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…
The simplest untempered tuning of the major scale is: 1 — 0 cents 2 — 204 3 — 386 4 — 498 5 — 702 6 — 884 7 — 1088 Here’s how that tuning compares with the equal tempered scale: The black numbers show the pitches of 12-tone equal temperament. They are equally…
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…