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.