Since I’ve already covered frequency content of scales and chords in the previous two posts, I’d like to move on to common chord progressions, and what better chord progression than the I-V-vi-IV progression that is literally everywhere (The Axis of Awesome even made a song about how common this progression is, titled the four chord song). What’s even more, there’s an entire wikipedia article for popular songs containing this progression. From the list, basically every song ever uses this progression.
This progression involves the I (tonic), V (dominant), vi (submediant), and IV (subdominant) scale degrees; in the key of C major this corresponds to C – G – Am – F.
Now, it probably makes things a little prettier to visualize it in terms of a logarithmic vertical axis, since musical pitch is exponential.
When we look at pitch with a logarithmic vertical axis it just looks exactly the same as things would if all we did was plot the number of intervals in a linear fashion. Finally, it may be beneficial to look at the progression in terms of pitch names.
There you have it, the most common chord progression in western music!
Get The Code
If you’re interested in checking out any of the super hacky code I’ve written for this, check out my github repo.
In part one I discussed frequency content of the standard chromatic and C Major diatonic scales. I’d like to move forward by discussing the frequency content of the C class of chords on piano.
Chords With Root Note C
If we take a look at the various C Chords in terms of frequency content, and format our plots such that each tick on the vertical axis coincides with the frequency content in semitone intervals, we obtain the following plot:
I’ll probably come back and re-organize this plot as the ordering of chords is a bit crazy since the python script I wrote just iterated through keys in a dictionary… but that’s a whole other story; for now it’s good enough.
From the above plot we can see that we are using A440 concert pitch, as A4 appears as 440 hz, and A5 appears as 880 hz, which is one octave higher.
Let’s take this same plot and change the vertical scale from frequency to number of semitone intervals.
From this plot it’s very easy to see that all of the C chords consist of a root note (in this case middle C), most have a major third as a second note, which is 4 semitones above the root, all the minor notes have a minor third as a second note, which is 3 semitones above the root, and nearly all have a perfect fifth as the third root, which is 7 semitones above the root. All of the numbered chords 7, 9, 11, and 13 have a minor seventh as the fourth note, which is 10 semitones above the root. All chords numbered 9, 11, and 13 have a major ninth as the fifth note, which is 14 semitones above the root, or a perfect octave plus a major second. All chords numbered 11 and 13 have an eleventh as the 6th note, or a perfect octave plus a perfect fourth, which is 17 semitones above the root, and finally both 13 chord have a thirteenth as the 7th note, or a perfect octave plus a major sixth, which is 21 semitones above the root.
Now, there’s one last easy thing we can do, and that’s to show the same plots in terms of the note contents of the chords, again, assuming we are in the fourth octave.
One thing about music theory that’s always bothered me is the disconnect between pitch interval and pitch frequency content. Hopefully this will become a little series where I attempt to easily bridge this gap with frequency domain plots.
Chromatic Scale
First things first, let’s take a look at the chromatic scale ranging from middle c (c’) to tenor c (c”), using scientific pitch notation ISO 16, with A 440hz concert pitch (A above middle C having a pitch of 440 hz):
Chromatic Scale Frequencies
It’s easy to see a few things from this plot:
The notes of the chromatic scale are A, A#, B, C, C#, D, D#, E, F, F#, G, G#.
Some people refer to the chromatic scale as the dodecatonic scale as it consists of 12 notes per octave. In this case tonic refers to tones, not the presence of a tonic. In 12-tone equal temperament each tone is separated by a 100 cent wide semitone interval, thereby making the chromatic scale nondiatonic with no tonic note.
Each interval of the chromatic scale consists of one semitone, or half step, indicated by the letter S.
Each of the intervals are made up of 100 logarithmic units of measure named cents. Aside: Like the decibel’s relation to intensity, a cent is a ratio between two close frequencies, therefore, the frequency range encompassed by a cent must be proportional to the two frequencies. The number of cents between two pitches, a and b, can be calculated by:
Mathematically, the pitch frequency increases exponentially as a function of number of intervals, as dictated by the following equation:
where is the output frequency, is the starting input frequency, and is the number of intervals from the input pitch.
Due to the fact that the chromatic scale is exponential, and doubles every 12 intervals, it results in a monotonically increasing linear plot when viewed with semi-log axes.
Intervals
Visualizing the chromatic scale on a semi-log plot makes understanding intervals very straight forward.
minor seconds: 1 semitone wide (100 cents, or 1 interval) major seconds: 2 semitones wide minor thirds: 3 semitones wide major thirds: 4 semitones wide perfect fourths: 5 semitones wide diminished fifths: 6 semitones wide (also called an augmented fourth) perfect fifths: 7 semitones wide minor sixths: 8 semitones wide (also called an augmented fifth) major sixths: 9 semitones wide minor sevenths: 10 semitones wide major sevenths: 11 semitones wide perfect octaves: 12 semitones wide
Diatonic Scale
Moving on from the chromatic scale, let’s look at the diatonic major scale (Ionian mode) in C.
Looking at the diatonic major scale in C we observe some readily apparent things:
The diatonic scale consists of 7 distinct notes per octave, A, B, C, D, E, F, G. Some people may refer to this as being heptatonic as it contains 7 tones.
The intervals between the pitches of the chromatic scale are structured in a manner to separate the semitone (half step) intervals as much as possible, being separated from each other by a series of two or three tones (full steps).
The notes of the diatonic scale are made up of 8 degrees with the following names: 1) Tonic (key note) 2) Supertonic 3) Mediant 4) Subdominant 5) Dominant 6) Submediant 7) Subtonic (leading tone) 8) Tonic (Octave)
The tonic note is the first scale degree of a diatonic scale. Similar to how a root is the reference note of a chord, a tonic is the reference note of a scale. As the name implies, the diatonic scale consists of two tonic notes, with the second being one octave higher than the first.
Another characteristic of the diatonic scale is the presence of a subtonic note, which is the seventh scale degree. This is a note that resolves to a note one semitone higher or lower, meaning that it moves from dissonance, or an unstable sound, to consonance, a stable sound. The seventh scale degree has a strong affinity for, and leads melodically to, the tonic.
Moving On
Further on in this series I will plot other scales and modes, as well as common chords, chord progressions and songs in terms of frequency vs time.
is the output frequency, 
is the starting input frequency, and 
is the number of intervals from the input pitch.
