One Measure for the Region of Influence of a Dyad

One clever method to measure the (relative) region of influence of a dyad is to find the largest region containing that dyad and no other simpler dyads. For example, in the region between 4/3 and 3/2, there is no simpler dyad than 7/5.

Suppose the interval in question is A/B, and we want to find the region of influence. We start with the extended euclidean algorithm, A*D+B*C=1, where we attempt to find the smallest integer solutions for C and D. The farey pair generating A/B is then determined to be the absolute values of C/D and (C-A)/(D-B), which are the extreme bounds of the region of influence.

The computed region has strange implications for music theory though - an interval might have a larger region of influence in the positive-cents direction, but a smaller region in the negative-cents direction.

We can correct this by one of two methods:
1.) Determine the upper-half and lower-half regions of the dyad, and take some sort of mean (e.g. arithmetic or geometric). Let this be the width of both halves.
2.) Choose the minimum or maximum of the upper/lower half to be the width of both halves.

The above methods will guarantee that our interval A/B is in the absolute center of the region of influence. However, only the minimum method will guarantee that A/B is the simplest interval in the region of influence.

Ryan Avella

Great idea! I was originally going to write that this might sync up well
with "Harmonic max-Entropy" (aka HE with the Renyi Entropy extension and
with a set to infinity), but this is actually something different and quite
clever.

It would be a good idea to set up an HE calculation that assigns interval
regions these sort of widths rather than the usual mediant-to-mediant
widths, and then uses these probabilities to seed Harmonic Renyi Entropy
with varying settings for a. I'd love to see what this looks like as N
approaches infinity, for both the "Farey" (should be Weil) or "Tenney"
bounded sets.

It would also be nice to see if we can determine some sort of rough rule
that these interval widths follow: for mediant-to-mediant widths they're
mostly 1/sqrt(n*d), but I'm not sure what they'd be here.

Mike

On Sat, Jun 8, 2013 at 11:30 AM, Ryan Avella <domeofatonement@yahoo.com>wrote:

> **
>
>
> One clever method to measure the (relative) region of influence of a dyad
> is to find the largest region containing that dyad and no other simpler
> dyads. For example, in the region between 4/3 and 3/2, there is no simpler
> dyad than 7/5.
>
> Suppose the interval in question is A/B, and we want to find the region of
> influence. We start with the extended euclidean algorithm, A*D+B*C=1, where
> we attempt to find the smallest integer solutions for C and D. The farey
> pair generating A/B is then determined to be the absolute values of C/D and
> (C-A)/(D-B), which are the extreme bounds of the region of influence.
>
> The computed region has strange implications for music theory though - an
> interval might have a larger region of influence in the positive-cents
> direction, but a smaller region in the negative-cents direction.
>
> We can correct this by one of two methods:
> 1.) Determine the upper-half and lower-half regions of the dyad, and take
> some sort of mean (e.g. arithmetic or geometric). Let this be the width of
> both halves.
> 2.) Choose the minimum or maximum of the upper/lower half to be the width
> of both halves.
>
> The above methods will guarantee that our interval A/B is in the absolute
> center of the region of influence. However, only the minimum method will
> guarantee that A/B is the simplest interval in the region of influence.
>
> Ryan Avella
>
>
>

Hi,

in the past I came up with a ternary variant of the Stern-Brocot tree that centers around sqrt(2) - here's the subtree between 1/1 and 2/1: http://www.abload.de/img/tsbtreerep5h.png

You can get this structure by simply rearranging the nodes of a Stern-Brocot tree. I think this is exactly what you're looking for if you can express it algorithmically.

And here's a related diagram by David J. Finnamore that shows the relationship to Partch's odd limit: http://www.elvenminstrel.com/music/tuning/reference/orderly-filling-of-pitchsp.html

I can give you more data on the subject later, but currently I'm unable to access my main data hard disk because of a broken SATA cable (so it's nothing serious).

Best
-Gedankenwelt

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> The computed region has strange implications for music theory though - an interval might have a larger region of influence in the positive-cents direction, but a smaller region in the negative-cents direction.