The Golden Chord Progression.

Observing the values in this chart, if we were to align these notes in order of decreasing pitch, so: C4, A3, G3, F3, one can see that the difference between the integral values between every one of these notes is roughly 0.0002, and this roughly constant difference between areas can be demonstrated in Fig. 7:

Essentially chords are a group of three or more notes played simultaneously. When modelling the frequency graphs of a chord, one must understand the sine function of a chord.

To demonstrate this, we will start by plotting chord I (C chord) and its function. Firstly, you need the three notes in the chord: C4, E4, G4, with respective sine functions:𝑓(𝑥) = sin(524𝜋𝑥), 𝑔(𝑥) = sin(660𝜋𝑥), 𝑧(𝑥) = sin(784𝜋𝑥).

Reflecting on the data we have found, Chords I, Chord V, and Chord VI all have the exact same period of exactly 0.5 seconds, but Chord IV, the final chord in the progression has a period of 1 second – exactly double. All the chords share roughly the same amplitude in the range of 2.9-3.0.

The reason why this chord progression ends in a sonically pleasing way could be attributed to this difference in period for the final chord. This “sonically pleasing” feeling stems from something called cadence, which is "a melodic or harmonic configuration that creates a sense of resolution [finality or pause]"iii at the end of chord progression. Why would the period of the chord be correlated to this though? Well, a longer period for the last chord indicates that it would take longer to complete one full oscillation of the chord sound wave, which would mean that harmonically, there would be a difference that the listener can feel. One can hear this harmonic shift in certain songs: For example, the shift from one line to the next by the singer will have this shift, otherwise known as the completion of a “phrase.^{” iv}

There are different kinds of cadences, but the one employed here is called the plagal cadence, which stems from the transition between the subdominant (chord IV) and tonic (chord I) triad in a chord progression, which is the transition between the final and first chord in this progression. This cadence is commonly used in religious hymns and is generally one of the more satisfying cadences to the ear.

However, even though analysis of these chords as a whole provided the correlation with a mathematical insight into the progression, that does not mean that their components, the notes themselves, cannot be analyzed in depth.

\(𝑠(𝑥) = sin(524𝜋𝑥) + sin(660𝜋𝑥) + sin(784𝜋𝑥)\)

Their cumulative function or the C Chord sine function is shown below in Fig. 4

Using this general form, we can start finding the sine function equation for the notes in the key of C major that are being used. To start us off, I found the function of middle C, C4 (262Hz):

𝑓(𝑥) = sin((𝜈)(2𝜋)(𝑥))
𝑓(𝑥) = sin((262)(2𝜋)(𝑥))
𝑓(𝑥) = sin(524𝜋𝑥)

Following that method, the rest of the notes will be written in a table:

So far I have found a certain correlation between the root notes of these chords. However, even though I have found this correlation, I wanted to solidify more of what I found through more data. I was interested in the role that the other two notes in the chords played when it came to their musical correlation. So, using the method applied to calculate the definite integral of the root notes, I have made a chart with all the areas under the curve for each note in play in each corresponding chord, as well as a scatter plot graph in the analysis section.

Sound in general is a frequency, a vibration of molecules that humans can hear – the average human ear “can detect sounds in a frequency range from about 20 Hz to 20 kHz.” As so, sound can be modeled as a wave through a sine graph. A sine function with all its transformation is as stated and shown below in

Fig.1, using Desmos graphing calculator:

𝑓(𝑥) = 𝑎 sin(𝑘(𝑥 − 𝑑)) + 𝑐

Chords are made up of 3 notes, as mentioned earlier, the root note (1^st note,) the 3^rd/minor 3^rd (2^nd note, with minor or natural depending on whether it is a major chord or minor chord,) and a perfect 5^th (3^rd note.) The root notes of chords actually are the skeleton of the chord – all the other notes are built around them, so the most important and defining note of a chord is a root note – it is also usually the defining note in each of the four chords throughout the melody (so if one just played the root notes by themselves, you could hear the exact same melody – chords provide depth to the melody.) Since they are the baseline of this chord progression/melody, they themselves can be analyzed further.

Our current root notes for our chord progression are C4 for a root of Chord 1, G3 for Chord V, A3 for Chord VI, and F3 for Chord IV. Those four notes have sine functions that we determined earlier, and those can be observed in more detail and analyzed further.

Starting with the C4 note, to find the area under the curve, we must take the definite integral of the sine function from x = 0 to x = 0.0019084, as shown in Fig, 6

Personally, I have always been exposed to music from an early age – my parents were big jazz and R&B fans. I ended up starting to learn how to play guitar in the fourth grade, which I continued for 5 years, while learning music theory, In grade 8, I became exposed to musical production – the use of a Digital Audio Workstation such as Image Line’s Fruity Loops Studio to make music. This was a turning point in my life, as I discovered what I believe to be my dearest hobby to this day.

It is apparent that a certain music genre is always naturally topping the billboard charts – pop. Many of these big pop songs are written using this wonderful chord progression: I–V–vi–IV and in its other inversions, such as V–vi–IV–I, vi–IV–I–V (a more pessimistic variant, compared to I–V–vi–IV,) and IV– I–V–vi. ⁱ

All these Roman numerals might seem confusing, but what they indicate is which chords - depending on their degree - are played in which order in their key. For the sake of simplicity, all of my further work will be done in the key of C major, the major key with no flats and sharps. The corresponding chords are: I = C chord, V = G chord, vi = A minor chord, and IV = F chord.

Whereas there is a musical explanation through degrees (subdominants, median, tonic, etc,) on how these chords progress and sound, they can also be fundamentally understood mathematically – the most basic example of this is musical harmony: simply put the ratio between two notes determines whether two notes are harmonic or not. This will be elaborated upon further.

This chord progression, “I–V–vi–IV” is very popular. Analyzing this “golden chord progression” mathematically through formulas and sine wave graphs can really, really help music producers such as myself understand the reason WHY this progression works: WHY is it so enticing, and HOW to apply it in my own music. That bring to the crucial guiding question of this exploration What is the correlation between these four chords mathematically? What is it that causes chords to seamlessly lead into each other, yet provide a climactic experience for the listener? How is this shown through the sinusoidal graphs of these sounds?

As a music producer, this is crucial information to me – such knowledge could bestow upon me a newfound and deep understanding of such workings – the literal sound waves that shape success.

Generally, the aim of this exploration is to explore possible factors, re-occurring or not, which can be attributed to any trends in these four chords. More specifically, through the analysis of the chords’ and corresponding root notes’ sine waves and their attributes, (such as their amplitudes and periods, and areas under curves,) I want to explore any kind of constant or correlation between the four chords. I will first discuss the attributes of a sine graph of a single chord. From there, I shall take each chord’s root notes and observe their attributes. Then, leading to finding the area under the curve of half of a period for said graphs - which should be correlated to a frequency - and calculating the differences in areas between the root note sine functions, comparing all those values to one another. Finally, I shall summate my research and discuss the possible different factors of the graphs and equations on the harmonic nature of the “I–V–vi–IV” progression. I will be using Audacity to generate reference graphs for my experimentation, but the numbers used in my calculations will be the generally accepted frequency values for notes – and for the sake of accuracy, I will keep note frequencies to three digits. I will also use Desmos for a graphical representation of the chords and their corresponding root notes, and to find the zeroes, and max/min values of each graph.

Observing this function, one can start deriving its characteristics. The minimum is at y = -2.9897, and the maximum is at y = 2.9897. We can find the amplitude:

\(\frac{max-min}{2}\)

\(\frac{(2.9897)-(-2.9897)}{2}\)

a = 2.9897

Generally, with the rest of the chords, we will see the amplitude roughly triple that of one note. This is because each of the notes has an amplitude of 1, and when they superimpose the amplitudes of the peaks add up together. However, the amplitude isn’t exactly 3, because the peaks in each of the notes don’t align perfectly with one another on the x – coordinate axis – see Fig. 3. Because of this, the maximum value achieved through the superimposition of amplitudes cannot be the sum of all maximums – but as we saw, it is pretty close to it.

I found the period, by measuring trough to trough, in this case, x=0.3387 to x=0.387, resulting in a period of exactly 0.500 seconds.

So, using this method, all four chords and their respective stats are in the table below: