The Math of Music

On the face of it the title seems counter-intuitive. After all mathematics is pure intellectual reasoning, while music appeals to our senses and elicits an emotional response. But for us to respond to music there needs to be a fundamental underlying mathematical structure to the sounds and silences. Otherwise it is just noise. Dissonance, disharmony, discordance.

Pythagoras is the first in recorded history to attempt an understanding of this relationship. Legend has it that while walking past a forge he heard the sounds of hammers striking. Most of these rings were random vibrations, but a few sounded pleasant together. A heavy hammer emitted a lower note than a lighter one.  A hammer half the weight on the same anvil produced the same note as the heavier one – except higher. “Eureka!” (although Archimedes was born over two centuries later). We now call this an octave.

This is true of any resonating instrument – columns of air, bowls of water and importantly, strings.  All sound is vibration frequency, measured in hertz. The octave can be further divided, but the sounds are harmonious only if the frequencies are related in small number ratios 2:1, 3:2, 5:4 etc.

Thus the musical scale is a series of related notes that sound well together (their frequencies are small integer ratios) and create a mood or a feeling.  Let us take the the major scale for example  – the most common scale in western music. Solfège “Do re mi fa sol la ti do” (from the famous song in The Sound of Music) or the Indian equivalent “sa re ga ma pa dha ni sa”.

In a Pythagorean scale the frequency ratios between the notes of a major scale are:

Do Re Mi Fa Sol La Ti Do
Sa Re Ga Ma Pa Dha Ni Sa
1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1

 

Melody: In its simplest form, different combinations of just seven notes of a scale (major, minor etc.) create almost all the melodies composed ever. If this surprises you, consider that twenty-six letters of the English alphabet create over a million words in the language.

Harmony: Instead of playing notes of a scale one after the other to create a melody, harmony is the simultaneous sounding of related notes. By playing  behind, beneath, and around the melody, harmony supports it and gives the music texture and mood.

Rhythm: A beat is what we tap our foot to when we listen to music – sub-divisions of time. Like our heartbeat it is regular and tempo is the pace of this regularity. For example “fast” tempo is 120 beats per minute (BPM) and this is the average heart rate of a healthy young adult on the club dance floor! This can be further divided equally by 3 (triplets), 4 (quarter notes) 8 (eighth notes). All integers.

Of course the mathematics and physics of music can take a lifetime to learn and understand. And the aesthetics develops over millennia.

Musical perfection is mathematical, but musical inspiration is divine!”

Some reference links:

http://jackhdavid.thehouseofdavid.com/papers/math.html

Su, Francis E., et al. “Music Math Harmony.” Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

http://www.mathscareers.org.uk/article/harmony-numbers/

https://www.youtube.com/watch?v=2_-l3ByVj9k

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