The Complication with Consonance:

and why the frequency ratio theory isn’t completely correct

The frequency ratio theory is the most widely held belief among musicians of why certain combinations of sounds are consonant 1. This theory holds in most cases. For instance, the C below middle C (I’ll call C4) and the E above it (E4) sound good together. The frequency of middle C is 261.63 Hz and the frequency of E is 329.63 Hz. The C to E frequency ratio is about 4:5. If you tune to the key of C (perfect just-tuned ratios), the frequency of E becomes 327.03 Hz. Here’s how a just-tuned C and E sound together (these are simple sine waves). Pretty consonant, right? Well, let’s try C5 and C5#, a perfect just-tuned ratio of 11:12. As we expect, by the frequency ratio theory of consonance, this does not sound as good as the smaller whole number 4:5 ratio of C and E.

However, here’s another very similar example with a very different outcome: G3# and C4. Like the previous example of C & E these notes are 4 half steps away from each other and, again, make a 4:5 ratio. Let’s compare that to A2 sharp and B2. This, like the earlier example of C6 and C#, are just one half-step apart and make an 11:12 ratio. Click on the last two links again and compare them to the two links in the previous paragraph (*important note: This demonstration requires low audio distortion – you’ll need high quality speakers and you need to play it at low volume to minimize distortion). Which sounds better, more consonant? You’ve probably noticed that the reverse is now true. The 11:12 ratio sounds better than the simpler 4:5 ratio (or at least you can tell the difference between the relative consonance in this example and the one in the previous paragraph).

Why does the frequency ratio theory not always hold true? Unfortunately, there’s no easy answer to this question. In fact, there are many competing consonance theories2. Ohio State University lists 14 theories here. The frequency ratio theory is the first theory on the list. There is also the Periodicity Length theory that is similar to the frequency ratio theory, but is based on the length of the repeating pattern the combined tones create. Some of the other theories are based around the human ear, synaptic signals and our brain’s perception of those signals. Some are based on human learning and/or our cultural expectations of music. Even more theories are based on repeating patterns of sharp differences in volume called ‘beating’ that are unpleasant to our ears (You can hear this effect with this example of 100 & 110 Hz).

Some of the more interesting theories are based on harmonics. Harmonics are higher frequency waves that blend into the main/fundamental frequency that your ear perceives as the pitch. In fact, without these higher frequencies a basic main frequency tone doesn’t sound that pleasant. Here’s an example, an A4 sine wave (220 Hz). It sounds better when the higher frequency harmonics of an instrument are added in. Here's an example of a 220 Hz sine wave with added harmonics modeled after an organ.

Harmonics can be explained with a few simple equations (equations in this section are taken from ‘Fundamentals of Physics’ by Halliday, Resnick and Walker 3 ). For a stringed instrument the length of the string, the thickness of the string, and the tension the string determine the frequencies created by that string. The wavelengths created, though, are only dependent on the length of the string:

Wavelength = 2 × StringLength / n

The ‘n’ variable is the ‘harmonic number’ which starts at 1 and goes up by whole numbers from there. A harmonic number of 1 means the entire string is waving back and forth like this: You can see that this only creates half a wave which is why the wavelength is multiplied by 2 in the previous equation. Higher harmonics are produced with different patterns on the string:

2nd harmonic, n = 2: 3rd harmonic, n =3: 4th harmonic, n = 4, has 4 humps and it continues from there…

*note: harmonics are also sometimes called overtones. The 1st overtone = 2nd harmonic, 2nd overtone = 3rd harmonic, etc.4

The frequency/pitch of a wave on a string is dependent on the velocity of the wave traveling along the string:

Frequency = WaveVelocity / WaveLength

Substituting the WaveLength equation up above we get:

Frequency = WaveVelocity / ((2 × StringLength) / n)

And re-arranged:

Frequency = WaveVelocity × n / (2 × StringLength)

*note: the WaveVelocity depends on the tension you put on the string (increasing tension increases the frequency/pitch) and the linear density of the string (thicker strings are used for the lower pitch ranges):

VelocityOfWave = sqrt( tension / LinearDensityOfString )

We can see that the frequency goes up along with the harmonic number ‘n’. If the first harmonic was 220 Hz, the 2nd harmonic would be 440 Hz, the 3rd harmonic would be 660 Hz, etc. Different instruments produce varying degrees of these harmonics, which gives the instruments varying sounds. You can also change the harmonics produced by very lightly touching the string at specific points – for instance, you can see by looking at the 1st-3rd harmonic pictures above that if you lightly place your finger in the middle of the string the 1st and 3rd harmonics (and all other odd harmonics) would not be produced because their waves don’t come to a point there (You can try this on a guitar string by very lightly touching the string at the 12th fret – but don’t press on it so hard that it touches the fret and all waves are dampened). Real instruments have very complex mixtures of these harmonics that vary throughout the note. Because of the complexity involved, there are many different theories in this area. Many of these theories are based on the overlap of harmonics that create the dissonant ‘beating’ effect (talked about earlier).

The sound waves in this article were produced by a free program from Aspire Software: ToneGen (this program is provided as freeware and comes with no warranty of any kind, and no support). This program allows you to create accurate frequency tones, with optional harmonics. It outputs a wave (.wav) file. It also allows you to combine tones together to experiment with different combinations.

Using ToneGen:

To create a simple sine wave:

Put in the first column ‘sine.wav’ – without the quotes.

Put in the ‘Main Volume’ column the peak height of the wave – This can go up to 32767, but that’s loud – I suggest 10000.

Put in the ‘Main Frequency’ column the frequency of the note in Hertz. For instance, for the A below middle C this would be 220.

Click on the ‘Make Wave File’ button, type in the name of the wave file to create and hit OK.

To create two tones blended together:

Create the first tone as above.

Click on the second row where it says ‘blank’ and replace it with ‘sine.wav’.

Create the second tone with a different ‘Main Frequency’, such as 275.

Click on the ‘Make Wave File’ button, type in the name of the wave file to create and hit OK.

Explanation of the columns:

Wave file name:

If this is ‘blank’ the row is ignored.

If this is ‘sine.wav’ a sin function is used to create the wave.

If this is a valid wave file name, the program assumes there is one wave in the file and it repeats that wave to create the tone (instead of using a sine wave).

Main volume:

Level can go from 1 to 32767.

AM Modulation (optional):

If this is a valid wave file name, the program uses this wave to modulate the amplitude of the wave it creates. When this wave goes up, the output wave’s amplitude goes up and vice versa. For instance, if this wave’s value was zero, there would be no change in amplitude at that point. If this wave’s value was 32767, the amplitude would be doubled at that point.

FM Modulation (optional):

If this is a valid wave file name, the program uses this wave to instantaneously modulate the frequency of the wave it creates. When this wave goes up, the output wave’s frequency goes up (creating shorter wavelengths) and vice versa. For instance, if this wave’s value was zero, there would be no change in frequency at that point. If this wave’s value was 32767, the frequency would be doubled at that point.

Harmonic #:

Like the ‘Main Volume’ column, these values can go from 1 to 32767. These indicate the strength of the harmonic signal indicated. Remember, though, that higher frequencies need less volume to create the same loudness as a lower frequency.

Phase #:

This value can go from 0 – 2*PI (where PI = 3.1415926). For instance, to create a triangular wave you could enter 9000 for the main volume, 1000 for ‘harmonic 3’ and 360 for ‘harmonic 5’, and then enter 3.1415926 in the ‘phase 3’ column (for a sharper triangle you can continue this MainVolume/n2 pattern for the odd harmonics, alternating 0 and PI for the phase).

Explanation of edit boxes at the bottom:

Duration (in seconds): can be a fraction.

Main Freq starts at harmonic N: If you want the main frequency to start at a higher harmonic, you can increase this from the default of 1.

Volume Adjust: The default is 1.0 for no volume adjust. This adjusts the volume of the main frequency and all the harmonics at once. This is useful when you’ve added a lot of harmonics and start to get wave clipping because the values go over 32767.

References:

1. Consonance and Dissonance - Frequency Ratio Theory: http://www.music-cog.ohio-state.edu/Music829B/ratios.html

2. Consonance and Dissonance – The Main Theories: http://www.music-cog.ohio-state.edu/Music829B/main.theories.html

3. “Fundamentals of Physics”, Halliday, Resnick and Walker. Pages 389-90,379.

Another good site on harmonics: http://en.wikipedia.org/wiki/Harmonic