Or one could assign the ten digits to some other group of ten notes. Or, since the common scales of pentatonic, diatonic, and chromatic have five, seven, and twelve notes respectively, one could represent π in base 5, 7, or 12, so that each digit would correspond to a unique note in the corresponding scale, octave differences aside. The example below shows a diatonic rendition of π in base 7 (3.0663651...) in which register is freely chosen. (In base 7, .066 is quite close to .1, which is 1/7 in base 10. This is another way to see that π is very close to 3 1/7, a well-known rational approximation.)
You can find multiple examples of such representations around the internet, such as here. The resulting music sounds as one might expect: even though the digits of π are not random, the melody sounds more or less as if it were randomly generated.
However, as with so many random or apparently random phenomena, the appearance of randomness in this sequence does not preclude identifications of design. For example, one could find multiple digits in a row, such as the series of six 9's in a row that starts with the 762nd digit in the decimal representation of π. Or one could find an incremental series: 0123456789 occurs first at the 17387594880th digit. Or one could find one's birthday (mm/dd/yy, or dd/mm/yy, or otherwise) within the sequence. Any such series would be even more remarkable if π began with them, which, in the last case above, it would for someone born on this day four years ago (if one allows 3 to substitute for 03).
Aspects of design could also be identified through musical conventions. 999999 would create a distinctive sound if played as a melody -- repeated notes -- as would 0123456789 -- straight through the scale -- if adjacency of digit corresponded to adjacency within the scale. However, other aspects of design are more particular to music. For example, one distinctive design of Western classical music -- found especially in keyboard accompaniment patterns -- is a succession of evenly spaced notes whereby pairs of notes separated by a fixed time length (labeled as n notes below) are no more than a step apart in the prevailing scale, simulating smooth voice leading in multiple virtual parts. Below are some diatonic examples. The numbers below each note show the number of diatonic steps the note is away from the note that occurred n notes before it. The series of +2, +3, +3 below the excerpt from Schumann's music shows exceptions to the stepwise relations.
As shown with the first example above, π base 10, when realized as diatonic steps on and above middle C, starts with such a design with two notes in between: the next-adjacent notes F-G, D-D, and G-A are no more than a second apart from one another. This design is more infrequent with more notes in between: the first such design with three notes apart starts at digit 24 and the first with five notes apart starts at digit 28 and overlaps with the previous, as shown below. The first such design with four notes apart does not happen until digit 502, assuming the notes are in fixed registers.
Another distinctive musical design is a progression of triads, one of Western classical music's most privileged harmonies. In the second example above, which is in base 7, the first triad between successive notes is a B triad representing the digits 3, 6, 1, and 3, which begin at order position 13. However, there is no different triad immediately before or after this B triad, so there is no triadic progression. The first such triadic progression in π base 7, shown below, starts at the 696th note.
The preceding exposition provides the context for what makes the following so remarkable. Here are the opening 17 notes of π base 12, realized as notes in the chromatic scale in which 0 = C, 1 = C# or Db, and so forth. I use the digit B for 11 in base 12; it so happens that this number also stands for the note B when C is assigned to 0.
If one considers the whole-number portion of this number -- the 3 -- as a "before-the-beginning" pickup note, then π base 12, represented by chromatic notes, begins with both a triadic progression and an arpeggiated design that simulates smooth voice leading. The next such series of notes derived from the π base 12 sequence that has both of this properties does not start until the 5763rd note.
Moreover, this triadic progression is between two minor triads -- C-sharp minor and A minor -- whose roots are a major third apart and in which the "higher" triad in the minor-third relation is more like tonic -- in this case, because it comes first. As I discuss here with regard to its use in motion pictures, such a progression has been associated with villainy and the shadowy in a lot of Western music.
The four-sharp diatony of what follows, and how standard the implied harmonic progression is among the first seventeen notes, is also quite remarkable. For a Western classical musician, the opening triadic progression and its immediate continuation might as well be the equivalent of starting the fractional portion of π with 999999.
