Strange Rank 3 DE theorem that should work but doesn't

Let's assume that a rank-3 DE scale can be described by a scale
imprint such as abacabd. We'll assume that a is the smallest step
size, b is second-smallest, etc.

The largest chroma transforming a into d must be equal to the sum of
the two smaller chromata transforming it into b and c. This is a
direct implication of the reason why rank-3 scales have 4 step sizes
at all, which is that you have three chromata you can alter each
interval by - the first chromatic unison vector, the second chromatic
unison vector, and the sum of the two.

So for any imprint such as abacabd, for the scale to actually be
rank-3, e.g. for it to be constructible by three generators, we would
expect the following statement to hold:

d-a = c-a + b-a, thus

a+d = b+c

Sounds good, OK, let's test it: we'll look at the scale dbca. Let's
assume we tune this scale to 4:5:6:7, which is a rank-4 scale that
happens to have 4 step sizes, but requires 4 generators. However, if
we make b+c = a+d, then that's equivalent to tempering out 50/49, and
now the octave is constructible by two tritones. So far so good.

But I'm not seeing how it's supposed to work for the scale abacd. Try
6 7 6 8 9 - a+d = b+c = 15, but how the heck do you reduce this to
three generators?

Anyone have any insight into this case?

I think what's going on is that the above identity needs to hold for
-every- interval class in the MOS, not just the steps, and that there
might be more going on than that. I also have a conjecture that any
scale that meets the above property will either end up having one
interval class < 4, or perhaps having one or more Rothenberg
"ambiguous" intervals.

-Mike

Mike wrote:

>But I'm not seeing how it's supposed to work for the scale abacd. Try
>6 7 6 8 9 - a+d = b+c = 15, but how the heck do you reduce this to
>three generators?
>Anyone have any insight into this case?

Without looking at this example, it can be noted that not every
imprint of a,b,c,d is rank 3.

>I think what's going on is that the above identity needs to hold for
>-every- interval class in the MOS, not just the steps,

Yes, this is one way to understand why not every imprint is rank 3.

>I also have a conjecture that any
>scale that meets the above property will either end up having one
>interval class < 4, or perhaps having one or more Rothenberg
>"ambiguous" intervals.

Yes, I believe that's correct. Not every interval class can
have exactly 4 step sizes. I think that for a few scales -
identified by Keenan - every interval class can have exactly 3.
But most rank 3 scales will have mean variety > 3 and < 4.

-Carl