Mathematics of Musical Notes

Introduction

The fundamental element of music is sound. Each sound has a pitch, which is somehow related to Mathematics. When we talk about musical sounds we actually consider the certain pitch of sounds that can be used in musical pieces. Sounds that are able to make feeling to the listeners. Human feelings may be of different kinds; good, bad, sad, romantic, erotic, despair, anger etc. So does the music notes. A single musical note has a clear holy pitch of sound. For example, the note C has the pitch 261.63 Hz, that means, a violin string vibrates 261.63 times in a second when a violinist plays the note C. Similarly, when the note A is played the string actually makes 440 waves per second that tells us the pitch of note A is 440 Hz.

Diatonic Scale

The great old master Mathematician Pythagoras calculated the diatonic octave of musical notes. We all know there are twelve different notes in an octave. Actually, for completing the cycle of notes, an octave includes the thirteenth note, which is the double pitch of the first key note. For example, the octave of C starts with the key note C (261.63 Hz) and ends at the same note C (523.25 Hz). In Indian Music system, the starting and ending note is called Sa; octaves suitable for human vocals are called ‘udara’, ‘mudara’ and ‘tara’. So, the note Sa of ‘mudara’ octave starts the journey through other notes and ends at the Sa of ‘tara’. In traditional diatonic scale, originally standardized by Pythagoras, the pitches of notes are shown in the Table-1. My Bengali readers may read one chapter “Sur, Swar o Somomel” (Tune, Note and Harmony) of my book published in 2007.

Table-1: Calculation of traditional pitches of musical notes

C D E F G A B C
Pythagorian 264 297 330 352 396 440 495 528
9/8 X 5/4 X 4/3 X 3/2 X 5/3 X 15/8 X 2 X

 

Previously, the pitch of C was used as 264 Hz. 9/8 times of C is the note D, 5/4 times of C is the note E and so on. The notes in between were also calculated accordingly. The gaps between every pairs of the notes were not same. We followed this calculation system for couple of thousand years! Many great musical masterpieces were created that we had been enjoying for years including the Fur Elise. The tunes are sometimes so vivid to our memory so that we can feel the music when we are alone even in a pin drop silence. The notes are pitched now in a system of linear calculations for every higher notes. All the musical instrument manufacturers are following this standard system and the composers and musicians are also tuning their instruments in the standard pitch system of today. You may read another article of mine ‘Is the Old Fur Elise Mistuned Today?

Chromatic Scale

In a very simple word, pitch of each note is just 1.059463094 times of the pitch of immediate last note. The pitches of musical notes follow the same pattern unlimitedly upward or downward. How many octaves possible does not matter. So, where does 1.059463094 come from? This is the twelfth root of 2. Because, in each octave the ending note is the double of the starting key note by their pitches. The note C starts with 261.63 Hz and ends at 523.25 Hz. Pitch of other notes comes with their relative positions. Say, the positional value of C is 0, the positional value of next note C# is 1, positional values of next note D is 2, of D# is 3 and so on. Pitch of C# will be calculated as 2.61.63 multiplied by 1.059463094 to the power 1.  Pitch of D is calculated as 2.61.63 multiplied by 1.059463094 to the power 2. Pitch of D# is calculated as 2.61.63 multiplied by 1.059463094 to the power 3, and so on. The Table-2 shows the full octave.

Table-2: Calculation of current pitches of musical notes

Power 0 1 2 3 4 5 6 7 8 9 10 11 12
Note C C# D D# E F F# G G# A A#/Bb B C
Traditional 261.63 279.07 294.33 313.95 327.03 348.83 367.91 392.44 418.6 436.04 470.93 490.55 523.25
Current 261.63 277.18 293.67 311.13 329.63 349.23 370 392 415.31 440 466.16 493.88 523.25

 

The general formula for calculating the pitch of a note is given below.

Pitch of nth note = (Pitch of key note) x (2^(1/12))^(n-1)

OR,

Pitch of nth note = (Pitch of (n-1)th note) x 1.059463094. Here, 2^(1/12) = 1.059463094.

 

This system of calculations makes the Chromatic scale of musical notes that relates to the Chromatic Circle of notes. Chromatic Circle is the arrangement of musical notes in a circle so that the musicians can use it as a base of making chords with harmonious notes, using family chords in relation to the key chord, and changing key positions for a musical composition relatively.

Conclusion

Everyone may not notice the little difference in pitches of same musical note used in a masterpiece of hundred year back comparing to that of today. Yet, we feel the same thrill in an old musical piece even today admitting the fact that there is difference. Chromatic Circle helps musicians to make and play musical pieces in harmony; based on relative notes and relative chords. I will write on Mathematics of Chromatic Circle in my next article.

 

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