Odd-Limit and Dyadic-Odd-Limit Questions

Lets suppose that L(S) is the odd-limit of a chord S. DL(S) we will call the dyadic odd-limit of a chord S (the largest odd-limit of the multiset of dyads). For example, L(10:12:15) = 15, and DL(10:12:15) = 5.

Now I have some questions concerning chords with 3 or more notes:

1.) Is it possible to have an *otonal* chord where DL(S) < L(S)?
2.) Is it possible to have an *ambitonal* chord where DL(S) < L(S)?
3.) Is it possible to have a *utonal* chord where DL(S) = L(S)?

Mike told me that he thinks that case #1 and case #3 are impossible, but I don't know how to go about proving that. #2 is not possible for triads, but I don't know if it is possible for any other cases.

If anyone knows how to answer these questions, you are welcome to do so.

Ryan

On Mon, Feb 20, 2012 at 7:29 PM, Ryan Avella <domeofatonement@yahoo.com>
wrote:
>
> Lets suppose that L(S) is the odd-limit of a chord S. DL(S) we will call
> the dyadic odd-limit of a chord S (the largest odd-limit of the multiset of
> dyads). For example, L(10:12:15) = 15, and DL(10:12:15) = 5.
>
> Now I have some questions concerning chords with 3 or more notes:
>
> 1.) Is it possible to have an *otonal* chord where DL(S) < L(S)?
> 2.) Is it possible to have an *ambitonal* chord where DL(S) < L(S)?
> 3.) Is it possible to have a *utonal* chord where DL(S) = L(S)?
>
> Mike told me that he thinks that case #1 and case #3 are impossible, but I
> don't know how to go about proving that. #2 is not possible for triads, but
> I don't know if it is possible for any other cases.

No, I didn't say that. Before I go off talking about what I did say,
can you tell me how you're defining "otonal" and "utonal" above?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> No, I didn't say that. Before I go off talking about what I did say,
> can you tell me how you're defining "otonal" and "utonal" above?

I defined otonal and utonal the same way I defined them in my thread about ambitonal chords. If a chord has a smaller odd limit than its inverse, it is otonal. If it has a larger odd limit than its inverse, it is utonal. If the odd limits of the chord and its inverse are equal, they are ambitonal.

Ryan

On Mon, Feb 20, 2012 at 8:31 PM, Ryan Avella <domeofatonement@yahoo.com>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> > No, I didn't say that. Before I go off talking about what I did say,
> > can you tell me how you're defining "otonal" and "utonal" above?
>
> I defined otonal and utonal the same way I defined them in my thread about
> ambitonal chords. If a chord has a smaller odd limit than its inverse, it is
> otonal. If it has a larger odd limit than its inverse, it is utonal. If the
> odd limits of the chord and its inverse are equal, they are ambitonal.

OK, so then the answer is:

1) Yes - 3:7:15:21 is an example

2) Yes - 3:5:9:15 is an example

3) Yes - 10:12:15 is an example

What I'd originally said was that I didn't think there was an example
for #1, so I was wrong about that. I thought that all such examples
would be either ambitonal or utonal.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> OK, so then the answer is:
>
> 1) Yes - 3:7:15:21 is an example
>
> 2) Yes - 3:5:9:15 is an example
>
> 3) Yes - 10:12:15 is an example

Oh, okay sweet. So it is possible, then. But #1 and #2 are both impossible for triads, I suppose. That makes sense, since the outermost dyad must be reducible.

#3 doesn't look right though. The dyadic odd-limit (5) is smaller than the odd-limit (15), though I was looking for a utonal chord where DL(S) = L(S).

Ryan

On Mon, Feb 20, 2012 at 11:13 PM, Ryan Avella <domeofatonement@yahoo.com>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> > OK, so then the answer is:
> >
> > 1) Yes - 3:7:15:21 is an example
> >
> > 2) Yes - 3:5:9:15 is an example
> >
> > 3) Yes - 10:12:15 is an example
>
> Oh, okay sweet. So it is possible, then. But #1 and #2 are both impossible
> for triads, I suppose. That makes sense, since the outermost dyad must be
> reducible.
>
> #3 doesn't look right though. The dyadic odd-limit (5) is smaller than the
> odd-limit (15), though I was looking for a utonal chord where DL(S) = L(S).
>
> Ryan

Oh, sorry, I wasn't reading carefully for #3. The answer for 3 is no, because:

Let S* be the inverse of S. Some initial lemmas:

1) DL(S*) = DL(S), because all of the dyads in S and S* are exactly the same.
2) DL(S) <= L(S), meaning that the odd-limit of any chord can never be
less than its dyadic limit.

I'll leave proving these up to you; I think if you think about them
for a second you'll see they make sense.

Now assume that some utonal chord U exists such that DL(U) = L(U). Thus

3) DL(U) = L(U) (assumption we're testing)
4) L(U) > L(U*). (your definition of utonal)
5) DL(U) > L(U*) (substitute 3 in 4)
6) DL(U*) > L(U*) (substitute 1 in 6)

But, DL(U*) can never be greater than L(U*), because because of #2,
for any chord S, DL(S) must be less than or equal to L(S). So we've
arrived at a contradiction, and no utonal chords can exist where DL(S)
= L(S).

-Mike