A Seventh Symmetry of Music Perception

8 December, 2008
The seventh symmetry of music is frequency-dependent phase shift invariance.

Five or Six Symmetries of Music Perception?

In my book What is Music: Solving a Scientific Mystery? I categorise the five symmetries of music perception, and a "possible sixth" symmetry:

  1. Pitch Translation Invariance
  2. Time Scaling Invariance
  3. Octave Translation Invariance
  4. Time Translation Invariance
  5. Amplitude Scaling Invariance
  6. Pitch Reflection Invariance

(I now treat the sixth symmetry, pitch reflection invariance, as a definite symmetry, because it can be considered to apply to consonance as a function of two pitch values.)

As it happens, I have recently realised that there is a seventh symmetry. I even mentioned it my book, but I failed to state that it was a symmetry.

Because this seventh symmetry is a symmetry of most (but not quite all) aspects of sound perception, it may not tell us all that much about music perception, but it is worth adding to the list. In this respect, it is similar to time translation invariance and amplitude scaling invariance.

And the Missing Symmetry is ...

The seventh symmetry is frequency-dependent phase shift invariance. In other words, music perception is invariant when shifting the phase of different frequency components of sound by different amounts.

(An frequency-independent phase shift, where the shift is given as an absolute period of time, is the same as a simple time translation, so this is already included in the symmetry of time-translation invariance.)

Phase Shifting in Stereo

The main role that phase perception plays in human hearing is when the perception of sound direction is determined as a function of relative phase between left and right ears.

If follows that human hearing does have some dependence on phase.

In as much as relative left/right phase shift affects our perception of sound, it will have two major effects on the perception of music:

Phase Shifting of Transient Sounds

Frequency-dependent phase shift may also have some effect on the perception of transient sounds and changes to sounds, for example the beginning of a musical note, whereas it would have less effect on the perception of timbre of slowly varying components of sounds.

(The above analyses are somewhat speculative, and the best way to determine how much, if at all, music perception is affected by frequency-dependent phase shifts, is to apply transforms to actual musical recordings, and see if there is any significant perceptual effect.)


Like all the other symmetries of music perception, frequency-dependent phase shift invariance is not exact. While there is no limit to the absolute amount of phase shift (which never exceeds π radians anyway), there is a limit to how large a phase shift there can be over a very small frequency range.

This is because large phase shifts over small frequency ranges will be equivalent to moving substantial components of sound forward or backwards in time, and such movements will necessarily have a noticeable effect on perceived musicality. (From a mathematical point of view, this is equivalent to stating that human hearing can be considered as a Fourier analysis over a time window which is somewhat smaller than the duration of an item of music.)

Residual Relevance

As stated above, if invariance under frequency-dependent phase shift is an invariance of sound perception in general, then it doesn't tell us anything in particular about music perception.

However, in as much as temporally coded neural networks may be observed, or hypothesised, and in as much as those temporal codings are not invariant under frequency-dependent phase shift, and music perception is invariant under those phase shifts, we would have to conclude that music perception does not depend directly on those temporal encodings.

My constant activity pattern theory of music states that music perception is entirely a function of the location of neural activity, averaged over periods of time somewhat longer than the time periods involved in temporal coding. So this theory is consistent with the observation that the perception of music is mostly invariant under frequency-dependent phase shifts, and where it does vary, the effect is indirect, for example if phase shifts affect the perceived grouping of sounds or components of sound.