Yahoo Tuning Groups Ultimate Backup tuning A summary of Mario's progression of cells for regular temperament people

Basically, Mario has discovered that 612-EDO is a really good 5-limit
tuning, and his progression of cells hints at some temperaments
supported by 612-EDO. This is pretty remarkable to do by hand and
without the aid of any temperament finders or what not.

Mario's progression of cells is a 612-note scale which is made up of
three step sizes, called M, J, and U.
They are:

M = |-15 8 1>
J = |101/4 -13 -2>
U = |12 -21/2 2>

Furthermore, these cells are aggregated into larger entities which
I'll call "metacells." These metacells are

S = MMJJMM = |-19/2 6 0>
V = MMJJUU = |89/2 -31 2>
R = MMJJ = |41/2 -10 -2>

Then Mario's progression of metacells is
SSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRS,
making up 106 metacells in total. You can generate the full
progression of cells from this by replacing each S with MMJJMM, each V
with MMJJUU, and each R with MMJJ, and then you can turn that into a
list of monzos using the values given above.

In addition to this, Mario places the additional constraint that for
every cell C, the cell C+358 must be exactly 3/2 higher than C, and
the cell C+254 must be exactly 4/3 higher than C. Mario claims that
this is already the case in his progression of cells as tuned to
(contorted) JI, but in reality this constraint necessarily implies the
following unnamed temperament:

http://x31eq.com/cgi-bin/rt.cgi?ets=306p_612&limit=5

This tempers out |485 -306 0> and yields a 612-note MOS which is two
chains of 306 fifths separated by a |-69 45 -1> counterschisma. This
should perhaps be Pizarro temperament (I will wait for him to
confirm). If you temper the progression of cells by this comma, a bit
of poking around will quickly lead you to find that more tempering is
required to make all of his desired mappings for important 5-limit
ratios work out (monzismic temperament turns up a lot), collapsing the
whole structure to the rank-1 612-EDO temperament. But the |485 -306
0> temperament is still notable for basically being implied
point-blank by his dictum that 358 cells is always 3/2 and 254 cells
is always 4/3 no matter where in the progression you are.

If Mario's condition of having 3/2 and 4/3 everywhere is relaxed, the
progression of metacells is itself a scale which is epimorphic under
the contorted 106-EDO patent val. Since this scale has only three step
sizes, we know that this can be treated as a MODded version of a
hexagon tiling the lattice with unison vectors |54 -37 2> (defining
monzismic), |-30 16 2> (the square of the schisma, defining
Helmholtz), and |24 -21 4> (defining vulture).

The full progression of cells itself is a 612-note scale which is
epimorphic under the 612-EDO patent val. Like the progression of
metacells, this can also be interpreted as a MODded hexagon, this time
with the three unison vectors being |161/4 -21 -3> (the fourth root of
the atom, defining atomic temperament), |27 -37/2 1> (the square root
of the monzisma, defining monzismic), and |-13.25 2.5 4> (the fourth
root of the kwazy comma, defining kwazy).

To sum it up, it looks like the |485 -306 0> is an (indirectly)
important comma in Mario's scheme, which we should perhaps name after
him.

As a last note, Mario also has a version of this progression with an
additional SS at the end, making up 108 metacells for 624 cells total,
and making it repeat at a ~624\612 step stretched octave of |101/4
-26 -4> which Mario likes (he's called it the 'toctave'). From a
regular temperament standpoint, this fact is irrelevant because it
doesn't change the structure of the temperament much.

-Mike

On Mon, Nov 5, 2012 at 4:05 AM, Mike Battaglia <battaglia01@...> wrote:
>
> Then Mario's progression of metacells is
> SSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRSSSSSSVSRS,
> making up 106 metacells in total. You can generate the full
> progression of cells from this by replacing each S with MMJJMM, each V
> with MMJJUU, and each R with MMJJ, and then you can turn that into a
> list of monzos using the values given above.

For those interested, here's a scale in monzo form of the progression
of 106 metacells:

|-9.5 6 0>
|-19 12 0>
|-28.5 18 0>
|16 -13 2>
|6.5 -7 2>
|27 -17 0>
|17.5 -11 0>
|8 -5 0>
|-1.5 1 0>
|-11 7 0>
|-20.5 13 0>
|-30 19 0>
|14.5 -12 2>
|5 -6 2>
|25.5 -16 0>
|16 -10 0>
|6.5 -4 0>
|-3 2 0>
|-12.5 8 0>
|-22 14 0>
|-31.5 20 0>
|13 -11 2>
|3.5 -5 2>
|24 -15 0>
|14.5 -9 0>
|5 -3 0>
|-4.5 3 0>
|-14 9 0>
|-23.5 15 0>
|-33 21 0>
|11.5 -10 2>
|2 -4 2>
|22.5 -14 0>
|13 -8 0>
|3.5 -2 0>
|-6 4 0>
|-15.5 10 0>
|-25 16 0>
|-34.5 22 0>
|10 -9 2>
|0.5 -3 2>
|21 -13 0>
|11.5 -7 0>
|2 -1 0>
|-7.5 5 0>
|-17 11 0>
|-26.5 17 0>
|-36 23 0>
|8.5 -8 2>
|-1 -2 2>
|19.5 -12 0>
|10 -6 0>
|0.5 0 0>
|-9 6 0>
|-18.5 12 0>
|-28 18 0>
|-37.5 24 0>
|7 -7 2>
|-2.5 -1 2>
|18 -11 0>
|8.5 -5 0>
|-1 1 0>
|-10.5 7 0>
|-20 13 0>
|-29.5 19 0>
|-39 25 0>
|5.5 -6 2>
|-4 0 2>
|16.5 -10 0>
|7 -4 0>
|-2.5 2 0>
|-12 8 0>
|-21.5 14 0>
|-31 20 0>
|-40.5 26 0>
|4 -5 2>
|-5.5 1 2>
|15 -9 0>
|5.5 -3 0>
|-4 3 0>
|-13.5 9 0>
|-23 15 0>
|-32.5 21 0>
|-42 27 0>
|2.5 -4 2>
|-7 2 2>
|13.5 -8 0>
|4 -2 0>
|-5.5 4 0>
|-15 10 0>
|-24.5 16 0>
|-34 22 0>
|-43.5 28 0>
|1 -3 2>
|-8.5 3 2>
|12 -7 0>
|2.5 -1 0>
|-7 5 0>
|-16.5 11 0>
|-26 17 0>
|-35.5 23 0>
|-45 29 0>
|-0.5 -2 2>
|-10 4 2>
|10.5 -6 0>
|1 0 0>

=============

And here's a scale in monzo form of the full progression of 612 cells:

|-15 8 1>
|-30 16 2>
|-4.75 3 0>
|20.5 -10 -2>
|5.5 -2 -1>
|-9.5 6 0>
|-24.5 14 1>
|-39.5 22 2>
|-14.25 9 0>
|11 -4 -2>
|-4 4 -1>
|-19 12 0>
|-34 20 1>
|-49 28 2>
|-23.75 15 0>
|1.5 2 -2>
|-13.5 10 -1>
|-28.5 18 0>
|-43.5 26 1>
|-58.5 34 2>
|-33.25 21 0>
|-8 8 -2>
|4 -2.5 0>
|16 -13 2>
|1 -5 3>
|-14 3 4>
|11.25 -10 2>
|36.5 -23 0>
|21.5 -15 1>
|6.5 -7 2>
|-8.5 1 3>
|-23.5 9 4>
|1.75 -4 2>
|27 -17 0>
|12 -9 1>
|-3 -1 2>
|22.25 -14 0>
|47.5 -27 -2>
|32.5 -19 -1>
|17.5 -11 0>
|2.5 -3 1>
|-12.5 5 2>
|12.75 -8 0>
|38 -21 -2>
|23 -13 -1>
|8 -5 0>
|-7 3 1>
|-22 11 2>
|3.25 -2 0>
|28.5 -15 -2>
|13.5 -7 -1>
|-1.5 1 0>
|-16.5 9 1>
|-31.5 17 2>
|-6.25 4 0>
|19 -9 -2>
|4 -1 -1>
|-11 7 0>
|-26 15 1>
|-41 23 2>
|-15.75 10 0>
|9.5 -3 -2>
|-5.5 5 -1>
|-20.5 13 0>
|-35.5 21 1>
|-50.5 29 2>
|-25.25 16 0>
|0 3 -2>
|-15 11 -1>
|-30 19 0>
|-45 27 1>
|-60 35 2>
|-34.75 22 0>
|-9.5 9 -2>
|2.5 -1.5 0>
|14.5 -12 2>
|-0.5 -4 3>
|-15.5 4 4>
|9.75 -9 2>
|35 -22 0>
|20 -14 1>
|5 -6 2>
|-10 2 3>
|-25 10 4>
|0.25 -3 2>
|25.5 -16 0>
|10.5 -8 1>
|-4.5 0 2>
|20.75 -13 0>
|46 -26 -2>
|31 -18 -1>
|16 -10 0>
|1 -2 1>
|-14 6 2>
|11.25 -7 0>
|36.5 -20 -2>
|21.5 -12 -1>
|6.5 -4 0>
|-8.5 4 1>
|-23.5 12 2>
|1.75 -1 0>
|27 -14 -2>
|12 -6 -1>
|-3 2 0>
|-18 10 1>
|-33 18 2>
|-7.75 5 0>
|17.5 -8 -2>
|2.5 0 -1>
|-12.5 8 0>
|-27.5 16 1>
|-42.5 24 2>
|-17.25 11 0>
|8 -2 -2>
|-7 6 -1>
|-22 14 0>
|-37 22 1>
|-52 30 2>
|-26.75 17 0>
|-1.5 4 -2>
|-16.5 12 -1>
|-31.5 20 0>
|-46.5 28 1>
|-61.5 36 2>
|-36.25 23 0>
|-11 10 -2>
|1 -0.5 0>
|13 -11 2>
|-2 -3 3>
|-17 5 4>
|8.25 -8 2>
|33.5 -21 0>
|18.5 -13 1>
|3.5 -5 2>
|-11.5 3 3>
|-26.5 11 4>
|-1.25 -2 2>
|24 -15 0>
|9 -7 1>
|-6 1 2>
|19.25 -12 0>
|44.5 -25 -2>
|29.5 -17 -1>
|14.5 -9 0>
|-0.5 -1 1>
|-15.5 7 2>
|9.75 -6 0>
|35 -19 -2>
|20 -11 -1>
|5 -3 0>
|-10 5 1>
|-25 13 2>
|0.25 0 0>
|25.5 -13 -2>
|10.5 -5 -1>
|-4.5 3 0>
|-19.5 11 1>
|-34.5 19 2>
|-9.25 6 0>
|16 -7 -2>
|1 1 -1>
|-14 9 0>
|-29 17 1>
|-44 25 2>
|-18.75 12 0>
|6.5 -1 -2>
|-8.5 7 -1>
|-23.5 15 0>
|-38.5 23 1>
|-53.5 31 2>
|-28.25 18 0>
|-3 5 -2>
|-18 13 -1>
|-33 21 0>
|-48 29 1>
|-63 37 2>
|-37.75 24 0>
|-12.5 11 -2>
|-0.5 0.5 0>
|11.5 -10 2>
|-3.5 -2 3>
|-18.5 6 4>
|6.75 -7 2>
|32 -20 0>
|17 -12 1>
|2 -4 2>
|-13 4 3>
|-28 12 4>
|-2.75 -1 2>
|22.5 -14 0>
|7.5 -6 1>
|-7.5 2 2>
|17.75 -11 0>
|43 -24 -2>
|28 -16 -1>
|13 -8 0>
|-2 0 1>
|-17 8 2>
|8.25 -5 0>
|33.5 -18 -2>
|18.5 -10 -1>
|3.5 -2 0>
|-11.5 6 1>
|-26.5 14 2>
|-1.25 1 0>
|24 -12 -2>
|9 -4 -1>
|-6 4 0>
|-21 12 1>
|-36 20 2>
|-10.75 7 0>
|14.5 -6 -2>
|-0.5 2 -1>
|-15.5 10 0>
|-30.5 18 1>
|-45.5 26 2>
|-20.25 13 0>
|5 0 -2>
|-10 8 -1>
|-25 16 0>
|-40 24 1>
|-55 32 2>
|-29.75 19 0>
|-4.5 6 -2>
|-19.5 14 -1>
|-34.5 22 0>
|-49.5 30 1>
|-64.5 38 2>
|-39.25 25 0>
|-14 12 -2>
|-2 1.5 0>
|10 -9 2>
|-5 -1 3>
|-20 7 4>
|5.25 -6 2>
|30.5 -19 0>
|15.5 -11 1>
|0.5 -3 2>
|-14.5 5 3>
|-29.5 13 4>
|-4.25 0 2>
|21 -13 0>
|6 -5 1>
|-9 3 2>
|16.25 -10 0>
|41.5 -23 -2>
|26.5 -15 -1>
|11.5 -7 0>
|-3.5 1 1>
|-18.5 9 2>
|6.75 -4 0>
|32 -17 -2>
|17 -9 -1>
|2 -1 0>
|-13 7 1>
|-28 15 2>
|-2.75 2 0>
|22.5 -11 -2>
|7.5 -3 -1>
|-7.5 5 0>
|-22.5 13 1>
|-37.5 21 2>
|-12.25 8 0>
|13 -5 -2>
|-2 3 -1>
|-17 11 0>
|-32 19 1>
|-47 27 2>
|-21.75 14 0>
|3.5 1 -2>
|-11.5 9 -1>
|-26.5 17 0>
|-41.5 25 1>
|-56.5 33 2>
|-31.25 20 0>
|-6 7 -2>
|-21 15 -1>
|-36 23 0>
|-51 31 1>
|-66 39 2>
|-40.75 26 0>
|-15.5 13 -2>
|-3.5 2.5 0>
|8.5 -8 2>
|-6.5 0 3>
|-21.5 8 4>
|3.75 -5 2>
|29 -18 0>
|14 -10 1>
|-1 -2 2>
|-16 6 3>
|-31 14 4>
|-5.75 1 2>
|19.5 -12 0>
|4.5 -4 1>
|-10.5 4 2>
|14.75 -9 0>
|40 -22 -2>
|25 -14 -1>
|10 -6 0>
|-5 2 1>
|-20 10 2>
|5.25 -3 0>
|30.5 -16 -2>
|15.5 -8 -1>
|0.5 0 0>
|-14.5 8 1>
|-29.5 16 2>
|-4.25 3 0>
|21 -10 -2>
|6 -2 -1>
|-9 6 0>
|-24 14 1>
|-39 22 2>
|-13.75 9 0>
|11.5 -4 -2>
|-3.5 4 -1>
|-18.5 12 0>
|-33.5 20 1>
|-48.5 28 2>
|-23.25 15 0>
|2 2 -2>
|-13 10 -1>
|-28 18 0>
|-43 26 1>
|-58 34 2>
|-32.75 21 0>
|-7.5 8 -2>
|-22.5 16 -1>
|-37.5 24 0>
|-52.5 32 1>
|-67.5 40 2>
|-42.25 27 0>
|-17 14 -2>
|-5 3.5 0>
|7 -7 2>
|-8 1 3>
|-23 9 4>
|2.25 -4 2>
|27.5 -17 0>
|12.5 -9 1>
|-2.5 -1 2>
|-17.5 7 3>
|-32.5 15 4>
|-7.25 2 2>
|18 -11 0>
|3 -3 1>
|-12 5 2>
|13.25 -8 0>
|38.5 -21 -2>
|23.5 -13 -1>
|8.5 -5 0>
|-6.5 3 1>
|-21.5 11 2>
|3.75 -2 0>
|29 -15 -2>
|14 -7 -1>
|-1 1 0>
|-16 9 1>
|-31 17 2>
|-5.75 4 0>
|19.5 -9 -2>
|4.5 -1 -1>
|-10.5 7 0>
|-25.5 15 1>
|-40.5 23 2>
|-15.25 10 0>
|10 -3 -2>
|-5 5 -1>
|-20 13 0>
|-35 21 1>
|-50 29 2>
|-24.75 16 0>
|0.5 3 -2>
|-14.5 11 -1>
|-29.5 19 0>
|-44.5 27 1>
|-59.5 35 2>
|-34.25 22 0>
|-9 9 -2>
|-24 17 -1>
|-39 25 0>
|-54 33 1>
|-69 41 2>
|-43.75 28 0>
|-18.5 15 -2>
|-6.5 4.5 0>
|5.5 -6 2>
|-9.5 2 3>
|-24.5 10 4>
|0.75 -3 2>
|26 -16 0>
|11 -8 1>
|-4 0 2>
|-19 8 3>
|-34 16 4>
|-8.75 3 2>
|16.5 -10 0>
|1.5 -2 1>
|-13.5 6 2>
|11.75 -7 0>
|37 -20 -2>
|22 -12 -1>
|7 -4 0>
|-8 4 1>
|-23 12 2>
|2.25 -1 0>
|27.5 -14 -2>
|12.5 -6 -1>
|-2.5 2 0>
|-17.5 10 1>
|-32.5 18 2>
|-7.25 5 0>
|18 -8 -2>
|3 0 -1>
|-12 8 0>
|-27 16 1>
|-42 24 2>
|-16.75 11 0>
|8.5 -2 -2>
|-6.5 6 -1>
|-21.5 14 0>
|-36.5 22 1>
|-51.5 30 2>
|-26.25 17 0>
|-1 4 -2>
|-16 12 -1>
|-31 20 0>
|-46 28 1>
|-61 36 2>
|-35.75 23 0>
|-10.5 10 -2>
|-25.5 18 -1>
|-40.5 26 0>
|-55.5 34 1>
|-70.5 42 2>
|-45.25 29 0>
|-20 16 -2>
|-8 5.5 0>
|4 -5 2>
|-11 3 3>
|-26 11 4>
|-0.75 -2 2>
|24.5 -15 0>
|9.5 -7 1>
|-5.5 1 2>
|-20.5 9 3>
|-35.5 17 4>
|-10.25 4 2>
|15 -9 0>
|0 -1 1>
|-15 7 2>
|10.25 -6 0>
|35.5 -19 -2>
|20.5 -11 -1>
|5.5 -3 0>
|-9.5 5 1>
|-24.5 13 2>
|0.75 0 0>
|26 -13 -2>
|11 -5 -1>
|-4 3 0>
|-19 11 1>
|-34 19 2>
|-8.75 6 0>
|16.5 -7 -2>
|1.5 1 -1>
|-13.5 9 0>
|-28.5 17 1>
|-43.5 25 2>
|-18.25 12 0>
|7 -1 -2>
|-8 7 -1>
|-23 15 0>
|-38 23 1>
|-53 31 2>
|-27.75 18 0>
|-2.5 5 -2>
|-17.5 13 -1>
|-32.5 21 0>
|-47.5 29 1>
|-62.5 37 2>
|-37.25 24 0>
|-12 11 -2>
|-27 19 -1>
|-42 27 0>
|-57 35 1>
|-72 43 2>
|-46.75 30 0>
|-21.5 17 -2>
|-9.5 6.5 0>
|2.5 -4 2>
|-12.5 4 3>
|-27.5 12 4>
|-2.25 -1 2>
|23 -14 0>
|8 -6 1>
|-7 2 2>
|-22 10 3>
|-37 18 4>
|-11.75 5 2>
|13.5 -8 0>
|-1.5 0 1>
|-16.5 8 2>
|8.75 -5 0>
|34 -18 -2>
|19 -10 -1>
|4 -2 0>
|-11 6 1>
|-26 14 2>
|-0.75 1 0>
|24.5 -12 -2>
|9.5 -4 -1>
|-5.5 4 0>
|-20.5 12 1>
|-35.5 20 2>
|-10.25 7 0>
|15 -6 -2>
|0 2 -1>
|-15 10 0>
|-30 18 1>
|-45 26 2>
|-19.75 13 0>
|5.5 0 -2>
|-9.5 8 -1>
|-24.5 16 0>
|-39.5 24 1>
|-54.5 32 2>
|-29.25 19 0>
|-4 6 -2>
|-19 14 -1>
|-34 22 0>
|-49 30 1>
|-64 38 2>
|-38.75 25 0>
|-13.5 12 -2>
|-28.5 20 -1>
|-43.5 28 0>
|-58.5 36 1>
|-73.5 44 2>
|-48.25 31 0>
|-23 18 -2>
|-11 7.5 0>
|1 -3 2>
|-14 5 3>
|-29 13 4>
|-3.75 0 2>
|21.5 -13 0>
|6.5 -5 1>
|-8.5 3 2>
|-23.5 11 3>
|-38.5 19 4>
|-13.25 6 2>
|12 -7 0>
|-3 1 1>
|-18 9 2>
|7.25 -4 0>
|32.5 -17 -2>
|17.5 -9 -1>
|2.5 -1 0>
|-12.5 7 1>
|-27.5 15 2>
|-2.25 2 0>
|23 -11 -2>
|8 -3 -1>
|-7 5 0>
|-22 13 1>
|-37 21 2>
|-11.75 8 0>
|13.5 -5 -2>
|-1.5 3 -1>
|-16.5 11 0>
|-31.5 19 1>
|-46.5 27 2>
|-21.25 14 0>
|4 1 -2>
|-11 9 -1>
|-26 17 0>
|-41 25 1>
|-56 33 2>
|-30.75 20 0>
|-5.5 7 -2>
|-20.5 15 -1>
|-35.5 23 0>
|-50.5 31 1>
|-65.5 39 2>
|-40.25 26 0>
|-15 13 -2>
|-30 21 -1>
|-45 29 0>
|-60 37 1>
|-75 45 2>
|-49.75 32 0>
|-24.5 19 -2>
|-12.5 8.5 0>
|-0.5 -2 2>
|-15.5 6 3>
|-30.5 14 4>
|-5.25 1 2>
|20 -12 0>
|5 -4 1>
|-10 4 2>
|-25 12 3>
|-40 20 4>
|-14.75 7 2>
|10.5 -6 0>
|-4.5 2 1>
|-19.5 10 2>
|5.75 -3 0>
|31 -16 -2>
|16 -8 -1>
|1 0 0>

-Mike

A summary of Mario's progression of cells for regular temperament people

🔗Mike Battaglia <battaglia01@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗martinsj013 <martinsj@...>

🔗Mike Battaglia <battaglia01@...>

🔗Mike Battaglia <battaglia01@...>