Explaining Music Notes Consonance with Wave Functions

[JohnnyGui]

Hello all,

First of all, I am aware that dissonance and consonance between pitches also depend to an extent by culture and musical origin but there also seems to be some degree of objective perception among people that can be explained scientifically. Also, I'm very new to this so I could be understanding and doing this the wrong way.

I've been trying to understand why 2 musical pitches would sound consonant or dissonant when played simultaneously by looking at their wave functions. Thus, I'm trying to explain consonance or dissonance by using the wave functions that are created by the 2 different frequency pitches.

I've read so far that:
1. Consonance decreases more and more as the ratio between 2 pitches approaches 1, with the exception of a ratio of exactly 1, which is obviously a unison.

2. Psychoacoustic experiments show that consonance between 2 pitches decreases rapidly as the frequency ratio between the 2 pitches starts to increase from 1, it then reaches a nadir at about one semitone difference, and then reaches nearly 100% consonance again near a minor third.

I've tried to find an explanation in the wave functions that satisfies these 2 "laws". I've drawn 2 sin functions that represent 2 pitches, each with a different frequency (for example, sin x and sin 3x).

First, I've tried to see if the frequency of how often the two waves intersect each other (and thus amplifies each other) in a certan time determines the degree consonance of the 2 pitches. This interpretation however, means that a frequency ratio of 3 is more consonant than a frequency ratio of 2 (octave) since sin 3x intersects a sin x wave more often. However we know that an octave difference (e.g. f and 2f) sounds more consonant together than a 3 times higher frequency (f and 3f).

Then, I've thought that maybe the quantity of how much the 2 waves deviate from each other in a certain time duration also determines the consonance. I've tried to quantify this deviation by calculating the area difference between the 2 wave graph lines in a certain time duration, using integral functions. The more area deviation there is, the more dissonant 2 pitches might sound. This gave me an interesting result:

- When increasing the frequency ratio between 2 pitches a and b in steps of Equal Temperament (¹²√2ⁿ), the area deviation, and thus dissonance, increases until b is the fourth note from a (¹²√2)⁴) and it then decreases until it has an area deviation of 0 when b is an octave higher than a.

- When increasing the frequency ratio between 2 pitches a and b in steps of the Pythagorean Tuning, the area deviation, and thus dissonance increases until b is a semitone apart from a and then decreases again. The interesting thing here is that, since Pythagoras considered the ratio 3/2 as the most consonant frequency ratio after an octave (2/1), a frequency ratio of 3/2 between a and b gives me an area deviation of almost 0 (4 x 10^-13).

This area deviation explanation satisfies more or less the second law mentioned above about consonance. However, it has its flaws.
1. Since calculating area also differs with the range of x that you choose, so does the area deviation between 2 pitches. The area deviation thus changes depending on how wide you choose the x range that you want to calculate the area of. This means that consonance is also dependent on how long you're listening to 2 pitches (time duration). This clashes with the fact that the degree of consonance is actually time-independent, from what I've read.
2.The area deviation also differs when you play the second pitch at a later time and not simultaneously with the first pitch. The second wave then shifts along the x-axis giving another area deviation. This says that consonance also depends on the time difference when 2 notes are played.

This whole area devation thing might thus be a wrong interpretation to explain consonance with wave interference. So could also be my understanding of the mentioned laws. If so, are that any other explanations of wave functions that show why 2 notes might be consonant or dissonant?

 

[Stephen Tashi]

Read about the phenomena of "beats" between two waves of different frequencies. As I recall, sounds that most people consider dissonant have beats of relatively rapid frequency and sounds that are consonant have beats with a relatively lower frequency. Look at the shape of the waves involved.

 

[JohnnyGui]

Read about the phenomena of "beats" between two waves of different frequencies. As I recall, sounds that most people consider dissonant have beats of relatively rapid frequency and sounds that are consonant have beats with a relatively lower frequency. Look at the shape of the waves involved.

Thanks for your answer. I've read a bit about beats and it is characterized by an interaction of sound waves, whose combination gives rise to alternating periods of constructive and destructive interference which causes "jarring" or "roughness".

It has apparently something to do with the fact that, if a range of frequencies is small enough, it vibrates the same particular part of the basilar membrane in the cochlea ("critical band") .

However, an interesting article shows that apparently, perception of consonance and dissonance is independent from the forming of beats. One of several arguments is that a study showed that participants with congenital amusia (a deficit in melody processing) showed smaller differencces between pleasure ratings of consonant and dissonant chords compared to the control group (participants without amusia). However, the ratings between the 2 groups did not differ in roughness ratings. So people with amusia exhibit abnormal consonance perception but normal roughness perception.

There are several arguments which are discussed that support that roughness is not really tied to consonance. It's quite an interesting read. The article is called: A biological rationale for musical consonance by D. L. Bowling and D. Purves.