The Dissonance Curve and Applet 

2 tones of moderate loudness, sounded simultaneously, are said to be dissonant if the result is unpleasant to listen to, and consonant if they blend well together, and are pleasing to listen to. The explanation of what causes dissonance (specifically sensory dissonance) is the beats produced by a harmonic of the first tone, when it is near, but does not coincide in frequency with a harmonic of the second. However, if the beats are very slow or extremely fast, the effect is not so unpleasant. There is an interesting discussion of this and more in [Pierce 1983 p.76], including the human ear, and the unpleasantness of dissonance resulting from 2 different sounds transmitting along the same nerve fibres. There is also the confounding effect of musically trained individuals being sensitive to what intervals they expect to be consonant or dissonant, based on past experience. This is omitted from the mathematical model below.

Let us now consider tones with no harmonics higher than the first, i.e. pure sine waves. Fix tone 1 at frequency f, and denote the frequency of the other tone by g. As g varies upward from f, there is a region of roughness or critical bandwidth where there is dissonance; this dissonance dies down as g continues to increase. The length of the critical bandwidth depends on the base frequency f, and Pierce gives the rule of thumb that it extends to about the minor 3rd, that is 6/5 f. The strength of the dissonance sensation is a smooth bump function, reaching a single maximum at about 1/4 of the length of the critical bandwidth.

What we now have is a weight function for the amount of dissonance between 2 pure tones. [Sethares 1997] gives an excellent discussion dissonance curves, the experiments of Plomp in the 1960s, and provides optimal curve fitted parameters from the data of Plomp and Levelt, which we shall use. [Benson 2002] also provides an informative discussion.

We can now give a mathematical expression for the dissonance weight function as:

d( f, g, Af, Ag ) = Af Ag [e -0.84Q - e -1.38Q ]
where Q =
( g - f)
0.021 f + 19

where Af is the amplitude of f, Ag is the amplitude of g, and g >= f.

Then for standard musical tones with harmonics, we can compute the total dissonance by summing the dissonance of each pair of harmonics. We represent the significant harmonics of tone 1 by the array
F = ( f1, f2, f3, f4, f5, f6 ) where fi := i * f1
and tone 2 by G = ( g1, g2, g3, g4, g5, g6 ) , with corresponding arrays for the amplitudes of each individual harmonic.

We then loop over each element of array F, and for each element of F we loop over array G, summing up the result of plugging fi and gj into d( f, g, Af, Ag), with the warning that f is always the minimum term of {fi, gj}, and g is always the maximum. The following applet draws the curve as g1 varies over the octave of f1.

Dissonance Curve Applet

The relative intensity of the first 6 harmonics of a piano wire, struck at 1/7 its length, are taken from [Helmholtz 1877 p.79], and they have the approximate ratios:
Plucked:       (1.0   0.8   0.6   0.3   0.1   0.03)

Soft hammer: (1.0 1.9 1.1 0.2 0.0 0.05)
Medium hammer: (1.0 2.9 3.6 2.6 1.1 0.2 )
Hard hammer: (1.0 3.2 5.0 5.0 3.2 1.0 )

view the source code
If you save it to disk, change the .txt to .java before you try to javac it.

Discussion of Results for scales 

The above applet's diagram tells us how well the frequencies blend with the base frequency, which we always call C. The reader should systematically experiment with increasing base frequencies and harder hammers ( the harder the hammer, the more prominence of the upper harmonics). A number of interesting phenomena emerge, and we shall discuss some of them, while wondering which are artifacts of Plomp's model, or Sethares' parameters, or the author's own implementation.

Some general observations (by no means all):

  1. The greatest consonances occur at both ends of the scale, and at G = 3/2 f in the middle. On either side of these are maxima of dissonance.
  2. At low frequencies, dissonance reigns with only a few shallow peaks. At high frequencies ( say 2112 Hz ) the curve is in general much lower. Going too high is probably physically unrealistic.
  3. Soft hammers damp harmonics and bring out few minima; harder hammers bring out many more, especially at higher frequencies.
  4. Minima at F, G, and A persist across the various frequencies and hammers, and they are the deepest. Others come and go.
  5. It is probably not meaningful to be concerned about what is the exact lowest point in a wide valley, or to claim a minima location to match a fraction if the distance is too great (say 5 cents?). This may explain away some theoretically unattractive minima like 9/7 and 12/7 for 264Hz, soft hammer, but experimentation reveals many other anomalies!
  6. E = 5/4 and E-flat = 6/5 appear sometimes, but are fairly high in dissonance. D = 9/8 rarely appears, except at high frequencies, and B = 15/8 never does.

Implications for scales 

Let us take as our baseline middle C = 264Hz and the Medium hammer. Any scale we derive in the primary octave would just be transposed to upper and lower octaves. Recall that the Just Intonation C scale has frequency ratios

C major: (C-D-E-F-G-A-B) = [ 1 9/8 5/4 4/3 3/2 5/3 15/8 ]

C minor: (C-D-Eb-F-G-Ab-Bb) = [ 1 9/8 6/5 4/3 3/2 8/5 9/5 ]

We would like to show that these choices are optimal in some fundamental sense. This will lead us into functional harmony, which is not well understood by the author, but we'll go as far as we can. We defined the C major scale from the notes of the triad chords C, F, G based on harmonics. Here we ask what that means in terms of the amount of dissonance.

The C major triad is certainly very consonant with C, therefore it is in the scale. The F chord is then a great choice because its gets both the other consonance heavyweights F and A. Why not use A major if A is so good? The problem is the almost maximum dissonance around C# ( and no leading tone property to C).

How do we justify the G chord being part of the scale? G is excellent with C alone, but B and D are terrible, they do not even appear as local minima; 3 octaves up at 2112Hz, we get a deep but wide minima near D and a bit higher than B-flat ( but never B!). Perhaps this very weakness can be construed as a strength, though. The notes of the G chord serve two functions:

Therefore, we have found that the C major scale to contain the notes with the following properties:
  1. All the tones with maximal consonance against C are included.
  2. The notes form 3 distinct major triads.
  3. Intervals between notes, while different sizes, do not contain any gap larger than 204 cents.
  4. The notes can form at least one dissonant chord G7C = (G-B-D-F) which "resolves" to C major (or minor).
What we have not established is any reason why the interval size should matter, except for the common sense reasons that they shouldn't be extrememly small, so as to be readily distinguished. We have also not precisely defined the leading tone property, or explained the mechanics of how one chord resolves into another. Therefore we can't say why there shouldn't be other notes also included in the scale. We may vaguely conjecture that the leading tone property is somehow the "frustrated expectation" of the listener to hear the tonic, and instead hearing something close.

Idea: It would be interesting to see if one could construct a mathematical function to indicate what chord ( if any) a given chord should resolve to, perhaps a "dissonance metric" to give the shortest path to a satisfactorily consonant chord. Also, by plotting the dissonance value of chords, could one study classical music in a dynamical systems approach as trajectories winding around on dissonance surfaces, and interacting with minima, like a particle in a potential energy well?

We may reasonably postulate the following as another musical axiom, and justify it by the lack of general acceptance of 20th century atonal music:

Resolution Axiom: A critically important function of harmony in music is the "tension" of dissonant harmony resolving into the "relaxation" of consonance, usually the tonic chord.

For the minor scales, we again have the dissonance minima at C, F, G and A; we form the minor triads for C,F and G ( Am is already in C major ). With the Fm we lose the strong consonance of A with C and instead have Ab. Historically, people had trouble accepting a minor chord ending a composition, but it sounds perfectly acceptable today. This fact could play havoc with creating the above mentioned chord resolution function.

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