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.