Mathematics of Chromatic Circle

Introduction:

Musical tune comprised of certain notes can express human feelings of different kinds; good, bad, sad, romantic, erotic, despair, anger etc. Combination of notes is responsible to express the mood. A single musical note has a clear holy pitch of sound. For example, the note C has the pitch 261.63 Hz and the pitch of note A is 440 Hz. Details of notes in terms of pitch can be read with my article Mathematics of Musical Notes. Al least, three notes together can make a chord, sometimes called Triad. In this article, formation of family chords is explained with the Circle of Fifths.

Chord:

Chords are of two basic kinds; Major and Minor. Major chords, based on the main key notes, are formed as 0, 4, 7. Here, 0 indicates the main key note or the root note. Next number 4 indicates the next fourth note (Semitone) on the scale. And, 7 indicates the seventh note in that same scale. Table 1 shows the positional numbers of different notes in both Western and Indian Music systems. In Indian Music system, a major chord will be SA-GA-PA based on the same logic in the west. The chord is to be called SA-Major. In western Music, C Major chord is made with the three notes C-E-G. Other major chords like C# or D or E or F# or on any other note can be made accordingly. Similarly, the minor chords are formed on the counting 0, 3, 7, where 0 is the root note, 3 is for third note and 7 is for the seventh note in the scale. In Indian Music system the minor chord, called SA-Minor, will be made as SA-ga-PA. Likewise, the C Minor chord is made with the notes C-D#-G, and other minor chords on different notes can be made in the same way. Every musical piece is composed on a root note on which the Home Chord is made. The whole piece of music pivots over that home key and flows through other related chord. Other chords in harmony are called the family chords o the home chord.

Table 1: Notes and their positional numbers

SA re RE ga GA ma MA PA da DA ni NI SA’
0 1 2 3 4 5 6 7 8 9 10 11 0
C C# D D# E F F# G G# A A#/Bb B C

Circle of Fifths:

Family chords have to be in harmony with the home chord. The music presenters, such as singers and instrumentalists, sometimes struggle to find the right chords in no time while they are on the show. Western music experts have made a circle of notes to represent the relative positions and thereby find the related chords on a mathematical calculation. The calculation is simple and based on a basic algebra toll called Modulo. It gives us the remainder after dividing one whole number by another. That means, number1 MOD number2 = remainder, when number2 divides the number1. For example, 42 MOD 12 = 6. Details on this calculation technique can be found in my video Modulo Operation.

Table 2 shows the full list of notes with their new positions after calculations with two steps. In the table, the third row is the positional numbers starting from the root key number 0. The next row of numbers has come after calculating each number by seven. The immediate next row of numbers is with the remainders for each case when every number is divided by twelve. Then the notes are repositioned according to their numerical values. SO, G comes after C, then D, then A and so on.

Table 2: New positions of notes for making the Circle of Fifths

SA re RE ga GA ma MA PA da DA ni NI
C C# D D# E F F# G G# A A#/Bb B
0 1 2 3 4 5 6 7 8 9 10 11
0 7 14 21 28 35 42 49 56 63 70 77
0 7 2 9 4 11 6 1 8 3 10 5
C G D A E B F# C# G# D# A#/Bb F
SA PA RE DA GA NI MA re da ga ni ma

Based on this new order of notes, the Chromatic Circle is formed, which is actually the Circle of Fifths and sometimes also called as Circle of Fourths. Because, in Diatonic Scale, we can start from the note C and rotate clockwise with the fifth higher note every time for getting the next note. If we rotate counter-clockwise, the notes will come as every fourth higher note on the current one. Figure 1 shows the Circle of Fifths with the Western notes. Notes in red letters at outer position of circle is put in this order. The notes in green are explained in the later section. This figure is dedicated to help make the family chords based on the home chord, which is on any note and on major chord frame. That means, we can make directly the family chords of C-maj or D-maj or of any other major scale. We can also use the same diagram if we understand the rule of making family chords for also the minor scale besides that for the major scale.

450px-Circle_of_fifths_deluxe_4.svg

Figure 1: Circle of Fifths

Family Chord Formation:

The Figure 1 helps musicians to form family chords easily and changing the main key while performing a number of musical pieces composed in various scales. In the figure, the outer notes are shown as they were repositioned by mathematical calculation. Let us suppose, we want the family chords of C-maj scale, we will take the left and right notes as major chords. So, F-maj and G-maj are two family chords. The next three notes on the right side are D, E, and A. So, D-min, E-min and A-min are also the family chords of C-maj. And the next immediate right note is B, which indicates B-dim is another family chord of C-maj. Here, B-dim the diminished chord on the note B, which is formed with the serial number values 0, 3, 6. Therefore, family chords of C-maj are the seven chords altogether: F-maj, G-maj, D-min, E-min, A-min and B-dim. When ascending through higher notes in music is considered, the order of chords for a major scale always comes like maj-min-min-maj-maj-min-dim. Finally, the family chords of C-major scale are:

C maj D min E min F maj G maj A min B dim

The notes presented in the inner side of the circle is just turned the circle three times left because of finding the next notes in one place.

For making the family chords for the home chord C-Minor, we need to use this diagram in a little different way. Now the first three outer notes will be considered as minor chords, such as, C-min, F-min and G-min. Diminished chord will come right after that. So, D-dim is the family chord of C-min. And then, the next three notes will be flattened down by one note (Semitone) and these will be major chords. D#-maj (Eb-maj), F#-maj (Ab-maj) and Bb-maj are the other three family chords of C-min. When ascending through higher notes in music is considered, the order of chords for a minor scale always comes like min-dim-maj-min-min-maj-maj. Finally, the family chords of C-minor scale are:

C min D dim Eb maj F min G min Ab maj Bb maj

In exactly same way, family chords of any other home chord, in both major or minor scale, can be found easily by using this Circle of Fifths.

Conclusion:

Note pitch and octave formation are same in the Indian Music system like those are used in the West. Musical pieces are different in musical taste and human mood comparing to these two systems of music. But the theory of chords and family chords applies quite similar in popular music genre. In my next article I will write about how the Circle of Fifths can help make Family Chords in Indian Music System.

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