self TC (was re: open problem)

; original TC => lattice TC
; new proposal => self TC

>>The scale I am most familiar with -- the diatonic scale in 12tET -- is
>>remarkable in that a mode change and its corresponding key change differ
>>the same number of notes, in opposite directions. Example: the dorian mode
>>of Cmaj, re-rooted on C as is done in the diamond, has 2 flats (3rd and
>>7th). Transposing Cmaj by its second degree gives Dmaj, which has two
>>sharps (also 3rd and 7th).
>
>Isn't that totally obvious?

I think so. Does this mean that CS and self TC are the same? In
particular, since all of a scale's modes share the same diamond, CS is a
property of scales. But all of a scale's modes do not share the same
square set, so is self TC a property of modes or scales? That is: if one
mode of a scale has self TC, do all modes?

>I mean, taking the second mode is completely equivalent to transposing
>_down_ by the interval from tonic to second degree. What scale could
>possibly not have this property?

That transposing a scale by any ratio and its reciprocal would not change
the accidentals the same amount in opposite directions? None. So I
retract my statement that it is remarkable in the diatonic scale.

-Carl