Then you decided to straw man me to say I didn't get some arithmetic as if I had no comparison of rational temperament with 12ET? Bull_shit.Similar to how 360 degrees are divided well by many integers, 12 semitones line up nearest to the desired integer fractions.
If you're going to ask this question I assume you've done the math?
frequency = root * (2 ^ 1/semitones) ^ offset
Then you need to find the ratio between notes, from offset = 0 to semitones.
Then find the LCM for those fractions.
What do you feature you actually showed the class with that shite there? That you're a maths guy and they're not?
What I said can be researched, the history of 12 to an octave and 7 taken from that has to do with decisions according to rational temperament, a history of some centuries and people having musical reasons behind some choices. The twelth root of 2 didn't enter into it until it did, relatively late in the game, by necessity. However things were built in China in antiquity that were, statistically speaking 12 equal inside the 2:1.
In terms of western music: to obtain a system more useful [so it was ultimately believed by enough musicians] for modulation to twelve keys, which is a musical convention which was not always in place, a certain development of music thought over the ages. Over the not-trivial period this was worked on, many musicians found something was lost in the bargain. It had become the thing to do to have more keys/more variety and ultimately 'all twelve' and according to a system of ratios, the relationships got more and more problematic as per distance from "C Major". One should know this, and I frankly think setting forth one system is probably out of ignorance. <It's been a while since you checked it to see just how close you'd come to ET> rather reinforces that impression.
This is In The West; not everywhere, not by a long shot. Why did the Greeks settle on twelve after a time? Maybe it's cosmic, maybe they had the beautiful premise, maybe it just works for the ear in some way that's hard to define. But why seven of that is another matter still.
Like my rhetorical question, why would the Chinese go with six of twelve - quite like this seven but one omisssion, but that's the ear they have - and not the seven. Where there are instruments built to approximate seven equal to an octave, do they have a different mathematics do you think? Caucasian, please.
You have a premise you like but it isn't what happened, and isn't better really than the OP
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