When I started exploring the extended lattice beyond the central 12 notes, the first note that was really new to me was the 10/9 major second, also called the minor or lesser whole tone. Now I call it the 2-. The lattice extends forever in all directions. When you continue multiplying and dividing, generating new notes beyond the boundaries…
As I’ve analyzed my songs on the lattice, and written new music using it as a tool, I have found that I have a certain palette of notes in my mind, a territory of the lattice that I can hear and think with. The notes in this portion are distinct individuals for me. Each one…
Yesterday, I described a simple way to hear, and more importantly, feel, the difference between equal temperament and just intonation, by singing Frère Jacques over an open G chord with ET major thirds, and then over a G chord that has only roots and fifths in it. The second half of the experiment is called singing over…
A closer look at the untempered 12-note scale reveals some interesting patterns. Here’s the scale on the lattice again. This time, I’ve colored the notes as follows: Red = notes from the central, Pythagorean row, or spine, of the lattice. They are generated by multiplying and dividing by 3. Green = notes from the next row…
The familiar 12-note scale is also called the chromatic scale. There are many ways to generate an untempered version of such a scale. The oldest way seems to be Pythagorean tuning. When you multiply a frequency by 3, you get a new note, an octave plus a fifth above. Multiply by 3 again, and you get…
In harmonic space, the clearest name for each note is its ratio — 5/3, 3/2, 4/3, etc. Precise and unambiguous. But the ratios don’t give a very good idea of how to go about playing or singing those notes. There are a few instruments, such as the left hand keyboard on the accordion, that are…
Musical notes can be mapped onto many different spaces. The two I find useful so far are: — Harmonic space, the space of the lattice, organized by harmonic connections (ratios of whole numbers). — Melodic space, the space of the scale, organized by pitch, or frequency. Both maps show the location of a note relative…
Throughout my musical education, I’ve been taught that music happens in a linear space. This is the space so beautifully laid out on the piano keyboard. Music teaching is organized around scales. In most Western music, the full scale consists of twelve notes, equally spaced. Other scales, such as the seven-note minor and major scales,…
In one of my favorite passages in Harmonic Experience (p. 104-105), W.A. Mathieu points out that by the time Mozart came around, equal temperament was well enough established that a D# and an Eb could be thought of as the same note. So when Mozart wanted his melody to go back and forth between an Eb and…
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…