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Introduction I have chosen to investigate the mathematical properties that surround music; more specifically, classical piano. I have always had a deep appreciation for music, not only as a listener, but also as a novice pianist. I enjoy listening to the great works of Bach, Chopin and Beethoven, as well as many others. Recently, I have become interested in how a composer can constantly and consistently produce such beautiful compositions. It was a mystery to me how Beethoven, whilst becoming deaf, could still compose some of the most beautiful music of all time. It came as quite a surprise that mathematics is a strong governing force in the “quality” of music. Hence, I aim to explore and understand how mathematics relates to music in terms of harmony. I will focus in on the aspects of mathematics in relation to harmony in classical piano solely. Although harmony is only one of the main aspects of music (the two others being melody and rhythm), it is still very complex and interesting on its own. This paper will be split into three parts. The first will cover a graphical representation of notes and chords, examining what makes certain notes sound harmonic in groups. The second part will investigate Pythagoras’s contributions to music as it relates to ratios, and hence the development of the harmonic series. The third and final part will look into probability.
Part 1: Graphing Notes and Chords with Sine Waves
A piano, similar to most other instruments, is built on the foundation of the octave (series of 12 semitones ending on a note twice the frequency of the first note’s tone). There are 12 semitones (smallest interval in western classical music) in each octave. Each tone inclusive is considered to produce a unique sound. As a result, each unique semitone (not a repeated tone in a new octave) is given a representational letter. The octave the note is located in is represented by a subscript number (ex: A4). On the piano, the scale range of the white notes is A, B, C, D, E, F and G going left to right. The black notes are named according to what key the music is in and what white note they relate to. They are given the symbol ♯ (sharp or above) and ♭ (flat or below.) Here is a graphical representation of this:
Elliot Falkner
IB Mathematics SL
6/14/2016
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Image Source: http://music.lovetoknow.com/music-genres/printable-piano-keyboard-layout
Each semitone, or note, will produce a sound with a set frequency. This frequency is a measure of pitch and is reported in Hertz or cycles per second (Hz). Each frequency, audible or inaudible to humans, can be graphed in a two-dimensional plane as a sinusoidal wave. We will be trying to observe a pattern in harmonic notes and chords through the ratio of periods of each graph. The first step to visualizing these frequency patterns is to graph a single note. We will begin with A4, a note that has a frequency of 440 Hz. The equation for A4 is: 𝑦 = 𝑠𝑖𝑛(440 ∙ 2𝜋𝑡)
Graphing as a sine function allows us to visualize the oscillation of the string, 2𝜋 gives us the period interval which represents one second, 440 represents Hz and 𝑡 is our variable for time. To observe the important aspects of the graph and to find the correct period ratios we must scale down each equation by the same factor.
𝑦 = 𝑠𝑖𝑛(
𝑍 ∙ 2𝜋𝑡) 440
We have divided the Hz value (let Z = Hz) in the sine function by 440 in order to make the function for note A scale to 𝑦 = 𝑠𝑖𝑛(2𝜋𝑡). I have chosen to make A4 the “normal function” as we will build our harmonic chords of note A4. We will apply this same transformation to each note’s equation in order to graph and observe proper ratios. Hence we achieve the graph of the equation for note A4:
Elliot Falkner
IB Mathematics SL
6/14/2016
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𝑦 = 𝑠𝑖𝑛(2𝜋𝑡) Before we graph our first harmonic chord we must note a few basic ideas. The wavelength or period of the function is directly related to pitch: the shorter the wavelength, the higher the pitch; the longer the wavelength, the lower the pitch. Here is a graphical representation of A major (Notes A4, C♯, E and A5).
These are the equations that created this graph of A major. They were derived by inputting 𝑍 Hz values into the general equation 𝑦 = 𝑠𝑖𝑛 (440 ∙ 2𝜋𝑡). 440
A4: 𝑦 = 𝑠𝑖𝑛 (440 2𝜋𝑡) 659.25 2𝜋𝑡) 440
E: 𝑦 = 𝑠𝑖𝑛 (
554.37 2𝜋𝑡) 440
C♯: 𝑦 = 𝑠𝑖𝑛 (
880
A5: 𝑦 = 𝑠𝑖𝑛 (440 2𝜋𝑡)
From the graph we can see that all the functions intersect at the origin and come very close to intersecting each other again at 𝑡 = 1,2,3 𝑎𝑛𝑑 4. To better observe their relationship, we can look at the relative ratios of periods compared to the benchmark of A4, as seen in the following table. Note A4 C♯ E A5
Semitones from A 0 4 7 12
Elliot Falkner
Frequency 440 554.37 659.25 880
Period 1.00 0.794 0.667 0.500
Ratio to A (1.00/period) 1.00 1.26 1.50 2
IB Mathematics SL
Ratios to A as Fractions 1⁄1 63/50 3/2 2/1
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In this harmonic chord we can see that each note that has harmony with A has a relatively simple, whole number fraction ratio with exception of C♯: however, C♯ has a ratio to A very close to 1.25 which gives another simple fraction ratio, 5/4. We can make this exception because the graph of C♯ comes very close to intersection at intervals 2 and 4. These nuances are crucial to observe in mathematical representations of music because values are not exact, which makes proximity very important. Let’s take another harmonic chord and see if the same patterns emerge. This time we will use the chord A minor (Notes A4, C, E and A5)
440 2𝜋𝑡) 440
A4: 𝑦 = 𝑠𝑖𝑛 (
659.25 2𝜋𝑡) 440
E: 𝑦 = 𝑠𝑖𝑛 (
523.25 2𝜋𝑡) 440
C: 𝑦 = 𝑠𝑖𝑛 (
880 2𝜋𝑡) 440
A5: 𝑦 = 𝑠𝑖𝑛 (
Note
Semitones from A
Frequency
Period
Ratio to A (1.00/period)
A4 C E A5
0 4 7 12
440 523.25 659.25 880
1.00 0.840 0.667 0.500
1.00 1.19 1.50 2
Elliot Falkner
IB Mathematics SL
Ratio to A as a Fraction 1/1 119/100 3/2 2/1
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In A minor we can see very identical results as in A major. Harmonic notes tend to have simple period ratios. The case of C, similar to C♯, has a period ratio very close to 1.20 that would result in a simple fraction ratio of 6/5.
To test the truth of the conjecture that harmonic chords have simple whole number ratios when comparing period values, we will test the appositive, a dissonant chord. The dissonant chord is A4, B, D♯ and F (it does not have a given name as it is not a frequently used chord).
440 2𝜋𝑡) 440
A: 𝑦 = 𝑠𝑖𝑛 (
D♯: 𝑦 = 𝑠𝑖𝑛 (
Note A B D♯ F
Semitones from A 0 1 6 8
622.25 2𝜋𝑡) 440
Frequency 440 493.88 622.25 698.46
Period 1.00 0.890 0.707 0.630
493.88 2𝜋𝑡) 440
B: 𝑦 = 𝑠𝑖𝑛 (
698.46 2𝜋𝑡) 440
F: 𝑦 = 𝑠𝑖𝑛 (
Ratio to A (1.00/period) 1.00 1.12 1.41 1.59
Ratios to A as Fractions 1⁄1 28/25 141/100 159/100
In the dissonant chord, we do not get simple, low, whole number fraction ratios as we do in the harmonic chords. We cannot justify rounding simplification of the fractions either as
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IB Mathematics SL
6/14/2016
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the graph shows no common intersection of the waves. From this, we can conclude that, indeed, the period ratios of different frequencies are a telling sign of them being harmonic; simple is more harmonic, complex is more dissonant. This should make sense to us as humans. Most people prefer when things are neat, clean and simple. This is what Pythagoras predicted when he investigated the mathematics of music and this glorification of the simple has led to the creation of the western classical style. To explain cases of C and C♯ we can say that the note is less harmonic to A than other notes with simpler fraction ratios; however, it is very close to perfect harmony given its proximity to a simple fraction ratio and common intersect point on the graph. This may be the mark of a unique chord or the note that makes the chord stand out to the listener. While a perfectly harmonious chord may sound good, it does not stand out the way a slightly off chord may sound, it is unusual, it is different and it creates a unique feeling. To summarize, harmonic notes in chords will generally have a simple period ratio to the fundamental note. Notes close to a simple period ratio can give the chord a special sound and character; however, if they are too complex or have periods far from the fundamental they will sound with dissonance.
Part 2: Pythagoras and the Harmonic Series
The observations from the previous section can be backed up by statements by a famous Greek mathematician, Pythagoras (570 BC - 495 BC). Pythagoras, known widely for his theorem involving lengths of a right triangle, had a strong relationship with music. Pythagoras was fascinated with the lyre, a common string instrument in Greek music. First, he observed that when two strings of same length, tension and thickness were plucked they produce a consonant sound. Second, he found that strings of different lengths produced different pitches and generally dissonant sounds. Lastly, he realized that certain different string lengths produced a consonant sound in pair: the relationship between these notes he called, an interval. These intervals can be seen as the harmonic note combinations from the previous section. Their names have been derived from the number of semitones they sit from the fundamental. From this he determined the octave, the perfect fifth, the fourth and so on. Each one of these consonant pairings was so because the length of the strings was in a simple ratio: octave, 2:1; fifth, 3:2; fourth, 4:3; etc. These ratios are reversible and apply to any type of string, hence they are relative ratios. Also, string length relates inversely to frequency which allows for the conclusion that frequency should have the same ratio relationship with harmony. Although Pythagoras knew that he could keep using larger integers and continue the pattern, he also wanted to keep his mathematics simple. He believed the simpler, the more correct.
Elliot Falkner
IB Mathematics SL
6/14/2016
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Fast forward to today, we now have a harmonic series that models Pythagoras’ observations. The formula for the series looks like this:
∞
∑ 𝑛=1
1 𝑛
First let’s make some observations. In this series we have 𝑛 going up to infinity which means that we do not have at any point, the “complete” series. The series begins with the 1 term and increases by one in the denominator for subsequent terms. What we do not 1
know yet is if this series is divergent or convergent however, we can investigate this. We will begin by writing out the first few terms.
𝑆 = 1+
1 1 1 1 1 1 1 + + + + + + +⋯ 2 3 4 5 6 7 8
We can see that each term is smaller than the previous one but is it small enough to converge? We will check this by constructing a new series. One that has each term less than or equal to the value of the corresponding term in the harmonic series. If each term is smaller than the sum will be smaller as well. If our second series diverges, we can conclude, by the comparison test, that the harmonic series diverges as well. To construct our new 1 series, we will make each term the largest power of 2 less than or equal to the corresponding term in the harmonic series. The reason for picking one half is purely ease and simplicity, any other number less than one and greater than zero could have been used 1 however, it is easiest to visualize a pattern 2 . This proof variation was created by French mathematician, Nicole Oresme.
1
We begin with the first term of the harmonic series, 1. The largest power of 2 less than or equal to 1 is 0. Hence, the first term of our new series is 1. The second term of the harmonic 1 1 1 series is 2. Again, the largest power of 2 less than or equal to 2 1
1
is 1. Hence, the second term of our new series is 2. The next term of the harmonic series is 3. 1
1
1
The largest power of 2 less than or equal to 3 is 2. So, the next term of the new series is 4. We continue doing this to each term till we begin to see our series take form. Here is what the new series looks like:
Elliot Falkner
IB Mathematics SL
6/14/2016
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1 1 1 1 1 1 1 𝑁 =1+ + + + + + + +⋯ 2 4 4 8 8 8 8
As we constructed it, each of the terms in our new series is a positive value and less than or equal to its corresponding term in the harmonic series. Now let’s see if this series diverges. We can start by simplifying our series.
1 1 1 𝑁 =1+ + + +⋯ 2 2 2
If we are to continue adding terms of this new series, we will see that the sum of each part 1 1 1 1 1 1 1 of the series adds to 2. 2 terms of 4 add to 2, 4 terms of 8 add to 2, 8 terms of 16 add to 2, and so on. If a series adds the same value to itself an infinite amount of times than that series will diverge as such is the case for our new series. Therefore, since our new series is divergent and is less than the harmonic series, we can assume, from the comparison test, that the harmonic series diverges as well.
∞
∑ 𝑛=1
1 =∞ 𝑛
The harmonic series is used all over in music. It is constantly being studied in theory and used to improve systems of composition, performance and tuning. But what specifically does this divergence mean to music and harmony? In fact, its divergence shows us a lot about the world of music. We understand from it that simplistic relativity is a never ending pattern that can always find harmony. The idea that this series carries on without end may not seem of utmost importance in the real world but, it allows us to understand why our finite patterns work. For example, it helps us understand why certain notes have special relationships with others well beyond what the human ear can recognize. It also allows us to understand why a piano can never truly be “perfectly” tuned. There are many other practical applications of the harmonic series and its properties of divergence in music and well outside of it and those deserve another entire paper of explanation. We know that this special series extends to well beyond what humans are even capable of thinking about and even its practical justifications do not define its reason. If we can take away anything, it is most interesting, philosophically, to know that at every extreme, one can still find abundant harmony. Elliot Falkner
IB Mathematics SL
6/14/2016
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Part 3: Probability of Harmony
For the last part we are going to go back to the piano. If you had no understanding whatsoever of piano or music and you randomly played 3 notes in one octave, what is the probability that you play a harmonic chord? First we have to recognize that each 3 note combination, A through G♯, can be played in a major or a minor chord. These chords have the same top and bottom notes but they differ by 1 semitone in the middle. Also, since the probability of playing any chord is theoretically equal, we want to calculate the probability of playing a major or minor chord comparative and multiply that probability into the number of chords that are consonant. Lastly we must understand that a single note cannot be played more than once. Here is a table that represents how to find notes in a chord:
Image Source: www.jonweinberg.com/music/Alt_Chord_Finder_Chart.pdf
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IB Mathematics SL
6/14/2016
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We start with the probability of playing the correct base note for any one specific chord. 1 Since there are 12 semitones and only one is valid, our probability is 12. Next, the probability that we play the correct middle note for the chord. This time there are 2 valid notes since there are minor and major variations of each chord. Also since we have already played the base note, we now can only choose from 11 notes. Our middle note probability 2 therefore becomes 11. The last probability we need is the top or final note. Since we have used 3 notes, only 9 remain which gives us nine in the denominator. Since there is only 1 1 valid top note we are given the final probability of 9. Since these events are dependent, we can do simple multiplication to find the probability of one specific minor or major respective chord:
𝑃=
1 2 1 × × 12 11 9
𝑃=
1 594
Since this is the probability of only 1 major or minor chord, we need to multiply this probability by the number of possible major/minor chords.
𝑃=
1 × 12 594
𝑃=
2 99
This probability demonstrates that the likelihood of randomly playing a harmonic 3 note chord is very low. Theoretically, it would take near 50 tries to play a correct consonant 2 combination as 99 × 50 > 1. This makes utmost sense as most unexperienced piano players struggle initially to improvise. However, as they begin to learn the math and system behind creating harmony: such as octaves, perfect fifths and fourths, partials, etc. their improvisation begins to improve substantially. If anything this shows that music is truly a fine art and one where specific values and numbers are involved deeply behind the scenes.
Elliot Falkner
IB Mathematics SL
6/14/2016
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Conclusion
We have briefly explored sine waves and how they model the frequencies produced by different tones on a normal 88 key piano. We observed that simpler ratios of periods between notes produced a more consonant sound. We also saw that small variations in perfect harmony are ever present in some of the most popular chords, perhaps an indication of a musical personality. We also observed in our graphical exploration that Pythagoras’ claims of simple being better resonate in the world of music. Not only do simple ratios allow us to enjoy music, they also appear in beauty, nature and the mathematics of everyday life. However, an important thing to consider about the model used in this investigation is that ideal sine waves were used. String imperfections of any other type of interference was not accounted for which likely yield a different set of results. It would be interesting to take a look at more realistic waves to see if they have the exact same harmonic behavior. Adding on to our previously discussed patterns, we incorporated the idea of a series that models the relationship between frequency and period length. This we saw as the harmonic series. We were able to determine its properties, especially its divergence. From this property we understand that although seemingly simple in the finite world, basic patterns such as the Harmonic series can extend into more complex realms with logical reason. Also, the Harmonic series lends to the question of how many other natural phenomena are dictated by simple patterns in mathematics? Finally, we explored probability. The likelihood that at random one could play a harmonic or consonant chord. We found this probability to be quite low, unsurprisingly. However, we note that if one truly understands the patterns behind the music, the probability can be greatly and consistently increased. We see this apparent in the many works of Beethoven, who although going deaf in his middle ages, he was still able to produce incredibly beautiful compositions that incorporated perfect harmony and a strong sense of emotion. This exploration helped me to understand more about series and limits, probability and the behavior of sine graphs. In abstract, I have a new understanding of how mathematics relates to reality in simplicity and importantly, in imperfection. For one to further these mathematical investigations, need not look farther than music theory. This ever evolving science uses the power of patterns described in mathematics to achieve a deeper understanding of music. One could explore the area under our harmonic series if it was modeled in a curve using integrals. One could combine the psychology of piano note choosing and get a more realistic understanding of consonant probability. There are many facets in the mathematical regions of music which we have yet to scrape the surface of and they are only waiting to be discovered.
Elliot Falkner
IB Mathematics SL
6/14/2016
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Bibliography
www.sscc.edu/home/jdavidso/music/musicnotes/musicnotes.html -
Graphs of frequencies
www.aboutscotland.com/harmony/prop.html -
Pythagoras’ history and musical involvement
www.phys.uconn.edu/~gibson/Notes/Section3_2/Sec3_2.htm -
Pythagorean interval information
scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf -
Comprehensive detail of various harmonic series divergence proofs
en.wikipedia.org/wiki/Octave -
Octave explanation
en.wikipedia.org/wiki/Semitone -
Semitone explanation and examples
www.phy.mtu.edu/~suits/notefreqs.html -
Frequencies of musical notes
Graphs Courtesy of Desmos
Elliot Falkner
IB Mathematics SL
6/14/2016
