Toward a Geometry of Paradox
Every so often, a modest technical question becomes the doorway to a larger meditation. My own began with music, which is often where philosophy hides when it wishes to be discovered without announcing itself. I had been comparing the tidy symmetry of equal temperament with the unruly contours of real acoustic frequencies. In the world of equal temperament, the octave is sliced into twelve identical wedges, each semitone occupying its allotted thirty degrees like a well‑behaved citizen of a rational republic. But the moment one consults the actual Hertz values, the geometry rebels. The perfect fifth lies almost exactly halfway through the octave in frequency, the major third sits near a quarter of the way, and the seventh falls not at the neat three‑quarter mark but somewhere slightly off to the side. The shapes formed by these bearings are not the crisp polygons of a Euclidean diagram but the living asymmetries of a natural organism. They are certainly far from what you would get if you wrote the notes of the chtromatic octave on a clock face with C at 12 o clock, C# at 1 oclock. That system gives us D# and not E at the three o clock position (or fifteen minutes past the hour if you prefer to look at it that way) at the half hour/six o’clock position you have F# when in hertz terms the G is half way between two Cs.
To see exactly what I mean and how far apart the two systems are, see the table below,
| Note | Hz (approx) | ΔHz from C | Fraction of octave | Degrees | What the degrees would be in equal temperament | Δ in deg | How many minutes past the hour (rounded) in reality | How many minutes past the hour in the usual diagrams of music theory |
| C(4) | 262 | 0 | 0 | 0.0 | 0.0 | 0.0 | 0 | 0 |
| C♯ / D♭ | 279 | 17 | 0.065 | 23.4 | 30.0 | -6.6 | 4 | 5 |
| D | 294 | 32 | 0.122 | 45.0 | 60.0 | -15.0 | 8 | 10 |
| D♯ / E♭ | 314 | 52 | 0.198 | 71.5 | 90.0 | -18.5 | 12 | 15 |
| E | 327 | 65 | 0.248 | 89.3 | 120.0 | -30.7 | 15 | 20 |
| F | 349 | 87 | 0.333 | 119.8 | 150.0 | -30.2 | 20 | 25 |
| F♯ / G♭ | 367 | 105 | 0.401 | 144.3 | 180.0 | -35.7 | 24 | 30 |
| G | 392 | 130 | 0.496 | 178.6 | 210.0 | -31.4 | 30 | 35 |
| G♯ / A♭ | 419 | 157 | 0.599 | 215.7 | 240.0 | -24.3 | 36 | 40 |
| A | 436 | 174 | 0.664 | 239.1 | 270.0 | -30.9 | 40 | 45 |
| A♯ / B♭ | 471 | 209 | 0.798 | 287.2 | 300.0 | -12.8 | 48 | 50 |
| B | 491 | 229 | 0.874 | 314.7 | 330.0 | -15.3 | 52 | 55 |
| C(5) | 524 | 262 | 1 | 360.0 | 360.0 | 0.0 | 60 | 60 |
Irreconcilably far apart, yet both systems work. Equal temperament, for all its artificiality, gives us a musical language of extraordinary flexibility. It allows us to modulate, transpose, and wander freely through keys without the tuning collapsing under our feet. Harmonic tuning, for all its irregularity, gives us the physics of resonance, the way a choir instinctively adjusts its intervals until the chord locks and the air itself seems to vibrate with agreement. They contradict each other if we insist on forcing them into the same frame, but the contradiction dissolves when we recognise that they are describing different aspects of musical reality. One is a map; the other is the terrain. The map is not false because it simplifies, and the terrain is not incoherent because it refuses to be simplified.
Another example of how perception is far from physical reality is the way we see things as solid regardless of the fact that we know that most of he space between subatomic particles is completely empty and that mass bearing matter is so spread out that when it is concentrated together you can get a black hole and eventually a singularity. We are looking into what should be an almost empty space but as light itself is made to the appropriate wavelength, this is not what we see, but we see matter the way we need to see it. Seeing it like this makes it hard to believe how empty most of space is, not just outer space but even inside a diamond, and yet we seem to be able to accept that both these ways of viewing reality are true at once.
Calvinism vs Arminianism
Once I had seen this, I began to notice the same structure in places where I had not expected it. The long quarrel between Calvinism and Arminianism, so often conducted with the heat of a family dispute, suddenly looked less like a contest of doctrines and more like a pair of coordinate systems. One describes God’s initiative, the other human response. Each is internally coherent, each illuminates something essential, and each becomes distorted when made to swallow the other whole. The paradox is not a flaw in the theology but a sign that the reality it seeks to describe is too rich to be captured by a single model. We are creatures who experience both divine sovereignty and human agency, and any account that denies either does violence to the lived texture of faith.
Just as the sinner coming to Christ maybe perfectly unaware ibn his course of coming that he is being led there by the movement of the Holy Spirit, who is, in a way unique to this new Christian as an individual, breaking every barrier down, so what seems to be going on – the sinner making all the effort – is not necessarily what is really going on, just in the way we believe we are hearing equal temperament in music whereas what we are actually hearing in hertzometric, physical terms is far from that. There are scriptures enough that talk both of man’s agency in believing, coming, repenting, working and scriptures enough that tell us it is not of him that willeth nor of hjim that runneth, but of God that sheweth mercy, but somehow many Christians (with notable exceptions such as C.S. Lewis in his essays) see incapable of understandingf that here too there are separate, apparenbtly mutually exclusive models of how to understand something which are both true. Why Christians find this hard to accept when it is clear that God has made a world in which Einstein can correctly say that God does not play dice and Niels Bohr can equally correctly show that He does, is probably a function of intelligence, and the good news here is that people can work on their intelligence, (as well as it being a gift).
Creation vs Evolution
The same pattern appears again when we consider the origins of the world. Scientific models, grounded in continuity and observable processes, trace the unfolding of natural laws across deep time. The theological account of creation speaks instead of divine purpose, of a world summoned into being by a word rather than assembled by a mechanism. And the old idea of apparent age — that creation may have been brought forth in a state of functional maturity (Omphalism as initially discussed by Philip Henry Gosse in 1857) — adds a third dimension, not as a deception but as a recognition that beginnings are not always gradual. These frameworks seem contradictory only when we demand that they answer the same question. Once we allow each to speak in its own register, the tension becomes intelligible rather than adversarial. Mechanism, meaning, and initial condition are not rivals; they are layers.
What unites these reflections is the recognition that our perception is limited. Our ears, our minds, our interpretive habits are shaped by the constraints of our present condition. We cannot compose music using a Hertz‑based geometry because our biological hearing is not tuned to such crystalline symmetry. But the very fact that we can imagine it hints that our current sensory world is only a partial view of a richer reality. Christian hope has always included the renewal of perception — the idea that the resurrection does not merely restore life but transforms the very faculties by which we apprehend beauty. If that is so, then the harmonies we know now may be shadows of a more perfect music, one in which the asymmetries of the present are gathered into a higher order we cannot yet hear.
What began as a technical inquiry into tuning thus becomes a small philosophical ascent, an anabasis from the particulars of sound into a broader reflection on how truth often comes to us in pairs of models that cannot be collapsed into one another. Equal temperament and harmonic tuning, Calvinism and Arminianism, scientific cosmology and creation theology: each pair forms a kind of binocular vision. Depth appears only when both eyes are open. The satisfaction we feel in recognising this pattern is not the thrill of novelty but the quiet sense that reality is larger than any single framework, and that contradiction, far from being the enemy of coherence, is sometimes its most faithful guide.
This is of course down to the fact that our Creator is a glorious Being, not easy to understand because his thoughts and ways are so much higher than our own. He who comes to the Lord must believe that He is, and that He is a rewarder of those who diligently seek Him. One of the greatest of these rewards can be a rich seam of wisdom and understanding, to those who have ears to hear.
Red Rising
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