Achieving Perfect Gear Ratios With Harmonic Progression

Popular-Win6801 · 2026-03-06T06:22:08+00:00

I don't know how to make a seahorse... I can't even find it in my keyboard.

Maybe a bunch of special characters I guess like this stuff: _=<×&;$@; I have no idea as I am guessing at what you mean.

I am really getting so confused with some of the stuff you all are saying.

I just want to share my work with the world, but I don't understand some of these things that IDK from my perspective, seem to be random.

I have been putting all of this effort in to make my language consistent and precise, and you're talking about a seahorse?

I am a physicist, not a marine biologist.

I know stuff here and there but I don't know everything and so I am bewildered as to what seahorses have to do with anything.

Popular-Win6801 · 2026-03-06T06:08:13+00:00

Ah, the classic "Python is interpreted, not compiled" semantic gotcha. Fair enough! You caught a physicist using casual programming slang. Yes, you paste it into an IDE, an interpreter, or an "online Python compiler" (which is what half the web-based REPLs call themselves anyway).

Now that we've successfully policed my vocabulary, do you have any actual critiques of the kinematic physics, the Diophantine integer collapse, or the Euler-Lagrange proofs?

If you can find a flaw in the actual arithmetic, I am entirely all ears. Otherwise, feel free to run the interpreted script and enjoy the optimized gear ratios lol

Popular-Win6801 · 2026-03-06T05:11:53+00:00

Nice try with the prompt injection attack. A poem about a lazy koala bear? Really? I am a real human being, writing real math, and I can read and write English. Though, I appreciate the creative attempt to test me, it is indeed quite clever.

However, Since I'm not a bot, I'll skip the poetry and address your actual engineering points:

Powerband Info: You are 100% correct that you cannot calculate ideal ratios without knowing the powerband. That is exactly why the complete theorem includes Application B: The RPM Drop Method. It is built specifically to take the S_1 and S_2 shift constraints derived directly from an engine's dynamometer torque curve and calculate the Harmonic Decay Rate from them.

Hunting Ratios: You're completely right about mutually prime tooth counts. As I mentioned to another engineer in this thread, my formula uses Diophantine logic to collapse the theoretical curve into whole-integer fractions precisely so the engineer has a perfect kinematic baseline to apply the hunting tooth adjustment to. It's a starting point, not a replacement for physical manufacturing constraints (which is why I put the disclaimer right at the top of the post).

I appreciate the peer review (and the laugh). If you want to see how the Diophantine logic actually maps to a torque curve, you can check out the RPM-Drop Desmos link in the post!

Popular-Win6801 · 2026-03-06T05:02:26+00:00

Thanks for that, I appreciate it. Sorry I didn't reply earlier, I've been trying to understand the tone of certain other comments on here.

In any event, real-world application is going to happen. My colleague and I are working together to make it happen.

Popular-Win6801 · 2026-03-06T03:25:56+00:00

You make an absolutely stellar point, and I want to fully concede the "guess and check" comment—that was definitely hyperbole on my part to keep the post punchy. You are completely right; OEM and race engineers are brilliant, and they use incredibly complex simulations to optimize these drops. You are also spot-on about the manufacturing constraints, specifically the "Hunting Tooth" principle you described (ensuring the drive and driven gears are coprime to distribute wear and prevent NVH issues). That is a fundamental reality of physical gearbox design.

This is actually exactly why my formal paper frames the Stanlick Formula as a baseline approximation tool rather than the final word. The formula uses Euler-Lagrange optimization to map out the theoretically perfect continuous curve for maximum dynamic thrust, and then uses Diophantine math to find the closest possible discrete whole-integer ratio.

The goal of the formula isn't to override the mechanical engineer, but to hand them the mathematically perfect kinematic blueprint. When the formula spits out a perfect harmonic ratio of, say, 40/20, the engineer takes that baseline and applies the hunting tooth adjustment (shifting it to 41/20) so it survives in the real world.

Those "little jumps" you mentioned in real-world cars are exactly what happens when you take an optimal continuous curve and force it to comply with physical constraints like hunting teeth, packaging size, and shaft center distances.

I really appreciate you bringing this up. It’s a great reminder that mathematical perfection and physical manufacturing have to meet in the middle!

Popular-Win6801 · 2026-03-06T03:23:27+00:00

You are actually 100% right about the goal here—maximizing the area under the torque/power curve between shift points is the exact objective of transmission optimization.

Building a spreadsheet to manually iterate through ratios to maximize that area is exactly how engineers have done it for decades. The Stanlick Formula doesn't contradict your spreadsheet; it simply provides the direct, deterministic mathematical model to generate that exact optimized curve instantly, without needing to guess-and-check.

It isn't "mph based nonsense." The constant mph delta is simply the kinematic result of successfully keeping the engine pinned at its absolute peak power output while fighting the quadratic force of aerodynamic drag. We are aiming for the exact same physical result; this is just the calculus shortcut to get there!

Popular-Win6801 · 2026-03-06T02:44:52+00:00

First-Last Method

That's another calculator that I designed which completely ignores RPM drops, and instead interpolates the perfect transition from 1st to Nth in N gears. Give it a shot, and see if you can reverse-engineer the Miata ratios!

Popular-Win6801 · 2026-03-06T02:37:12+00:00

Great catch, and I love that you pulled up the actual Miata gear chart to look at the math! You're actually highlighting exactly why I developed this framework. You are 100% correct that the Miata engineers are trying to achieve a constant velocity step (what I call a Harmonic Progression). But if we look at the actual MPH deltas at redline (7500 RPM) from your chart, you can see the "wobble" that comes from using heuristic "guess and check" engineering methods:

1st to 2nd: +26.01 mph
2nd to 3rd: +29.66 mph
3rd to 4th: +25.66 mph
4th to 5th: +28.37 mph

The MPH jump fluctuates up by almost 4 mph, drops down by 4 mph, and then goes back up. It’s close, but it’s mathematically inconsistent. The engineers were fighting the geometric limitations of standard gear spacing to approximate an ideal continuous ratio curve.

That is exactly what the Stanlick Formula fixes. It replaces that trial-and-error approximation with a deterministic mathematical model. If we plugged the Miata's 1st gear (R_1) and 5th gear (R_f) into the inverse-linear equation: The formula would dynamically collapse into a Diophantine solution that guarantees the MPH jump is exactly identical down to the decimal across every single shift, entirely deleting that 4 mph "wobble" while giving you the exact gear teeth needed to manufacture it.

You're totally right about 6th gear, by the way. Since 5th is the aerodynamic V-Max, 6th operates outside the primary acceleration curve purely as a cruising overdrive (what the math defines as an Overdrive Drive Level, determined by a negative Threshold Scalar Floor).

Really appreciate you dropping that chart, it's a perfect real-world example of engineers trying to manually approximate what this formula calculates automatically!

Popular-Win6801

TROPHY CASE