.
A Special Note on Frequency Measurement: There are a variety of tools available to make the necessary frequency measurements. Primarily we need a tool with a digital readout with at least 4 digits of precision. Current smart phones provide inexpensive access to a number of applications that meet this criterion. These apps often provide a full set of features for evaluating frequency and energy. When we make these measurements we are trying to write down the single best number that represents the frequency of the note. This is however a moving target as most notes start sharp and drift flat. Other notes can oscillate, and other times they just act weird. The problem is to be consistent in how we pick the right number. Making multiple measurements is highly advisable. At this point practice makes perfect, or pretty good. A second option is to use an audio spectrum analyzer. A spectrum analyzer has the advantage that it gives the spectrum of the notes rather than its time signature. From the spectrum we can directly read the frequency. The disadvantage is its cost and time to make measurements.
Figures 3 - 8 show the results for the six individual strings of my latest lutherie effort TH 3.1.75 (Fig 25). a mid-size steel string guitar with Tilt Action Neck (Fig 14), Split Saddle Bridge (Fig 16) and Split Saddle Nut (Fig 17). The aggregate results are shown in Fig 2. The individual graphs will be used to make adjustments to our intonation system when we set up the guitar. The aggregate of these data will be used to calculate a set of statistical metrics that will help us understand how we are improving or not.
Setting up a Guitar
Among other things, the playability of a guitar is greatly influenced by the height of the strings above the fretboard (the action). Generally, the higher the strings are set above the fretboard the harder it is to play, the lower the strings the easier it is to play. The height of the strings also has an effect on the quality of the sound that the guitar makes. Set the strings too low and the sound gets thin and ultimately starts to buzz as they hit the frets. Make the strings too high above the fretboard and it sounds great but is too hard to play. Once the action is ultimately set, a refinement is achieved by controlling the longitudinal shape of the neck with the truss rod. Ideally there is a slight concave bow of about a millimeter or less at the 12th fret that helps to prevent buzzing. So, for each and every guitar and musician there should be a setup that makes the player most happy. Our objective is to find that happy medium and then make the guitar sound as good as we can with what we are given. Once we have adjusted the action of the guitar and it plays the way the owner would like it to play, it may or may not play in tune; probably not. What to do?
Fortunately, to improve the intonation, we can adjust (compensate) the scale length, defined as the distance from the nut to the bridge, by repositioning the nut and the saddle. It has been traditional to compensate a guitar by making the bass side of the saddle about 1/8 inch farther away from the nut and then further modifying the saddle for each string. This is the slanted saddle seen on most steel string acoustic guitars. This slanted saddle has the effect of increasing the scale length of each string and typically will produce a good but not great intonation. However, good may not do it when we are trying to build a high performance guitar. So we also have the option to adjust the position of the nut. This is a bit more difficult than the saddle adjustment, but not impossible. Nut Compensation has typically been done by fixing the type of strings that will be used, measuring the stiffness and action accurately and precisely, and then using that information and a theoretical formula to calculate the amount of compensation needed for each string. A unique nut and saddle must next be crafted to fit these calculations. Once manufactured the nut and saddle are not adjustable and must be replaced if alterations are necessary. Your results may vary and predictions from the model may or may not match the realities of the actual instrument.
There are a variety of ways to look at the data and extract useful information. Right now we have two objectives in this analysis. First, we would like to have a systematic method to optimize the intonation on an individual guitar, and second we would like to have a simple set of metrics that allows us to objectively compare two guitars or monitor the progress of an individual guitar as it is intonated.
Linear Equation Best Fit to Data: When we apply a linear least square fit (an Excel linear trend line) to a set of data we generate an equation that tells us the slope of the line and its Y-intercept, see Fig 9 for an example. These parameters directly correlate to the adjustments we have at the bridge and nut. The slope of the line is controlled by the position of the bridge saddle. If the slope is positive, the notes are getting sharp as they are played up the fretboard and the bridge saddle is too close to the nut. If the slope is negative the notes start to play flat and the bridge saddle is too far from the nut.
The second parameter we consider from the linear best fit equation is the Y-intercept. We can control this parameter by adjusting the nut saddle position. If the Y-intercept is negative, the intonation is flat and the nut is too close to the bridge. If the Y-intercept is positive the intonation is sharp and the nut is too far from the bridge.
Linear Plots and Body Resonances: When a note is played the top of the guitar vibrates and interacts with the vibrating string. This top resonance interacts with the vibrating string causing the string to play sharp if the top resonance is flat of the string resonance, flat if the resonance is sharp, and causes the note to split in two if the resonance is right on the string frequency. If this is happening and the signal is sufficiently strong, this phenomenon should present itself as a consistently flat set of notes that slowly transition to consistently sharp as they are played up the string. Figure 10 shows a plot where this might be happening around the G#... maybe.
Dot plots to investigate note compatibility: One of the desirable aspects of a pretty well intonated guitar is playing two notes on two strings at the same time and they play in tune. By changing the chart type from a line plot that emphasizes the relationship of notes on a single string to a scatter plot which removes that emphasis, we can easily see the relationship of equal notes on different strings. Figure 11 shows a plot where data points in a vertical line are equivalent notes and the distance between them is a measure of the frequency equivalence. For visual clarity the horizontal axis is the natural log of the frequency.
Displaced Dot Plots to Look for Fret Position Errors. Figure 12 is created by offsetting the scatter plots so that notes that are played on a single fret are now shown in a vertical line. By examining the data point placements it should be possible to see systematic fret placement errors. If the note errors drift systematically from sharp to flat or visa-versa it may be caused by a fret placed at an angle. If a single note or set of adjacent notes are sharp or flat, it may be possible that the fret is miss crowned. If all of the notes play sharp or flat it may be possible that the fret is misplaced or the crown is off center.
Reduced Plot Playing Area (Practical Playing Area): It may be desirable to optimize the intonation over the set of notes most likely to be played. Figure 13 shows a plot where the highest notes on the lowest strings have been ignored and not plotted. These are notes that are least likely to be played. With a reduced number of notes it will be easier to get the intonation right.
Statistics: The following are a set of statistics that help describe the distribution of errors. We can use these to compare different instruments, or to follow the progress of intonation improvement as we set up a guitar.
Mean: commonly called the average.
Median: the number where half the population is above and half below.
Standard Deviation: a measure of the spread in the data.
Kurtosis: a measure of how much data is in the wings of the distribution.
Intonation Quality Q=Frequency/Delta-Frequency = 100/(RMS Percent Errors)
The quality of an acoustic instrument has many elements with the sound it makes one of the most important. Intonation or how well it is in tune is a major component of sound quality. Needless to say, a well-tuned guitar sounds better than one out of tune. To enable our efforts I have develop a set of technologies and a system that is compatible with an acoustic guitar. The Portland Guitar Pretty Good (PGPG) Intonation System works Pretty Good, not perfectly, but pretty good. The basic physics and mathematics of the system is well known and has been for a very long time. This new system is flexible and adaptable and can be changed as the guitar matures and changes. The system should be applicable to other stringed instruments. The real opportunity here is crafting a new class of instruments that raise the level of quality by several factors, but a rearguard action can be taken to retrofit instruments already made.
Although not trivial to manufacture, the components for the system are straightforward, simple and intuitive. If getting to perfection is an admirable but fool’s errand it comes at the price of manufacture complexity, setup and maintenance, or else simplicity. The PGPG Intonation system works well for this purpose. It is easy to build if you have the right tools, is mostly intuitive, addresses intonation in-situ and respects the guitargeist of the instrument. But, it is much more complex than a straight line bone saddle and nut which works well enough for the majority of instruments. Using this system should help when the errors of the standard systems are not acceptable. If it has to be as right as possible however, this may be a way to get there. When the setup and maintenance steps are an acceptable burden then the results of this system may well be worthwhile. On the other hand, perhaps it is time we finally hear our guitars in tune. Even given the increased complexity of the PGPG system, so far it seems to be stable and robust and easy to implement. Only patience will tell if it will stand the test of time… or is that the test of tune… we shall see. My hope is that this system helps us enhance and enjoy the beautiful sounds these wonderful instruments make.
An Amusing Anecdote
When I was developing this technique I had finally put everything together and tuned up my new guitar. I then started to apply the PGPG intonation process and watched the graphs converge on the zero line. I kept working at it and started to get frustrated with the variation. I poked and prodded and re-measured every note a dozen times. I finally gave up and said that this is as good I can do I guess. I thought to myself that this works, but only to a degree. Then a brilliant idea, let’s check another guitar. So I grabbed my Collings D2H since if I had to face the music as, it were, this was the guitar to compare. I started to go through the process and got the first results. I finished up and looked at the graph and compared it to my new guitar and then went back and forth. Hmmm, maybe I’m on to something here. I promptly stopped working on my guitar and celebrated… Hazzah!
What’s Next, Follow-On Research and Development
Make measurements easier: One has to admit that measuring every note on the guitar several times is a tedious, boring time consuming process. It would be desirable to find a proxy for a full set of measurements, that is, what is the minimum number of measurements that need to be taken to get the same results as when a full set of measurements is used.
Reduce measurement error: Presently measurement error seems to be the limiting factor to improving the results. If we can increase the signal to noise level of the intonation measurements we should reveal the individual errors that the frets are introducing, and perhaps the resonance interaction will become more apparent. The fact that the notes pitch follows a trajectory that takes it from sharp to flat to sharp to whatever as it loses energy is a particular problem. To improve accuracy there are at least two paths to follow. One is to use a spectrograph to record the data. This increases the complexity of the measurement and can increase the time required for each measurement depending on precision. The results from a spectrograph are high quality and the method is worth pursuing although cost is prohibitive. The second path to pursue is to develop a transducer that can be lightly coupled to the strings. By using a high precision audio generator we can fret each note on the fretboard and then tune the generator to excite the fundamental of the string. This method has the advantage that the measurement is made at a steady state condition and should eliminate frequency wander. The disadvantage is that the coupling between the transducer and the string may alter the resonance slightly. This would need to be investigated.
A lot can be learned about the intonation characteristics of guitars by conducting a survey of instruments. What we don’t know is what we don’t know and it would be nice to shed some light on the subject. I would like to measure as many guitars I can get access to. It will be important to examine the gamut of instruments form off the shelf to custom high performance guitars so that we can see what works and what doesn’t.
Continue to develop the adjustable nuts into more elegant a sophisticated designs.
Spread the word about this new development.
And finally and most important, build more guitars.