Composer's note:The undertone series is a subharmonic sequence of partials that inverts the intervals of the overtone series. This idea has been around for a while, but it has had no practical application in music because it cannot be produced in any simple way by musical instruments. There is no problem producing the series in computer music, where any frequencies can be produced.In using the undertone series, the partials go down from the fundamental rather than up. In the overtone series, the number of partials doubles above each octave: there are two in the first octave above, four in the next, eight in the next, sixteen in the next, and so forth. The higher you go, the closer the partials become. In the lower registers, sine tones sound very muffled and indistinct. It is only when they get into the register of approximately middle C that they begin to sound like normal tones. In fact, most of the musical tones below this area consist of complex sounds that include many upper partials. With the undertone series, more and more partials are clustered into the lower registers, where they are more indistinct. This means that the "fundamentals" have to be in very high octaves in order to produce usable components, and that only fundamentals at least one octave above middle C are useful at all.In this work, there are many tones that originate one to two octaves above the highest note on the piano. Those notes are audible, but we don't usually hear them in music. For notes in the extreme high range, the piece uses only "upper" partials (which are actually lower), and notes in the lower range use only "lower" partials (which are in the higher range).The work was composed in 2019 and generated by the csound program. It is a companion piece to my composition Improvisation on the Overtone Series, written 43 years earlier.