Two Clarifications on Polytempic Polymicrotonality, Part LVII

Two Clarifications on Polytempic Polymicrotonality

Peter Thoegersen / zipcaustic.blogspot.com

Theoretical systems attract misreadings. Sometimes the misreadings are careless, sometimes motivated. Either way, precision is the best defense. Two points in my compositional system invite confusion and deserve explicit treatment.

I. C Zero Is Not a Master Fundamental

Each of the four independent tuning systems in my polytempic polymicrotonal work is described beginning from the note C. A reader encountering this for the first time might assume that C functions as a tonal center — a master fundamental around which the four systems are hierarchically organized, a hidden key of C governing the whole.

That assumption is wrong, and the error lies in confusing a measurement reference with a structural authority.

C in my system is what I call C zero: the synchronization pitch, the common origin from which four independent tuning systems launch. It is where I begin to describe the tunings — to express interval relationships in cents, to communicate the structure in conventional notation. It is not what the tunings point toward. After the zero coordinate, each system goes its own way, operating within its own intervallic logic, bonded to its own independent tempo. C zero does not govern those systems. It precedes them.

The analogy is straightforward: four clocks set to 12:00 at the same moment do not share a master clock. The synchronization is a starting condition. Once running — each at its own rate — the 12:00 is simply a common origin, not an ongoing relationship.

Similarly, C zero is a launch condition, not a governing principle. The system has a common origin. It has no common authority. These are not the same thing, and the distinction is the whole point.

The claim that my system has no master fundamental stands. C zero is the evidence for that claim, not against it.

II. Convergence Does Not Negate Polytempo

Polytempic streams running in ratios such as 3:4:5:7 will, by the mathematics of those ratios, periodically share a common downbeat. These convergence nodes are a predictable property of commensurate ratios. A critic might argue that such convergence implies a master tempo — that the music snaps back to a governing pulse at the moment the streams align.

This argument mistakes a mathematical property for a structural condition.

Convergence is what happens when you multiply independent rates by their common multiple. It is a consequence of the arithmetic, not evidence of hierarchy. Two runners lapping each other on a track do not thereby run at the same speed. Their periodic coincidence in space is a function of their ratio — it does not cancel the fact that one runs faster than the other.

The same logic applies to polytempic streams. The streams are always running at different speeds. The convergence node is a single point, a momentary coincidence, embedded within continuous divergence on either side. The music does not cease to be polytempic at the moment of convergence and resume being polytempic afterward. Polytempo is not a condition that switches on and off. It is the continuous reality of simultaneous independent tempos, and convergence is simply what that reality looks like at one particular instant.

This is why periodic polytempo and non-convergent polytempo occupy different regions of the same continuum rather than different categories. The defining condition is independent speeds. That condition holds throughout — at points of divergence, at points of convergence, and everywhere between. The periodicity of convergence does not flatten the tempos into one. It only means the ratios are simple enough to share a common multiple.

Consider the reductio: even if all four streams shared a downbeat at the opening of every measure — the simplest possible convergence — within each measure the four systems would still be running simultaneously at different tempos. The experiential reality of independent concurrent speeds would remain fully intact. The convergence frequency does not alter the structural independence of the streams.

Convergence is a feature of the mathematics. It is not a concession to a master system. Polytempo is not diminished by periodicity.

This has a practical implication worth stating plainly. A composer who chooses simpler ratios — tempos whose relationship is immediately perceptible to a casual listener, whose convergence nodes arrive frequently and audibly — should not be penalized by having their work characterized as non-polytempic. Accessibility of ratio does not disqualify the work. It places the work at the simpler end of the polytempic continuum, which is still the continuum. The continuum runs from the most elementary ratio relationships to the most complex and remote, and every point along it is polytempic by the same criterion: independent simultaneous tempos with no master pulse. Where a composer chooses to work on that continuum is an aesthetic decision, not a structural one.

These two clarifications address surface-level readings that could obscure what the system actually claims. C zero is a reference, not a root. Convergence is arithmetic, not hierarchy. The independence of the four streams — each tuning system bonded to its own tempo, with no master fundamental and no governing pulse — is not qualified by either.

Peter Thoegersen is the originator of polytempic polymicrotonality. His theoretical work is published by Jenny Stanford Publishing (2022, 2024). Scores are archived at the Internet Archive; recordings are available on Bandcamp, SoundCloud, and New World Records.