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 correct 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:

(Source: William A. Sethares)

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 applied to inharmonic gong 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.

Also, for more than 2 tones, the Sethares algorithm ranks minor and major chords as equally consonant, which is not the usual interpretation. So Erlich and Monzo interpret Sethares' number as only a measure of "roughness" and require "dissonance" to include both "roughness" and "tonalness", where major chords are more consonant because they are closer to the root of a harmonic series (4:5:6) while minor chords are farther away (10:12:15). I don't know of a way to quantify that for arbitrary frequencies, though.

Answer 2

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. E.g.:

    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 3

Yes, there is a way. There's some brilliant research by Norman D Cook looking specifically the acoustical properties of triads. What he does is sum various partials of any three tones, and maps them onto 3-dimensional space. To make triads fit on 2-D space he calculates the interval difference between the first and second note, and puts that on one axis, the difference between the second note and the third on the y-axis. He then performs the mathematical calculations for various properties such as consonance, tension, modality and instability. He places this on what he calls the triadic grid, where M stands for major triad, m for minor, s for sus, and a for augmented. This theory models how we perceive those chords in the order in which we know them to be consonant, but that we so far have failed to explain mathematically. Here is one of Cook's papers on the topic.

The paper is reasonably dense, if you're not into mathematics and music theory I'd suggest watching this video of him discussing his theory. It's revolutionary, I think, because this answer has, I think, so far eluded everybody. It seems like it should be based on how ratios act purely mathematically, but it's actually more complex than that, hence why he calls it a psychophysical model of harmony perception. https://www.youtube.com/watch?v=CrmnaiyS5EE[![N. D. Cooks video on his musical model of dissonance for triads]2]2

Answer 4

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 5

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 6

just a couple of additional comments:

1. To compute a measure of dissonance one should take into account harmonics, i.e. compute all the pairwise contributions to the measureand sum them up (not too hard to do).
2. For chords of more than two pitches you just sum up all the pairwise contributions to the measure, fundamentals and harmonics.
3. Dissonance decreases with distance: octave equivalence does not really work in this respect. A minor second is more dissonant than a minor nine that is in turn more dissonant than a minor 17.

Answer 7

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 8

The problem with Tenney’s Harmonic Distance formula and other formulas is that for higher primes and composites in ratios that sound close to simple ratios they’ll be measured as having a larger HD for instance 3001/2000 sounds incredibly close to a perfect fifth but has a very large HD even though it would be consonant. Here’s a paper on harmonicity where the formula works for many situations but probably has the same problem: https://www.mat.ucsb.edu/Publications/Quantification_of_Harmony_and_Metre.pdf

Answer 9

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 10

I would answer this from a pure music theory perspective, rather than from a signal processing or EE or audio engineering perspective. The problem when you go down that road (IMO) is that you get into all sorts of signals that can be observed on a oscilloscope, or run thru an FFT, that are not musical at all, it might be the sound of the metal to metal screeching of a rail car on the tracks. In musical terms all kinds of semi-tones that are not found on the diatonic or chromatic scale.

From a musical theory perspective, dissonance is measured by the amount of ALTERED notes in the chord. Altered notes are simply the b5, #5, b9, and #9. In C major these are Gb, G#, Db, and D# respectively. Notice these are all a 1/2 step above and below the 5 and the tonic (root) of the chord. The root and the 5 are the most consonant sounding notes, so by playing notes just 1/2 step away we get the dissonant sound.

Go to a piano or guitar, and play a chord using only these notes - G, C#, G#, C - the 5, b9, #5, and root, in ascending pitch. This is about as dissonant as you'll want to hear.

Answer 11

Maybe relevant after nine years :-) How to sort for pitch similarity?

The method described above rests on the assumption that two pitches with ratio $a/b$ sound consonant, if $a,b$ are small numbers.

I used the function $k(a,b) = \frac{\gcd(a,b)^2}{ab}$ to measure the simplicity of the ratio, and hence the assumed consonance. If the function takes a value near one, then the two pitches sound consonant (again you have to subjectively weight for volume and duration) otherwise, if it takes a value near zero they sound dissonant.

Since you want to use the computer, this link to a notes dataset in vector format could also be helpful:

https://www.kaggle.com/musescore1983/measuring-note-consonance-with-psd-kernel

Answer 12

Yes there is a new method I have developed:

It is based on a Dissonance Index based on the Modulator-Carrier equation, and the number of peaks and amplitude variations of the chord resulting sound wave (including the Modulation effect)

An increase in the number of amplitude peaks and their variations produces more stress and an unpleasant sensation for the ears. The method includes some considerations on the Temporal Resolution and the Temporal Modulation Function.

Based on this new concept, a new music scale has been constructed with all the best consonant two-notes chords extracted from Brocot sequences, and it can be extended to any number of notes, so a table with all the most consonant chords (up to 6 notes) for that scale is included. It mus be remarked that it can be extended to any number of notes.

This new scale is not based on the ancient Octave myth, but just on scientific foundations.

Very brief explanation: It is necessary to look for another way to define and measure consonance, and from above consonance, which must be the fundamental principle of music, shall never prevail neither the geometric ratio as the generating principle, nor the ancient 'Octave myth' as the ruling principle for dividing the whole realm sounds and stablishing laws about their equality, nor the concept of flats and sharps as a result of the incongruences in the construction of the scales, nor the equality of the intervals between the ratios to fulfill the requirements of musical instruments.

We all have listened to many compositions that we find very pleasant, but we have also heard classic pieces plenty of dissonant and unpleasant chords, and not precisely because they have been deliberately constructed or executed to represent some emotional state in the composition, but simply because they are very dissonant chords that inhabit those scales and other musicians prefer not to use in their musical pieces.

This work aims to find a scientific way to create an index for measuring dissonance. In this way, we will work initially with the geometry of two-sounds input, later, we will comment on how to extend this analysis to any number of sounds.

The following image is very important for several reasons,

here you can see various wave types, corresponding to the ratios 7/1, 3/1, I mean from top to bottom, 7/1, 3/1, 5/2, 2/1 which is the octave, 4/3, and 9/8.

Which one do you think has the most uniform appearance? The ratio 3/1 looks more uniform because its amplitude peaks are all the same, there is no deformation or wave modulation, and the peak distribution is totally uniform.

If we go from 9/8 upwards, we can see that 9/8 is a modulated wave, and that happens when the frequencies are very close to each other. The modulation effect decreases as we go up toward the octave, and at that point the modulation has not yet disappeared, so this means that the ratio 2/1 --the Octave chord-- belongs to the modulation zone, 2/1 is certainly another modulated wave.

Continuing upwards, the modulation tends to disappear until we reach the ratio 3/1, where there is no longer any trace of it, the wave is totally uniform.

From 3/1 upwards, the wave will always have that sinusoidal shape, it seems to be delimited or confined by two imaginary envelopes, as can be seen in the image above, two parallel sinusoidal envelopes that bring kind of a constant thickness to the wave.

On the other hand, in the zone above the ratio 3/1, there is not the same variety of waveforms as in the lower zone. In the case of the ratio 3/1, those envelopes seem to become into two parallel straight lines.

As we go back from 3/1, down toward 1/1, those imaginary sinusoidal envelopes appear to shift relative to each other, and the phase shift continues until it produces the typical packet shapes (modulation) and those envelopes are no longer parallel but symmetrical to each other as in the case of 9/8.

Two conclusions can be drawn from the above image:

Firstly, there are two definite zones that divide the chords domain, the upper zone above the ratio 3/1, and the lower zone below 3/1 which we might call the modulation zone.

Thus, the ratio 3/1 is a true milestone for the definition of chord consonance, even more important than the octave.

The second conclusion is that there are two mixed types of consonance within the interval [1,3] the first one ruled by the number of peaks and their amplitudes, and another dissonance introduced by the wave packing, that is, the modulation.

Considering all that, the immediate action should be to construct, a Dissonance Index based on the value of the chord frequencies, in this way, a programming code was developed to generating irreducible rational numbers according to the well-known Brocot sequences, which are used as frequency ratios in the Carrier-Modulator equation, to finally get the required parameters for the dissonance index.

By using such a Dissonance Index we will choose the most consonant ratios, to create the scale called 'Tríplice', which is based exclusively on the most consonant two-notes chords, a very different principle from that used in traditional scales.

Of course, some values from this new scale will match with some ratios from custom musical scales because there are obvious consonances, however, others do not.

Follows the equation of the sum of two sinusoidal waves

expressed as the product of a Modulating function cosine and another sine function called the Carrier, whose frequency is the average of the frequencies of the two-sound chord

(The resulting wave is represented by the thickest curve)

Next to the resulting wave, the Modulator and the Carrier waves are included: Cosine and sine.

The most important element here is the ratio of one half of the Modulator period to the whole Carrier period.

In the case of the ratio 3/1, one half modulator period embraces just one carrier period, and this is important because it explains why the ratio 3/1 is kind of a boundary limit for chord types, as the ratio descends from 3/1 towards 1/1, one half modulator period will span many carrier periods. On the other hand, in the superior zone that corresponds to all the ratios greater than 3/1, the opposite occurs, the carrier period is always greater than one half modulator period.

Now the question is: How does the modulator act on the carrier to generate the resultant wave? For the ratio 3/1, and within the fundamental period of the resulting wave, the Modulator and the Carrier intersect at the same time on the horizontal axis, on both sides of that point the modulator changes its sign and since it is the product of the cosine by the sine, then the whole carrier peak also changes its sign.

Besides, the Modulator also scales the Carrier peak amplitude however, without modifying its shape, (see notes) and finally as a result such an uniform waveform.

In the case of the ratio 2/1, the carrier does not coincide with the modulator when it intersects the horizontal axis, and at that point, because of its sign change the modulator modifies the shape of the carrier wave, and a new additional peak is created in the resultant wave. Therefore, the modulator does not only scale the Carrier peak but modifies the carrier waveform (see notes).

As I already said for ratios closer to 1/1, one half modulator period spans many carrier periods, and so it just gradually packs their amplitude peaks according to the cosine function, that is, as in the case of 9/8 that we saw earlier.

At this point it is important to mention the issue about the ears accuracy to decoding and processing the input, I mean the ears Temporal Resolution and the Temporal Modulation Function, among other topics, by which specialists in this matter agree that for low frequencies it becomes easier for the ear to perceive the difference in modulation amplitude, while for high frequencies this type of perception decreases, for that reason, in the case of low-frequency chords the modulation dissonance can be perceived to some extent, and it must be noted that this occurs even in the case of the octave, but not for ratio 3/1.

Thus, the expressions "Pure" and "Perfect", among many others commonly used in music should not be used to bring any relevance to any ratios below 3/1. For ratios 4/3 and 5/4 you don't need very low frequencies to perceive the modulation dissonance.

All this is fully detailed in the book and for the case of consonance comparison between different chords a table with the corresponding parameters is included, In the images shown on the screen note that all the carrier peaks match the resultant wave peaks.

The ratio between one-half Modulator period and the whole Carrier period, which is shown as the ratio 'r' in the above image, facilitates to know how the waveform is generated, it also allows to calculate the number of peaks and their amplitudes.

It confirms that the ratio 3/1 is a milestone which represents a very special point for the ratio ‘r’, moreover, it brings to light the boundary between two zones, two chord categories or waveform types. It is interesting to look at the carrier peaks distribution within the intervals [2,3] and [1,2].

The following image shows the y axis, that represents the number of carrier peaks, and the x-axis representing the frequency ratios.

The name: Harmonies Garden is just an abstraction, due to the similarity of that image to a forest or a garden. That image shows many curious properties. mainly when zooming and detailing at the bottom of any of the bushes that appear in there, of course, if I am allowed to call them that way.

A programming code was developed to generate the Brocot sequences, and by processing the parameters of the resulting wave as well as by using the Carrier-Modulator equation, a geometric dissonance index is calculated.

This geometric dissonance index is denoted using the acronym Idm and relies on various factors.

As you can see the Idm appears in the last column of the table of which just a small section is shown here. Besides, there are two Idm-distribution graphs for the intervals [1,2] and [2,3] which for the sake of better visibility appear separated on the screen.

Just notice, that they should be joined together since we are dealing with the whole scale [1,3].

Those two graphs allow us to choose the ratios with the lowest dissonance indexes, I mean, those ratios shown next to the lowest vertices in both graphs.

The y-axis indicates the dissonance index, and the x-axis represents the corresponding frequency ratios.

Those are the chosen ratios for the new scale called: The Tríplice, of course, everyone is free to add ratios from other vertices or eliminate others according to any criteria.

The point is to stablish a limit for the dissonance index. It is also included the Modulator-Carrier equation for more than two sinusoidal waves, so the dissonance index can be computed for any number of sounds.

In that case, the expression has basically the same structure, it has several modulating cosine functions that multiply to a carrier sine function, and by using that equation, a similar procedure can be easily followed to calculate the dissonance index.

Another way to determine the dissonance is by grouping the sounds in two-note sets, and then calculating the sum or average for all those sets.

It should also be noted that in order to make comparisons between harmonics it is necessary to do spectral analysis and that might represent any inconvenience for some people. Thus, It is commonly argued that in order to determine consonance, the waves whose harmonics do not have matching nodes are dissonant, however, the problem is that when comparing chords, if all the nodes of the maximum-amplitude harmonics do not match, then it becomes very difficult to determine which is more dissonant and to what extent. With the modulation dissonance index, there is not such problem.

Note: Applications for Structural Analysis (Structural Engineering)

Finally, although the consonance subject is always oriented towards arts, it might be also of some importance for engineering sciences. For a structural or a mechanical engineer, the vibration modes of a structure, which are usually called 'harmonics' in music, are the backbone of structural analysis, then, if in addition to taking into account the resonance phenomena, as well as the participation of the masses in the forces applied to the structure, due to those harmonics, as commonly done in structural analysis, then it would also include in the analysis what I might call the consonance of masses or the music of the structures, then the dissonance index Idm could be used to infer in advance the displacement demand on the structure.

Remind that a consonant chord has a small number of amplitude peaks, less wave geometric complexity, low demand for the ears, and minimum energy dissipation demand.

You can find full information on this method at: https://numbermusicrevolution.com/musicrevolution/

as well at: