More on miracles

As I posted over on tuning-math, the "miracle" phenomenon is the
second of a class of similar such miracles, the first of which is the
meantone, and the third of which has a generator of around 35.4
cents. The miracle generator lies between 16/15 at 111.7 cents and
15/14 at 119.4 cents, analogously to the way the meantone lies
between 10/9 and 9/8. Scales constructed by iterating the miracle
generator are therefore analogous to scales constructed by iterating
the meantone; however what is actually done in that case is to take
advantage of the fact that the numerator of 9/8 is a square, so that
9/8 = (3/2)^2 2^(-1), and then iterate the 3/2 within the 2. If we
were to proceed in an analogous manner with the miracle generator, we
would take advantage of the fact that the numerator of 16/15 is a
square, and factor it as 16/15 = (4/3)^2 (5/3)^(-1); we would then
look for a miracle fourth in analogy to the meantone fifth to iterate
within 5/3, producing mean semitone systems.

To get the meantone fifth, we select a value for the meantone m of
m=sqrt(5/4), which gives us pure thirds, and then take the fifth to
be sqrt(2m). To get a miracle fourth, we can similarly select a value
s for the mean semitone of sqrt(8/7) = 115.6 cents,
giving us a miracle fourth of sqrt((5/3)s); however of more
significance we can require instead that we have pure octaves, which
we will obtain if we set s = (12/5)^(1/13) = 116.6 cents, very close
to the 72-et value of 2^(7/72) = 116 2/3 cents, as well as to the
value 116.7 attributed to Keenan and Secor on Joe Monzo's web page.
This in turn leads to a miracle fourth of sqrt((5/3)s) = 500.5 cents
which we can interate within a major sixth.

The 72-et approximates 4/3 by 2^(30/72) and 5/3 by 2^(53/72); if we
want MOS scales within 53 divisions of 5/3, we can look at the
semiconvergents to 30/53, which give us 4/7, 5/9, 9/16, 13/23, and
17/30. If we look for instance at 16 steps out of 53, we get steps of
size 5252522525252252 which repeat within an interval of repetition
5/3. If we prefer octaves, we can easily extend the pattern until we
reach an octave, and repeat within that; in this way we would get
525252252525225252525, which has the slightly irregular feature of
two steps of 5 in sequence. If we pick representative intervals
approximated by these steps, we get

1-21/20-15/14-9/8-8/7-6/5-11/9-5/4-21/16-4/3-7/5-10/7-3/2-32/21-
14/9-18/11-5/3-7/4-16/9-15/8-40/21-(2)

which gives us some idea of the resources of this scale, in either of
its forms. If we compare it to Blackjack, we have instead
525252525252525252522; we have again 10 5's and 11 2's, but
distributed differently.

If anyone wants to try this out, the version within 5/3 is given by

! 16-miracle.scl
!
16 out of 53 within 5/3 miracle scale
16
!
83.43006824
116.8020936
200.2321622
233.6041876
317.0342560
350.4062819
383.7783092
467.2083755
500.5804036
584.0104712
617.3824974
700.8125646
734.1845918
767.5566184
850.9866863
5/3

The version inside an octave is

! 16-miracle-oct.scl
!
21 out of 72 within 2/1 miracle scale, from 16-of-53 within 5/3
21
!
83.33333333
116.6666667
200.0000000
233.3333333
316.6666667
350.0000000
383.3333333
466.6666667
500.0000000
583.3333333
616.6666667
700.0000000
733.3333333
766.6666667
850.0000000
883.3333333
966.6666667
1000.000000
1083.333333
1116.666666
2/1

--- In tuning@y..., genewardsmith@j... wrote:

/tuning/topicId_28290.html#28290

> As I posted over on tuning-math, the "miracle" phenomenon is the
> second of a class of similar such miracles, the first of which is
the
> meantone, and the third of which has a generator of around 35.4
> cents. The miracle generator lies between 16/15 at 111.7 cents and
> 15/14 at 119.4 cents, analogously to the way the meantone lies
> between 10/9 and 9/8. Scales constructed by iterating the miracle
> generator are therefore analogous to scales constructed by
iterating the meantone

Thank you, Gene, for bringing these discoveries over to this list...

I guess we had discussed before the relationship of 31-tET, with, I
believe, a generator of 116.129 cents or 1/3 comma meantone with the
generator of the miracle family 116.667.

31-tET is, of course, also a fine scale for just harmonies, minor
thirds in particular, so it was interesting to see how such a small
variation of the "generator" could produce so many scales containing
just intervals...

I think I'm getting this right...

________ _______ _________
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> I guess we had discussed before the relationship of 31-tET, with, I
> believe, a generator of 116.129 cents or 1/3 comma meantone

31-tET <-> 1/4-comma meantone
19-tET <-> 1/3-comma meantone

> with the
> generator of the miracle family 116.667.
>
> 31-tET is, of course, also a fine scale for just harmonies, minor
> thirds in particular,

Major thirds in particular?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_28290.html#28531

> --- In tuning@y..., jpehrson@r... wrote:
>
> > I guess we had discussed before the relationship of 31-tET, with,
I
> > believe, a generator of 116.129 cents or 1/3 comma meantone
>
> 31-tET <-> 1/4-comma meantone
> 19-tET <-> 1/3-comma meantone
>
> > with the
> > generator of the miracle family 116.667.
> >
> > 31-tET is, of course, also a fine scale for just harmonies, minor
> > thirds in particular,
>
> Major thirds in particular?

Thanks, Paul... for correcting that slip....

_________ _______ _________
Joseph Pehrson