Question for Dave Keenan: Classic Mediant

Dave,

Hello!

I'm wondering if you would kindly explain the history and origins of
the use of the Classic Mediant?

Thanks,

Jacky Ligon

Hi Jacky -- Dave K. is away on vacation, so hopefully I can be of help here.
My guess is that you're reading too much into something. Where is the
original reference that led you to ask this question?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Hi Jacky -- Dave K. is away on vacation, so hopefully I can be of
help here.
> My guess is that you're reading too much into something. Where is
the
> original reference that led you to ask this question?

Paul,

I began to study his and Margo's paper "The Noble Mediant", after you
brought it to my attention a while back when I asked you
about "maximal beating" between simple ratios. I have just been
wondering about the history and origins of this, and am curious to
know if it's a neologism of Dave's.

Anyway, I'm finding it an extremely valuable concept. I may post some
of my findings from tinkering with the Classic and Noble Mediants.

Thanks,

Jacky Ligon

Jacky wrote,

>I began to study his and Margo's paper "The Noble Mediant", after you
>brought it to my attention a while back when I asked you
>about "maximal beating" between simple ratios. I have just been
>wondering about the history and origins of this, and am curious to
>know if it's a neologism of Dave's.

Dave and Margo used the term "Classic Mediant" to distinguish it from "Noble
Mediant". The term is usually simply called the "Mediant", and comes from
number theory. It importance is that if you have two adjacent fractions in a
"reasonable" series, by which I mean fractions p/q and t/u such that

q*t - p*u = 1

then the mediant, defined as r/s where

r/s = (p+t)/(q+u)

is the simplest fraction between p/q and t/u. Also, the mediant fits right
into the "reasonable" series, since

q*r - p*s = (q*(p+t)) - (p*(q+u))
= p*q + q*t - p*u - p*q
= p*q + 1 - p*q
= 1.