Tetrachords: Building the perfect instrument (pt.3)

In order to understand the connection between the three ancient tetrachords and the layout of the modern keyboard, we’re going to have to assume that we also understand all the math behind the entire western tonal system. At the very least, we need to believe that 12 unique pitches fit equally within the interval of an octave; this implies that the distance from any note to its nearest neighbor is the same distance as any other note to its nearest neighbor. The distance from one pitch to its nearest neighbor is called a semitone; the combination of two semitones is called a tone.

A tetrachord is a four-note scale that’s built around 6 of those 12 tones. If we were to think of this as a puzzle, the object would be to arrange those 6 notes using a a combination of ‘tone’ and ‘semi-tone’ intervals so that a four-note scale is formed; that is the tetrachord. Given these conditions, we can begin to see that there can be only a very limited number of possible tetrachords.

Without making this more complicated than need be, let’s color-code the pieces to our puzzle with four white pieces to represent the final tetrachord and two black pieces to be used for distinguishing the semi-tones from the tones. Sound familiar? The images below show the three possible tetrachord formations as they appear on the keyboard in their simplest form. They are easy to see and easy to play.

lydian_on_lydian

The Lydian tetrachord, more commonly referred to as a major tetrachord. Its construction is tone-tone-semitone.

dorian_on_dorian

The Dorian tetrachord, commonly referred to as a minor tetrachord. Its construction is tone-semitone-tone.

phrygian_on_phrygian

The Phrygian tetrachord. Its construction is semitone-tone-tone.

It’s interesting to look at these three images and make some simple observations. We can see that the Lydian and Phrygian tetrachords are inversions of each other while the Dorian tetrachord is evenly distributed. With that, can we make a case that any of these might be stronger than the others? or weaker? Here is a visualization of one way to answer those questions.

tetrachord_strength

Which tetrachord is the most stable? Which is the weakest? The larger squares represent the tone intervals for construction while the rectangles are the semitones.

Lots of inconclusive answers, and only more questions. Next, we’ll try to consider how our three tetrachords fit together and continue piecing together the rest of the keyboard.

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