How do you Measure Dissonance?

I think the scientific way of measuring dissonance has something to do with frequency ratios and other complicated stuff that I don't understand.

Realistically, one should say that probably most pianos that currently are standing around somewhere contain many microtonal intervals and are far from 12-tone ET :P

True, but idealizing things a bit...

I think the scientific way of measuring dissonance has something to do with frequency ratios and other complicated stuff that I don't understand.

frequency ratios is exactly what he's talking about - that's all interval analysis is.

Didn't Hindemith make a list of intervals from most to least dissonant?

He certainly did.

Didn't Hindemith make a list of intervals from most to least dissonant?

Well, from my understanding, he catergorized them into six different types of dissonance so it's a bit less straightforward than just a standard list. But yeah. I'd love to get my hands on his book. :sadtears:

Trying to quantify dissonance is an interesting idea, but it isn't really straightforward. You'll have to start smudging absolute values with subjective opinions. I havea friend back at UT who did a mini-experiment on this though.

He basically asked music majors when they heard dissonance in certain music and made a distribution plot with it and then tried to find the most common intervals that the music majors found dissonant. Then he looked for those same intervals when they weren't considered dissonant and did some more plots based on the context.

Some pretty hard math in there, but it's interesting.

What'd he do, regression analysis? That'd be the most straightfoward way...

i would just take the frequencies of the soundwaves, find their greatest common factor, and see how small it is. the smaller, the worse sounding.

I guess you are thinking a bit along the pythagorean approach, but the way you formulate it doesn't work. If you have a 10 Hz tone and a 20 Hz tone you have an octave, but the greatest common factor is 10 Hz, i.e. very low. The tritone between 5000 Hz and 7000 Hz on the other hand has a greatest common factor of 1000, i.e. a lot higher.

What'd he do, regression analysis? That'd be the most straightfoward way...

Sure, but in this case, it can't be done since it doesn't statisfy the axioms required for regression analysis furthermore it would have to be nonlinear and in which case would not be so straight forward. Keep in mind, it is multi-variate statistics. But anyways, the story is just there to illiterate the complex and sometimes impossible task of assigning what is dissonance.

I have taken a mild interest in the study of acoustics, and the relationship between math and music. And yes, at one point I tried to quantify dissonance, and it still confounds me to this day. The problem occurs because you have to quantify something that is very subjective. I personally feel that a minor third is more consonant than a major third. I'm sure there are people who would disagree. Also, the preparation and resolution of dissonances greatly affects the feeling of dissonance that the interval carries.

So the first thing you need to do to objectify dissonance is to remove all context. If you want to make a mathematical analysis of an interval, you must treat it as if there was nothing that came before and nothing after.

The second problem is the fundamental one: How do you define dissonance?

At first I thought that a Pythagorean approach was the most logical and objective approach since he defined musical intervals as simple ratios. He also noted that the intervals with the simplest ratios were the most consonant. Soooo, all you need do is find the lowest ratio of each interval and then add the two numbers of the ratio together and you would get a so called "dissonance rating". This would mean that a unison has a "dissonance rating" of 2, an octave would be 3, a perfect fifth would be 5 (kind of interesting :happy:), and a major third would be 145...

:huh:?

That can't be right.

It's not... It's the result you get if you define a major third as the dominant of the dominant of the dominant of the dominant of the tonic. The actual just definition of a major third is 5:4, which would give a "dissonance rating" of 9, which is a bit more reasonable.

But choosing between these two definitions is quite an arbitrary decision. They both approximate the same interval and both sound quite consonant. Yet they have widely different "dissonance ratings". In fact, this whole process is very arbitrary and is riddled with problems; the most prominent of which is the fact that this doesn't work at all for equal temperment :w00t:. You could try to analyze the sinewaves of the intevals, and what happens when you add or subtract them from each other but the same problem still continues: These are still based on ratios.

The whole problem begins when you try to objectify something that is subjective. There is no way to do this. So here is my very discouraging advice to you: Give up. Throw in the towel before you go insane. And it will drive you insane.

In the end, I can only leave you with one thing: a vast generalization, and over-simplification of how dissonance might function.

Dissonance functions like a cosine graph. The closer two notes are together, the more dissonant they are. The dissonance gradually decreases until it bottoms out and then starts to climb again in order to repeat the cycle over again. This happens because we classify the dissonance of an interval along with it's inversion, as you said.

There are two exceptions to this pattern: The point where the notes are closest together (the unison) is actually the least dissonant, and the point where the notes are furthest apart (the tritone) is the most dissonant (arguably). This means that the highest point of the graph is the lowest and the lowest point is the highest, which is encouraging... I think... :ermm:

So the first thing you need to do to objectify dissonance is to remove all context. If you want to make a mathematical analysis of an interval, you must treat it as if there was nothing that came before and nothing after.

No, that shouldn't be done. A lot of dissonance occurs because of context, it's almost like a permutation, but not really.

Dissonance functions like a cosine graph.

Umm I would hope not.

The whole problem begins when you try to objectify something that is subjective. There is no way to do this.

There are ways, the problem is that not everyone would agree it is the way to do it. However, if you can somehow manage to get the majority of people to agree, it could be considered good enough.

if you can somehow manage to get the majority of people to agree, it could be considered good enough.

Good enough's not good enough for me...

:dry: Wait... does that make any sense?

Good enough's not good enough for me...

:dry: Wait... does that make any sense?

Sure, but good enough is good enough for other people, which again is the point: If the majority of people say it's good enough, it's good enough. But anyways, that's all theory, the real point is, it's hard to do :D.

I'm sorry. It's late and I think I'm being funny. :P

But seriously, I'm going to try to focus for a while. :angry: <-- That's me focusing (I don't think it's working)

A lot of dissonance occurs because of context, it's almost like a permutation, but not really.

That's the point, if you want to judge dissonance objectively, you would need to take it out of context, because the right context can make anything sound dissonant. Not only that, but the same context may cause different listeners to react differently. Point is, context increases subjectivity.

How is like a permutation? I'd like to learn more about that.

Dissonance functions like a cosine graph

Umm I would hope not

Why not?

Alright, I'll admit that it is a very vague and unsubstantiated claim, but just look at the most common definitions of dissonance, and even the definition giving in the original post and you'll see the the minor second is considered extremely dissonant, the major second less dissonant, the minor third barely if even at all dissonant, the major third slightly consonant, and the Perfect fourth and fifth very consonant. The level of dissonance then starts to get higher and higher again as the intervals move toward the octave where the cycle repeats.

I'm not saying that it's a perfect cosine graph, it's just the closest analogy I could find. It could be a jagged saw-wave for all I know.

Dissonance functions like a cosine graph. The closer two notes are together, the more dissonant they are. The dissonance gradually decreases until it bottoms out and then starts to climb again in order to repeat the cycle over again. This happens because we classify the dissonance of an interval along with it's inversion, as you said.

There are two exceptions to this pattern: The point where the notes are closest together (the unison) is actually the least dissonant, and the point where the notes are furthest apart (the tritone) is the most dissonant (arguably). This means that the highest point of the graph is the lowest and the lowest point is the highest, which is encouraging... I think... :ermm:

You can only speak of "interval inversions" and of the tritone as the interval where the notes are "furthest apart" if you assume octave identity. But this is obviously a strong simplification. An octave is not the same as an unison. I actually have that problem with all the classical (such as the pythagorean) descriptions of dissonance: They focus on intervals where octave transposition is completely ignored.

Another major problem with the pythagorean system (apart from the ones you mentioned, i.e. different tuning systems) is that there's no clear definition of what "simple" means. Just adding the numerator and the denominator of the interval proportion and seeing how large it is doesn't even work without things like your example of using different major thirds or other intervals that are "similar". Take for example the interval between the third and tenth harmonic, i.e. a major sixth plus one octave (which would classically be considered to be consonant). 3+10=13. The tritone between the fifth and seventh harmonic however gives you 5+7=12, i.e. lower. The problem is just that there is no clear definition of what a "simple ratio" is.

As for the cosine thing: Would you, for example classify the interval that lies exactly between a major third and a fourth as consonant, as more consonant than a major third even? And even otherwise: If all you mean that the relationship between interval size and consonance "goes up and down", I'd advise against using such specific terms as "cosine".

Oh, and: "(the tritone) is the most dissonant (arguably)". Yes, I'd very much argue that.

That's the point, if you want to judge dissonance objectively, you would need to take it out of context, because the right context can make anything sound dissonant. Not only that, but the same context may cause different listeners to react differently. Point is, context increases subjectivity.

How is like a permutation? I'd like to learn more about that.

*stares and wonders if he is playing stupid*

If you want to make a comprehensive chart of what is dissonance and isn't you have to consider the cases were notes normally considered non-dissonant notes become dissonant and find out what factors encourage the dissonance. Doing anything else makes music linear and music isn't exactly linear, so any mathematical analysis that depicts music to be linear would have an inherent flaw and therefore be an incorrect model.

Thinking about it though, it isn't a permutation, more rather a recursion. I say this because the dissonance depends on where it is, much like the recursion value depends where it is.

As for the cosine thingy, Gardener got it.

Oh, and: "(the tritone) is the most dissonant (arguably)". Yes, I'd very much argue that.

See, I might not argue that. Like I said before, I find M2s much more dissonant, regardless of anything. But that;s where a lot of this breaks down - there comes a point when you're going to have people say "No, this is how it is..."

Prolly what DOFTS was talking about with survey results of music students is the most reliable way, because you have to have context to evaluate dissonance, even of pure intervals, and dissonance is a social value anyway.

First of all, I never said that I believed that the tritone was the most dissonant, I said that it arguably was. Personally, I think tritones sound cool and are certainly less dissonant than minor seconds. It's just too damn subjective, is the problem.

As for the cosine of dissonance, I did admit that it was:

a vast generalization, and over-simplification of how dissonance might function

If there ever was discovered a way to objectively measure dissonance I'm sure that the result would be a much more complex. But I still stand by the assumption that the intervals of the 12-tone Equal Temperment system fall more or less on a cosine-like curve.

And if you had read my first post carefully, you would have noticed that the whole purpose of the post was to discredit any sort of objective, arbitrary approach to dissonance analysis.

Here are some quotes that should remind you:

this whole process is very arbitrary and is riddled with problems

The whole problem begins when you try to objectify something that is subjective

Give up. Throw in the towel before you go insane. And it will drive you insane.

Just like it's driven me insane.

What are you saying? You're not insane.

Then why are you talking to yourself?

It's just the way I work out problems. Having a discussion with myself allows me to move beyond the circular logic of being trapped in one mind.

Why don't you just admit that some things can't be explained and quantified with logic and that's the real reason your mind runs in circles. It's not that you're "trapped in one mind".

But the greatest attribute of the human mind is the ability to break things down and define them in order to extrapolate possible future applications.

But if something is too subjective then it cannot be broken down since it varies from person to person and even from time to time.

But the ideal of science is that everything can be understood through logical and perhaps even mathematical means.

But maybe we can't really understand anything!

To not at least TRY to understand the universe is simply ingnorance!

To think we CAN understand is simply arrogance!

SCIENCE WILL EVENTUALLY EXPLAIN EVERYTHING!!

YOU JUST NEED MORE PERSPECTIVE!!

YOU JUST NEED MORE LOGIC!!!

LOGIC INHIBITS OUR INTUITION!!!

gently caress YOUR MOTHER!!!!!!!!!!!!!!!

*my head a-splode*

Methinks it's time for a cup of tea.

* wathces Fer explode*

That was neat.

Luckily I have a replacement head stuck so far up my butt that it wasn't damaged by the explosion.

...

I think that any positive contribution I can make to this thread ended with that last post. Thus, adieu.