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

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:

*
*

where

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 = ( f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6} )* where

and tone 2 by

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 f_{i} and g_{j} into
* d( f, g, A_{f}, A_{g})*,
with the warning that

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

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.

- 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. - 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.
- Soft hammers damp harmonics and bring out few minima; harder hammers bring out many more, especially at higher frequencies.
- Minima at F, G, and A persist across the various frequencies and hammers, and they are the deepest. Others come and go.
- 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!
- 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.

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-E ^{b}-F-G-A^{b}-B^{b}) = [ 1 9/8 6/5 4/3 3/2 8/5 9/5 ] *

We would

The

How do we justify the

- G, B and D fill in large interval gaps in (C-E-F-A-C)
- B has the
*leading tone*property to C ( cf. [Helmholtz 1877 p.285 ])

- All the tones with maximal consonance against C are included.
- The notes form 3 distinct major triads.
- Intervals between notes, while different sizes, do not contain any gap larger than 204 cents.
- The notes can form at least one dissonant chord
**G7**_{C}= (G-B-D-F) which "resolves" to C major (or minor).

**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
A^{b}.
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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