Category Archives: Music Theory

Lateral Music Theory #2: Tension, Instability

Continuing on from the earlier discussion on dissonance, there is another couple of numeric quantities that can help describe how chords are perceived. These are Tension and Instability. Instability matches the empirical perception of 3 note chords.

Tension is a word used to describe the ambiguity of how a chord seems to want to resolve. It is calculated from the relative size of the intervals of a three note combination. Specifically, from all the combinations of 3 tones from all the partials of the 3 fundamental tones.

TensionEqn

where:

ν is the product of the relative amplitudes of the three tones (0.0 to 1.0),

x and y are the interval sizes of the lower and upper intervals defined in semitones

α = 0.6 determines the steepness of the fall from maximal tension.

Total Instability is (I) is calculated from Tension (T) and Dissonance (D)

I = D + δT

where δ = 0.2

When all the partial combination D and T effects are summed in this way, the scores rank the sonority in a similar order to the empirically determined order.

You can calculate your own dissonance, tension, and instability for chords at a web page I wrote a while ago: here.

Lateral Music Theory #1: Dissonance

Dissonance is word used to describe how unpleasant something sounds; particularly in psycho-acoustic studies. In one article I read by Normal Cook, in the Music Perception Journal, it can be calculated numerically from the magnitude of tones that make up a sound.

The formula for calculating dissonance (D) from tones is

DissonanceEqn

where:

  • ν = the product of the relative amplitudes of the two tones (0.0 to 1.0),
  • x = the interval size defined in semitones.
  • β= -0.8 is the interval of maximum dissonance
  • β= -1.6 is the steepness of the fall from maximal dissonance
  • β= 4.0

When we plot this for 2 pure tones we get a nice little plot like this:

However, real instruments do not produce pure tones. They produce harmonics, that is, lots of extra tones that are roughly integer multiples of the frequency of the lowest tone. Interestingly, if you add up all the dissonances from all the pairs of tones between 2 “notes”, the plot starts looking a lot more interesting.

There are tantalising dips at important musical intervals; like the 5th, 4th, and octave.

I did my own plot, using the first 8 harmonics with similar magnitudes as a piano sound and got this plot:

This graph has some very interesting features. I have plotted the equal temperment semitone points with small grey lines. These miss the kinks in the dissonance plot slightly. But, when I plot the Just Intonation semitone points [1, 16/15, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8, 2] in red, they match the troughs and kinks almost perfectly.

I wonder what this graph looks like if we extend it into the next octave?

Interestingly, this calculation can be extended to cover any combination of any number of notes and is an interesting way to compare chords of any kind. The part that appeals to me is that a simple relationship between tones and their perception can generate parts of music theory, without requiring “the edifice of music theory”. This is Lateral Music Theory.

You can calculate your own dissonance for chords at a web page I wrote a while ago: here.

 

Missing Fundamental = 7th Chord

I came across a psychoacoustic phenomenon where humans perceive a tone that isn’t really there, called the “Missing Fundamental“. Many musical sounds are comprised of harmonics that are integer multiples of a fundamental frequency.

If you only have all the harmonics without the fundamental, then the brain hears the fundamental.

This gave me the idea of combining notes to achieve this same effect.

Assuming each note is comprised of a harmonic series, then the fundamental of the base note is twice the frequency of the desired “missing” fundamental.

We should be able to add notes at any frequencies where harmonics of the missing note should appear.

Fake Note Harmonics 1 2 3 4 5 6 7 8 9 10
Bass Note (C) 1 2 3 4 5
5th up (G) 1 2 3
10th up (E) 1 2
15th up (B) 1

These higher notes are not exactly the harmonic frequencies in equal temperment, but are close enough. (C, G (+2 cents), E (-14 cents), B (-32 cents)). Many instruments don’t have harmonics at precise multiples of the fundamental, and in Just Temperment the top note is a Bb.

So it is interesting that the C Major 7th, or C Dominant 7th chord could produce a sense of playing a G lower than the lowest note, when using this voicing.

However, the effect is probably better if the volume of the higher notes is reduced so that the amplitudes of the harmonics is more like a real instrument.

This effect is indeed audible if you listen really closely.