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The analogy between binary numbers and the relative duration of music notes

In ordinary mathematics, the concept of “amounts” (or “values”) is normally represented using ten symbols, known as the digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any amount represented using these symbols is in a “base-10” format (10 symbols).  The values represented with these symbols will be determined according to how the number is arranged.  For example:

903 = 9 x 100 + 0 x 10 + 3 x 1 = 9 x 102 + 0 x 101 + 3 x 100

In other words, the first-from-the-right position indicates the units (in the example: 3), the following number to the left indicates the tens (in the example: 7), the following indicates the hundreds (in the example: 9), the following number indicates the thousands, and so on.  Summarising, a base-10 number uses powers of ten (10n) for each of its ciphers (positions in the number).

Another way of representing a value is by using only 0 and 1 as symbols.  Numbers in this format are in “base-2” (2 symbols), also known as “binary numbers”.  Any value in base-10 can be represented in base-2 as shown in the following example:

903 = 1 x 512 + 1 x 256 + 1 x 128 + 0 x 64 + 0 x 32 + 0 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 1 x 1 =

= 1 x 29 + 1 x 28 + 1 x 27 + 0 x 26 + 0 x 25 + 0 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20 =

= 1110000111

 … therefore, 903 (base-10) is equivalent to 1110000111 (base-2).

 Using the same procedure, any fraction of a base-10 number can be transformed into a binary number using negative powers of two (2-n).  This idea, incepted during the 14th century and standardised in the 20th century, is the concept used for assigning the duration of the notes in, once Western, now Universal musical notation.  The unit of musical notes is the “whole note”.  The “minim” is half of a whole note; the “crotchet” is half of a minim, and so on.  In other words, two crotchets have the same duration as one minim; and two minims have the same duration as a whole note.  The following figure shows all of the musical symbols used in bagpiping and the comparison with the whole note.