We saw earlier that the group of three black keys was created by the combination of the lower and upper tetrachords. Where the Lydian, Dorian, and Phrygian tetrachords could be arranged so that a perfect fourth is made on the keyboard, the group of three black keys forces an additional semitone that results in the augmented fourth, or tri-tone; a whole-tone now separating each of the keys.
Within that tri-tone cluster exists two separate perfect fourth intervals. Scroll through the images below to see where they are and how our three primary tetrachords are created from them.
This may very well seem a bit redundant, or excessive, but we’re almost there with only two more possible places on the keyboard left to look. The purpose of getting through these tetrachords is to try and see the symmetry that exists on the piano and to see that the larger scales and modes are not uneven patterns of whole steps and half-steps but that they come from the combination of two equal parts. If this is new information to you, as a suggestion, find the tetrachords on the piano and play each of them. With the exception of the Locrian cluster of notes that surround the group of three black keys, each of the other tetrachords can be found in two different locations on the keyboard.