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Note to the reader: I wrote this article series over 10 years ago, and it has been here on my site all this time. Over the ensuing decade, I have continued to study guitar intonation in my work as a luthier, and have answered countless questions about these articles. I shared my experiences in a lecture at the Guild of American Luthiers Convention in 2006. A transcript of the lecture has been published in American Lutherie #92, which represents the current stated of my understanding on the topic. While the ideas presented in these online articles are still substantially sound, I strongly recommend reading the GAL article, it's a much more thorough explanation with much more practical application. You can get a back issue copy of AL#92 at the Guild of American Luthiers site.
This is the first of a short series of articles on intonation. Getting guitars to play in tune on every fret up the neck is one of the most common repair tasks, and one of the most critical parts of building new instruments as well. I hope you enjoy the articles. Meanwhile, please visit my website DoolinGuitars.com for a look at some guitars which not only play in tune all the way to the 22nd fret, but let you get there without learning new hand positions from the Kama Sutra...
The 1997 Healdsburg Guitar Maker's Festival included a seminar on intonation, a complex topic which always seems to generate confusion and spirited debate. While the gist of the seminar was simple - "intonate the nut" - questions also arose concerning temperament and stretch tuning. In this series of articles, I'd like to present an overview of these intonation topics, summarize the Healdsburg intonation seminar and to suggest a method for tuning guitars once they are properly intonated.
Out of Tune... With What?
To understand the intonation problems common to all instruments, let's first look at the origin of the twelve-tone chromatic scale and the diatonic Major scale we select from those twelve tones to make the vast majority of music. Why are there 12 keys (and 12 frets to the octave) and why do we use only seven notes to create the tonality for each of them?
Any stretched string vibrates naturally in whole-number divisions of its length, generating a fundamental frequency and a set of overtones whose frequencies are whole-number multiples of that fundamental. The relative strength of all these frequencies in combination over time is perceived as the timbre, or tone quality, of the sound.
The common guitar technique of playing harmonics simply dampens the fundamental, to force the string to vibrate in some whole-number division without vibrating as a whole. The frequencies of these notes are multiples of the open string frequency. Thus, the harmonic at the 12th fret divides the string in two equal parts (2 times the frequency), while the 7th fret harmonic divides it in three parts (3 times the frequency), the 5th fret in four parts (4 times the frequency), the 4th fret in 5 parts (5 times the frequency), and if you crank up the gain you can get further harmonic divisions between the frets up towards the nut.
The overtones also describe the seven notes of a scale, albeit spread over several octaves. This scale is similar to our major scale, but oddly enough has a flatted 7th and sharped 4th intervals. In effect, a single vibrating string is singing a whole scale of notes in its harmonic series, and when other strings play notes corresponding to these notes, the result is varying degrees of consonant harmony. Consonance is the result of simple ratios between the frequencies, while dissonance occurs between frequencies having a more complex ratio. Intervals which are perfectly in tune with the Harmonic Series are called Just intervals.
The distances between the notes in this naturally occurring scale happen to fall roughly into whole steps and half steps. If we build a harmonic series from each of these notes we'll fill in the spaces and generate a 12-tone chromatic scale. Approximately. Because it turns out that the half steps are not all the same size, the whole steps are not all the same size, and the half steps are not exactly half the size of the whole steps. This causes small, but definitely audible discrepancies between the pitch of overtones generated from different fundamental pitches.
The early history of music describes the discovery of these discrepancies and various attempts to cope with them. In fact, with each breakthrough advance in harmony usage, new adjustments, or temperaments, were devised to accommodate the increasing sophistication in harmony. In Intonation II, I'll describe the main temperaments and their corresponding musical periods. Meanwhile, I'll look for your suggestions and questions in my mailbox...
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