Ambitonal Chords

Yesterday Keenan and I were having a discussion about otonality and utonality on the Xenharmonic Alliance chat. We were talking about chords which aren't otonal/utonal, and which aren't necessarily Anomalous Saturated Supsensions either. An example of such a chord is 8:10:15, whose inverse is 8:12:15. As you can see, the inverse still has the same odd limit.

Anyways, we found a generalized pattern that seemed to hold true for such "ambitonal" chords. All of these chords (after removing factors of 2) have the form (A):(B):(BC), where A, B, and C are all odd numbers, B>A, C>1, and C does not equal A. There might be further restrictions on the variables but these are the only obvious ones I found.

Now of course, it is easy to see why (A):(B):(BC) is "ambitonal," since its inverse is (BC):(AC):(A). BC is the greatest odd factor in both the chord and its inversion, it is therefore neither otonal or utonal.

I'm afraid I don't have a wide enough background in mathematics to figure it all out though. The above pattern might accidentally report false negatives, but I wouldn't know how to go about testing that. Could someone here with a wider background in math help me out here?

-Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:

> I'm afraid I don't have a wide enough background in mathematics to figure it all out though. The above pattern might accidentally report false negatives, but I wouldn't know how to go about testing that. Could someone here with a wider background in math help me out here?

I'm not sure what you are asking; are you interested only in triads.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@> wrote:
>
> > I'm afraid I don't have a wide enough background in mathematics to figure it all out though. The above pattern might accidentally report false negatives, but I wouldn't know how to go about testing that. Could someone here with a wider background in math help me out here?
>
> I'm not sure what you are asking; are you interested only in triads.
>

I think triads would be a lot simpler. Though generalizing it to tetrads, pentads and whatnot would be cool too.

-Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:

> I think triads would be a lot simpler. Though generalizing it to tetrads, pentads and whatnot would be cool too.

The following discusses how to classify asses:

http://x31eq.com/ass.htm

One way of proceeding is simply to start from here, and look at subsets of the asses.

On Thu, Oct 27, 2011 at 7:11 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> > I think triads would be a lot simpler. Though generalizing it to tetrads, pentads and whatnot would be cool too.
>
> The following discusses how to classify asses:
>
> http://x31eq.com/ass.htm
>
> One way of proceeding is simply to start from here, and look at subsets of the asses.

I think Ryan's talking about asses that aren't saturated.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I think Ryan's talking about asses that aren't saturated.

Yes, that's the point. An ass is maximal in some odd limit, so a chord which is not maximal must be a subset of an ass.

On Thu, Oct 27, 2011 at 8:59 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I think Ryan's talking about asses that aren't saturated.
>
> Yes, that's the point. An ass is maximal in some odd limit, so a chord which is not maximal must be a subset of an ass.

Oh, sorry, I didn't see your very last sentence in the original
thread. Yes, just parts of asses, to which things can be added.

However, this is part of the reason why, for the "otonality
coefficient' stuff I worked out a while ago, that I used tenney height
instead of odd-limit. The otonality coeffcient was
log(complexity(1/chord)/complexity(chord)). If you define complexity
as the max odd factor of the chord, you end up getting a coefficient
of 0 for 8:12:15. But if you define it as the tenney height of the
chord, with all factors of 2 removed, you get that 8:12:15 is slightly
less complex than 8:10:15, so 8:12:15 is slightly more utonal and
8:12:15 is slightly more otonal.

Paul has countered that octave-equivalent TH isn't good because it
makes 7/5 more complex than 32/31. I'd also counter that, in that
case, TH itself isn't good because it makes 7/5 more complex than
31/1. And I'd also counter that using odd-limit to determine whether a
chord is more "utonal" than "otonal" isn't good because it makes
4:5:6:7:105 "ambitonal," despite that it's of considerably higher
complexity in the utonal series - 4:60:70:84:105.

-Mike

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Yes, that's the point. An ass is maximal in some odd limit, so a chord which is not maximal must be a subset of an ass.

Or a subset of a good old otonality or utonality.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > Yes, that's the point. An ass is maximal in some odd limit, so a chord which is not maximal must be a subset of an ass.
>
> Or a subset of a good old otonality or utonality.

Yes, but such a chord is unlikely to be ambitonal.

"genewardsmith" <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "Keenan Pepper"
> <keenanpepper@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "genewardsmith"
> > <genewardsmith@> wrote:
> > > Yes, that's the point. An ass is maximal in some odd
> > > limit, so a chord which is not maximal must be a
> > > subset of an ass.
> >
> > Or a subset of a good old otonality or utonality.
>
> Yes, but such a chord is unlikely to be ambitonal.

The original chords that were mentioned here -- 8:10:15 and
8:12:15 -- are obviously 15-limit otonal subsets. They
aren't anomalous chords in a lower limit because the 8:15 is
15-limit.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "genewardsmith" <genewardsmith@...> wrote:

> > Yes, but such a chord is unlikely to be ambitonal.
>
> The original chords that were mentioned here -- 8:10:15 and
> 8:12:15 -- are obviously 15-limit otonal subsets. They
> aren't anomalous chords in a lower limit because the 8:15 is
> 15-limit.

Yes, but they are subsets of the 1-3-5-15 MOS; a randomly selected otonal chord is not likely to produce an ambitonal result.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Yes, but they are subsets of the 1-3-5-15 MOS; a randomly selected otonal chord is not likely to produce an ambitonal result.

MOS? As in Moment of Symmetry?

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > Yes, but they are subsets of the 1-3-5-15 MOS; a randomly selected otonal chord is not likely to produce an ambitonal result.
>
> MOS? As in Moment of Symmetry?

Brain fart for ASS.

On Fri, Oct 28, 2011 at 7:30 PM, Graham Breed <gbreed@gmail.com> wrote:
>
> The original chords that were mentioned here -- 8:10:15 and
> 8:12:15 -- are obviously 15-limit otonal subsets. They
> aren't anomalous chords in a lower limit because the 8:15 is
> 15-limit.

If 8:10:15 and 8:12:15 are both otonal, then what about 8:10:12:15?
And what about 10:12:15?

I still suggest the best way to go is to compare a chord's complexity
with the complexity of its utonal inverse in lowest terms to make the
decision. For example, 10:12:15 is more complex than 4:5:6 =
1/(10:12:15), so it's utonal. But, I don't believe in the concept of
utonality anyway; to me there are otonal chords and dyadic chords.

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:
> On Fri, Oct 28, 2011 at 7:30 PM, Graham Breed
> <gbreed@gmail.com> wrote:
> >
> > The original chords that were mentioned here -- 8:10:15
> > and 8:12:15 -- are obviously 15-limit otonal subsets.
> > They aren't anomalous chords in a lower limit because
> > the 8:15 is 15-limit.
>
> If 8:10:15 and 8:12:15 are both otonal, then what about
> 8:10:12:15? And what about 10:12:15?

8:10:12:15 is the 1.3.5 Euler genus. As such, it's the
same as either an otonal or utonal chord, and so what we
called "symmetric". But it isn't an ASS because it isn't
anomalous: it looks like the 15-limit and the 8:15 is a
15-limit interval. Any subset including both 8 and 15 will
be both a 15-limit otonality and utonality.

10:12:15 is the 5-limit utonality.

Often an ASS is symmetric, and so the second S used to
stand for "symmetric". 3:5:9:15, or 6:9:10:15, is a
symmetric 9-limit ASS.

> I still suggest the best way to go is to compare a
> chord's complexity with the complexity of its utonal
> inverse in lowest terms to make the decision. For
> example, 10:12:15 is more complex than 4:5:6 =
> 1/(10:12:15), so it's utonal. But, I don't believe in the
> concept of utonality anyway; to me there are otonal
> chords and dyadic chords.

Otonal and utonal subsets are two ways of finding dyadic
chords (if that means what I think it means). They're the
only ways of finding dyadic chords up to the 7-limit, and
you might have assumed that all dyadic chords were
otonal or utonal subsets. The chords that break the rule
will be subsets of anomalous saturated suspensions by
definition. Anomalous because they break the rule, saturated
when you can't add an octave-equivalent pitch class without
raising the odd limit, and suspensions because it makes a
better acronym than "ASC".

Graham