Triad extensions begin with adding the 7th above the root. Like climbing a ladder built on thirds, the 9th, 11th, and 13th are the upper extensions that may be added to any chord. The scale degree represented by these extensions can be calculated by subtracting the number 7 from each extension. A 9th is the same note as a 2nd, the 11th same as a 4th, and the 13th same as a 6th.
Theorists do not generally call notes added to the triad extensions unless the 7th is present. If it is not present, they refer these vertical structures as added-note chords. The “add 2” chord is not referred to as a 9th chord without the 7th contained in the chord. Similarly an “add 6" chord is not referred to as a 13th chord. Chords with extensions above the 7th are sometimes referred to as tall chords.
Extended triads became popular in the music of the Impressionists in the late 18th and early 20th centuries. Among the composers in this style were Debussy, Ravel, and Faure. Extended triads are a hallmark of Jazz harmony, and upper extensions are added to most chords by arrangers, keyboardists, and guitarists. This practice gives Jazz its unique harmonic flavor. Look at the Lead Sheet Symbols guide in the Materials below. At a keyboard, slowly roll the extended chords from the bottom up to learn how to distinguish them aurally. Then try creating harmonies with extended chords beginning with just the 3rd and 7th above the bass note, and add the 9th and 13th within the octave. These extensions replace the root and 5th in the upper structure.