Not to hijack the thread, but I was stunned when you said that. I always assumed that most dojangs everywhere follow white, yellow-tag/stripe, yellow, green-tag/stripe, green, blue-tag/stripe, blue, red-tag/stripe, red, black-tag/stripe, black belt. I know some prefer brown to red, but I still thought they were in a minority.
So I'm stunned that the standard or as you wrote "seems" at least a very common configuration includes orange and purple belts... I've never seen a Taekwondoin wearing those colours.
It used to be like that, with a black piece of electrical tape on the ends to signify color belt ranking. In some instances it is still used for some of the higher color belts, like red or brown. There are also ranks like "senior blue", which has a red or black stripe running down the center of the length of the belt. Over the last twenty five years or so, color belts have evolved so that you get a different color belt for every guep promotion. Electrical tape is still used, but they are used to denote progress within a particular guep rank. For example, if you are going from green to purple, you may have up to three stripes or "tips" which are given to you at "tip tests". A tip test is sort of like a mini test. The material would be broken up in fourths (three tips) and when you have all the material, then you can test for the next color. I think it was started by EFC, an early martial art marketing firm, but a lot of schools have since adopted it, at least the more commercially successful ones.
I believe the alternating standard and non-standard color belt system evolved for tournament purposes. Color belts divisions are arranged via the standard colors of yellow, green blue and red, so it makes sense to have alternating or non-standard colors in between the standard colors, hence the yellow, orange, green, purple, blue, brown, red scheme.
Some schools add a lot of color belts including camo, senior blue, senior brown, senior red, bodan/danbo, so you get way more than the usual 8 or 9 guep ranks. In fact, numbers (I'm an 8th guep) is not really used in the US dojang for the most part. They just go by the color.
I'm talking in generalities and so each school has their own way. But this seems to be the basics.