This is a very interesting idea Kim, and I'm impressed with the amount of thought you've put into it.
I think you need to take a better look at other tuning systems, such as Just Intonation. Just Intonation is a good place to start because it has a fairly simple basis. There isn't just one version of it, but the idea behind it is that the interval between any two adjacent semitones is always the ratio of two fairly small integers.
You could say "Okay, a fifth is a step of seven semitones, and 3/2 it's very close to 7 x the twelfth root of 2, so I'll just assume it's close enough to be the same". The point is, it isn't the same.
I still struggle to wrap my head around the reasons for it, but it does work very well. A simple melody played in Just Intonation is subjectively more pleasant and emotionally satisfying than the same tune played in Equal Temperament. Yes, that is subjective, but it's also something that can be measured and quantified.
I'm sure Colin (The Bear) is right in that you can clearly see the difference between intervals in Equal Temperament and ones in others (lets just stick with Just Intonation, to keep it simple). Why would that be? The usual reason that's given is that it's about harmonics. The extra dissonance, and visible jaggedness, in Equal Temperament is simply because there is more harmonic content from combining the two waveforms.
I'm now struggling to remember how you calculate that combined waveform
🙂 I expect you could do a much better job.
If you were observing a waveform, but didn't have a way of calculating it, because you didn't know how it came about (in this case, the tuning system used to make an interval), one way to look at it would be as a Fourier Transform, or look at the observed waveform and break it down using
Fourier Analysis. The extra jaggedness Colin mentioned would be quantifiable as extra terms representing the higher frequency harmonics.
So, one observation is that your paper applies only to Equal Temperament, and not to any other temperament. Saying that the frequencies of the notes happen to be quite similar, so it applies well enough to both, does not work. I'm not trying to be critical, but that is what your paper is discussing - and it's interesting - and I still don't fully understand it
🙂 - but it doesn't apply to any other tuning system, so I personally think you should clarify that.
I'm also not sure about the assumption that the same dissonance applies, regardless of the frequencies of the notes involved. It's fine to say you're making a mathematical argument based on that assumption, but unless there's some way you can prove it, your hypothesis is based on that, so it's kind of on shaky ground.
The reason I think it might not be a safe assumption is that we have a limited range of pitches we can perceive. As we get older, the range gets smaller too. If I took the jagged looking dissonant interval that was shown on the scope, I might be hearing a lot of that jaggedness - or less of it, depending on where my ears cut off the upper harmonics. My left ear has very little treble response any more, so that would be the less critical ear, and probably hears less dissonance.
If I then raised the whole thing up a couple of octaves, squashing all the peaks together on the oscilloscope, I'd still be able to see the jaggedness, but those harmonics might well be out of my range of hearing, so I probably wouldn't hear as much dissonance, if any.
Mathematics definitely can be applied to music, and the way we perceive music - which is really all there is; if we didn't perceive sounds as music, they would still be sounds, but they wouldn't be music. I find it interesting that we can examine some of it in terms of mathematics.
A lot of people have the idea that they can't do maths, but can do music - or the other way around. Personally, I think we all have the capacity to do both, and they're both a fundamental part of what makes us human. Unfortunately, the way they are taught in schools can put people off one or both of them for life.
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