The Major Scale

The notes 1, 2, 3, 4, 5, 6 and 7, clustered at the center of the lattice, constitute a major scale. This tuning uses the smallest ratios (the ones with the lowest numbers) available for each position in the scale. It goes back at least to Ptolemy in the 100’s AD. I find it visually…

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Names

Musical nomenclature has been cobbled together over the centuries like a medieval city. Different systems leave their imprint in convention, later developments try to be compatible with accepted names, and the whole thing ends up confusing and contradictory. Take enharmonic equivalents, for example. G# and Ab are the same note on the piano, the black…

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Mixed Messages

The harmonic lattice can be divided into four quadrants. Northeast is pure overtonal energy. All these notes are reached by multiplication alone. Powers of 3 and 5 are in the numerators, and the denominators are all powers of 2. Southwest is pure reciprocal energy. You get there by dividing. All the 3’s and 5’s are…

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Compound Notes

Multiplying the tonic by 2, 3 and 5 creates the octave, fifth and third respectively. The ear hears these intervals very well. We can easily sing them. Each one has a feel, a sort of harmonic flavor, that makes a fifth a fifth and a third a third. It turns out that the ear can…

The Tonic Major Chord

The tonic is the center of the lattice. A drone note on the tonic establishes the center of that particular musical universe. Adding a major third and a perfect fifth (5/4 and 3/2) further reinforces the center and starts to carve out some territory on the map. This is the tonic major chord: In my…

The Lattice

In 1739, the great mathematician Leonhard Euler published something he called a Tonnetz, German for “tone network.” It looked like this: Euler’s Tonnetz organizes the notes into a matrix, instead of a scale. Moving down and to the left represents motion by an interval of a fifth (V) in musical space. Down and to the right…

Harmonic Space

Now to relate all this to the lattice in the video. Listening to music is like going on a journey. Most tonal music starts by establishing a center, or basic note, and a basic harmonic framework for the song, such as a major or minor mode. A few melody notes, and a beginning chord, and…

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The Tonic

The heart of the lattice is the note called 1. This note is the tonic. Almost all the music you hear — pop, rock, classical — has one note that is at the center, a master note against which all other notes are measured. That note is the tonic. It’s the Do of Do Re Mi….