What makes music so musical?
The first article of this series provided a methodology to analyze the frequencies of the major scale. In particular, what happens when two notes on the scale were combined. This experiment found that any note on the major scale had reasonable resonance with the fundamental note, but some resonances were better than others:
The second article did a similar analysis with the minor scale. Again, all the minor notes resonated with the fundamental, but in a different way to give us that different mood the minor scale brings.
The first article also discussed the 12 divisions between the fundamental and its octave. The table below show how these notes were used:
The major and minor scales used 10 of these 12 divisions or “notes.” So two notes did not make it into either scale, the C#/D♭ and the F#/G♭. And we should ask why.
All the notes in the major and minor scales found resonance with fundamental within 60 radians (or 0.04 seconds) of time. How will those other two notes resonate with the fundamental?
Combining C + C#
I would like the reader to try to predict the quality of resonance with these two notes. To do that, let’s go back to the C + D blend:
C# has a frequency between C and D. So what will this blend look like?
Here is the C + C# curve:
The two notes did not find resonance within 60 radians.
The three notes closest to the fundamental produce a flat spot within the cycle. As the curves moved from E♭ to D to C#, the flat spot gets flatter. In the C# curve, the flattest area occurs about 53 radians. I will claim that this is the midpoint of the C-C# cycle. This means the two curves will find resonance at 106 radians, a value which goes beyond the scale we used to compare other combined sounds. As well, it looks like this curve will have 18 peaks in each cycle, which is the most of all the blends so far in this analysis.
All the other blends of the major and minor notes had resonance before 60 radians. Of all the notes so far analyzed, the most peaks per cycle was 16, two peaks fewer that C + C#.
In other words, C + C# has had the worst resonance of the notes so far analyzed.
Combining C + F#
The other “12th root of two” note not used in the major and minor scale is F#. This note is between F and G, two notes that produced the second and third best resonance with C. Should not F# produce great resonance as well?
Well, here is the curve:
C+F# graph not available. For some reason, this platform will only put in the C+C# graph for C+F#. I have spent about an hour, trying all sorts of tricks to get the C+F# graph here. The platform has worked flawlessly up to now. And it worked flawlessly for the next two graphs. I cannot explain this. Time to move on to something else.
This curve surprised me.
My experience with music written in the Key of C is that the notes of C# and F# are almost never used in this key. We saw C# as having less resonance than the other notes, so it’s not hard to understand why this note is not used.
To apply similar logic, the infrequency of F# in the Key of C led me to hypothesize that it too would not find resonance in the 60 radians.
Let me explain this in a different way. The cosine curves of any two notes will eventually find resonance. But some notes resonate better than others. The quicker the fundamental and test notes find that their peak-to-peak meeting, the better the resonance.
The C+F# curve is showing resonance at 44 radians. I was expecting that the resonance would be found beyond the 60-radian limit of this graph.
I went back to my son’s piano. All the blends with the fundamental and each of the major notes produced nice sounds, especially the C + F and C + G. When I played C+C#, I got a blend that did not sound nice. But that is what I expected. When I played C+F#, I also got a blend that produced another cruddy sound.
Yet the above curve shows the C+F# has better resonance than C+D and C+B.
F# is still a “12th root of two” of C. So maybe being part of that pattern has some special magic I do not understand, as least graphically. My piano playing says C + F# makes more noise rather than music.
I’m at a loss as to why my graph is showing a good resonance.
Combining C + 339 Hz
On a stringed instrument, we can tune the strings to any frequency we want. However, music has adopted a standard frequency so that musicians have a common set of frequencies to work with. You might have heard “A440” before. This means the instruments are tuned such that A below Middle C vibrates at 440 Hz. For example, I had a tuning fork for my guitar which was A440. When I got my A string to resonant with the tuning fork, I would then use the A-string to tune the other five strings. With this tuning, my guitar would be in tune with almost all other instruments.
Now it is time to check out some notes that don’t belong in the “12th root of two.”
My first chosen note vibrates 339 Hz, which about half way between E and F. The frequency 339 Hz is not on the “12th root of two” for Middle C. Here is the combined cosine curve:
I see a possible resonance at 44 radians. If my hypothesis was correct, that resonance should have been way beyond 60 radians.
Combining C + 427 Hz
My next “not 12th root of two” note is 427 Hz. This is about halfway between A♭ and A. And here is this blend’s combined cosine curve:
Resonance seems to happen at 50 radians. Again I was expecting resonance beyond 60 radians.
C+D resonates at 50 radians; C+B resonates at 56 radians. These are worst resonances within the major scale.
C+C# resonates at >60 radians; C+F# resonates at 44 radians. These notes are still “12th root of two notes.” Both should have worse resonance than C+D and C+B. C# does, but F# has better resonance.
C+339 Hz resonates at 44 radians; C+427 Hz resonates at 44 radians. These notes are not mathematically related to C. They should have worse resonance than C+D and C+B.
Conclusion
My purpose of this series was to use cosine curves to visually show the concept of musical resonance.
I analyzed 14 notes. Twelve were “12th root of two” notes. Seven were major scale notes. Seven were minor scale notes. Four notes were not on the major or minor scales. Two notes were not mathematically related to the fundamental note.
My analysis of the major and minor scale notes is within my understanding of musical theory.
My analysis of notes not on the major or minor scales is counter to my understanding of musical theory.
All this musical math has already been done, maybe two or three centuries ago. I am not doing anything new. I was undertaking a certain mathematical challenge “to figure things out by myself.” But maybe the math is more complicated than I had anticipated.
In my first article, I mentioned some slight aberrations with the combined cosine curves. My next article will investigate these aberrations more fully. I will be getting into a little more math than I wanted to.
Published on Medium 2024
Understanding the Other Political Side
Addendum 2025
It's been a year since I published Mysteries of the Minor Scale. I was hoping to finish this project with at least two more articles.
The first would have been a more thorough mathematical analysis as to why C+F#, C+339, and C+427 resonated sooner than C+D and C+B.
The second article would have been the resonance between the three notes of a chord, e.g. C+E+G. This would have required a X-axis of about 300 radians.
And maybe there would have been another musical theory path for me to explore.
Alas, there seem to have been other life priorities. So this project remains unfinished at this time.
If I can extrapolate this third article into TDG thinking, sometimes our plans do not work the way we want. Even the TDG will not be perfect. But when things need to be changed, the TDG will be able to change them. Western democracy does not do this so well.
