Ludus Supertonalis
Ludus Supertonalis is a structured improvisation, inspired by (but much smaller in scale than) La Monte Young's The Well-Tuned Piano. All of the material stems from the overtone series of a single low F on the piano, except for the bass notes, which slowly proceed through all 12 chromatic pitches in a fixed order. The changing bass notes continually reframe the otherwise static harmonic content, drawing both performer and listener towards different sets of pitches at different times. The tone row in the bass begins on a low F, the fundamental of the overtone series, and subsequent pitches provide a departure from this home base. The final note of the bass row is C, the dominant to F and the pitch most closely related to F in the overtone series. Now the higher notes suggest C minor rather than F major; we have almost returned home, but not quite. The title, makeshift Latin for "play of overtones," is more than a passing nod at Paul Hindemith's Ludus Tonalis, as the central idea of Ludus Supertonalis is unintentionally quite similar to Hindemith's conception of chromatic tonality as related to the overtone series.
Ludus Supertonalis may be performed on a regular equal-tempered keyboard, or on a piano or synthesizer that has been retuned in just intonation. The notes of the keyboard should all be tuned to harmonics of F:
| pitch | overtone | deviation* |
| F | 1 | 0 |
| F# | 17 | +5 |
| G | 9 | +3.9 |
| Ab | 19 | -2.5 |
| A | 5 | -13.7 |
| Bb | 21 | -39.2 |
| B | 11 | -48.7 |
| C | 3 | +2 |
| C# | 25 or 13** | -27.4 or +40.5 |
| D | 13** or 27 | -59.5 or +5.9 |
| Eb | 7 | -31.2 |
| E | 15 | -11.7 |
* Deviation is measured in cents with respect to 12-tone equal-tempered tuning
** One of C# or D should be tuned to the 13th harmonic.
Alternately, if you are performing Ben Johnston's Suite for Microtonal Piano, you may use the same tuning to perform Ludus Supertonalis, transposed down a perfect fourth or up a perfect fifth, so that you use the overtone series of C. The idea of duplicating the alternate tuning of an existing piece sprung from an earlier (but as of yet unrealized) idea, to write music for prepared piano, using the same preparations as some other well-known piece, such as John Cage's Sonatas and Interludes. Retuning a piano, like preparing one, is a difficult and time-consuming enterprise, so piggybacking on the alterations involved in an another work offers many practical advantages.
