written by: sarah ning
graphics by: yunyi cui
Have you ever wondered why some songs put a smile on your face while others leave you lying on the floor crying? What if I told you that the emotions you feel from songs are caused by physics and math? Shockingly enough, there is more correlation between music and science than one might think!
Before we get into the scientific explanation, let’s define some musical terms:
Consonance: pleasant tone to the ear
Dissonance: unpleasant tone to the ear
Pitch: quality of sound caused by the rate of the various frequencies producing it
Low pitch results from a low frequency sound
High pitch results from a high frequency sound
Interval: difference in pitch between two notes
Interval ratio: the ratio of the frequencies of the pitches
Octave: the distance between one note and the next note with the same name
Ex. C3 to C4
We should also familiarize ourselves with the piano keyboard. White keys are given names lettered from C to B. The black keys that are in the middle of two white keys are given a ♯ (sharp) or ♭ (flat) symbol. For example, a C♯ is slightly higher in pitch than C, and D♭ is slightly lower in pitch than D. On a piano, a C♯ and a D♭ are the same note, and so for the purpose of this article, we will only look at the sharps, plus the G♭.
Each note has a different frequency, and thus a different pitch. This table shows the frequencies of notes in one octave, from low C3 to high C4.
Note Frequency
C3 132.000
C#3 137.500
D3 148.500
D#3 154.688
E3 165.000
F3 171.875
Gb3* 187.733
G3 198.000
G#3 206.250
A3 220.000
A#3 229.167
B3 247.500
C4 264.000
*Although F♯ and G♭ are the same note on the piano, this is not the case for all instruments. As shown in the frequency difference is 2Hz, making them sound very similar.
Interval ratios are determined by dividing the frequency of the higher note by that of the lower note. For example, if we divided the frequency of G by the frequency of C, the interval ratio would be 3:2, which is calculated from 198Hz/132Hz. Take a listen to all the intervals provided below, and make a table of which intervals you believe sounds pleasing (consonant), and which sound unpleasing (dissonant).
Interval Interval Ratio Interval Name
C3 to C3 1:1 Perfect Unison
C3 to C#3 25:24 Minor Second
C3 to D3 9:8 Major Second
C3 to D#3 6:5 Minor Third
C3 to E3 5:4 Major Third
C3 to F3 4:3 Perfect Fourth
C3 to F#3 45:32 Augmented Fourth
C3 to G3 3:2 Perfect Fifth
C3 to G#3 8:5 Minor Sixth
C3 to A3 5:3 Major Sixth
C3 to A#3 9:5 Minor Seventh
C3 to B3 15:8 Major Seventh
C3 to C4 2:1 Perfect Eighth/Octave
How does knowing the interval ratios help understand why some songs sound happy and some sound sad? Let’s take a look at the table you (hopefully) made of the consonant and dissonant intervals. Based on research, smaller ratio integers tend to sound more consonant, while larger ratio integers tend to sound more dissonant (1). Looking at table 3, the perfect intervals are the most consonant, since they have the smallest ratio integers. Diminished and augmented intervals have the greatest ratio integers, which explains why they sound more dissonant. So, how did you do?
Interval Consonance/Dissonance
Perfect 1, 8, 5, 4 Most Consonant Intervals
Major/Minor 3rd and 6th Imperfect Consonances
Major/Minor 2nd and 7th Sharp Dissonances
Augmented Dissonant
Diminished Most Dissonant
Below is a table of popular songs where the first two notes are the interval indicated. This can help to apply what we just learned about intervals with a familiar tune.
Interval Name Song
Minor Second Jaws Theme Song, Pink Panther
Major Second Happy Birthday
Minor Third O' Canada
Major Third When the Saints go Marching, Kumbaya
Perfect Fourth We Wish You a Merry Christmas, Amazing Grace
Augmented Fourth Simpson's Theme Song
Perfect Fifth Twinkle Twinkle Little Star, Star Wars Theme Song
Minor Sixth The Entertainer, We Are Young
Major Sixth My Bonnie Lies Over the Ocean
Minor Seventh Somewhere from West Side Story
Major Seventh Take on Me
Perfect Eighth/Octave Somewhere Over the Rainbow
Now, what is the scientific explanation behind consonance and dissonance? Consonance is related to a natural phenomenon known as the harmonic series, which is a set of frequencies consisting of a fundamental frequency, which the lowest note that is heard, and its harmonics, which are positive integer multiples of the fundamental frequency (2). The harmonics that immediately follow the fundamental frequency/first harmonic resonate the most. For instance, when A3 is played you hear its fundamental frequency of 220Hz the loudest, but you are also able to hear its harmonics which is each increase in 220Hz: A4 (440Hz), E5 (660Hz) ... (Figure 1). Each instrument has a unique combination of fundamental frequency and harmonics that are heard, which creates what is known as the timbre of an instrument. Timbre is what causes different musical instruments to produce distinct musical sounds, even when playing the same note at the same volume.
Figure 1: The harmonic series
It is not a coincidence that the intervals between the first few harmonics are perfect intervals, because consonant intervals will resonate the most with the fundamental frequency (1). In other words, the most consonant intervals are perfect because they appear earliest on the harmonic series.
Essentially, the harmonic series explains how consonance occurs as a result of natural musical ratios naturally present in tones. On the contrary, dissonance occurs when musical tones do not fit into the naturally present harmonic series (1). More specifically, consonance depends on how well sound waves match the pattern of the initial note’s frequency. If the second note is a harmonic of the initial note, this results in their sound waves reinforcing each other to sound pleasing to the ear. However, if the second note was not a harmonic of the fundamental frequency, the harmonic series of the two notes would clash, giving off a dissonant sound.
Figure 2: Consonant and dissonant sound waves
What is the benefit of knowing the science behind music? How can musicians apply this knowledge to their songs? Composers and musicians use the harmonic series to create the desired tone of their song. Perfect and major intervals commonly portray a happy mood, as they sound more consonant. On the other hand, minor intervals are typically associated with sadness, due to their dissonance. Diminished and augmented are used to create an unsettling sensation, often being implemented into jazz music. In classical music, dissonant intervals tend to be followed by consonant intervals, so that the endings of musical phrases sound more pleasing to listen to. Typically, the endings of songs will end on the 1st, 5th, or 4th note to create a perfect interval (P1, P8, P5, P4). Fun fact: perfect intervals were so desired in songs that in the Middle Ages, only perfect intervals could be used in composition, and other intervals were banned by the Roman Catholic Church!
Next time you listen to your favourite song, pay attention to some of the intervals you hear, and it might help you understand why the song makes you so happy (or sad, we don’t judge). Maybe even challenge yourself to try composing a song of your own to capture those feelings you wish to express. It is truly amazing to think how science is everywhere around us, even in the music we listen to every day! So if you ever forget to do your science homework, just tell your teacher that you were listening to music, because it technically is science.
References
1. Bowling DL, Purves D. A biological rationale for musical consonance. Proc Natl Acad Sci U S A. 2015 Sep 8;112(36):11155–60.
2. Harmonic Series [Internet]. [cited 2021 Feb 22]. Available from: https://cnx.org/contents/VhLlFFhd@20/Harmonic-Series
3. Just_Tuning [Internet]. [cited 2021 Feb 22]. Available from: https://www.sfu.ca/sonic-studio-webdav/handbook/Just_Tuning.html
4. Newco Shift | The Intersection of Marketing & Product Management [Internet]. Newco Shift. 2017 [cited 2021 Mar 15]. Available from: https://shift.newco.co/2017/06/14/the-intersection-of-marketing-product-management/
5. Muzoracle: The Tarot of Music - The Harmonic Series [Internet]. [cited 2021 Feb 22]. Available from: https://muzoracle.com/the-harmonic-series
Comments