Circle of Fifths Chart

And now, we present our customized version of the Circle of Fifths Chart. Unlike the usual textbook version, ours is arranged so that the dominant 7th chords resolve clock-wise.

We show the analogical similarity of the Circle of Fifths with the analog 12-hour clock and with the Gregorian calendar of twelve months. These are coincidence, but all three of these artifacts have deeply resonant roots in human culture, as does the number 12 itself, going back to ancient Babylonian culture of several thousand years ago. Consider the 12 signs of both the Western and Chinese Zodiac, the "12 Gates to the City of Heaven", and the 12 Apostles of Christianity. The astronomical patterns of twelve come from an approximation of the relationship of the phases of the moon to the solar year; the musical patterns come from imposing the artifact of tempered tuning on the relationships derived from the harmonic overtone series; that both give the number 12 is purely coincidence, but a beautiful and resonant coincidence for artists and musicians.

The Circle of Fifths with its 12 points has a relationship to purely modal and fixed-key musical systems that is analogous to the relationship of the Solar Calendar to the Lunar Calendar - it makes simplifying assumptions (through the use of modern tempered tuning) that permit us to easily use complex harmonies in all 12 of our keys at the expense of the harmonic accuracy and subtlety which is demanded by a system like that of the Indian Ragas, just as the solar calendar permits general reckoning on an annual basis at the expense of accurate tracking of the lunar cycle over its complete 52 year cycle.

We hope you will find this Circle of Fifths Chart a useful tool.

Basic Theory
Practical Application
History

Basic Theory:

"Western Music" (that is, music derived from the European classical music tradition which goes back through Bach and Palestrina to the Gregorian Modes) has used a system of 12 possible keys, as shown in the diagram above, since the time of Bach about 300 years ago, although the theory was in place 200 years before that. This musical language has evolved over the last thousand years, mostly in Europe (but also in the Americas and parts of Asia). It is clearly NOT the only musical language in current use in the world, and the Western Musical Language is set apart from these other languages and rendered incompatible with them by its choice of possible pitches - which is both strict and limited due to the modern use of tempered tuning - and by its methods of organizing those pitches into chords.

If this doesn't make sense to you, consider the difference between metric and English threads on nuts and bolts. If you have ever tried to thread a metric nut onto an English bolt, you will appreciate the difficulty which musicians from completely different musical cultures have in trying to play together. They have different sensibilities and different tolerances for pitch, and it requires that they each abandon the aesthetic subtleties of their respective arts and be tolerant of those elements of each other's styles that appear primitive.

The standard for pitch choice in the Americas is that the note "A" should be defined as "440 Hz", or 440 vibrations per second. (Other standard pitches have been used in various parts of Europe in the past.) The pitches of the remaining notes are chosen by a mathematical formula that divides the octave (an "octave" is the sound-distance between one note and another note which is double the frequency of the first, such as A-440 to A-880) into 12 equal intervals such that the frequency ratio between each consecutive pitch and the next is always the same, and the frequency of each pitch equals that of the previous pitch times about 1.05947. Pitches which occur between these 12 intervals, as further subdivisions, are not permitted (but some musicians do anyway, particularly guitarists bending strings).

Each half step (a semitone) up the equal tempered scale multiplies the previous note by the twelfth root of two. The twelfth root of 2, for those of us who are mathematically challenged, is somewhere between 1.05946 and 1.05947. You can verify this empirically if you don't understand the math, by multiplying any number by 1.05947, 12 times, with a calculator; the twelfth result will be a little more than double the first number (using 1.05946 comes out a little less than double.)

Example:

• A   440
• Bb 466.167
• B   493.8897
• C   523.261
• C# 554.3797
• D   587.349
• Eb 622.278
• E   659.285
• F   698.493
• F# 740.032
• G   784.042
• G# 830.669
• A   880.069 (this last should of course be exactly 880; the difference represents the inaccuracy of the decimal figure 1.05947.)

Using this calculation, E-659.285 is the tempered-pitch analog of the true pitch of the overtone series, which would be E-660 exactly, a ratio of 3/2 to the starting pitch.

The result of this method of choosing and limiting the possible pitches is that the true pitches derived from the Harmonic Overtone Series of a given key are approximated (in a way that is unacceptable in some other musical languages) but in such a way that the same set of 12 possible pitches results in an identical set of possible relationships in each of the 12 keys. Thus an interlocking system of a limited number of fixed pitches is obtained and a flexible (although slightly out of tune) harmonic system which otherwise would require at least twelve times the number of working pitches, and possibly an infinite number.

Practical Application:

Every musician who plays a musical instrument which uses the western system of 12 subdivisions of the octave can benefit by memorizing not only the circle of fifths but the Major and Minor scales derived from each of the 12 tones (as well as other possible scales as desired), the sets of chords which are derived from each of those scales, and the possible alterations of those chords. This system of scales and chords constitutes the grammar of western music. This grammar was almost fully developed by the time of Bach. Further extensions of the language in the 19th and 20th centuries have tended toward weakening the tonal centers represented by the Major and Minor scales, in favor of a system which is equally at home at any tonal center in the circle at any time, without, however, introducing additional pitches within the octave.

See our paper on The Guitarist's Easy Way to Memorize the Circle of Fifths.

The basic set of grammatical patterns which we recommend that every guitarist learn is as follows:

Major Scales:

1. C Major
2. F Major
3. Bb Major
4. Eb Major
5. Ab Major
6. Db Major
7. Gb Major / F# Major
8. B Major
9. E Major
10. A Major
11. D Major
12. G Major

Minor Scales: (The Harmonic Minor form is recommended for practical purposes)

1. A Minor
2. D Minor
3. G Minor
4. C Minor
5. F Minor
6. Bb Minor
7. Eb Minor / D# Minor
8. G# Minor
9. C# Minor
10. F# Minor
11. B Minor
12. E Minor

The triads and seventh chords derived from these scales, in every key:

Seventh Chords derived from each degree of the Major Scale

1. I Major Seventh
2. ii Minor Seventh
3. iii Minor Seventh
4. IV Major Seventh
5. V Dominant Seventh
6. vi Minor Seventh
7. vii Half-Diminished Seventh

Seventh Chords derived from each degree of the Harmonic Minor Scale

1. i Minor with Major Seventh
2. ii Half-Diminished Seventh
3. III Augmented with Major Seventh
4. iv Minor Seventh
5. V Dominant Seventh
6. VI Major Seventh
7. vii Diminished Seventh

At Guitar Vacation Retreats we teach music theory to guitarists.

History:

           The Circle of Fifths is an artifact arising from the study of music theory in the West, which is to say the European tradition of music theory which dates back to the 5th-Century BCE Greek mathematician and music theorist Pythagoras. Pythagoras studied the vibrating properties of strings — a subject of obvious interest to a thinking guitarist — and was the first to document the structure of the harmonic overtone series. He applied his findings about the overtone series to the tuning of a musical scale by a method which is still known as "Pythagorian Tuning". Pythagoras worked out the circle of fifths using the perfect ratio of 3:2, and such a circle of fifths does not come out exactly true — there is a "leftover" interval which is now called the "Pythagorian Comma". This situation is somewhat analogous to the inconvenience of having 365 - 1/4 days in the year - the extra 5-1/4 days which prevent the perfect division into 12 months of 30 days each must be dealt with somehow. It is a mathematical artifact of taking a perfect mathematical system — the overtone series — and using it to create an artificial mathematical system (somewhat like the Gregorian Calendar, which imposes 12 regular months over the top of 12 1/2 lunar months, and calls it a close-enough fit.) In Pythagorian tuning, this extra interval is tucked away in a corner by using a non-perfect fifth between G# and Eb, which makes certain chords sound quite sour, but this is compensated for by only playing in common keys. The modern method, known as "Tempered Tuning", distributes the Pythagorean Comma equally among all the intervals of a fifth between the 12 notes of the chromatic scale, by using a ratio of approximately 2.996:2 instead of 3:2 to tune the fifths.

The distinction between Pythagorean tuning and tempered tuning is not of great concern to guitarists, who are challenged to tune their guitars by whatever method works for them individually. The guitar, rather than being a "well-tempered" instrument, is a kind of "approximately tempered" instrument. It might be worth noting, though, that commercial electronic tuners give tempered pitches.

The guitar has complex issues with tuning. The frets and bridge are placed according to mathematical formula. However, the six strings are made of different materials and thicknesses, and each string responds differently to the slight stretching that occurs when fretting. On many guitars, this difference results in the first few frets of the B and G strings being quite sharp, which causes the common A Major and D Major chords in first position to be quite out of tune. The best remedy is to compensate both the nut and the saddle by adjusting the lengths of the strings - particularly the G and B - at both ends. The details of this have been thoroughly worked out by luthier Greg Byers. Since the nut is very rarely compensated by guitar builders, the next best remedy is to tune the G and B strings slightly flat, with the "A" on the 2nd fret of the G string and the "D" on the 3rd fret of the B string being slightly sharp to the octave harmonics of the A and D strings.

The system of 12 equal semitones which we have today in the west — which is not in use by older musical systems such as the Arabian Maqams, the Indian Ragas, and the classical musics of China, Japan and Indonesia — is an artificial system which has its origins in the mathematical musings of Pythagoras. However, it is now the true basis of the western system of harmony, as it came to its peak development in the music of Bach, in the chromatic harmony of the late 19th century, and in 20th century jazz. A serious music student will find it necessary to memorize the circle of fifths, the key signatures of the 12 keys, and the relationships of the relative and parallel major and minor modes. Without this background, it's really hard to truly understand chord progressions, and very difficult to transpose keys on the fly. It is of further value to know some general principles about the other classical musics of the world (see links above in this paragraph) in order to understand the limitations and biases of the western system.

*   *   * At Guitar Vacation Retreats we developed a practice several years ago of correlating the Circle of Fifths with the calendar, for study purposes. Every month we study a different key signature, rotating through the circle of fifths on a regular basis, and we use that major key and its relative minor for all our scales and exercises during that month. In this way, we give equal time to all the keys of our system. Our newsletter usually has a feature on the "key of the month".

Guitar Vacation Retreats is a one or two-week program of intensive guitar study and cultural experiences in San Miguel de Allende, Guanajuato, Mexico. About three hours north of Mexico City; the closest international airports are in Queretaro and in Leon, Guanajuato (BJX). We are somewhat biased toward classical / fingerstyle technique and nylon strings, but we teach a jazz-based harmony and theory with the twin purposes of (1) commercial professionalism and (2) musical communication with other musicians, and our interest in classical repertory takes a back seat to our evolving interest in bolero, tango and other styles which permit fluid interpretations, improvisation, and personal creativity.