I. Introduction

For years I have tried to learn to play the guitar by ear. My father could strum along to accompany the singing of popular songs, and that is what I wanted to be able to do. So, about 1967, I bought a Yamaha FG-75 steel-string guitar and enrolled in a Fairfax County Community College, Virginia, course on how to learn to play the guitar.

The course was taught by a degreed music instructor, who was well qualified. He taught the basics of how to read music and play simple songs. He taught the basic triad-based chords. My only criticism of him is that he disdained country/western music, which I enjoyed very much. He said disparagingly that there were basically only six different types of country songs. He was evidently unimpressed with the simplicity of country music. Although the fact that it has simple, catchy melodies, is understandable, and the lyrics are about real life (happiness, tragedy, love, unrequited love, folk songs, ballads) did not impress him, those are the very reasons I enjoyed it and wanted to learn to play the guitar.

I already knew how to read music from secondary school, where I played the trombone. With this background, learning to read music for the guitar was not difficult. The treble clef is used instead of the bass clef, but everything else is about the same (flats, sharps, note durations, key signatures, terminology). The big difference is that the trombone is a “one-note” instrument, whereas on the guitar (like the piano) many notes can be played at the same time. This represents a substantial increase in complexity. Furthermore, unlike the piano, in which each tone is represented by a single key, a particular tone can be played a number of different ways on the guitar (since there are six strings, each covering an octave or more). The fact that several notes may be played simultaneously means that the guitar, like the piano, can play chords (which, by definition, are simply a selection of several different notes played at the same time).

The additional complexity of the guitar means that it takes somewhat longer to master than a “one-note” instrument, but instruction once a week for one semester, plus some daily practice, is sufficient time to accomplish the basics. At the end of the course, I could play Valse Lente – a beautiful slow waltz with lots of pretty chords -- from the music. I could play a number of other songs, such as Malagueña, Marty Robbins’ El Paso, and Wheels, all from music. I could not play a new song by sight, but could master it with some work.

At this point, I knew how to play the guitar, but my knowledge was useless with respect to accomplishing what I wanted to do – accompany popular songs in any key by ear. Learning each new song required a lot of hard work and memory, and changing keys was out of the question. Since I soon realized that there was no way my then-knowledge of the guitar could help me realize my goal, I lost interest in the guitar. I would pick it up every few years, relearn Valse Lente or Malagueña or Wheels, and drop it again.

On a business trip to the Philippines, I came across some fine-sounding locally made guitars for about $100 (the Yamaha had cost $75 new), and I bought one. I liked playing it much more than the steel-string model that I had originally bought. It was a “classical” style guitar, with nylon strings instead of steel strings. Another difference between this one and the “country” style Yamaha that I had started with is that the neck is wider. These two differences make the classical (or “folk”) guitar substantially easier to play. The wider neck makes it easier to form the chords, particularly for someone with large hands. The nylon strings are easier to hold down than the steel strings, and your fingers do not get sore, as they do with steel strings. Also, it is easier to play “barre” chords (in which your index finger presses some or all of the strings at a fret). The only problem for some people, especially those who like country/western music, is that nylon strings do not produce the same (traditional) “sound” as metal strings.

About 15 years later, I took another “first course” in guitar, at Pima County Community College in Tucson, Arizona, but it was a repeat of the Fairfax County course, and brought me no closer to my desired goal. Then, in a Tucson restaurant one evening, I heard a superb guitar player, Jorge Lopez, who played exquisitely. He played in the classical style in which the thumb plays (picks) bass notes and the fingers play treble notes. He played mostly Mexican and Spanish pieces, which I liked very much. I asked whether he gave lessons, and he said that he did. I began lessons with Jorge.

Jorge did not use standard staff (stave) music notation, but “tablature” notation, in which each string to be pressed is indicated on a six-line staff (in which each line of the staff represents a string). After a while, I could play some beautiful Mexican and Spanish tunes. Jorge would write down the tablature representation of any song I requested, so I learned a number of other favorites as well. Once again, however, I could play only the version of the song that I read from the music, and changing keys was impossible. Memorizing songs was difficult and slow, so I was essentially “chained” to the sheet music. Despite much additional work and the ability to play many more pretty tunes, I was no closer to my goal of accompanying songs, and after some time I again lost interest. I produced a record (“Conquistador”) of Jorge’s favorite songs. Jorge moved to Toronto, where I imagine he still lives (and entertains). I was no closer to my goal of learning to accompany popular songs in any key by ear.

While I was living in Tucson, a group of friends started a small band at noontime, and I learned to play the baritone horn (similar to the euphonium). I had always thought that the baritone had a more mellow sound than the trombone, and in band arrangements it seemed to play the melody much more than the trombone did. I enjoyed that very much. We played a lot of marches by John Phillip Sousa and others. I then moved to Sierra Vista, where I joined the Cochise County Community College band. We played a wide variety of music, and that was extremely gratifying. Things changed, I moved from Sierra Vista, and the amount of music in my life dropped to near zero. Even my spare time while driving the car I spent listening to French and Spanish language tapes, rather than music on the radio or tape player.

My father was very musically talented. He could play the guitar (acoustic, electric, or “Hawaiian steel-string”), piano, organ, electric keyboard, accordion, violin, and bagpipes. He could play a “one-man-band” setup of a drum, harmonica, and guitar (simultaneously). He could play an ocarina, flute, recorder, a jaw harp, and a kazoo. He could even play a “saw.” He could play from music or by ear. He was a “caller” for square dancing. He enjoyed electronics, and built an electronic organ in the 1950s, years before they became commercial items. He built and played a marimba. Although not a perfectionist, he was good at music, and was often asked to lead local parades playing the bagpipes. He played in local groups, such as the Spartanburg “Rhinelanders” German-music band (accordion), and often played to entertain people in “old folks” homes (usually with the accordion or electronic keyboard).

Because he could play by ear so easily, and with so many instruments, I always assumed that he had some sort of natural “ear” for music. He always “knew” what chord to play next in a song. And not just chords -- he could play tunes (melody and harmony) on the piano or accordion by ear, without ever having seen the music. I could never do this, even with the “one-note” instruments that I had played regularly for years (trombone, baritone). With respect to chords (on the guitar), I could try several chords and guess which one sounded best, but I never had the foggiest idea of what chord to play next in a song. There is no way I could play a new piece by ear, on any instrument. I could play a song in chords if the chords were specified, but had considerable difficulty in figuring out which chord to play by myself. Dad showed me an article about the “circle of fifths,” and that it was a guide to determining the “progression” of chords in a song. This information was of limited assistance in determining which chord to play next in a song. It narrowed down the possibilities, but that was about all.

After having taken several courses in the guitar and still having no idea how to play a piece or accompany a song with chords by ear, I assumed that I had a “tin ear” that prevented me from playing by ear, even though I could carry a tune singing. I was chained to sheet music. I enjoyed playing in a band, and could learn to play pieces solo, but I resigned myself to the belief that I did not have the ability to play music by ear. I had mastered two instruments (trombone, baritone) and learned to play a few others (harmonica, bagpipe chanter, mandolin, electric keyboard). I had invested a lot of hours in practicing the guitar over the years, and learned two different methods of reading music (standard music notation and tablature). I had learned all the basic chords (major, minor, seventh, in “home” position and barred). In addition to the courses that I took on the guitar, I had bought a number of books on how to learn to play the guitar and on the theory of music. None of these sources gave any insight in how to play by ear. The fact that I never saw courses advertised on how to play music by ear supported my conclusion that this was a natural talent that I did not possess.

The main reason why I concluded that I could not learn to play by ear was that I could not sense what the next chord in a piece was. There were other reasons as well. In grade school, when the choir director asked the choir to sing a particular note (e.g., “Sing a middle C”), I never had any idea what tone to sing. In 1987, a friend of mine asked me to join a “barbershop-quartet” singing group. Since I enjoy music very much, I agreed. It was a disaster. Not only could I not intone specified notes identified by letter (e.g., “Sing a middle C,” as the choir director had years earlier requested), but I could not follow the sheet music even when I had heard the correct starting note. In other words, I could not sing written notes of a specified scale, given the first one – I could not sing from musical notation, as I could using a musical instrument. I had thought that I could, since I never had any problem singing the melody of hymns (even unfamiliar ones) in church by following the music. Evidently, however, I was being guided by the piano or organ. In the barbershop group, however, you are “harmonizing” – you are not singing the melody, but a sequence of seemingly unrelated notes that are in harmony with the seemingly unrelated notes that the other members of the group are singing.) Although I have heard a “middle C” hundreds of times, I have no memory of it and cannot reproduce it without hearing it again. In elementary school, I had learned to sing a major scale using the Italian note names (Do-Re-Mi etc.), so I did have a sense of how to spread eight notes over an octave (from one sound frequency to double that frequency). And I can “carry a tune” – sing melodies of songs from arbitrary starting points (keys).

So, based on my inability to reproduce middle C from memory, my inability to sing arbitrarily designated notes on a scale given the base (key) note, and my inability to play by ear with instruments that I had mastered, I concluded that I was simply unable to learn to play the guitar by ear. My “ear” was evidently not sufficiently well tuned to enable me to do this. I had no doubts about my intellectual abilities to learn music, or to learn anything. I earned a PhD in mathematical statistics from the world’s leading university in the subject, and I have self-taught myself many things, from auto repair to building computers to systems and software engineering. I have participated in many physical activities (tennis, golf, fencing, archery, judo, jiu-jitsu, Tae Kwon Do, SCUBA diving, cross-country running, skiing), and am in good health (except for one deaf ear in recent years). The fact that I am in good health and have been able to master about anything I wanted made it difficult for me to accept that I could not learn to play by ear, when others could. Perhaps that is why I kept trying, from time to time, for more than three decades.

Another fifteen years passed. I found myself in Botswana, on a two-year consulting contract. As luck would have it, at about the same time that my wife and I arrived, another man – Gordon Atkinson – and his wife arrived from Canada, also on a long-term contract. He was an accomplished guitar player and singer. He could strum chords to accompany any song, by ear, and in any key. Moreover, he played all the country/western, folk, and popular songs that I liked. Gordon said that there was no special secret to it, and that with a little practice I could learn to do it also. This was exactly what I was looking for! I would try one more time!

We became good friends, and in fact rented homes next door to each other. He was quite willing to show me his method of playing the guitar, and I resolved to learn it before my assignment was over. We generally practiced for a couple of hours one night a week, for several months. Over the Christmas season, I spent about an hour a day practicing. I can now strum chords to accompany many songs by ear (despite the fact that I am stone deaf in one ear). It is not simple, and I am not at all perfect at it yet, but it is quite possible to accomplish this with a modest investment in practice. This article will tell you how I did it.

(My learning to play the guitar by ear, and my writing of this article, are a direct result of the "Year 2000 Problem" (the inability of some computer systems to correctly recognize the millennium date change). I was Director of Management Systems for the Bank of Botswana (Botswana's central bank), and had to stay in Botswana over the millennium date-change period (December 1999 - January 2000) to address any date-change problems that might arise. In mid-December my wife left for Christmas vacation in the US for six weeks, and I had a lot of spare time. Before she left, I could not play a single tune by ear in an arbitrary key. As she departed, I told her that my goal was to be able to play ten songs by ear in a number of keys. When she returned six weeks later, I could play virtually any popular song in a variety of keys, and had written this article.)

This method involves only the playing of chords to accompany songs. I still have no idea how my father or others can play the melody as well as the harmony of songs by ear. So far as I am aware, that is a natural talent. It is possible that there is a secret to that, also, but I do not know it. As I mentioned, despite many years of playing two instruments, I have no idea how that is done.

One thing that always bothered me in the books on music theory (at least the books that I found and bought) is that they did not explain at all the “why” of the theory. They described chords (major chords, minor chords, seventh chords, augmented chords, diminished chords, and others) and the circle of fifths, but they offered no explanation as to why certain tones sound good together, while others did not, why a particular selection of chords is used in a song, and why some chords seem to naturally follow others. They did not explain why attention centers on major scales and minor scales, or why musical notation is so complicated (with flats, sharps and keys), or why particular keys are used, or why the basic chords involve just three notes. I have not been exposed to the academic field of the theory of music, and perhaps the problem was that I was buying music theory books in popular-music stores. Perhaps there are books on the theory of music (in universities) that do explain the answers to the many questions that I had, but they are definitely not in the popular press.

I like to understand the reason behind what I do, and just accepting the rote pronouncements of the music theory that I read was very unsatisfying. Recently, I spent some time figuring out the “whys” of music theory. There are very good reasons for the tenets of music theory, and it is very satisfying to know why things are the way they are. Before explaining how to accompany songs with chords, I will explain this theory. I have not seen this theory anywhere else, but I have not researched the topic and this material is probably available from somewhere (but not in music stores!). (I do not have ready access to specialized reference materials here in Botswana, so I have not checked this out.) The explanation is not difficult to someone who has some basic knowledge of physics (sound, wave motion, harmonics) and mathematics (the exponential and logarithmic functions). It is because of this mathematical/physical description of music theory that I refer to this method as one directed to mathematicians and physicists. If you have no interest in understanding music theory, you can still learn to play the guitar by ear – you will just not understand why it works.

II. What to Look for in a Guitar

The reason why I bought a Yamaha guitar was the good reputation of the firm. I had been aware that Yamaha made some of the finest pianos in the world, and I assumed that they made good guitars. They certainly make good baritone horns.

There are lots of good guitars on the market. Listed below are basic features that you should look for or consider in selecting an acoustic guitar. I don’t know much about hardbody (solid electric) guitars, and have no comments on them. You do not need to spend a lot of money for a good guitar. I have bought several guitars that sounded great and played well, and never spent over $150 for one (although that was some time ago – I bought the Yamaha in 1967). If you buy a used guitar, check carefully for cracks in the wood or separations in the joints, or strange vibrations that might suggest these. Nicks, dents, and scratches all affect the sound quality, so do not buy one in poor physical condition unless the sound is good and the price is good. The age of a guitar doesn’t matter that much – they do not wear out like cars.

There are good buys available in private sales, because a lot of people try to learn and then give up. Ask why the seller is selling the guitar. If in doubt, have a local music shop that does guitar repairs check out a used guitar, if you are buying it from a stranger and going to spend much on it. The main things are that it sounds good (resonant, not “dead”), has a “long-lasting” sound (i.e., the tone diminishes slowly after a string is plucked), can be tuned, and is easy to play (strings level and close to frets, but not so close as to "buzz"). Go to a musical instrument store that has a lot of guitars, and strum several, ranging from very expensive to inexpensive. You will find that you do not have to pay a lot to get one that sounds about as good as an expensive one. If you can't find a reasonably priced new guitar, check the newspaper want ads for a used one.

The list of what to look for in a guitar is as follows. For pictures, definitions, and more information, buy any book on guitar basics.

The fretboard (fingerboard) is made of very hard, nonwarping wood, such as ebony, rosewood, or walnut.
The soundboard is a “springy” wood, such as spruce.
The front and back of the guitar have many braces (interior wooden strips), fanning out across the (inside) surface. The braces strengthen the guitar, and they affect its tone. They may be examined with a small mirror, or by removing the strings and inserting your hand in the soundhole. They should not be loose or missing. They should be arranged in a symmetrical pattern, and neatly glued.
The grain of the front and back is fine near the middle of the guitar and wider toward the outer edges.
The frets are very hard nickel-brass alloy, with so much nickel that they are almost silver colored, not yellow, brassy colored.
The strings should be close to the frets, so that you do not have to press the strings down very far.
The frets are level, i.e., some are not noticeably lower or higher than others (or else the strings will “buzz” when certain frets are used. The distance between the frets and the strings increases as you approach the body of the guitar. The reason for this is obvious – when you press a string against one fret, the string should not vibrate against any other frets (closer to the body).
The neck is straight. On steel-string guitars there is a metal “adusting” rod (truss rod) through the length of the neck, which may be tightened to remove curvature from the neck (i.e., raise or lower the neck slightly, in case, over time, the neck warps slightly and the strings start to buzz the frets). There is an adjusting nut at one end or the other (at the tuning-key end or the body end), that can be turned with a hex (Allen) wrench or other wrench. If the adjustment is at the turning-key end, the adjustment nut is under a small cover held on by one or more screws. If the adjustment nut is on the inside of the guitar, it can be reached directly through the soundhole. Classical guitars use nylon strings, not steel stings, and do not require a truss rod (since the tension on the strings is much less).
The tuning machines (post, gear, and tuning key), or “machine heads” on the headstock (used to tighten the strings) should be of very hard metal (nickel alloy or chrome steel), rather than brass. If the guitar is a classical-style one, the string adjusting posts lie flush in the neck. If the guitar is a steel-string one, the string adjusting posts lie perpendicular to (i.e., project from) the neck.
If the guitar is a steel-string one, it is usually played with a pick (plectrum). To prevent damage to the guitar front (from the pick), a plastic shield is placed just below the soundhole. If you wish to play with a pick, buy several thicknesses of picks. The thin ones are easier on the strings. My dad used a thumb pick (a curved pick that fits over the first joint of the thumb), but I never saw the advantage in it. Perhaps it is a good idea if you are play steel strings for long periods, and strum with your thumb. It does produce a much louder sound on the bass notes than does the thumb. You may also use finger picks (that fit over the finger). The thumb and finger picks produce a somewhat different sound from the ordinary (flat, heart-shaped) picks. For rapid strumming, the ordinary pick is the simplest.
Acoustic guitars have hollow bodies, and soundholes to allow the air to freely flow in and out of the guitar when the strings are plucked, to achieve the loudest sound possible. (The sound comes mainly from the vibrating soundboard, not from the soundhole(s), and very little directly from the strings. Without soundholes, the soundboard could not vibrate as freely, since more energy would be lost in compressing the air in the guitar than in pushing it in and out of the soundhole(s), and the guitar would sound “dead”.) Acoustic guitars usually have flat fronts (soundboards) and backs. These guitars are called “flat-tops.” Steel-string models are often shaped like violins, with “landau” or italic-f design cuts for the soundholes instead of a round soundhole. These guitars are called “arch-tops.” In general, the larger the guitar body, the louder the sound.
The guitar is shellacked (lacquer), not varnished (which would penetrate the wood and affect the sound).
If you wish to play standing up, there should be a post to which to attach the neck strap on the bottom of the guitar. If you wish to attach the strap to the place where the neck is attached to the body of the guitar, there should be a post there. Otherwise, you will tie the strap at the end of the neck, just past the last fret.
Buy a tuning fork (E or G or D or B will tune open strings; A440 forks are easier to find) to tune your guitar. (You need just one tuning fork – you tune one string to it, and tune the other strings to that string. A440 is the A at the 5^th fret on the first (narrowest-gauge) string.) Pitch pipes are not reliable. Electronic tuners are great – fast and accurate – but they are not inexpensive.
Tune the guitar by tuning a single string with a tuning fork (or electric tuner), and then tune the other strings relative to that string (by matching the tone of the next higher string to the tone from the same note played using a fret on the next lower string). Check all the strings with an electronic tuner. The strings should all be in tune. Check the harmonic relationships of certain strings, if you know how to do that. (There are several methods of tuning a guitar: relative tuning, tuning according to E notes, tuning with harmonics, tuning to a central string. This article will not describe these different methods.) On solidbody electric guitars, the intonation of each string may be adjusted individually at the bridge saddle. Arch-top guitars may have a movable bridge and slanted saddle to permit adjustment of the intonation. The problem that this adjustment is designed to fix is when a fretted note on a string is not in tune with the second harmonic of the unfretted string. If the fretted note is flat compared to the harmonic, then the string is too long, and its length should be decreased by moving the bridge forward. On acoustic guitars, the bridge and saddle are fixed, and the saddle slanted. You are pretty much stuck with the intonation that is “designed into” the guitar, but changing the string gauges may help. It is difficult to tune light-gauge strings. The easiest-to-tune steel strings are of gauges .012 (inches), .016, .024 (wound), .032 (wound), .042 (wound), and .054 (wound). Normal-gauge nylon strings are .028, .032, .040, .030 (wound), .034 (wound), .042 (wound).
The guitar should sound great – a loud, clear, mellow tone that lasts for a long time. No discordant sounds, which may suggest a crack or joint separation. (It is assumed that good-quality strings are on the guitar.)
I have been told not to put steel strings on a classical-style (folk) guitar – that it may rip the bridge (where the strings are attached to the soundboard) out of the front of the guitar. The reason for this is that steel strings place much more tension on the bridge, and so it is anchored more firmly to the guitar front on a steel-string model. Willie Nelson does exactly this (puts steel strings on a classical guitar). Steel-string guitars are usually played with a pick, however, and classical (folk) guitars are usually played with the fingers. Without a “pick shield,” the front of your guitar will gradually become damaged (steel-string acoustic guitars have pick shields because they are usually played with a pick).
Check to see that all joints are solid (neck against body, sides meeting front and back).
It is convenient to have marks (“dots” or more elaborate inlaid designs, often of mother-of-pearl or ivory inlay) at certain positions along the fretboard (on the top of the fretboard on on its side) at certain key frets, such as the seventh and twelfth, or the third, fifth, seventh, ninth, twelfth and fifteenth. On a classical guitar, the twelfth fret is usually located where the neck meets the body. The twelve frets cover a full octave. On a steel-string (acoustic) guitar, the neck usually meets the body at the fifteenth fret. On a solidbody guitar, the neck may contain over twenty usable (reachable, playable) frets – almost two full octaves (24 frets). (It is much easier to play a long-necked guitar than a short-necked one. A short solid-body guitar may seem like a great idea for travelling, since it would fit inside a regular suitcase, but it is very difficult to play, except perhaps for a child.) On some guitars, part of the body is cut out below the fretboard out so that you can have a larger body (relative to the length of the neck) and reach more higher-pitch frets. (There is no need for cutouts on solidbody electric guitars, because the sound is captured by the pickups, and the sound quality has nothing to do with the size of the body (it is affected by the quality of the pickups). The body need be only large enough to hold the pickups and controls. Because the body may be smaller, the neck may be much longer, and hold many more frets. The quality of tones at high frets is not very good on a classical guitar, but it is quite satisfactory on an electric guitar. If you traveling a lot, an electric guitar with a small portable amplifier (e.g., a Pignose amp) is handy, because the body can be very small so that the guitar will fit in a suitcase.)
If you want to amplify the sound of the acoustic guitar through speakers, purchase an electrified ("electric") acoustic guitar. The incorporation of a pickup in the guitar when it is built does not detract from the quality of the sound. The alternative of attaching a pickup to a nonelectrified acoustic guitar is not as good. The advantage of an electrified acoustic guitar over an electric solidbody guitar is that you can play to yourself or a small group without amplification (no electricity or speakers). Without electrification, a solidbody electric guitar produces hardly any sound at all.
Rounded-body guitars look neat, but they are hard to hold.
Buy a classical-style guitar if you wish to play classical music or any other kind of music. Do not buy a steel-string model if you wish to play classical music – not only would it not sound right (metallic rather than mellow), but it is hard to play “barre” chords on a steel-string model, even with lightweight steel strings. Buy a steel-string guitar (narrow neck) if you intend to play only country/western music, and want the traditional steel-string sound. In a way, it is easier to strum with a pick on a steel-string guitar, because the strings are much tauter and closer together. Because the strings are a little closer together, however, it is a little harder to select particular strings with the pick.
Buy a hard-shell case if you will be carrying your guitar many places, a soft (vinyl or leather) case only if you will be keeping it home. Be careful not to knock the guitar. Each little scratch or dent changes the sound quality, usually for the worse.
If you will be traveling by airplane, so that you will have to check your guitar as luggage, buy a heavy-duty case (which looks more like a trunk than a hardshell guitar case).
If you buy or are given a guitar in poor physical condition and wish to restore or refinish it, remember to use shellac, not varnish. A water drop placed on a dent will raise it, unless the wood is torn.

III. Some Theory of Music

A. The Issue of How Many Tones and Notes to an Octave

1. Some Definitions, and The "Evenly Tempered" Scale

The first issue to address is why the musical scale (in Western music) consists of twelve tones (seven designated as “natural” notes plus five designated as “accidental” notes), and how those tones are defined (in terms of frequencies). First it is useful to define some terms. A “tone” is the sound produced by vibrating a medium (air, or a wire, or a bar, or a skin) at a particular frequency. A “note” is the name given to a tone. The terms “tone” and “note” are often used interchangeably. An “octave” is a range of tones from a specified tone to the tone of double that frequency. The word “octave” is derived from the Latin word, octo, meaning “eight,” and it was introduced when the frequency range from one frequency to its double was divided into eight tones (i.e., eight tones of that range were used as the basis for melodies). The word octave now refers to the frequency range from one frequency to its double, however, no matter how many tones that range is divided into.

At this point, I must point out that the definition of the word “tone” is different in physics and in music. It is also different in different countries. In this article, I will use the physical definition (which also corresponds to the lay concept). In physics, a “tone” is a sound of a specified frequency, such as the sound of the note A, which is a frequency of 440 vibrations per second (or 440 cycles per second, or 440 hertz, or 440 hz). In music, a (physical) tone (sound of a particular frequency) is referred to as a “note,” and the frequency of the note is referred to as the “pitch” of the note. A “semitone” (half tone) is the difference, or length of the interval, between two adjacent tones on the 12-tone chromatic scale (used in modern music, and to be define later); e.g., the difference between A and A#, or between B and C, is a semitone. (The scale is AA#BCC#DD#EFF#GG#A, with ascending letters representing higher frequencies.) A “tone” (in music) is the difference between next-adjacent notes, i.e., two notes with a note in between); e.g., the difference between A and B, or between B and C#, is a tone (or “full” tone).

By using the physical (and lay) definition of “tone,” I will say that the interval between two tones (sounds of two different pure frequencies) is so many hertz, or so many tones, or so many steps. For example, on the modern 12-tone musical scale, I will say that the difference (or length of the interval) between A and A# (adjacent tones) is one tone or one step, not a semitone or half-tone or half-step. Similarly, I will say that the difference between A and B is two tones (not a full tone or full step). It is understandable that music and physics (and common parlance) would not use the same term for such a fundamental concept as a pure-frequency sound, but it is unfortunate that they use the same word (tone) for quite different concepts (a sound of a given frequency vs. the interval between to sounds of different frequencies). In any event, no confusion should arise as long as this distinction is recognized, and it is known which definition I am using.

A “scale” is a specified set of tones over an octave (just as the twelve inches mark a one-foot ruler). Although there are twelve tones per octave in Western music, most Western musical pieces involve only a selection of eight of these tones. (I will use Western with a capital initial letter to refer to music derived from Greek civilization, and western with a lower-case initial letter to refer to country/western music of North America.) Most people know that an octave is a range of frequencies from a specified frequency to the frequency that is double that. But why has this range been divided into twelve tones (i.e., twelve tones have been selected as a basis for music), and how are the tones specified, and why are most musical pieces based on a subset of eight tones, and how are these tones selected?

The names of the twelve tones used in the Western music scale are A A# B C C# D D# E F F# G G# or A Bf B C Df D Ef E F Gf G Af. (I shall use the symbol # for sharp, as is standard, and the symbol f for flat, which is not.) These tone names are called notes. Tones an integral number of octaves apart have the same note names. Hence a particular note name refers to many different tones (e.g., A can refer to A110, A220, A440, A880, etc.). As mentioned, most Western musical pieces use are based on a particular subset of seven of these twelve tones. The particular selection of notes used for a song is called the “key.” It is also called a “scale.” For a piece written in the key of C, the notes that are used are A B C D E F G. For a piece written in the key of F, the notes are F G A Bflat C D E. I will say more about keys later on, and why most music is based on a scale of just eight notes. The scale of all twelve notes is called the “chromatic” scale. The scales of the subsets used as the basis for musical pieces (called "diatonic" scales) have various names, such a C, Aminor, Gmajor, and the like (we shall say more about the subset scale names later). The modifier “chromatic,” which relates to color, is used to refer to the complete scale since it contains a full range (or spectrum) and rich variety of tones. Since sound, like electromagnetic radiation, is produced by wave motion, the terms from wave theory (wavelength, frequency, octave, harmonic, spectrum) apply.

The fact that most (Western) musical pieces involve only eight tones over an octave may trace its origin to the simple (traditional) flute. Since the human hand has eight fingers opposed to the two thumbs, it is possible to cover up to eight tone-holes on a flute. This means that a simple flute (without a back hole) can play up to nine tones. (If you have played a flute, you know that by blowing harder you get a tone an octave higher, for each tone hole. Hence a simple flute with n holes can actually produce 2(n+1) tones, not just n+1 tones.) If the holes are equally spaced and appropriately distanced from the lip hole (or slot over which the sound vibration is created) so that the tones are evenly spread over a full octave (i.e., from a frequency to its double), then a wide range of melodies may be played. Melodies sound “complete” if they end on the same note as they began (or an octave higher or lower). Hence it is convenient to have the tone holes cover a complete octave, and maybe even an extra note. These considerations would suggest that an octave have at most eight or nine tones. For players of harps, it would be convenient to cover a full octave (from one frequency to double that) with eight fingers. Whatever the reason, the melodies of Western music evolved with most songs based on eight tones to the octave.

The tone holes (holes covered by the fingers) on a simple (whistle-type) flute are equally spaced. (On modern flutes, recorders, or tin whistles, they are not equally spaced, in order to generate a “major” scale – more will be said about this later.) From physics, it can be shown that the frequencies generated by equally spaced holes are equidistant on a logarithmic scale. The logarithm of the frequency associated with a tone hole is inversely proportional to the distance of the tone hole from the lip hole (hole near the lips over which sound passes to generate the vibration). That is, if d1 and d2 are the distances of two finger holes from the slot at the top of the flute (near the lips), then the frequencies f1 and f2 of corresponding to these two holes are related by the equations

f1 = a exp(-bd1)

f2 = a exp(-bd2)

or f1/f2 =exp(b(d2-d1)) ,

where a and b are constants (that depend on the size and geometry of the flute), and exp(.) denotes the exponential function. The negative sign occurs because the longer the distance of the hole from the slot, the lower the tone (frequency). If the distances di are expressed as ranging from one to two, then the constant b is in fact ln2 (where ln(.) denotes the natural logarithmic function), so the relationship may be expressed as:

f1/f2 = exp((d2-d1)ln2) = 2^(d2-d1) .

So, the ratio of the frequency (f1) of the highest note (d1=1) to that (f2) of the lowest note (d2=2) is

f1/f2= 2^(2-1) = 2 .

With the notes separated equally (physical distance of holes, or equal intervals on a logarithmic scale), the scale has a “balanced” character, and is called an "evenly tempered" scale. With a total of eight different tones spanning an octave, it is possible to construct a tremendous variety of melodies. (The reason why tones that are equidistant on a logarithmic scale sound physiologically to be an equal distance apart has to do with the design of the ear.)

2. More About the Evenly Tempered Scale

Above, it was stated that a flute’s having equally spaced holes produces tones on an “evenly tempered” scale, and it was stated that with this type of scale the notes are “evenly spaced” in a physiological sense. Let’s explore the concept of an “evenly tempered” scale a little deeper. An “evenly tempered scale” is defined as one in which the ratios of the frequencies of successive tones in the scale are all equal. That is, if f_idenotes the frequency of tone i and f_i+1 denotes the frequency of tone i+1, then

f_i+1/f_i = k,

where k is a constant. For an evenly tempered scale of 12 tones, the ratio of tones that are an octave apart (i.e., the frequency of the higher tone is twice the frequency of the lower tone) is hence

f_i+13/f_i = 2.

But since f_i+1/f_i = k for all I,

f_i+13/f_i = (f_i+13/f_i+12)(f_i+12/f_i+11)…(f₂/f₁) = k¹².

So k¹² = 2, or k = 2^1/12.

So, the ratio of the frequencies of successive tones on an evenly tempered scale of 12 tones is the twelfth root of 2, or 1.059463094.

The fact that the ratio of successive tones of an evenly tempered scale is constant means that the difference in the logarithms of successive frequencies is also constant, i.e.,

ln(f_i+1/f_i) = ln f_i+1 – ln f_i = ln k = k’,

i.e., the logarithms of successive frequencies of an evenly tempered scale are linearly (equally) spaced.

The “chromatic” scale of Western music is an evenly tempered scale. Why are we interested in evenly tempered scales? One reason is that the tones sound evenly spaced in a physiological sense. They are natural and pleasing to hear. The very important result of this property is that if you sing a song at different pitches (e.g., in different keys or octaves), the melody of the song sounds exactly the same! So, although human beings may know nothing of the technical definition of “evenly tempered scales,” they nevertheless use them naturally! In singing the same song at two different pitches, the ratios of the frequencies of corresponding tones of the two songs are exactly the same throughout the song. Or, the ratios of the frequencies at two different points in the song are the same, regardless of what pitch is used (e.g., bass, tenor, soprano, alto). A song is hence defined simply in terms of a series of ratios of the frequency of each tone of the song to the frequency of the starting tone.

The use of evenly tempered scales in music greatly simplifies the writing of music, since it does not matter what pitch (e.g., what octave) a musical instrument uses. It also dramatically simplifies the construction of musical instruments.

3. Guitar Fret Positions for an Evenly Tempered Scale

It was mentioned earlier that a flute with equally spaced holes produces tones on an evenly tempered scale. Now, let’s discuss the guitar. From physics, it may be observed that the fundamental vibrating frequency of a taut string under constant tension is inversely proportional to the length of the string. Let p_i denote the position of the i-th fret of a guitar fretboard (i.e., the distance of the fret from the bridge), and let g_i denote the frequency of the string, when the string is pressed to the i-th fret and plucked. Then