Potential new terminology-'MOS boundary'

Hey,

Maybe Gene or Monz can refine this.

I propose a new term for a (new?) concept-a 'MOS boundary',
which could be an 'lower MOS boundary' and an 'upper MOS boundary'

A 'lower MOS Boundary' (in cents) for any MOS scale size, call it
's', would be any fifth size, call it 'f', such that an
infinitesimally smaller fifth size, call it 'f-a', would not be a
MOS for that scale size 's'.

As an example, for a 12-note MOS, the lower MOS boundary to three
decimal places is 685.715 cents

An 'upper MOS boundary' would be the opposite boundary in the
opposite direction, so it would be 'f+a', etc.

It appears that the 'upper MOS boundary' for a 12-note fifth is 720
cents.

Maybe there are interesting things going on at the boundaries, and
maybe this is a useful theoretical concept. On might define for
instance, a 'boundary mean MOS' where one could figure the size of
fifth halfway between the boundaries, etc. Here the boundary mean
MOS fifth would be ~702.8575.

Useful, useless? I don't know ;)

Best,
Aaron

--- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:
> A 'lower MOS Boundary' (in cents) for any MOS scale size, call it
> 's', would be any fifth size, call it 'f', such that an
> infinitesimally smaller fifth size, call it 'f-a', would not be a
> MOS for that scale size 's'.
>
> As an example, for a 12-note MOS, the lower MOS boundary to three
> decimal places is 685.715 cents
>
> An 'upper MOS boundary' would be the opposite boundary in the
> opposite direction, so it would be 'f+a', etc.
>

Wow! I've been thinking about how to predict these very values, and yesterday the
pattern dawned on me. Mainly I was trying to draw out a chart of what regions of
generators produce MOSes of which sizes.

It seems that under certain conditions, the region between two intervals of value
2^(a/b) and 2^(c/d) [in your perfect-fifth case, 3/5 and 4/7] will always have a MOS
of size b+d! And it looks like 2^[(a+c)/(b+d)] gives you the generator for equal L and
s sizes. Whether that would be the 'boundary mean MOS' you had in mind or not, I
don't know.

Woah. I, like, get it! That much, anyway.

> Maybe there are interesting things going on at the boundaries,

At the boundaries the L steps are devouring the s steps. So between 700cents and
720 cents you have 5L+7s, but as you near 720 the L's are so big that they just want
to eat those little s's. Yep. You get the "superpythagorean" system, and then beyond.

Fun stuff!
Jacob

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> --- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:

> It seems that under certain conditions, the region between two
intervals of value
> 2^(a/b) and 2^(c/d) [in your perfect-fifth case, 3/5 and 4/7] will
always have a MOS
> of size b+d! And it looks like 2^[(a+c)/(b+d)] gives you the
generator for equal L and
> s sizes. Whether that would be the 'boundary mean MOS' you had in
mind or not, I
> don't know.

This is connected to the Farey sequence. If you take the 12th row of
the Farey seqence, previous to 7/12 we get 4/7, and after it 3/5, so
there is a 4/7 < 7/12 < 3/5 in it.

From a boundry point of view, if n is the denominator it isn't just n
which is important, but also 2n; if you take the 24th row of the Farey
sequence, you get 11/19 < 7/12 < 10/17. Between 11/19 and 7/12 we may
take the fifth to be meantone, and between 7/12 and 10/17 to be (a
rather broadly defined) schismic. 3/5 can't be taken as a lower
boundry for superpyth, but it makes a reasonable upper boundry; the
22nd row of the Farey sequence contains 10/17 < 13/22 < 3/5, and this
could be taken as the superpyth range, though the flatter side of it
might be a temperament which isn't very good with commas 2240/2187 and
6144/6125. The true superpyth range can be considered
13/22 < 42/71 < 29/49 < 16/27, where 42/71 is "copop"; optimal in some
sense in both the 7 and 9 limits.