I’ve described eleven notes now, and each one has a piano key to go with it, an equal tempered equivalent. The one remaining black key has a lot of names. It’s the note between the 4 and 5, right in the middle of the octave — the tritone, devil’s interval, flatted fifth, augmented fourth. In…
The last three notes (b6, b3 and b7) are related to each other. They all contain a reciprocal third. There is a family resemblance of sound and function. (They also all happen to be a little flat in equal temperament. On a guitar it’s a nice trick to bend them a little to sweeten them.)…
I’ve been listening to yesterday’s chord progression showing off the b7. I think it offers an excellent opportunity to hear the difference between equal temperament and just intonation. Equal temperament works by implying or evoking a note rather than playing it exactly. There are dozens of singable notes per octave; ET represents them all with just…
The farther we get from the center, the less consonant the notes are, when played against the tonic. Consonance is a whole subject. It’s generally spoken of as though it could be plotted on a scale, from consonance to dissonance. I think this is a big mistake. Consonance has more than one dimension. Trying to…
Here’s an interesting and perhaps misunderstood note. It’s a compound move on the lattice: down a third and up a fifth. Or up a fifth and down a third, it doesn’t matter what order. So the ratio is 3/5, or 6/5, octave reduced. The note is the minor third. I call it b3. It lives a…
There are four basic moves available on the lattice of fifths and thirds. They are: Up a fifth (x3) Down a fifth (÷3) Up a major third (x5) Down a major third (÷5) Each of these moves has its own harmonic flavor, and they can be combined to create new flavors. The major scale only…
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…
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…
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…
All the notes I’ve discussed so far are found above and to the right of the tonic, in the northeast quadrant of the map. These notes are generated by multiplication alone. What about the notes that are generated by division? These are found to the left and down on the lattice. The closer a note is…