Musical Modes, Their Associated Chords and Their Musicality

In this paper we present a mathematical way of defining musical modes and we define the musicality of a mode as a product of three different factors. We conclude by classifying the modes which are most musical according to our definition. 1. Musical scales and modes First we must define some concepts. In traditional western style of music there are 12 different notes in an octave. Two notes represent the same tone if they are at a distance of 12n semitones (a multiple of octaves apart). The 13 note restarts the cycle and it is the same tone as our first note (simply an octave higher). Loosely speaking a scale is simply an ordered sequence of different tones within the same octave. For instance CDEFGAB is the well known C major scale, ABCDEFG is the A minor scale, and EGAABD is the hexatonic blues scale in the key of E. The first note of the scale is said to be the key of the scale. Definition 1.1. For further considerations about scales we will associate every tone with a numerical value, namely its position in the chromatic A scale AABCCCDDEFFGG The lexicographic order of tones will be defined as the order induced by their numerical values defined as above. The precise mathematical definition of a scale is: Definition 1.2. A k scale is a circular permutation of k different tones in lexicographic order. To exemplify the definition let’s take 7 different tones in increasing order, namely ABCDEFG. This is simply the A minor scale. The next circular permutation of the same tones is BCDEFGA and is called B Mixolydian. The following is CDEFGAB and is called C major, and thus circularly permuting the notes of the A minor scale we obtain 7 different types of scales used in western music. It is well known that these scales have the same number of chords (harmonic triads). A song composed in the A minor key simply uses only the tones of the scale A minor. For instance a song written in the key of A minor will use the same tones as a song in the C major key. In order to understand what a musical mode is, let’s examine the following two scales. Take the first one to be CDEFGAB (C major) and the second one ABCDEFG(A major). Although they start on a different note