Theory & Impro This is exactly how dissonant each interval is

[Dibbs]

The assumption is just a question about the definition of the word "dissonance". It isn't just an assumption I've come up with, it is how we in general defines the word. My dictionary says that a dissonance is a: "musical interval or harmony that sounds unpleasant in a given context".

More on this here: Consonance and Dissonance - The Main Theories

 

[Dibbs]

Here's a guy using a similar approach to yours. http://arxiv.org/pdf/1306.6458.pdf

 

[Andy Hornblower]

Kim,

Though I didn't explain it very well, the need to look at it as a Fourier series is to do with the harmonics or overtones that a real world instrument has. Your model is two sine waves, which is fine, but there aren't many things that can produce that - only a synthesizer really, or maybe a Glass Armonica comes close.

The jaggedness Colin mentioned, when looking at two instruments playing together, comes from the overtone series or harmonics, being out of tune with each other. In Equal Temperament, there are more harmonics that don't coincide (or none of them coincide), and that's visible on the oscilloscope as a more complicated waveform.

If you were looking at a trace of two actual instruments, it would be appropriate to break down what you're seeing by Fourier analysis. Terms stretching to infinity isn't a problem. We're only concerned with the ones we could hear, so you only need to look at the those. Terms after that will probably get smaller and smaller anyway.

If you model two theoretical instruments playing together as a Fourier series - like your model of two sine waves but adding in the overtones - you also only need to consider ones within the normal range of human hearing. Anything beyond that isn't going to be important.

I can see the assumption has probably been used before. It is just an assumption though. When you start with something like that, then construct a proof, what you're saying is "This is true, if my assumption is true", but you don't have any way of proving the assumption, so where does that get you?

I think quite possibly it's a useful assumption, but it has limitations - for the reason I gave earlier; Human hearing has a limited range. Towards the ends of that range - especially the upper end, it probably stops working.

 

[Kim]

I'm not going to rewrite it, as I can see that everything I found out can be found if you just know how to search for it. I'll read everything I can find about the topic, and perhaps (oh.. good ol' Perhaps by Charlie Parker) I'll can do some research on it later in my education or when I've finished it.

About the consonance/dissonance between overtones of different instruments, which Andy mentions; it isn't taken in account, to keep everything as simple as possible. It's possible that I'm going to make a simulation program that can show how different intervals works together when the notes comes from different instruments. I think it could be interesting for arrangers, who arrange for several different instruments.

In general: thank you for you response and thoughts about the whole thing.

 

[Andy Hornblower]

Cool. Reading the existing theories might inspire you to come up with more ideas on it, Kim. No doubt there's more there to be looked into. Computer simulation wasn't available to the main theorists, so you may have an advantage there.

I wasn't aware before looking at Dibbs' link, but it seems Helmholtz was the main source for the theory of beats between harmonics.

Consonance and Dissonance - The Main Theories

I played the example of two sine waves playing an interval in Just Intonation vs two playing the same interval in Equal Temperament, and I couldn't pick out much difference, but if two of the strings on my ukulele are out of tune with each other in just the right way (or maybe in tune, but in ET) and I happen to strike just the wrong interval, it's really noticeable - quite unpleasant beating, that even someone as cloth eared as myself notices. That has to be the out of tune harmonics, it seems to me.

 

[Dibbs]

...
I played the example of two sine waves playing an interval in Just Intonation vs two playing the same interval in Equal Temperament, and I couldn't pick out much difference, but if two of the strings on my ukulele are out of tune with each other in just the right way (or maybe in tune, but in ET) and I happen to strike just the wrong interval, it's really noticeable - quite unpleasant beating, that even someone as cloth eared as myself notices. That has to be the out of tune harmonics, it seems to me.

Plomp and Levelt found that pure sine waves behave differently from sounds with harmonics. The details are in the Sethares link I gave above. Relating Tuning and Timbre

 

[Nick Wyver]

That has to be the out of tune harmonics, it seems to me.

It can be quite noticeable with sop saxes playing duets.

 

[Pete Thomas]

I don't deny that the tuning system makes a difference, but I don't think its overall influence is as important.

It's true that it only applies to equal temperament, but it still gives an overall picture. A minor second is very dissonant in both just and equal temperament, and a perfect fifth is very consonant in both temperaments.

I tend to agree with this although I totally understand the relevance of bringing just intonation into the discussion.

e.g. after listening to just 3rds, equal 3rds suddenly sound a bit dissonant.

 

[Kim]

I tend to agree with this although I totally understand the relevance of bringing just intonation into the discussion.

e.g. after listening to just 3rds, equal 3rds suddenly sound a bit dissonant.

Yes, there's a little difference with most of the intervals. There's one interval where it (mathematically) makes a huge whether you use equal or just intonation. The diminished 5th.

This is a plot of diminished 5th in just intonation (582.51 cents).

And here it is in equal (600 cents).

The first one has the oscillation period 5, and the second one has the oscillation period 29 (12 depending on your tolerance). If determining consonance/dissonance only based on periodicity (as I tries to do here), a diminished 5th in just intonation should be much more consonant than with equal temperament.

 

[Kim]

Cool. Reading the existing theories might inspire you to come up with more ideas on it, Kim. No doubt there's more there to be looked into. Computer simulation wasn't available to the main theorists, so you may have an advantage there.

Yes, it is indeed really exciting. The article Dibbs came up with is very interesting, and the references therein are very interesting and useful articles too.

Here's a guy using a similar approach to yours. http://arxiv.org/pdf/1306.6458.pdf

I've already bought a book that associates physics and psychophysics with music, which is also referred to in the article.

 

[Andy Hornblower]

The minor second dischord fits in with the idea of out of tune harmonics. In Just Intonation, the interval depends on which minor second you mean - because each one is slightly different.

For Just, you have to start from a root note. In a major key, that's the root note of the key. For a minor key, I've read it's taken from the relative major, but I'm not sure why, or if that's correct.

Anyway, in one possible version of Just Intonation, the first semitone interval, or minor second, from the root note (lets say C to C#) is 16/15. Using the same idea as before, tone one is f plus its overtones (2f, 3f, 4f...) and tone two is 16f/15, plus its overtones.

The first time two of those overtones concide is the 16th overtone of tone one, with the 15th overtone of tone two (16f = 15x16f/15), so all the lower overtones don't match. Depending on how high the note is, we might not even be able to hear the overtones that do match.

On the same definition of the intervals in Just Intonation (there are other definitions), the diminished fifth from the root note is 45/32 - so the problem is the same; where overtones do coincide, for the two notes, they're very high, and lots of lower overtones exist that can clash / beat / be noticeably out of tune with each other.

The concordant / less discordant intervals are things like perfect fifths (3/2, so the second and third harmonics of the two notes can line up), perfect fourths (4/3 - so the third and fourth harmonics can line up)... and so on.

I'm going by the list of Just ratios given in the fifth paragraph, on this page:

www.chrysalis-foundation.org/just_intonation.htm (a good explanation of the concept)

In the case of any real world instrument we're used to, or any synthesized one with a rich enough timbre, the discord / dissonance is easy to spot. I'm still wondering if it's clear at all with just two sine waves and no overtones / harmonics, or if discord just arises from the clashing harmonics.... Maybe it depends who you ask, I don't know.

 

[mizmar]

Thought I'd drop this here.
The Physics of Dissonance

It's a popsci video; but goes above, beyond, left ... in a bit of a different direction. Quite interesting.

View: https://youtu.be/tCsl6ZcY9ag

 

[Pete Effamy]

Thought I'd drop this here.
The Physics of Dissonance

It's a popsci video; but goes above, beyond, left ... in a bit of a different direction. Quite interesting.

View: https://youtu.be/tCsl6ZcY9ag

The usual “non-musical “ theory being explained using non-musical / non real world examples.

What’s the thing about dissonance anyway? For some people, a major 7 chord is horribly dissonant and unpleasant. For others it’s melancholic or beautiful. Probably depends even more on how it’s voiced too, or arranged in an ensemble or full orchestra, and what harmonies precede or follow.

What about jumping to the famous Rite of Spring part - what is it again; an F chord over an E ? - anyway, it’s dissonant. For many people, it’s not unpleasant, but as exciting as orchestral music gets.

Music.

 

[mizmar]

Fair enough, given he titles the video "dissonance" - but I think it's actually more about consonance and how that can vary in different situations and traditions. A whole lot better than the cod-science "it Pythagoras, innit" usually trotted out....
... And a lot more respect for cultural and traditional issues.

 

[Pete Effamy]

Fair enough, given he titles the video "dissonance" - but I think it's actually more about consonance and how that can vary in different situations and traditions. A whole lot better than the cod-science "it Pythagoras, innit" usually trotted out.

But sine waves are used. We all listen to instruments playing together with not only their fundamentals, but all their overtones and interactions / sympathetic vibrations etc

And, given different circumstances, do we perceive the same dissonance as a dissonance every time ?

The word dissonance usually means “ugly” for most. But, there are a myriad of examples in classical / film music and jazz where most listeners will probably refer to hearing beautiful sounds / music and not even realising the dissonance involved.

 

[mizmar]

But sine waves are used

Yes. Which is normally done very naively - this takes the analysis to a much better level. Shows how that makes the usual story of ratios simplistic etc. I thought that came across well... Maybe not.

And, given different circumstances, do we perceive the same dissonance as a dissonance every time ?

No. And chappie says so - says there's a whole issue of physiology , psychoacoustics and culture...
But, yeah, fair enough, in 15mins he doesn't cover everything.

 

[Pete Effamy]

Yes. Which is normally done very naively - this takes the analysis to a much better level. Shows how that makes the usual story of ratios simplistic etc. I thought that came across well... Maybe not.


No. And chappie says so - says there's a whole issue of physiology , psychoacoustics and culture...
But, yeah, fair enough, in 15mins he doesn't cover everything.

Tbf, I saw about 90s. Just doesn’t resemble anything I hear or studied.

 

[mizmar]

Tbf, I saw about 90s.

You don't say 😉

Just doesn’t resemble anything I hear or studied.

No. Nor should it.
Personally, I'm quite against this things people do of leaning heavily on "physics explanations " for all the reasons you give.
But people do. And this video is actually quite a good antidote - at least compared to much of the stuff out there.

 

[Pete Effamy]

You don't say 😉

No. Nor should it.
Personally, I'm quite against this things people do of leaning heavily on "physics explanations " for all the reasons you give.
But people do. And this video is actually quite a good antidote - at least compared to much of the stuff out there.

What I’d like to see are musical examples. Music can be a magic trick - you can play a semitone dissonance on its own, then bury it in a chord voicing. Has dissonance disappeared ?

 

[mizmar]

What I’d like to see are musical examples. Music can be a magic trick - you can play a semitone dissonance on its own, then bury it in a chord voicing. Has dissonance disappeared ?

Chuck one up ... You don't need physics to explain that.