a list of lists

the 6 symmetries of music

26 August 2005
Each symmetry of music is defined by a set of transformations, which, when applied in some way to music, leaves some perceived aspect of that music at least approximately unchanged.
1. Pitch Translation Invariance The musical quality of a tune is essentially unchanged if it is transposed into a different key.
2. Time Scaling Invariance This corresponds to playing music faster or slower. Although speeding up or down has more effect on musical quality that raising or lowering pitch, we can still recognise the equality of two versions of the same rhythm played at different tempos.
3. Octave Translation Invariance This is a subset of Pitch Translation Invariance, since a translation by an octave is a particular amount of pitch translation. But octave translation invariance applies to more things. For example, individual notes within a chord or bass line can often be translated up or down by an octave without significantly affecting the musical quality of a musical item.
4. Time Translation Invariance This says that the musical quality of a tune is much the same now as it is at a later time. Along with the next symmetry, it is not particularly a musical symmetry, and would be regarded as applying to all sound perception.
5. Amplitude Scaling Invariance This says that the musical quality of a tune is much the same whether played louder or quieter. This isn't quite true, because we do enjoy our favourite music more if it is played more loudly. But apart from that the quality of music is not much changed by changes in volume.
6. Pitch Reflection Invariance This applies specifically to the consonance relation between notes. For example, the degree of consonance between A and C is the same as the degree of consonance between C and A. This symmetry does not imply that music can be played "upside down", no doubt because there are aspects of melody perception that are more than just a function of consonance relationships between notes. However, there is an apparent "upside-downness" in the preferred choices of home chord in the diatonic scale (consisting of the notes A, B, C, D, E, F and G). The two preferred choices are A minor and C major, which are indeed mirror images of each other within the scale, reflected about the note D.
The general concept of symmetry plays an important role in many areas of science, including classical mechanics, quantum mechanics and crystallography. It is perhaps surprising that until now there has been no attempt to classify and analyse all the symmetries relevant to music perception.

It is not that some of these symmetries have not been studied, but they have been studied without the benefit of a unified concept of symmetry. For example:

  • Octave Translation Invariance is studied in terms of "pitch class", which is really what you have left after the symmetry is factored out. The phenomena of Shepard tones and the tritone paradox both have to do with octaves and octave translation invariance.
  • Pitch Translation Invariance is implicit in the term "relative pitch", which is sort of the opposite of "absolute pitch". Absolute pitch seems interesting, because few people have it, but a bit of thought shows that the implementation of absolute pitch perception is relatively trivial, and it is the implementation of pitch perception invariant under pitch translation which is quite non-trivial to explain.
  • Time Translation Invariance seems quite trivial, perhaps because one can explain it by assuming no more than the tendency of a system to return to a standard initial state in the absence of stimuli.
  • Amplitude Scaling Invariance also seems trivial, but it may not be so trivial to implement, especially given that louder sounds don't just cause greater activity in the same sets of neurons as softer equivalents, rather they actually cause greater activity in larger sets of neurons. Identification of the equivalence between louder and softer versions of a sound may require a significant amount of experiential learning to achieve.

For each musical symmetry, we can ask:

  • what is the mechanism of that symmetry?, and,
  • what (if anything) is its purpose?

The most interesting observation about pitch translation symmetry and time scaling symmetry is that there appears to be a "symmetry of symmetries" consisting of a series of analogies between the two symmetries:

  • Both symmetries imply the existence of a calibration mechanism. In the case of pitch translation the calibration mechanism is the identification of harmonic intervals. In the case of time scaling the mechanism is the identification of very simple ratios (i.e. 2 or 3) between pairs of time periods. Each of these identifications is suitably independent of the corresponding set of transformations.
  • In each case the calibration mechanism occurs prominently in music, i.e. consonant intervals occur between notes, and beat periods are nested within each other and related to each other by multiples of 2 or 3.
  • Both of these symmetries have a plausible purpose within speech perception. For pitch translation it is that different people might speak within different pitch ranges. For time scaling it is that different people speak at different speeds (and also the same people speak at different speeds under different circumstances).
This symmetry of symmetries may give some clue as to the nature of the musical universal, the one mysterious aspect of music which distinguishes what is musical from what is not musical.

More details can be found in my book What is Music? Solving a Scientific Mystery, especially in Chapter 9: Symmetries.