Is there a way to measure the consonance or dissonance of a chord?

Question

I know that if I played C and B together they would be very dissonant compared to if I played a G or C one octave up. Is there a quantitative way to describe that sort dissonance?

Edit: I understand that smaller ratios between the frequencies of the notes means they are more consonant. My question is how to express that. Notes with the ratio of 2:1 are very consonant, and 15:16 are dissonant. Saying that smaller ratios are more consonant is very qualitative, I'm looking for a quantitative way to represent consonance.

Basically, what I'm trying to do is figure out a way to describe to a computer how consonant or dissonant two or more notes are. I want to be able to put in 1:2 or 15:16 and have a programmatic way to determine which one is more dissonant.

Answer 1

Yes, there are ways to measure it, though there are many different algorithms claiming to be more accurate than the others. This formula by Vassilakis is recent (2007).

These measure "roughness", which is similar to dissonance. (Dissonance is basically roughness, but weighted towards certain intervals due to cultural conditioning, which is obviously hard to measure quantitatively.) For two sine tones, roughness vs frequency difference looks like this:

For more complex signals, made up of multiple tones:

The roughness of signals corresponding to spectra with more than two sine components is calculated by summing the roughness of all sine-pairs in the spectrum.

For tones with harmonic spectra, the net effect of the roughness between all the harmonics present produces graphs with notches of consonance at intervals that we're familiar with, like 3:2 perfect fifth:

The black curve is from the older Plomp-Levelt 1965 paper, with this description:

We assume that the total dissonance of such an interval is equal to the sum of the dissonances of each pair of adjacent partials ... these presuppositions are rather speculative ... In this way, the curves ... were computed for complex tones consisting of 6 harmonics. ... shows how the consonance of some intervals, given by simple frequency ratios, depends on frequency.

(So the Plomp-Levelt curve is based on summing up the roughness of adjacent partials while Vassilakis sums "all sine-pairs". (Sethares wrote me and says the "adjacent" thing is just because computational power was limited in the 60s, comparing every pair is more appropriate.))

Further descriptions of this curve can be found in Marc Leman - Foundations of musicology as content processing science (which also talks about deriving the slendro and pelog scales from the same algorithm but with inharmonic instruments) and Plomp and Levelt's Hidden Ratio

The blue curve is from Sethares Relating Tuning and Timbre, which uses this MATLAB calculation, also based on the Plomp-Levelt curves. (And here's my Python translation.) Here's a MATLAB-based app that uses the 2007 Vassilakis model to also calculate the same curve for 6 harmonics (and has the M3 as more consonant than the m3).

You can see the two curves disagree on whether the m3 or M3 is more consonant. I'm not sure if this is due to calculating only adjacent partials vs all partials or if the partials have different amplitudes or what. Of course, real instruments produce lots of variation in their harmonic spectra, even playing the same note on the same instrument, so these curves are all inherently approximations. Here's a plot I made of violin vs clarinet, showing that the M3 is more consonant when the violin is playing the higher note, due to clarinets producing mostly odd harmonics.

Answer 2

There's a short answer and a longer, more complicated answer; I'll just give the short answer here along with the barest basics of the long answer.

The short answer is: Yes, there is, sorta. If you take the ratio of the frequencies of the two pitches, you'll get some fraction in lowest terms. The smaller the numbers in that fraction, the more consonant the interval. For example, two pitches in unison have a 1:1 ratio. An octave has a 2:1 ratio. A perfect fifth (such as C to G) has a 3:2 ratio, etc. Matthew does a good job in his answer explaining why ratios with smaller numbers sound more consonant than ratios with larger numbers.

But this is all made more complicated by temperament, which is the way in which pitches are tuned relative to each other. See, suppose you tune your A to 440 Hz and then start tuning the other notes relative to that A, using the whole-number ratios as a guide. You'll tune E at 660 Hz, for example. For the first few notes, everything will sound great, but it won't be too long before you start to hear some strange intervals. Some intervals have nice, whole-number ratios, but then others that you'd think should sound good, like the major third from Eb to G, sound really bad. To make a long story short, it turns out to be impossible to tune all twelve chromatic notes using whole-number ratios of frequencies and have everything come out right. Mathematically, it just can't be done.

So you have to make some compromises somewhere. There are many, many different ways to make such a compromise, and I won't detail them here. But for the last two hundred fifty years or so, we've settled on a tuning system known as Equal Temperament. In this system, you start with a reference pitch (e.g. A440), and then the frequency of every other note is 2^n/12, where n is the number of half-steps above the reference pitch.

In this system, none of the intervals will have whole-number ratios. But all intervals are consistent (some would say consistently imperfect), and so it allows you to play in any key. It's an effective compromise, but you lose the purity of true whole-number ratio intervals. And so the short answer I gave above turns out to be only sort-of correct, because the consonant intervals will have ratios that are almost, but not in fact actually, nice small whole-number ratios.

Answer 3

Yes. It has to do with the ratio of their frequencies. Essentially, the smaller the numbers involved the better.

The perfect unison, with a 1:1 ratio (e.g., C played with the same C), has perfect consonance. C to the next G has a 2:3 ratio; the perfect fifth is the next most consonant. The minor second (e.g., C to C#) is the most dissonant in Western scales with a frequency ratio of 15:16.

What this represents is how often the sound waves "match up". Every third cycle of a C matches with every second cycle of a G, and vice versa; i.e., the peaks of the waves occur at the same time every two cycles (or three cycles, depending on which note you choose as the base). This is often! So overall, your ear perceives the sounds as being in sync and melodious. In contrast, waves that match up infrequently, such as the minor second with only the 15th (16th) cycle matching, are largely out of sync and therefore dissonant.

The mind is strange, and what one perceives as dissonance is not necessarily what another would perceive as dissonance. That said, the closest you'll get to an absolute, objective measure is the base 2 logarithm of the Least Common Multiple of the the sides of the ratio. I.e., lg(LCM(15, 16)) = lg(240) ~= 7.9. This is about 3 times more than lg(LCM(2, 3)) = lg(6) ~= 2.6. Neatly, lg(LCM(1, 1)) = lg(1) = 0, so this also reflects the fact that the perfect unison has no dissonance. Interestingly, Euler seemed to think the LCM was the way to do this as well¹.

(Note that LCM(x, y) = x*y for fully reduced ratios; e.g., 2:3 rather than 4:6.)

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[1]: Knobloch, Eberhard (2008). Euler Transgressing Limits: The Infinite and Music Theory. Quaderns d’Història de l’Enginyeria, IX, 9-24. Available online: http://upcommons.upc.edu/revistes/bitstream/2099/8052/1/article2.pdf

Answer 4

Using the empirical formula A+B divided by AB where A and B represent the frequency ratio of the two notes of that interval seems to give an absolute measure of the magnitude of the degree of consonance as follows

Unison-frequency ratio 1:1 yields a value of 2

Octave-frequency ratio 2:1 yields a value of 1.5

Perfect 5th-frequency ratio 3:2 yields a value of 0.833

Perfect 4th-frequency ratio 4:3 yields a value of 0.583

Major 6th-frequency ratio 5:3 yields a value of 0.533

Major 3rd-frequency ratio 5:4 yields a value of 0.45

Minor 3rd-frequency ratio 5:6 yields a value of 0.366

Minor 6th-frequency ratio 5:8 yields a value of 0.325

Major 2nd-frequency ratio 8:9 yields a value of 0.236

Major 7th-frequency ratio 8:15 yields a value of 0.192

Minor 7th-frequency ratio 9:16 yields a value of 0.174

Minor 2nd-frequency ratio 15:16 yields a value of 0.129

Although the formula used is empirical , the results adhere remarkably closely to the accepted order of degree of consonance of the harmonic intervals within an octave

Answer 5

You wrote "Chord" but spoke only about two notes combinations, and the answer is already marked. Also don't forget that we are speaking about music here. What is measured is not the physical tone but the perception of the listener.

I wanted to answer nevertheless that for 'real' chords with three or more members the simple math does not apply anymore. The psychological component becomes more important. An augmented chord (c e g#) is (imperfect) consonance for all combinantions, yet it is perceived as one of the most dissonant chords.

Also for two note combinations it matters in which octave you play them. Dissonant intervals are less dissonant if you play them one octave apart while consonant intervals are less consonant when played apart, both psychological effects.

There are other parameters other than the pitch that may take over. Even with just tuning intervals on the lower end of our hearing spectrum do not sound nearly as "good" as in the middle range. Our ear is not linear, it has it preferred bands in the audio spectrum, where the spoken language is.

Answer 6

Take all the theory with a grain of salt. You may love hot peppers or hate hot peppers.

You may think the major 7th chord (for instance C E G B) that ends many jazz compositions and Darius Milhaud's "La Création du Monde" is the most beautiful consonance imaginable, far more interesting than a plain triad. Or you may think it's the most horrible dissonance.

Consonance and dissonance can be defined objectively, as you see in the other answers, but most people think of them as subjective terms. As such, they depend on the ears, taste and history of the listener.

Answer 7

All of the answers above are wrong, unfortunately.

There is common confusion, when talking of consonance and dissonance, between cultural aesthetics vs objective harmony.

Maximal consonance is equivalence, at all factors of two of a given fundamental within discriminable range - which is about 8 octaves for most people (although our range is usually quoted as 10 octaves from ~20Hz to ~20kHz, the upper two are actually pitch ambiguous, because they're only sensed by lower-tuned hair cells, which are only specialised up to about 5kHz).

The 1:1 unison cannot strictly be regarded in terms of harmonic consonance and dissonance, since it is an interval of zero - it only has amplitude relationships, not a wavelength relationship; perhaps cancelling, or adding 2db, at the other extreme, but it does not belong in the realm of harmonic properties, since it has none. It's just the same frequency, overlaid, perhaps with up to an 180° phase offset. But that's nothing to do with "harmonic" interplay, which by definition denotes a property between two or more distinct tones that is purely emergent, and not a property of the individual constituent stimuli. To wit, "tonal harmony" concerns the properties of non-zero intervals!

The first interval being composed of two discrete frequencies in consonance is the octave, at 2:1, followed by all further factors of two of that fundamental. Thus, what we've been calling "consonance" is actually instead this paradoxical property of equivalence.

Dissonance is just its absence.

All further consonances and dissonances are but degrees of inequivalence.

The most consonant intervals are those resolving to the smallest temporal integration windows - those tones whose differing wavelength cycles repeat together most regularly. Hence, 2:1 resolves every alternate cycle of the upper tone, 3:2 every third, 4:3 every fourth, and so forth.

Dissonance is not caused by 'beating' - pure tone octaves still sound equivalent, and thus remain maximally consonant, and while pure tone fifths beat noticeably, 3:2 is still the second most consonant (or rather, 'least different') interval after the octave.

The octave is primal because it is the simplest frequency ratio possible - all other intervals are by definition more complex, and thus resolve less frequently.

This matter has nothing whatsoever to do with taste, culture or any subjective factor - it's a cross-species, universally consist property of mind. There is no harmonic consonance and dissonance - only sameness and difference - or, more accurately, degrees of difference, against an objective baseline of "zero difference", AKA equivalence.

Cultural aesthetic, style and taste are certainly subjective, but shouldn't be described in terms of consonance and dissonance if these words are to have rigorous definitions for objective tonal properties.

And even then, strictly speaking, consonance and dissonance shouldn't be so named themselves - there's only equivalence, and its magnitude, which peaks at factor-of-two synchrony. Tritones are just not very equivalent.

Unlike consonance, dissonance has no maximum value, because it's infinitely variable - a signal may be infinitely complex (regardless of its actual information content of course). But it may only be finitely simple, and this simplicity converges to 2:1 equivalence.

This is all perfectly empirical and verifiable by anyone. It's so simple, there is no mystery. None of the rote learners are thinking in clear terms on this issue. This sticks in my craw as this is obviously important stuff - 'equivalence and difference' whatever it actually is, is telling us something significant about how we assign value to information resolved via frequency analysis processes.. in other words it's also the stuff of language and higher thought, besides everything else... an objective "zero point" of "no difference" at all factors of two throughout our psyche.

Specifically, we hear discrete tones - higher and lower, yet the relationship between them - their harmonic sonority - registers as "no difference".

"Equal". "Same". "No difference". It is a value of metadata. A value we assign. It is internal, not external (the tones themselves, the stimuli, are by definition different pitches) - octaves touch a nerve for us. An informational zero point for meta-data, against which we're evaluating and encoding all the auditory information we process.

That, my friends, is what consonance is. It correlates to thermogeometric equillibria in a randomly self-organising network; an informational ground state, of minimal entropy.

Cultural aesthetic is "anything goes" - with the single reservation that all tonal systems must be derived from subdividing the octave, even without the progenitor being aware of the fact. Octaves can only sound equivalent, to everyone. Everything else is simply a matter of taste and refinement, but shouldn't be conflated with harmonic consonance and dissonance (such that they are).