ple, if Conlon Nancarrow had required himself and his music to serve the limitations of the score, the “notation” of a piece like Study No. 40a (canon e/pi) or Study No. 33 (canon 2:sqrt(2)) would have been impossible since the polyphonic parts of these pieces essentially never coincide in time. If the score is predicated on coincidences in time, any piece predicated on the lack of such coincidence cannot rely on the score as traditionally understood. (see Figure 13)
Figure 13: Nancarrow's "Study No 33" for player piano, which contains two musical strata at two different tempos in the ratio of 2 against the square-root of 2. Since the square-root of two is an irrational number, this tempo is simply an approximation, but essentially these two musical strata should theoretically never coincide in time, since 2 cannot be equally divided by the square-root of 2. This goes against all conventions of score notation, making any conventional score notation somewhat absurdist. Here, as designed, this piece is realized by a player piano, since a human performance is extremely difficult, if not impossible, if the piece is to be at all a close approximation.
These innovations in pitch, rhythm, and time notations, perhaps ironically, lead to the third and last innovation in musical communication I will address herein: non-prescriptive notation. Since the precision innovations of the score suggested the limitations it places on our ability to express free and incommensurate musics, the next innovation I wish to address is the return to non-prescriptive systems of musical notation. Thus far, this essay has been entirely concerned with the development of greater and greater levels of prescriptiveness in musical encryption. Perhaps 1000 years has finally lead us full circle in the 20th and 21st centuries, as some composers have realized that the representation of some sonic ideas cannot be encoded with pitch-rhythm-time prescriptions.
In the 20th century, composers, perhaps realizing many of the issues addressed above, sought to eschew the millennium of prescription and return to what might be aptly called “new neumatic notation.” The 20th century witnessed many examples of the abstraction of notation, returning to “pictographic” suggestions of sound, often open to much interpretation by the performer. A stark connection can be drawn between the neumatic notation of the Middle Ages and the “new neumatic notation” found in Earle Brown’s String Quartet (1965), clearly referencing this less prescriptive musical encryption method (see Figure 14). Other examples of this return to “suggestive” rather than prescription notation include (among many others) Karlheinz Stockhausen’s Zyklus (1959), Morton Feldman’s King of Denmark (1964), John Cage’s Song Books (1970), Krzysztof Penderecki’s De natura sonoris (1966), and John Zorn’s Cobra (1984). (see Figure 15)
Figure 14: The first page from Earle Brown's "String Quartet" from 1965.
Figure 15: The first page from Morton Feldman's "King of Denmark" from 1964.
If I may, in conclusion I wish to address one further development in the encoding of music, which will, by association, return us to our beginning with Herbert Brün, master of electronic music in the 20th century. This innovation, which you might have already guessed by now, is so significant that it might even supersede all of the above innovations, for it, in itself alone, needs no form of “musical” encryption in order to deliver to us the musics that we love and study; this discovery and its consequent innovations have even the power to encode a music that has no other means of notation: the Fourier Transform and the development of the “wave file”.  The ability to encode music as purely mathematical summations of sine waves and the second ability to unpack that data and audiate it with as few as one vibrating membrane through the computational power of the Fourier Transformation has perhaps shaped our musical world more today than anything else. Recorded sound that can be played anywhere at any time given the audio equipment necessary – whic