Music By Numbers

In this paper we present a mathematical way of defining musical modes, we derive a formula for the total number of modes and define the musicality of a mode as the total number of harmonic chords whithin the mode. We also give an algorithm for the construction of a duet of melodic lines given a sequence of numbers and a mode. We attach the .mus files of the counterpoints obtained by using the sequence of primes and several musical modes. 1. Musical scales and modes First we must define some concepts. In traditional western style of music there are 12 tones in an octave. The 13th note restarts the cycle and is considered the same note as our first note (simply an octave higher). Therefore we can assign numerical values to the notes as follows: A = 1 A = 2 B = 3 C = 4 C = 5 D = 6 D = 7 E = 8 F = 9 F = 10 G = 11 G = 12 This gives us a basis to relate musical concepts with mathematics. With this relationship defined, in its most general sense a melodic line simply becomes an extended sequence of numbers. Most frequently, our choices for which notes are acceptable is dictated by what we call a scale. Definition 1.1. Definition: A k−scale is an increasing sequence of k integers from one to twelve. 1 ar X iv :1 10 3. 05 96 v1 [ m at h. C O ] 3 M ar 2 01 1 2 MIHAIL COCOS SHAWN FOWERS Examples: (1) Amin = (1, 3, 4, 6, 8, 9, 11). These numbers will correspond to all the natural tones starting in A and hence we form what is known as the A minor scale. (2) A#ins = (2, 3, 7, 9, 12). These numbers correspond to the tones A,B,D,F ,G to form the A In Sen scale. Definition 1.2. Two scales with the same number of tones S1 = (n1, n2, · · ·nk) and S2 = (m1,m2, · · ·mk) are said to be equivalent iff mi − ni = mj − nj for all 1 ≤ i, j ≤ k. Note: (a) The set of tones whithin a scale define which notes we are allowed to choose when writing a piece of music. (b) Our definition of a scale will not cover all of the musical scales but will have an equivalent representant. Definition 1.3. The set of all equivalent k−scales is called a k−mode. Our first goal is to find out how many modes there are. Since the mode of a scale is determined by the spacing between the notes, regardless of which note we begin our scale with, we can assume without loss of generality that each mode begins on A or 1, and then you choose k − 1 out of 11 of the remaining tones in order to produce a mode, therefore we have: Theorem 1.4. The number of k−toned modes is Ck−1 11 . Crunching the numbers we can calculate that there are a total of 2048 possible modes. Taking into account that any givenmusical scale can start on any of our twelve tones, we multiply the number of modes by 12 to determine that there are 24, 576 different scales! Not every one of these scales is musically practical, however. Not very often do we find a song that is composed entirely using tones 1 through 7, for example. A song composed in this manner would sound rather dissonant to the typical ear. To mathematically understand why, we should take a look at what is known as a harmonic chord. Definition 1.5. The interval between two tones is defined as the absolute value of the diference between their numerical values. If the interval is one we call it a semi-tone. Definition 1.6. A harmonic interval is an interval of 3, 4, 5, 7, 8, or 9 semitones. Western music has evolved to where these intervals are considered generally more pleasant to the ear than other intervals between notes. These intervals form the building blocks for major and minor chords. Definition 1.7. A harmonic subset of a scale S is a subset of tones in S such that the interval between any two of its tones is a harmonic interval. Theorem 1.8. The maximum number of tones in a harmonic subset is three.