Ennealimmal temperament

family name: ennalimmal, 72&99
period: 1/9 octave, or 133 1/3 cents. Very nearly 27/25
generator: 36/35 or 21/20

5-limit

name: ennealimmal
comma: ennealimma = 2/(27/25)^9 = |1 -27 18>
mapping: [<9 15 22|, <0 -2 -3|]
5-limit generator: 250/243
poptimal generators: 68/1665, 37/906
TOP period: 133.3322145
TOP generator: 49.0061286
TOP mapping: <1199.98993 1901.97096 2786.29033|
MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1503, 1665

7-limit

name: ennealimmal
wedgie: <<18 27 18 1 -22 -34||
mapping: [<9 15 22 26|, <0 -2 -3 -2|]
poptimal generator: 43/1053, 61/1494, 49/1200
(from the last, exactly 49 cents is a poptimal generator.)
TOP period: 133.3373752
TOP generator: 49.0239856
TOP mapping: <1200.03638 1902.01266 2786.35030 3368.72378|
TM basis: {2401/2400, 4375/4374}
MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1053

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

I forgot to include the 9-limit:

family name: ennalimmal, 72&99
period: 1/9 octave, or 133 1/3 cents. Very nearly 27/25
generator: 36/35 or 21/20

5-limit

name: ennealimmal
comma: ennealimma = 2/(27/25)^9 = |1 -27 18>
mapping: [<9 15 22|, <0 -2 -3|]
5-limit generator: 250/243
poptimal generators: 68/1665, 111/2718
TOP period: 133.3322145
TOP generator: 49.0061286
TOP mapping: <1199.98993 1901.97096 2786.29033|
MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1503, 1665

7-limit

name: ennealimmal
wedgie: <<18 27 18 1 -22 -34||
mapping: [<9 15 22 26|, <0 -2 -3 -2|]
7-limit poptimal generator: 43/1053, 61/1494, 147/3600
(from the last, exactly 49 cents is a poptimal generator.)
9-limit poptimal generator: 131/3231, 206/5067
TOP period: 133.3373752
TOP generator: 49.0239856
TOP mapping: <1200.03638 1902.01266 2786.35030 3368.72378|
TM basis: {2401/2400, 4375/4374}
7-limit MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1053
9-limit MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 783, 1395

Using 441 for the 7-limit and 612 for the 9-limit is reasonable

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> family name: ennalimmal, 72&99
> period: 1/9 octave, or 133 1/3 cents. Very nearly 27/25
> generator: 36/35 or 21/20
>
> <etc. -- snip>

thanks, Gene!

i'll have this in the new version of the Encyclopedia
when it's complete and uploaded.

-monz
http://tonalsoft.com
microtonal music software

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> family name: ennalimmal, 72&99
> period: 1/9 octave, or 133 1/3 cents. Very nearly 27/25
> generator: 36/35 or 21/20
>
> 5-limit
>
> <snip>
> MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1503, 1665
>
> 7-limit
>
> <snip>
> MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1053

after 612, does the MOS really go to 1503 in 5-limit
and to 1053 in 7-limit, or is there a typo there?

-monz
http://tonalsoft.com
microtonal music software

"monz" <monz@tonalsoft.com> writes:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > family name: ennalimmal, 72&99
> > period: 1/9 octave, or 133 1/3 cents. Very nearly 27/25
> > generator: 36/35 or 21/20
> >
> > 5-limit
> >
> > <snip>
> > MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1503, 1665
> >
> > 7-limit
> >
> > <snip>
> > MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1053
>
>
> after 612, does the MOS really go to 1503 in 5-limit
> and to 1053 in 7-limit, or is there a typo there?

1503 has to be a typo; consecutive MOSes can't differ in size by
factors of two or more. The size of MOS #n+1 is the size of MOS #n
plus the number of large steps in MOS #n.

- Rich Holmes

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > family name: ennalimmal, 72&99
> > period: 1/9 octave, or 133 1/3 cents. Very nearly 27/25
> > generator: 36/35 or 21/20
> >
> > 5-limit
> >
> > <snip>
> > MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1503, 1665
> >
> > 7-limit
> >
> > <snip>
> > MOS: 18, 27, 45, 72, 99, 171, 270, 441, 612, 1053
>
>
> after 612, does the MOS really go to 1503 in 5-limit
> and to 1053 in 7-limit, or is there a typo there?

Good spot! It's a typo; from the semiconvergents to 68/185 (generator
within period for 1665) you get

18, 27, 45, 72, 99, 171, 270, 441, 612, 1053, 1665

441+612=1053, so it's a compromise, making it a nice 7-limit choice
(in fact, poptimal.)