Pepper ambiguity

On Wed, 29 Mar 2006, Gene Ward Smith wrote:
> > We may define the p-limit Pepper ambiguity, for any odd prime p, as
> > the maximum of the ratios of the errors of the nearest approximation
> > to the members of the p-limit tonality diamond to the next nearest. In
> > the 5-limit, that means we look at the ratios of the errors for the
> > nearest approximations to 3/2, 5/4 and 5/3 to the next nearest. The
> > concept was proposed by Pepper Keenan.
>
> Sorry, this should say "any odd number greater than one" not "any odd
> prime".
>
> We may define the n-limit Pepper ambiguity, for any odd number n
> greater than one, as the maximum of the ratios of the errors of the
> nearest approximation to the members of the n-limit tonality diamond
> to the next nearest. In the 7-limit, that means we look at the ratios
> of the errors for the nearest approximations to 3/2, 5/4, 5/3, 7/4,
> 7/5 and 7/6 to the next nearest. The concept was proposed by Pepper
> Keenan.

(Keenan Pepper, of course.)

Gene, three questions for you:
1. Does this belong on tuning-math, or can the non-
mathematical members get something out of it?

2. Is the PA the ratio between the two errors or
between the first error and the second interval?

3. Could you (on either list) please spell out an
example: eg, for 3/2, in the 7-limit, the nearest
approximation is ... and the next nearest is ...;
consequently the errors of each are ... and ....;
the PA is the ratio of ... to ..., that is, ....

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@...> wrote:

> Gene, three questions for you:
> 1. Does this belong on tuning-math, or can the non-
> mathematical members get something out of it?

I put it here as I thought it was more likel to catch Monzo's eye
here, but I don't see anything so mathematical about it as to preclude
it from being discussed here.

> 2. Is the PA the ratio between the two errors or
> between the first error and the second interval?

For each member of the tonality diamond D, you take

error of best approximation of D / error of second best approximation

The maximum of all of these ratios is the Pepper ambiguity.

> 3. Could you (on either list) please spell out an
> example: eg, for 3/2, in the 7-limit, the nearest
> approximation is ... and the next nearest is ...;
> consequently the errors of each are ... and ....;
> the PA is the ratio of ... to ..., that is, ....

Let's say you take 12edo. The best approximation to 3/2 has an error
of 1.955 cents, and the second best has an error of 98.045 cents. The
ratio is 1.955/98.045 = 0.01993, and this is therefore the 3-limit
Pepper ambiguity for 12. For the 5-limit, you need to look at the
corresponding ratios for 5/4 and 5/3 also, and take the maximum,
obtaining 0.1854. In the 7-limit, you need to look also at the ratios
of 7/4, 7/5 and 7/6, obtaining 0.4954. The ambiguity is always between
zero and one.

Hi Gene,

On Thu, 30 Mar 2006 Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@...> wrote:

> > Gene, three questions for you:
> > 1. Does this belong on tuning-math, or can the non-
> > mathematical members get something out of it?
>
> I put it here as I thought it was more likel to catch Monzo's eye
> here, but I don't see anything so mathematical about it as to preclude
> it from being discussed here.
>
> > 2. Is the PA the ratio between the two errors or
> > between the first error and the second interval?
>
> For each member of the tonality diamond D, you take
>
> error of best approximation of D / error of second best approximation
>
> The maximum of all of these ratios is the Pepper ambiguity.
>
> > 3. Could you (on either list) please spell out an
> > example: eg, for 3/2, in the 7-limit, the nearest
> > approximation is ... and the next nearest is ...;
> > consequently the errors of each are ... and ....;
> > the PA is the ratio of ... to ..., that is, ....
>
> Let's say you take 12edo. The best approximation to 3/2 has an error
> of 1.955 cents, and the second best has an error of 98.045 cents. The
> ratio is 1.955/98.045 = 0.01993, and this is therefore the 3-limit
> Pepper ambiguity for 12. For the 5-limit, you need to look at the
> corresponding ratios for 5/4 and 5/3 also, and take the maximum,
> obtaining 0.1854. In the 7-limit, you need to look also at the ratios
> of 7/4, 7/5 and 7/6, obtaining 0.4954. The ambiguity is always between
> zero and one.

Excellent! So the 3-limit PA of 12-EDO is very small, which
indicates that one can very easily distinguish the best
approximation to 3/2 from the second-best; one will not be
confused as to which note to choose. Contrariwise, the 7-
limit PA is quite high, indicating that distinguishing the best
approximations to the ratios involving 7 will be harder.

That seems informative. however, rather than taking the
maximum of these ratios, wouldn't it be more informative to
take their geometric mean?

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@...> wrote:

> That seems informative. however, rather than taking the
> maximum of these ratios, wouldn't it be more informative to
> take their geometric mean?

It would be different; I don't see why it would be more informative.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@...> wrote:
>
>
> Hi Gene,
>
> On Thu, 30 Mar 2006 Gene Ward Smith wrote:
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@> wrote:
>
> > > Gene, three questions for you:
> > > 1. Does this belong on tuning-math, or can the non-
> > > mathematical members get something out of it?
> >
> > I put it here as I thought it was more likel to catch Monzo's eye
> > here,

Sorry guys, i've been preoccupied with other stuff and
haven't been following any of the tuning lists lately.
I'll get around to this when i can ... send me a private
email if you don't see anything after about a week.

-monz
http://tonalsoft.com
Tonescape microtonal music software