Keys from a Coffee Cup

This week I wrote code in Julia to create visualizations of the series of chords I wrote about before. I’ll go into the programming process later this post. First, the mathemusical stuff. Let’s have a look at a minor chord, and then I’ll explain what we’re seeing.

This is all based on an article I referenced before, and the concept of a “critical bandwidth of the human ear,” which is well studied. However, my programming is preliminary and rough, so these images are by no means definitive. I hope to refine and improve them in the future, and using a more complete and accurate theory of critical bands.

The images each show the overlaps in the harmonic series above a three-note chord. The brightness indicates completeness of overlap and the colour indicates which notes’ series are overlapping: red for the root, green for the variable note M, and blue for the fifth. Therefore a magenta stripe indicates an overlap between the root and fifth, yellow is between the root and M, and cyan is between M and the fifth. A white stripe is where all three overlap, and a dim or murky stripe is where overlap is partial. It should be noted that this is based on my programming, and if I had made the underlying function more or less sensitive, it would change what we define as “partial,” and I will have to do so if I want to refine these. As it stands, the direct overlap between notes two semitones apart is very dim, kind of at the threshold of being an overlap. Under their hypothesis, these murky stripes in these pictures are what we hear as dissonance.

The major chord shows something interesting, a white stripe near the top. And what is the top? 60 notes up. If you were to play the chord near the middle of a piano, the top end here would be near the end, which is also close-ish to the end of the human audible range.

Here’s a highly dissonant chord:

I have made a collection of these images on their own page. I find them fascinating, but they are less and less meaningful past M=6. I hope to find to way to visualize four-note chords and higher.

Now, for the programming aspect. Julia is a wonderful programming language that should appeal to all mathematicians and computer scientists for being about as easy to write and read as Python, yet compiling code that runs at incredible speed (like C and Fortran). I wrote my code by playing around in the Julia command line and collecting what worked in microsoft notepad, saving the program with file extension “.jl” instead of the usual “.txt” of windows text documents.

using ImageView, Colors, Images

mean(x...) = sum(x)/length(x)
# I couldn't be bothered to find the appropriate library or function

freq(x) = 2^( x/12 )
# This converts from note-space to frequency-space. For our purposes it doesn't really matter where we start, what matters is that going up N semitones corresponds to multiplying the frequency by 2^(N/12)

harmonic_weight(x) = (( cos(2pi*freq(x)) + 1 )/2 )^20
# This function makes a little bump around each harmonic of the root (x=0). We can shift this function to some other note's harmonics by subtracting the relative position of that note from x in the input

# Next is the worker function. We provide it three notes and it makes a colour-coded image of the overlaps in their harmonics
function harmony_vis(note1, note2, note3, resolution=0.025, distance=48, do_filter = true)
	x_space = 0:resolution:distance
	R = harmonic_weight.(x_space.-note1)
	G = harmonic_weight.(x_space.-note2)
	B = harmonic_weight.(x_space.-note3)
	Y = R.*G
	M = R.*B
	C = G.*B
	
	im = do_filter ?
		RGB.([R.*mean.(Y,M), G.*mean.(Y,C), B.*mean.(M,C)]...) :
		RGB.([R,G,B]...)
	imsquared = [im for i in 1:length(im)]
	return vcat(transpose.(imsquared)...)
end

Here is what using that looks like:

This week, I discovered a nice little musical idea:

If you’re a musician, you may not find that particularly interesting, although, isn’t it unusually consonant? Let’s see what it is and where it comes from. Last time, we saw that in typical consonant chords, there are patterns in how the overtones of all three notes simultaneously overlap. In the cases of the sus2, sus4, and minor chords, the pattern is itself a harmonic series. This seems to indicate that the sounds of these chords each contain the sound of a related higher note.

Chord	Position of related note relative to the root
Minor	two octaves and a fifth up (31 semitones)
Sus2	three octaves and a major second up (38 semitones)
Sus4	three octaves and a perfect fifth up (43 semitones)

At my digital piano, I investigated how it sounds to play these chords and the related notes. To me, it does sound like a consonant relationship. I didn’t record any of that but I recommend trying it yourself if you are interested.

More interesting to me was what I saw above a major chord:

There are three almost evenly space tones of increasingly close overlap, at positions 38. 47, and 55. To make that more concrete, let’s choose a key. Above a C major chord, the related notes are D, B and G, which happen to be the notes of a G major chord. In the key of C, we call C major the One chord (I), and G major the Five chord (V). Moving from I to V is a standard way to create tension, and going back from V to I releases that tension, and is called a perfect cadence. But what we’ve found here is a that a Five chord played way high up should sound like it’s an aspect of the One chord. This suggested to me the existence of the smooth-sounding cadence you heard earlier.

I got another useful idea in this article. Different tones within a sound stimulate different regions of the basilar membrane within the inner ear. If two tones stimulate partially overlapping regions, dissonance is perceived, and if they stimulate fully overlapping regions, consonance. The key is that natural sounds are composed of a series of tones, and the interaction between two sounds will have many overlaps.

Inspired, I improved on an interactive graph I’ve been working on in Desmos graphing calculator. Below is an example depicting how the overtones of a note relate to standard tuning. The units of the x-axis represent positions on a standard piano, so 1 to the right is a semitone up and 12 to the right is an octave. The solid black vertical lines represent the a root note and the octaves above it, and the peaks are the root’s overtones.

The more complex version of the graph shows some interactions within a chord. In the video below, we examine a chord with root note at x=0. The dashed vertical line is the chord’s fifth, and the dotted one is a movable note I’ll call M, which can be animated to change the quality of the chord. The different-coloured peaks represent overlaps in overtones: the green ones are from the root and M, the red ones are from M and the fifth, and the blue ones are from the root and the fifth. You’ll notice that the blue peaks are constant, since they come from the two unchanging notes (which, on their own, comprise a power chord). Finally, the black peaks are overlaps of tones shared by all three notes.

The overlap explanation is compelling here, but perhaps is an incomplete story. I notice that the most audibly dissonant chords (M = 1, 6, or 8) have a partial black peak under the first blue one, then a region where M interacts with just one of the power chord’s notes, and then chaos. On the other hand, the most consonant chords have their own series of black peaks. For next time, I’ll make recordings of what these chords sound like and explore some more of what these graphs might be telling us.

I started my investigation by watching some videos on Youtube. This is how I learned much of what I know about music theory, and I find it can make information quickly and easily understood.

Here is one which does a good job explaining the idea I am investigating. The creator explains how to find the first few overtones of a note on a keyboard, and how to use them to play particularly consonant major chords.

A commenter said that notes of a minor chord share a common overtone. I investigated on my own keyboard and found this is true: two octaves and a perfect fifth above the root is an overtone shared by all three notes. I noticed that by dropping the chord’s third down an octave, we can make this overlap happen an octave lower, and it creates a very nice sound. In an example of how confusingly music theory uses and reuses numbers as names, what I made there is also called a “drop 2 chord,” because it drops the third, which is the second highest note. (It’s also the second note up. It’s called the “third” because it’s the third note of a major scale starting at the root 🙄.)

Next I looked for articles on Google Scholar. I found this one, which is fairly compelling and explains the history of main scientific hypotheses for our experience of consonance and dissonance. I also started reading this one, but didn’t get very far through it before I was distracted by an idea from it. In audacity, I messed around with some sounds to see what pitch I perceived when a sound wave was played containing a periodic gap.

the audio clips played are as follows:
440 sine: a sine wave at 440 Hz
gapped sine: the same sine wave but with a gap of half a period inserted every four cycles
98 string: a combination five sine waves in decreasing intensity and increasing pitch, meant to roughly resembling the overtones of a string with fundamental frequency 98 Hz. They are 98, 196, 294, 392, and 490 Hz. I made this because I noticed that pattern in the spectrum of the gapped sine wave.
saw 440: a sawtooth wave at 440 Hz
gapped saw: same sawtooth wave, gapped the same way as earlier
saw 391: a sawtooth wave at 391 Hz, which is theoretical frequency of the “long pattern” that comprises the four cycles and a gap. Notice it is also very close to the third harmonic above 98 Hz. If my method of introducing a gap in the sound had been more exact, I expect those numbers would have been identical.

Here is another video I skimmed. I didn’t really like the presentation and knew most of the content already. However, for someone who is new to music theory or its associated math, it could be a good crash course.

Finally, I found a this music stack exchange thread which indeed places the minor chord within a harmonic series. I haven’t quite understood where that fits on a keyboard yet but I expect to at some point.

A warning: this first post is not educational. I won’t fully explain the meaning of the things I’m saying. Understanding it all will require knowing a little music theory.

I’m interested in the relationship between the overtone series and common chord types in western music. A major chord actually appears in root position within the overtone series of the note two octaves below the chord’s root, while the minor chord is a little more mysterious.

Questions to investigate:

• Can a minor chord be related back to a base note the same way?
• If not, what is it closest to?
• What overtone patterns are there between chords in common chord progressions?
• What can these relationships tell us about the nature of harmony and dissonance?
• Can we use the answers to any of these questions to help us compose music?

In addition to doing research online and asking knowledgeable people for guidance, I would like to investigate the math myself, using technology, and hopefully create some useful educational resources.