TC

Okay, I'd like to make one of my questions regarding this thread more clear.

Imagine some scale, S. Imagine multiplying each note in it by some factor,
F. Call the new scale S_f. Now see if any notes are shared between the
two scales. If they are, see if they occupy the same scale positions in
their respective scales. If they do, write...

S -TC-> S_f

If they don't, write...

S -/TC-> S_f.

Now, the main question is:

IF S -TC-> S_f AND S -TC-> S_g, can S -/TC-> S_(f-g)?

It is easy to see that this is the same as asking...

1. If a scale is TC by each of its notes, are each of the resulting scales
TC with eachother?

...or...

2. If a scale is TC by each of its notes, is it TC by each of the notes in
its diamond?

...or...

3. If a scale is TC by the intervals through which it is connected on a
rectangular lattice, is it TC by the intervals through which is is
connected on a triangular lattice?

-Carl

P.S. Contrary to what I implied in a previous post, this question has
nothing to do with: If a given mode is TC, will all modes be TC? We'll get
to that question next time.

I don't see what is gained in talking about specific modes in this context.
The TC-related properties make no reference to a root, so the scales could
be very well be left as a set of octave-repeating pitches without a
particular tonic note. All you're really doing is rotating the scale in
various ways and comparing the rotated configuration with the original one.

>I don't see what is gained in talking about specific modes in this context.

When you take a cross set of a scale and itself, you only multiply by the
intervals measured up from the root. So each mode of a scale has a
different cross set, right?

>The TC-related properties make no reference to a root, so the scales could
>be very well be left as a set of octave-repeating pitches without a
>particular tonic note.

I agree.

>All you're really doing is rotating the scale in various ways and comparing
>the rotated configuration with the original one.

Right, but not necessarily with eachother. That's what today's question
was all about.

-Carl

Carl Lumma wrote,

>When you take a cross set of a scale and itself, you only multiply by the
>intervals measured up from the root. So each mode of a scale has a
>different cross set, right?

Different only up to a transposition, which shouldn't matter. Say you take
the cross set of a scale with itself. Then you do the same for a mode of the
scale starting on a pitch a ratio r above the original root. That is
equivalent to transposing the original scale by 1/r. The resulting cross set
will be the same as the original cross set, transposed by 1/r�. (Can you
read that ASCII? That's supposed to be 1/(r^2).