interesting property concerning meantone intervals

i've discovered an interesting property about
1/4-comma meantone which i haven't seen mentioned
before.

this concerns specifically a 12-tone chain of
1/4-comma meantone, which may be described as
the "8ve"-invariant set of pitches determined by
generators 5^(_p_/4), where _p_ = -3...+8.

i've found that the entire set of intervals that
can be found in this scale may be described as
the "8ve"-invariant set of intervals determined
by generators 5^(_i_/4), where _i_ = -11...+11.

i was wondering if this could be generalized to
the set of pitches 5^(_p_/4), where _p_ = a...b,
and the set of intervals 5^(_i_/4), where
i = (a-b)...(b-a).

and how about generalization to other forms of
meantone? to other types of scales in general?
my guess is that it has been written about before,
but my math-challenged brain missed it.

... ?

-monz
"all roads lead to n^0"

Ok, I'm not going to say that you shouldn't be surprised,
otherwise Johnny Reinhard will start laughing.

It is true for all meantone or Pythagorean generated scales.
Suppose _g_ is the size of the generator, and _a_ the size of
the octave. Then you can express each pitch as
x g + y a, where x in your case is in -3..8, and y such that
the pitch is in the range of one octave.
The intervals are two pitches subtracted, and the result
has the same form, say x'g + y'a.
So if the range of x is -3..8 then the range of the differences
of two x's is -3 - 8 .. 8 - -3 = -11 .. 11.

Manuel

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> i've discovered an interesting property about
> 1/4-comma meantone which i haven't seen mentioned
> before.
>
> this concerns specifically a 12-tone chain of
> 1/4-comma meantone, which may be described as
> the "8ve"-invariant set of pitches determined by
> generators 5^(_p_/4), where _p_ = -3...+8.
>
> i've found that the entire set of intervals that
> can be found in this scale may be described as
> the "8ve"-invariant set of intervals determined
> by generators 5^(_i_/4), where _i_ = -11...+11.

i'm sorry -- isn't this completely obvious?

> i was wondering if this could be generalized to
> the set of pitches 5^(_p_/4), where _p_ = a...b,
> and the set of intervals 5^(_i_/4), where
> i = (a-b)...(b-a).
>
> and how about generalization to other forms of
> meantone? to other types of scales in general?
> my guess is that it has been written about before,
> but my math-challenged brain missed it.
>
> ... ?

if a scale has a generator g, and its pitches are described by g^p,
where p=a...b, then its interval classes will of course be g^i, where
i = 0...(b-a), and if you don't use inversional equivalence, you must
include the inversions of all of those intervals, hence g^k where k =
(a-b)...(b-a). it follows immediately from the definition of interval.