lattice TC (was re: open problem)

; original idea (TC of connecting intervals) => lattice TC
; new proposal (TC of square set) => self TC

>>1) Should we require TC by all of the intervals needed to connect the
>> scale, or simply by any one of them?
>>
>>2) What would be the preferred lattice type here, triangular or
>> rectangular? I would think that connectedness should always be defined
>> on a triangular lattice. My guess for defining TC would be to use
>> rectangular if we answer "all" to #1, triangular if we answer "any".
>
>I would suggest the opposite.

I'm not sure what opposite means here. But I've changed my mind -- we
should use triangular for everything, no matter what.

>I see I should amend my statement from before. Connectedness plus propriety
>implies coherence (if the propriety is strict, so is the coherence). I was
>tacitly assuming connectedness.

So all connected proper scales are TC? But which kind of connected? Only
1-D? If not, by which connecting intervals are they TC? This should
answer #1 for us.

-Carl

Carl, all strictly proper scales are TC.

>Carl, all strictly proper scales are TC.

TC by what?

1. Any interval?
2. Any group of intervals?
3. Any group of intervals by which it is connected?
etc.

-C.

There is no way to transpose a strictly proper scale so that a changing note
hops over an unchanging note.

On Tue, 2 Nov 1999, Carl Lumma wrote:
> TC by what?
>
> 1. Any interval?
> 2. Any group of intervals?
> 3. Any group of intervals by which it is connected?
> etc.

It seems to me, if I've understood correctly how you've defined it, that
in most cases it would be most meaninful to consider TC shorthand for
"TC by all intervals within the scale itself".

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "How about that? The guy can't run six balls,
-\-\-- o and they make him president."

>It seems to me, if I've understood correctly how you've defined it, that
>in most cases it would be most meaninful to consider TC shorthand for
>"TC by all intervals within the scale itself".

You mean only those intervals measured up from the root, or measured from
anywhere? If you mean the latter, then your suggested definition is
equivalent to CS (constant structureness).

I prefer the former. But if the answer to this question...

IF [S -TC-> S_f] AND [S -TC-> S_g], must [S -TC-> S_(f-g)]?

Example, S= 1/1, 5/4, 3/2:
IF [S -TC-> S_5/4] AND [S -TC-> S_3/2], must [S -TC-> S_6/5]?

...is "yes", then the former is the same as the latter, and TC is just CS.
If the answer to the question is "no", there will be scales whose modes
vary with respect to the former definition, and only scales whose modes all
meet the definition will be CS. I wrote...

>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?

But it isn't true! I was right the first time- if the answer to the $1
question is "yes", then all modes will be TC if any are TC. If the answer
is "no", different modes of the same scale could vary with respect to the
former definition.

-Carl