Tetrachords: Where did you come from? (pt.6)

We were able to build a replica of a piano keyboard by taking the inseparable version of our Lydian, Dorian, and Phrygian tetrachords and layering them a second time. The result was the creation of three heptatonic (seven-note) modes comprised of a lower and upper combination of equal tetrachords separated from each other by a tone. Something else happened, too: now there is a group of three black keys.

Part of our goal in creating our perfect instrument was to make something that would make step-wise intervals easy to see and also easy to play. That’s the reason for having white keys and black keys; they provide a visual and tactile reference. Remember that the sonic distance between each key is equal. The layout of the keyboard can distort that notion.

The white keys are set up to represent the uneven order of tone and semi-tone intervals that we used to build our tetrachords, and in turn expand to build the larger modes. This new group of three black keys presents a bit of a problem, sort of, since the four surrounding white keys are separated from each other now only by whole-tone intervals. This total distance has grown by a semi-tone with the addition of the new key. Previous patterns were made up of six keys, but this cluster is made up of seven and so our pattern is no longer called ‘perfect’. This new interval is called an augmented fourth or tri-tone. We also have a new tetrachord: Locrian.

The Locrian tetrachord as it appears on the white keys of the piano's keyboard. This tetrachord is made of three who-tone intervals whose sum is equal to an augmented 4th.

The Locrian tetrachord as it appears on the white keys of the piano’s keyboard. This tetrachord is made of three whole-tone intervals whose sum is equal to an augmented 4th.

final_upper_lower_locrian_whole_tone

A whole-tone scale is created shown here by combining lower and upper Locrian tetrachords, separated by a whole tone.

I referred to the tri-tone as a problem because it doesn’t fit perfectly into the tetrachords or modes. The image below illustrates an upper and lower combination of Locrian tetrachords separated from each other by a whole-tone. This is the same logic that we used to create the heptatonic scales with the other tetrachords, but completely different results have occurred. This pattern splits the keyboard into half; 12 notes separated into two equal groups of six. The absence of any semi-tone intervals makes this a mode with only one other possible transposition.

final_upper_lower_locrian_whole_tone_transp

As explained in the previous example, this is the other possible whole tone scale formation.

In the next example, an upper and lower combination of Locrian tetrachords are separated now by the interval of a semi-tone. The problem here is that rather than the two halves combining to meet at the octave just as the other tetrachords had, instead they span to reach an augmented octave.

final_upper_lower_locrian_augmented_octave

The combination of two Locrian tetrachords separated only by a semitone is problematic. The distance from the first to the last note is no longer that of an octave, but instead an augmented octave.

There is much more going on with tri-tones than what is stated above, but that’s probably enough to start with for now. We’re going to need to look at that three note cluster again, and from a different perspective, to see what else is happening there.

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