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A Chord Formula is a Musical Recipe

A chord formula is musical numbering system that gives us a way of describing a chord without referring to specific notes. This is done by enumerating the intervals that make up a chord. For example, a Major 7th chord is made up of four notes and three intervals. A chord formula spells out the harmonic structure using the intervals rather than the actual note names.

The Numbering System
The numbering system that gives us these formulas is based on the intervals that make up the Major scale. The intervals of this scale are the yardstick we use to measure the distances between notes that make up chords.

To understand where this numbering system came from lets look a the Chromatic scale. The Chromatic scale is made up of every possible note you can play between an octave on Western instruments. It consists of 12 notes each a half step (also known as a semitone) apart. It’s the scale you get when you play a piano key and then the next 11 keys up the scale without skipping any keys.

The Chromatic scale is the 12-note scale that includes every possible note that exists between an octave.

There are many scales that can be created from these 12 notes. Over the centuries different cultures with different musical tastes have left us with about a dozen scales that are popular today. The most popular of all is the Major scale. It is made up of only seven of these 12 possible notes.

The notes that make up the Major scale are not equally spaced within the octave. Some of the notes are two semitones apart while others are only one semitone apart. The unequal steps and placement of these notes is what gives this scale a bright sound and a solid finish. If you were to play the Major scale, you would hear a beginning, a middle, and an end. This scale is used in a great majority of the songs we hear today.

The Chromatic scale is made up of 12 notes that are spaced a semitone apart. The Major scale is made up of 7 notes with some of them spaced a semitone apart and some space two semitones (whole note) apart.

Centuries ago, before the Chromatic scale had been invented, the notes that make up the Major scale were all that anyone really cared about. Instruments existed that could only play these notes. It is not surprising that this scale ended up being the benchmark by which all scales and chords are measured.

At some point, the first note of the major scale become known as the root, and the other notes we just numbered according to their position in the scale – second, third, fourth, fifth, sixth, and seventh. This system allowed musicians to tune their instruments without knowing what the pitches were. In fact, there was really no agreement as to what pitch each note should be until fairly recently. However, everyone knew what the major scale should sound like and so tuned accordingly.

Since musicians tend to be creative people, even back then they experimented with different kinds of sound for their music. They created new scales. In order to play these new scales they added strings to their instruments and holes in their flutes. But some of the notes of their new scales did not exist in the Major scale but fell in between the notes numbered, 1-2-3-4-5-6-7. (The number “1″ is the same as the root.)

Fortunately, the Chromatic scale helped them arrange and space these notes in a systematic way. Musicians now started paying attention to how many semitones each note was from the root. This meant that the third note of the Major scale is four semitones away from the root. The fifth note of the Major scale is seven semitones from the root. The seventh note of the Major scale is 11 semitones from the root. These harmonic distances are known as intervals. In other words, an interval is the distance between notes.

The Major scale is the scale is made up of intervals by which all others intervals are measured.

As the pattern of notes that makes up the Major scale became the yardstick that measures the distance between any two notes. The distance between notes were given names. For instance, distance between the root and third note of the Major scale became called a “third.” The interval of a fifth is the distance between the root and the fifth note of the scale. The seventh note of the scale which is 11 semitones from the root is called a seventh. These names have been expanded upon to delineate them from intervals that were named later.

The notes of the Major scale represent the intervals that are used to measure the distances between notes.

Scale Formulas
If you wanted to convey to someone the intervals that make up the Major scale you could write it this way: 1-2-3-4-5-6-7. This is a scale formula and you use it to “calculate” which notes notes go together. You can overlay this formula over the Chromatic scale to create the Major scale in every key.

The Major scale pattern is the same no matter what note the scale begins on.

When you overlay the Major scale pattern over the C Chromatic scale it results in the C Major scale. This same pattern can be overlaid the C# Chromatic scale to create the C# Major scale. This same process can be used to create the Major scale in every key.

Measuring Other Scales
As mentioned earlier, there are other scales used in music. The Natural Minor scale happens to be one of these. The spacing of its notes creates a musical sound that is often described as a somber and ungrounded. If we compared the notes of the Natural Minor scale to the Major scale it would look like this:

The Minor scale patters is different from the Major scale pattern. The 3rd, 5th, and 7th notes of the Minor scale are a semitone lower than their Major scale counterparts.

When we use the Major pattern as our yardstick, we find that some of the notes of the Minor scale fall between the Major scale notes. To fix this we need to add some fractions to our yardstick. The intervals of the Major scale are like inch-marks on our ruler. But now we need to add some “in between” marks to measure the intervals of the Natural Minor scale. The third, sixth, and seventh note of the Minor scale are a semitone less (lower)  than their Major scale counterparts.

To accommodate these “in-between” intervals someone came up with the idea of using sharps (#) and flat (b) symbols to indicate the difference.  The sharp (#) adds a semitone to a note’s pitch while the flat (b) diminishes a note by a semitone. In the diagram below you can see that the third note of the Major scale is two whole steps from the root. Meanwhile the third note of the Natural Minors scale is one-and-a-half steps from the root.

The notes of the Minor scale are measured according to the intervals established by the Major scale.

Since the sharp (#) adds a semitone to a note and a flat (b) diminishes a note by a semitone, it is possible for an interval to be referred to in two ways. A sharp five (#5) is the same as a flat six (b6).

Finally, when we add the sharps and flats to the yardstick, we have a complete “ruler” that we can use to measure the distance between any two notes. Each interval has a name (or two) associated with it. Intervals that are derived from the Major scale pattern are called major or perfect. The other intervals are referred to as minor, diminished, or augmented.

All intervals get there names based on the notes of the Major scale. Even the notes in between are indicated as alterations of the Major intervals.

Minor Scale Formula
Using our complete ruler we see that the Natural Minor scale has the following intervals: 1-2-b3-4-5-b6-b7. We can now overlay this formula on the Chromatic scale to come up with the notes of the Minor scale in whatever key we want.  The diagram below shows C, C#, and D Minor scales.

The Natural Minor scale formula can be overlaid the Chromatic scale to create the Minor scale in different keys.

Minor Scale Formula
Now for a little review. The Major scale pattern follows this formula: 1-2-3-4-5-6-7. The Natural Minor scale pattern follows this formula: 1-2-b3-4-5-b6-b7. Every scale you encounter can be expressed using this formula system. The formula is the same and but the root note is what determines the key.

Chords
Now that you know how to work with scale formulas, you can apply the same concept to chords. Chords are in essence small scales. Chords are often made up of three or four notes but can have up to seven. For now, we will just consider the three and four-note chords.

Consider this three-note chord known as C major (not to be confused with C Major scale). It is made up of these notes: C-E-G. This triad major chord can be described as a formula just as a scale can. C is the root. The second note of a major chord is always a major third from the root. The third note of the chord is an interval of a fifth from the root. That makes the chord fromula for a major chord this: 1-3-5.

When this formula is applied to the Chromatic scale this is the result:

Chords are like mini-scales. Formulas can be used to create them. The root may change but the formula will stay the same.

The names given to chords are based on the intervals that make them up. Unfortunately, the name of a chord can be confusing if you don’t already know the formula for the chord.  In the above examples we have Cmin7, Cmin(maj7), Cmaj7, and C7. If you didn’t know the formula for these chords, you might not know if the fourth note of these chords is a major or minor seventh interval. What makes this even more confusing is that there is not a grand agreed upon naming system. Therefore, there are several “correct” names for chords and ways to write them.

Chord names may vary but the formulas do not. That is why understanding chord formulas and how to apply them is so important. Below are chord formulas for several common chords. Along with them are their names and ways to write them.

Notes:
* Sometimes you will see a minor(major) chord written this way. Actually, a sharp 7 (#7) would be an octave interval. However, in this context the sharp is reminding you that the seventh is major, like in a major 7th chord.

** Often you will see a diminished 7th formula written this way: 1-b3-b5-6. You could even call this chord a minor 6th flat five (min6b5). However, the double flatted 7th is technically the more correct way to refer to this interval. The general principle to remember here is that a double flatted note is two semitones lower, 6 = bb7.

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