Prime Weighted Errors and Complexity

I've just finished my long awaited treatise on weighted errors and the like. I'll probably change it tommorow, but why not have a look now?

http://microtonal.co.uk/primerr.pdf

Graham

At 02:44 PM 6/28/2006, you wrote:
>I've just finished my long awaited treatise on weighted errors and the
>like. I'll probably change it tommorow, but why not have a look now?
>
>http://microtonal.co.uk/primerr.pdf
>
> Graham

Everything made sense up to equation 4, and all of a sudden I
feel like I'm taking crazy pills. How is the Tenney-weighted
equal to the "size of that interval in just intonation"? That
seems more like no weighting. And then it's 1. How the hell
is it 1?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Everything made sense up to equation 4, and all of a sudden I
> feel like I'm taking crazy pills.

My problems start with equation 1, which doesn't say whether frequency
ratios or log frequency ratios is being used to measure interval size.

But, onward!

Gene Ward Smith wrote:

> My problems start with equation 1, which doesn't say whether frequency
> ratios or log frequency ratios is being used to measure interval size.

Good call! I'll have a look at the introduction today.

Graham

Carl Lumma wrote:
> At 02:44 PM 6/28/2006, you wrote:
> >>I've just finished my long awaited treatise on weighted errors and the >>like. I'll probably change it tommorow, but why not have a look now?
>>
>>http://microtonal.co.uk/primerr.pdf
>>
>> Graham
> > > Everything made sense up to equation 4, and all of a sudden I
> feel like I'm taking crazy pills. How is the Tenney-weighted
> equal to the "size of that interval in just intonation"? That
> seems more like no weighting. And then it's 1. How the hell
> is it 1?

The weighting of prime intervals is equal to the interval size. At least by my definition of "weighting" which I'll have to look at. It leads to intervals with the highest weight having the least importance, so I should probably call it a "buoyancy factor" instead. The buoyancy of an interval is proportional to its size. Otherwise, larger prime intervals (or more complex intervals in general) are naturally heavier.

The Tenney-weighted size of a JI prime interval is 1. For an octave, it's log(2)/log(2). For a 3:1, it's log(3)/log(3). For a 5:1, it's log(5)/log(5). And so on. What else did you think Tenney weighting was?

Graham

>>>I've just finished my long awaited treatise on weighted errors and the
>>>like. I'll probably change it tommorow, but why not have a look now?
>>>
>>>http://microtonal.co.uk/primerr.pdf
>>>
>>> Graham
>>
>> Everything made sense up to equation 4, and all of a sudden I
>> feel like I'm taking crazy pills. How is the Tenney-weighted
>> equal to the "size of that interval in just intonation"? That
>> seems more like no weighting. And then it's 1. How the hell
>> is it 1?
>
>The weighting of prime intervals is equal to the interval size. At
>least by my definition of "weighting" which I'll have to look at. It
>leads to intervals with the highest weight having the least importance,
>so I should probably call it a "buoyancy factor" instead. The buoyancy
>of an interval is proportional to its size. Otherwise, larger prime
>intervals (or more complex intervals in general) are naturally heavier.

Of primes only.

>The Tenney-weighted size of a JI prime interval is 1. For an octave,
>it's log(2)/log(2). For a 3:1, it's log(3)/log(3). For a 5:1, it's
>log(5)/log(5). And so on. What else did you think Tenney weighting
>was?

Of primes only.

-Carl