A map of starling space

I made a chart of ET's with the comma 126;125, and drew lines on it to illustrate some of the landmark starling temperaments. I couldn't find a name for <<1, -8, -26, -15, -44, -38]] ("Number 114"), so I decided to call it "grackle" (a name that originally referred to a kind of starling, and still does in some parts of the world).

http://www.io.com/~hmiller/png/starling-map.png

Note that I haven't bothered to limit the list to 7-limit consistent ET's, or to include ET's that might be compatible with starling in some mapping other than their closest approximation to 3, 5, and 7.

The temperaments illustrated are:
#114 grackle <<1, -8, -26, -15, -44, -38]]
#2 meantone <<1, 4, 10, 4, 13, 12]]
#49 diaschismic <<2, -4, -16, -11, -31, -26]]
#31 tripletone <<3, 0, -6, -7, -18, -14]]
#17 diminished <<4, 4, 4, -3, -5, -2]]
#112 muggles <<5, 1, -7, -10, -25, -19]]
#27 kleismic <<6, 5, 3, -6, -12, -7]]
#30 valentine (formerly quartaminorthirds) <<9, 5, -3, -13, -30, -21]]
#29 myna (formerly nonkleismic) <<10, 9, 7, -9, -17, -9]]

Any major starling-related temperaments that I'm missing?

The original starling map, like most of the others we've seen, is based on graphing the deviations from 3/1 and 5/1. Possibly a more useful map for a 7-limit temperament family is one that shows the deviations from 7/1. So I created a second map.

http://www.io.com/~hmiller/png/starling-map2.png

You can see that valentine and myna look good on this chart, while kleismic (which is a good 5-limit temperament) isn't so good in the 7-limit. Another benefit of this kind of chart is that you can use it to look for new temperaments, like the more familiar 5-limit charts

e.g. http://www.io.com/~hmiller/png/et-scales.png

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> I made a chart of ET's with the comma 126;125, and drew lines on it to
> illustrate some of the landmark starling temperaments.

Great! This is something I've suggested Paul do from time to time; I
didn't know you were up to this level of graphic design.

> Any major starling-related temperaments that I'm missing?

Here's a list of 21 starling temperaments, ordered in terms of
increasing Graham-TOP badness:

[4, 4, 4, -3, -5, -2]
[[4, 6, 9, 11], [0, 1, 1, 1]]
4 5.871540 93.944647

[6, 5, 3, -6, -12, -7]
[[1, 0, 1, 2], [0, 6, 5, 3]]
6 3.187309 114.743119

[10, 9, 7, -9, -17, -9]
[[1, -1, 0, 1], [0, 10, 9, 7]]
10 1.171542 117.154200

[9, 5, -3, -13, -30, -21]
[[1, 1, 2, 3], [0, 9, 5, -3]]
12 1.049791 151.169891

[1, 4, 10, 4, 13, 12]
[[1, 2, 4, 7], [0, -1, -4, -10]]
10 1.698521 169.852100

[3, 0, -6, -7, -18, -14]
[[3, 5, 7, 8], [0, -1, 0, 2]]
9 2.939961 238.136875

[7, 9, 13, -2, 1, 5]
[[1, -1, -1, -2], [0, 7, 9, 13]]
13 1.610469 272.169318

[7, 4, -2, -10, -23, -16]
[[1, 1, 2, 3], [0, 7, 4, -2]]
9 4.771049 386.454969

[2, -4, -16, -11, -31, -26]
[[2, 3, 5, 7], [0, 1, -2, -8]]
18 1.267597 410.701556

[11, 13, 17, -5, -4, 3]
[[1, 3, 4, 5], [0, -11, -13, -17]]
17 1.485250 429.237250

[17, 18, 20, -11, -16, -4]
[[1, 6, 7, 8], [0, -17, -18, -20]]
20 1.127925 451.169872

[5, 1, -7, -10, -25, -19]
[[1, 0, 2, 5], [0, 5, 1, -7]]
12 3.148011 453.313549

[8, 1, -13, -17, -43, -33]
[[1, -1, 2, 7], [0, 8, 1, -13]]
21 1.116045 492.175800

[1, -1, -5, -4, -11, -9]
[[1, 2, 2, 1], [0, -1, 1, 5]]
6 14.789095 532.407425

[3, 5, 9, 1, 6, 7]
[[1, 2, 3, 4], [0, -3, -5, -9]]
9 6.584324 533.330263

[8, 13, 23, 2, 14, 17]
[[1, 2, 3, 4], [0, -8, -13, -23]]
23 1.024522 541.972271

[1, -8, -26, -15, -44, -38]
[[1, 2, -1, -8], [0, -1, 8, 26]]
27 1.021376 744.583261

[5, 8, 14, 1, 8, 10]
[[1, 2, 3, 4], [0, -5, -8, -14]]
14 4.143252 812.077399

[4, 9, 19, 5, 19, 19]
[[1, 1, 1, 0], [0, 4, 9, 19]]
19 2.289528 826.519647

[1, -3, -11, -7, -20, -17]
[[1, 2, 1, -2], [0, -1, 3, 11]]
12 7.214635 1038.907446

[0, 5, 15, 8, 24, 21]
[[5, 8, 12, 15], [0, 0, -1, -3]]
15 5.665687 1274.779472

> The original starling map, like most of the others we've seen, is based
> on graphing the deviations from 3/1 and 5/1. Possibly a more useful map
> for a 7-limit temperament family is one that shows the deviations from
> 7/1. So I created a second map.
>
> http://www.io.com/~hmiller/png/starling-map2.png

You could also graph the deviation from the 3 and 5 of planar starling
in some tuning; if you use the rms values, the 3 comes to 1899.984
cents, the 5 to 2789.270 cents, and the 7 follows from 7=125/18. Using
this as the origin makes more sense, really.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> > I made a chart of ET's with the comma 126;125, and drew lines on
it to
> > illustrate some of the landmark starling temperaments.
>
> Great! This is something I've suggested Paul do from time to time;

I must have missed those; I remember a request to plot according to
the generator mapping.

> Here's a list of 21 starling temperaments, ordered in terms of
> increasing Graham-TOP badness:
>
> [4, 4, 4, -3, -5, -2]
> [[4, 6, 9, 11], [0, 1, 1, 1]]
> 4 5.871540 93.944647
Dimisept in my paper

> [6, 5, 3, -6, -12, -7]
> [[1, 0, 1, 2], [0, 6, 5, 3]]
> 6 3.187309 114.743119
Keenan in my paper

> [10, 9, 7, -9, -17, -9]
> [[1, -1, 0, 1], [0, 10, 9, 7]]
> 10 1.171542 117.154200
Myna in my paper

> [9, 5, -3, -13, -30, -21]
> [[1, 1, 2, 3], [0, 9, 5, -3]]
> 12 1.049791 151.169891

> [1, 4, 10, 4, 13, 12]
> [[1, 2, 4, 7], [0, -1, -4, -10]]
> 10 1.698521 169.852100
Meantone in my paper

> [3, 0, -6, -7, -18, -14]
> [[3, 5, 7, 8], [0, -1, 0, 2]]
> 9 2.939961 238.136875
Augene in my paper

> [7, 9, 13, -2, 1, 5]
> [[1, -1, -1, -2], [0, 7, 9, 13]]
> 13 1.610469 272.169318
Sensisept in my paper

Gene Ward Smith wrote:

> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >>I made a chart of ET's with the comma 126;125, and drew lines on it to >>illustrate some of the landmark starling temperaments.
> > > Great! This is something I've suggested Paul do from time to time; I
> didn't know you were up to this level of graphic design.
> > >>Any major starling-related temperaments that I'm missing?
> > > Here's a list of 21 starling temperaments, ordered in terms of
> increasing Graham-TOP badness:
> [7, 9, 13, -2, 1, 5]
> [[1, -1, -1, -2], [0, 7, 9, 13]]
> 13 1.610469 272.169318

Semisixths. I should add that one at least. Some of the others are ones that I've seen before, but don't know much about.