Re: David Clampitt: question about trivalent scales with even numbers of notes

Hi David,

I see you are still posting to this list.

Saw your earlier interesting post on trivalent scales, which
I enjoyed.

Have you noticed the discussion of Dan Stearn's conjecture
that every trivalent scale with an even number of notes
includes the interval 1/1 2^(1/2) as one of the intervals
in one of the interval classes.

I.e. that one of its modes has to include the mid-point.

I wonder if anyone has proved this.

If so, would mean that there are no completely rational trivalent
scales with an even number of notes, as Dan commented
- quite a strong result!

(you can get a six note trivalent scale with only two irrational notes).

Also, I did a search of the SCALA archive,
and there were many trivalent scales, also many
that have four intervals of every size, and some with
more than that - could be a useful resource
for first tests for conjectures about them.
(see tuning files under Robert Walker)

Robert

Dear Robert,

I was not aware of this interesting conjecture. Almost all of my study of trivalent scales involves pairwise well-formedness (which entails trivalence). All pairwise well-formed scales have odd cardinality, and there are pairwise well-formed scales for all odd numbers greater than or equal to 3.

I'll try to think about the even cases.

Thanks,

David

>Hi David,
>
>I see you are still posting to this list.
>
>Saw your earlier interesting post on trivalent scales, which
>I enjoyed.
>
>Have you noticed the discussion of Dan Stearn's conjecture
>that every trivalent scale with an even number of notes
>includes the interval 1/1 2^(1/2) as one of the intervals
>in one of the interval classes.
>
>I.e. that one of its modes has to include the mid-point.
>
>I wonder if anyone has proved this.
>
>If so, would mean that there are no completely rational trivalent
>scales with an even number of notes, as Dan commented
>- quite a strong result!
>
>(you can get a six note trivalent scale with only two irrational notes).
>
>Also, I did a search of the SCALA archive,
>and there were many trivalent scales, also many
>that have four intervals of every size, and some with
>more than that - could be a useful resource
>for first tests for conjectures about them.
> (see tuning files under Robert Walker)
>
>Robert
>
>
>
>
>
>
>
>
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