Scale. Wanna' split hairs? It's all a gosh-dang'd numbers game. The TRUE interpretation is only available to those of us mathematicians in the know.
For instance, a lot of depends on which side of the "pond" you are on. In America, true O-scale would be (or maybe should be, considering the distance between the rails?) considered as 1/48 scale. Or, 1/4" = 1'-0". Which has its origins based on English measurements. Not even opening the 1-1/4" (scale 5'-0") between the rails debate. But Proto 1/48 scale addresses that issue. Half-O, or HO, therefore should really be 1/96 scale, or basically, 1/2 the scale of O-scale, which would also be 1/8" = 1'-0". But something funny happened on the way to the forum.
The English adopted the metric system, and as a result, their O-scale became 7 mm to the foot. Or, rounding off to the English adopted system, 1/43.5. Therefore, half that size became HO scale, or as commonly known, 3.5mm/foot (half of 7 mm/foot), a.k.a. "1/87" scale, which pretty much stuck. HO scale is 3.5 mm/foot, like it or not. It may be the only one long-time true scale, to which all other standards are measured from. Which is kind of a conundrum. Shouldn't 3.5 mm equal something metric, like say a meter, or a decimeter, or even a centimeter or something like that? But no, the English decided to let 7 mm, and half or that, 3.5 mm (METRIC measurements) equal one FOOT (an ENGLISH measurement), thereby screwing up model train scales for all eternity ever after!
And let's consider N-scale, 9 mm between the rails, or 1/160 scale. Which is kinda' sorta' half of HO scale. Or thereabouts. Doing the math, we find that N-scale is also not exactly 1/4 the size of O-scale, as can be arbitrarily assumed, but more precisely 3-1/3 times smaller than American O-scale, which of course is not the current English O-scale, but IS the current world-wide N-scale accepted measurement, except in Japan, where it's somewhere closer to 1/150 scale, or maybe 1/144 scale, or maybe somewhere around that. It's kinda' like playing "Whack-a-Mole" at the circus carnival. You're never quite sure where that next head is gonna' pop up, you just have to try your best to hammer it when it does!
If that doesn't blow your minuscule minds, it ought to. It blows my minuscule mind! Which is why I like O-gauge, as opposed to O-scale. It's more of a suggestion than an absolute. I remember taking pre-calculus back in high school (a long time ago). In retrospect, I think it was even more understandable than model railroading scales!
And more food for thought - you want my honest opinion? As far as I'm concerned, there all only 2 (two) TRUE model railroad SCALES - Proto 1/48 (1/4" = 1'-0") for one, and S-scale (3/16" = 1'-0") for the other. You wanna' know why? It's because they are based on strictly English measurement scale-downs, and NOT English-to-metric conversions with their resultant rounding off's to the closest inch or millimeter or whatever.
So, how's the batteries in your calculator? Hope they're fully charged. You'll need 'em if you're gonna' try and figure out model railroading scales to the Nth degree!