How do you Measure Dissonance?

I'm somewhat working on a way to quantify dissonance.

I have a few things so far:

A basic categorization of dissonance/consonance by intervals

(measured by half steps between notes):

• PerfectConsonance = 0, 7, 12 = octaves and fifths
  
• ImperfectConsonance = 3, 4, 8, 9 = minor and major 3rd, minor and major 6th
  
• SharpDissonance = 1, 11 = minor second and major seventh
  
• SoftDissonance = 2, 10 = major second and minor seventh
  
• Perfect4th = 5 = well, um, a perfect 4th?
  
• Tritone = 6 = diminished fifth
  

I built that list from this: Music Theory/Consonance and Dissonance - Wikibooks, collection of open-content textbooks

A way to determine the inverse interval

(Un?) Surprisingly, each dissonance value is the inverse of itself. The inverse of a minor second is a major seventh (and incidentally, a SharpDissonance)... the inverse of an ImperfectConsonance is an ImperfectConsonance... etc.

Actually, a perfect 5th and a perfect 4th are inverses, but that's the one exception.

So now I have intervals mapped to dissonance, and I can determine the inverse interval...

A way to determine all intervals for a note cluster

Next, I have a way to take a note cluster (a chord, or otherwise), and determine the full set of interval pairs created by all combinations of those notes. Not going to get into the algorithm here, but I have the ability to do it.

Thus, I can take a note cluster, determine all intervals produced by that, and come up with an "average" dissonance produced by all the intervals.

What I'm missing... the relativity of "dissonances"

The critical piece I'm missing, is that I don't really have a good grasp of "how" dissonant each interval is.

Right now, I'm assuming that a Tritone (dim5) is more dissonant than a SoftDissonance (M2, m7), and less dissonant than a SharpDissonance (m2, M7). Is that a correct assumption?

Likewise, I'm assuming that a perfect 4th is more dissonant than a perfect consonance (octave, 5th) [duh?], but less dissonant than an ImperfectConsonance (m3, M3, m6, M6). Is that correct?

Also, it'd be nice to have some way of quantifying that not only is a tritone more dissonant than an imperfect consonance, but that it's (say) 3 times more dissonant.

I haven't been able to find any research into this, although I suppose I could probably come up with a quantitative way of measuring harmonics produced by the tone-pairs in an equal-tempered scale... but I'd rather not have to do that.

Any suggestions, or references, or just, telling me how it is?

(Note: this is entirely for personal use, just trying to come up with some informal music analysis tools.

Also, if anyone wants any more details, feel free to ask... I'm just excluding them right now, to focus on this particular piece of the puzzle.)

Instead of classifying it as notes and intervals, it may be more fruitful to include all shades of a note, given a center frequency. So, calculate it for mean-tone and just tuning, as well, if its for analysis purposes, given changes in temperament.

And then to cover computer music, you'd have to use a host of different parameters to justify dissonance.

While admirable, it might be better served to use one's ear instead of a rubric.

[*]SharpDissonance = 1, 11 = minor second and major seventh

[/i]

I'm afraid I have to disagree with you there. From personal experience, I've written several pieces where major 7ths are heard as consonance. Hell, when composing in a major key, I rarely actually use a typical I chord. It almost always has a Major 7th extension. And it's my personal favorite interval so yeah... I definetly wouldn't qualify it as "sharp" dissonance. Maybe soft... maybe.

I'm somewhat working on a way to quantify dissonance.

Quantify dissonance? Impossible.

You can devise your own opinions regarding the hierarchy or levels of dissonance...but you can not define them in a universal way.

EVERYTHING contributes, and will affect how a dissonance is perceived: orchestration, tessitura, voicing, dynamics, listener experience, melodic context, rhythms...

...

Don't bring this pseudo-scientific mentality into MY HOUSE!! This is ART, dammit!!

The problem with your approach is that you're basing it on a list that doesn't state its foundations properly and mixes them up quite a bit. The distictions this wikibook list makes are in no way scientific, but are a wild mix of traditions and hypotheses. Otherwise you couldn't say that a 5th is a "perfect consonance": "Perfect" would mean there isn't anything more consonant, but what about an octave then, or an unison?). Not even to mention the question what tuning etc. we're talking of. The differences are gradual, not distinct and depend on many factors such as the aural training of the listener. (Some people will hear a slightly out-of-tune note better than others, for instance. And a pianist may even hear a fifth in equal temperament as "more consonant" than a just fifth. I won't even go into the fact that many instruments don't produce harmonic overtones, especially in certain cultures such as Gamelan music, but also church bells etc., which changes the whole thing fundamentally. Or the fact that the degree of "perceived dissonance" is highly dependant on the register of the pitches. Or the huge role cultural and personal experiences play in all this. The list already mentions that certain chords "are considered" consonant in Jazz that aren't in classical music. That shows how pointless the whole list is in the first place for any serious deductions: It's just a listing of certain arbitrary cultural examples without much explanation. That's probably fine if you're totally new to music and want to get an idea what people are talking about when they say "perfect consonance", but that's about it.

If you really want to measure such things (which is a highly dubitable thing in itself), then at least base it on a clear foundation, rather than a random list. :P

Okay this is for the INITIAL post (not Gardener's) -

I leave you with questions -

What determines the hierarchy? Frequency, that is how often is the interval heard? Duration - how long is the interval held? a cluster pedal tone will sound consonant after awhile - eg a drone . Direction - in palestrinian counterpoint descending sixths are considered "unstylistic" -- reason being is the direction an interval moves can create subtle variation of dissonance and consonance. For example, what provides more consonance - a falling P5 or ascending? This later parameter is more dependent on context. So -

What is the context? This is more what type of instruments/voices and acoustical space. A series of P5, M3 will sound like mush if played very fast on an organ in a hall with a tonal decay of 7 second (like St John The Divine's space in NYC). It may not be the most extreme in dissonance but it would create the same greyness (eg "imperfect dissonance") if you had clusters and tritones played at a slower rate.

SSC in the shout box mentioned Schoenberg's alternate understanding of dissonance and consonance as sonance - wherein hierachy and context develop or is created to set up a set of "dissonances" and "consonances" . I agree with this view of "tonality" considering the past 1,000 years of Western art/liturgical music

In response to nirvana...typically a major 7th chord contains the inner members before the seventh occurs, masking the seventh.. so a root position closed Cmaj7 chord would be CEGB. Therefore youd have that major third on top. However, of course a maj 7 can sound fairly consonant, or at least not harsh in a passing tone type situation. Simple answer to the opening question is that dissonance is determined by its surroundings. Bernstein has written with the tritone ending up as a more consonant place and his music is rarely considered harsh. Also keep in mind register, the higher up these intervals occur, the more potential to sound consonant there is. The notes cde played on the bottom of the piano are much harsher and muddier than at the top of the piano...basic overtone theory..

Nirvana's and ben's points show another weakness. A M7 puts you at the same PC as a descending m2 - when you look at octave displacement, stuff gets nuts.

I understand the want to quantify, but without obscene amounts of studying, you end up with half-truths.

Why would you even need to measure dissonance?

Why would you even need to measure dissonance?

So one can have a quantifiable way to analyze pieces of music objectively? You can't tell me that that ability wouldn't be useful...

Achieving that? A-WHOLE-NOTHER STORY

By the way, I use this Dissonancometer:

The red plug goes into the lower note, the black one in the higher. The higher the number, the uglier the measured interval. Highly useful!

Dude, I heard about those.

They're crazy, I used one once in a masters-level course, but I just kept hitting the down button to make it less.

The only consonant interval is a unison in pure tones (and maybe the octave). Everything else contains dissonance.

Go from there and may god help you.

Heh, well, that got quite an interesting set of responses.

I don't really have the time right now to even attempt to sufficiently reply. (I'll do that later)

As far as instruments, I'm really asking in the context of piano music, so the 12 pitches available, and standard piano tuning.

I was doing some visual analysis of piano-roll views of music (which is functionally just a spectrum analysis of the primary tones in relation to time)... and by visual analysis, I mean, hearing something, and going "I wonder what this looks like"... and vice versa. And I was like... "huh, I wonder if I can come up with a way to visualize dissonance too".

I'm just toying around with this; I don't even want to pretend I'm dealing with absolutes. By coming up with a number, it at least gives me a way to visualize dissonance over time, which I can see being useful, even if the number isn't even remotely perfect. (It's just a starting point that I can further tweak, to fit what I consider correct.)

Art and science (although, it's more, music and math) aren't mutually exclusive... it's only when you start to limit one by the [limited] achievements of the others that I find problems.

(For instance, should you not play a sound because you can't notate it? I personally think notation is secondary to the actual music... Notation shouldn't be normative, nor should any sort of analysis.)

Math is just a tool... it's neither subjective nor objective in itself, and I'm using it in a very subjective manner. Quantifying something doesn't mean it's anywhere near an objective value... all I really care about is that it's fairly consistent.

A good example is loudness... you can say something is louder or softer than something else, both subjectively and objectively. However, there's not really any absolute loudness or softness, especially when you consider perceived loudness (which varies by pitch), versus actual sound intensity. But having a number... it's at least something to work with.

The really core question that I'd like answered is this: Which interval is most dissonant, a diminished 5th, a major 2nd, or a minor 2nd? And which is the least dissonant of those three?

And secondly, in which tonal/harmonic theory system is that considered the case?

*chuckle* I'm not writing your essay for you.

;)

*chuckle* I'm not writing your essay for you.

;)

What essay?

Now there's the problem with actually asking a serious question around here.

I'm a software developer, by profession; I used to be a somewhat serious pianist, and I've been improvising for years, and I'm just wanting to merge some things together. Why would I sit down and analyze all the chords in a piece of music when I can write a piece of software to do it?

All I'm really asking for is to be pointed in a direction, any direction. Someone else has a different theory on dissonance and consonance? Who? Where do I look?

What I'm working off of is what I saw in that wikibooks link, and also from what I've been able to gather from the book "Serial Composition" published by Oxford.

So far, the results of my software seem to be consistent with the examples provided in the latter, but I'm wanting something a little more "real", than me just punching some numbers in to make it look right.

Sorry, I just took a bit of offense to being considered a student writing an essay.

The really core question that I'd like answered is this: Which interval is most dissonant, a diminished 5th, a major 2nd, or a minor 2nd? And which is the least dissonant of those three?

And secondly, in which tonal/harmonic theory system is that considered the case?

A m2 is the most dissonant, going by normal tonality, I think, then b5 then M2. However, I find M2s really dissonant, for whatever reason.

Scoping it down to just piano is good - is it just struck keys you're looking for or all possible sounds with a piano?

(hence the "wink" smiley)

Yeah, I'm going to stick to struck keys.

At least for the moment, I want to be able to take a midi score and analyze that directly. (Notes, and pedal markings.)

Granted, there's going to be all sorts of details to get into, like measuring dissonances created by the actual notes that are sounding (held notes too); and taking into account the decay of those sounds, as well as things like "perceived" dissonance, such as a sudden jarring modulation, even if it modulates from something very consonant, to something else very consonant.

This is just a starting point. It's nothing more than an experiment right now.

I don't really want to be the one coming up with the theories, I just want to subjectively test other people's theories and my own hypotheses... see what I like, what I don't like, and how those line up with what I personally perceive when I hear things.

How are you doing this? I'm sure Max/MSP or pD could help a lot! ::plugs mah boyz::

What'd I'd honestly do is set up a program that takes the MIDI and analyze one singular score from the Baroque era. The reason for Baroque is that it is the most clear setup for tonality, and the dissonances are clear and easy to find. Starting there, have it analyze the score for all the MIDI parameters and then compare.

Sorry... lost my brilliant train of thought into the huge ravine of puppies.

A m2 is the most dissonant, going by normal tonality, I think, then b5 then M2. However, I find M2s really dissonant, for whatever reason.

Scoping it down to just piano is good - is it just struck keys you're looking for or all possible sounds with a piano?

I'd second this. In common practice tonality anyway. Another one I consider dissonant is the parallel fourth. Not because it's painful to the ears but it's the aural equivalent of hitting a brick wall in my opinion. It's almost unsettling in a way. I'd consider it more dissonant than a major second.

Yeah, I've heard the parallel 4th as dissonant, but I'd wouldn't agree. Parallel 4ths to me is exactly like parallel 5ths, just inverted.

Jonathan,

Don't have time to read all the responses (usually I'm in 100% agreement with Gardener :))

Now a few extra thoughts:

1. I'm pretty sure that CDb is definately more consonant than CC Half # (quarter tones). The most dissonant thing I've heard is some pitifull people trying out in X Factor. Because they are so bloody dissonant in themselves! Because they defy the tonal 12 tones, in a tonal song and go towards quartertones or worst, without being able to control it! :D

So define if you're going for the set 12 pitches, or more than that. If this is towards academia you'll find plenty of dissagreement towards that point exactly! Just some warning.

2. The other VERY important thing, which I'm NOT sure if it's been brought up yet, is the environement.

You could, of course, make a list with the 12 intervals. After all it's not THAT hard; It's only 12!!!! (hehe). Problem is that half of them depend on the environement. A perfect 4th is neither consonant or dissonant. Which, by your idea makes the perfet 5th the same. The interval in thirds (maj and min) are quite lovely all the time, but if they are following a devilishly difficult melody, look at what you get: BLIAH (of course I love such things, but anyways).

3. Why do you want to quantify dissonance? And why does it need to be 'universally' accepted? Why not instead make ANY kind of list by yourself and then use the list to quantify BOTH dissonance and consonanse in a piece of yours, thus giving excuse to what you do. If you present a work that has 5/8 consonance and 3/8 dissonance then you've got a solid idea for a form, no matter if it was wiki (thus non academic) based, or some book by Cook, or anyone else who have done tons of experiments and wrote plenty of essays.

Some quick ideas :)

1. I'm pretty sure that CDb is definately more consonant than CC Half # (quarter tones). The most dissonant thing I've heard is some pitifull people trying out in X Factor. Because they are so bloody dissonant in themselves! Because they defy the tonal 12 tones, in a tonal song and go towards quartertones or worst, without being able to control it! :D

See, I've rarely seen quarter-tone shifts that couldn't have been explained as articulation. The quarter-tone stuff I've seen is used in a way to "correct" problematic consonances and dissonances for the scale - at any rate, since he's talking about a piano, I'd assume some flavor of 12-tone ET, since there needs to be an artificial limit in order to maintain a sense of scope.

What you're talking about is an intrusion of improper quarter tones into a largely semitonal structure - those quarter tones were abberations in the same way that a tritone would be in a largely consonant piece.

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