Corner clipper commas

Suppose c is a comma (not assumed greater than one) whose denominator
is a power of two. It follows the numerator is odd, and so defines the
genus. The comma will be a scale note, either just greater or just
less than 1, and will define with 1 the only step of this size. If c
is a strong comma, this step will be the smallest one in the genus. If
we remove it, thereby clipping the corner of the genus, we get a scale
which can be interesting and sometimes epimorphic; the same remark
applies to removing the 1 and transposing (or equivalently, inverting
the scale.) Hence the clippers come in pairs.

Some five-limit corner clippers are 135/128, giving the
Ptolemy/Zarlino diatonic, 2025/2048, giving a 14-note clipper,
16875/16384, giving the epimorphic 19-tone Scott scale, 32805/32768,
giving an epimorphic 17-note clipper, the semicomma (orwell comma),
giving an epimorphic 31-note clipper. The atom is a corner clipper
comma, leading to a scale of 1104 notes which I gave up checking to
see if it was epimorphic since it was taking up time.

>Suppose c is a comma (not assumed greater than one) whose denominator
>is a power of two. It follows the numerator is odd, and so defines the
>genus.

Which genus?

>The comma will be a scale note, either just greater or just
>less than 1, and will define with 1 the only step of this size.

Huh?

>If c is a strong comma, this step will be the smallest one in the
>genus. If we remove it, thereby clipping the corner of the genus,

What's a corner? Are you deleting a pitch? Don't quite know what
a step is.

-Carl

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

> >Suppose c is a comma (not assumed greater than one) whose denominator
> >is a power of two. It follows the numerator is odd, and so defines the
> >genus.
>
> Which genus?

If n is the numerator, genus(n).

> >If c is a strong comma, this step will be the smallest one in the
> >genus. If we remove it, thereby clipping the corner of the genus,
>
> What's a corner? Are you deleting a pitch? Don't quite know what
> a step is.

Right, I'm deleting corner, though in my next posting it won't
necessarily be a coner. If we take the divisors of n, then we remove
either n or 1, and reduce to the octave, to get the clipped genus.

>> >Suppose c is a comma (not assumed greater than one) whose denominator
>> >is a power of two. It follows the numerator is odd, and so defines the
>> >genus.
>>
>> Which genus?
>
>If n is the numerator, genus(n).

And, pinch me, I even know what that is!

-Carl