User:Eliora/Concoctic scale: Difference between revisions

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 On the other hand, in [[25edo]], stacking 18\25 will lead to maximum evenness scale of 7 note "black keys", and stacking 7\25 will result in a 18-note scale of "white keys". This is the EDO that only has the scale through the second formula.
+=== Observations ===
+A scale that is of the form (n/2+1)\n, where n is divisible by 4, is always orthoconcoctic. 12edo diatonic is also an example of such.
+It can be shown as follows:
+Let k = n/4 and rewrite the expression as (2k+1)\4k;
+(2k+1)^2 = 4k^2 + 4k + 1;
+k^2 is divisible by 4 and k and thus by 4k;
+k being divisible by 4k is self-explanatory.
+Therefore the remainder of +1 means that such a scale will always be orthoconcoctic. This type of scale, when used in keyboard making, produces two bundles of white keys whose numbers of black keys inside of them are 1 number apart, and so are the numbers of white keys themselves.
 == List ==