Back when I was teaching my first music theory class as a graduate student, I was caught completely off guard by a student's complaint. We were talking about the difference between simple meter and compound meter, which I regarded as a very basic topic. In one the beat is divided into two equal parts, and in the other the beat is divided into three equal parts. The exemplars for each of these would be 4/4 and 6/8. The quarter note beat in 4/4 is divided into two eighth notes, and the dotted-quarter beat in 6/8 is divided into three eighth notes. Easy, right? Well, this student, who was quite bright, said that 6/8 wasn't really a compound meter, because the eighth notes themselves didn't divide into three equal parts. I thought he was making a common mistake of assuming that eighth notes were the beat in 6/8, but he wasn't . He expected that if the beat is divided into three equal parts, then each subsequent lower level in the metric hierarchy would also divide by threes. This is actually a very rare phenomenon, not even covered by normal metric notation. There is no time signature that indicates a beat that will be divided into 9 equal parts, much less 27 or 81 parts. The basic assumption in Western notated music is for beats to be grouped by 2's or 3's (or some combination thereof) and to be divided by 2 or 3, but that all lower levels of subdivision are by 2.
I never really thought about this lack of symmetry, until I was recalling an argument about chord and key relationships. Petr Janata and his colleagues have done some interesting work on brain imaging and tonal sensitivity. But at one conference he presented a torus mapping of key relations, playing along a chord sequence that modulated from one key to the next. This mapping was meant to represent brain activity, showing how the brain interprets different modulations as close or far. As expected, the modulations followed a circle of fifths pattern, but what I found quite disturbing was that mode was not taken into account. At one point the progression went from C minor to G major. I pointed out to him that this was a very distant key relationship in music theory, which he found surprising. After all, C minor chords go to G major chords all the time. We then started a debate about the difference between chord relationships and key relationships, with Carol Krumhansl coming up to keep us from punching each other. (Not really, but it was an interesting and sprited debate, including the aforementioned Dr. Krumhansl). Carol did the seminal work on tonal hierarchies, using probe tones to determine the closest cognitive notes to a given tonic, and helped develop a key-finding algorithm that might mimic how our brains determine what key we are in. What I realized that I knew to be false, but that these scientists had assumed to be true, was that the hierarchy of relationships between notes and chords would extend to keys without change.
Why is it that Western music* does not encourage complete symmetry among all levels of a given feature? Could it be like the asymmetric design of the diatonic scale, working as a signpost to help us identify where we are? The assymmetry between keys and chords could help distinguish the scope of relations we are perceiving, so we don't get the two mixed up. Likewise with meter, to help identify the beat more clearly.
*If anyone knows about any other music traditions that do include non-binary metric symmetry or note/harmony/tonality symmetry, let me know.