152-tET

Graham, did you come across anything relating to 152-tET in your
searches? I notice it's near-just, and also contains 76-tET, which
contains many linear temperaments (meantone, paultone, double-
negative-7-limit-tetradecatonic-thingy . . .)?

In-Reply-To: <9jqfk1+ifbe@eGroups.com>
Paul wrote:

> Graham, did you come across anything relating to 152-tET in your
> searches? I notice it's near-just, and also contains 76-tET, which
> contains many linear temperaments (meantone, paultone, double-
> negative-7-limit-tetradecatonic-thingy . . .)?

No. I was originally only considering ETs with less than 100 notes. I've
expanded that now, but it doesn't seem to be 13-limit consistent, and is
too high to be considered for the other limits.

I did notice 72 came up a lot in the higher limits. Here's what you get
by combining the two:

3/28, 16.2 cent generator

basis:
(0.125, 0.013529015588479165)

mapping by period and generator:
([8, 0], ([13, 19, 23, 28, 30], [-3, -4, -5, -3, -4]))

mapping by steps:
[(152, 72), (241, 114), (353, 167), (427, 202), (526, 249), (562, 266)]

unison vectors:
[[-3, -1, 0, -1, 0, 2], [1, 0, 6, -4, 0, -1], [-7, 1, 0, -3, 4, 0], [1,
10, 0, -
6, 0, 0]]

highest interval width: 6
complexity measure: 48 (56 for smallest MOS)
highest error: 0.004556 (5.467 cents)

11-limit:

highest interval width: 6
complexity measure: 48 (56 for smallest MOS)
highest error: 0.001072 (1.286 cents)
unique

7-limit:

highest interval width: 5
complexity measure: 40 (48 for smallest MOS)
highest error: 0.001044 (1.253 cents)
unique

Graham

--- In tuning-math@y..., graham@m... wrote:
>
> I did notice 72 came up a lot in the higher limits. Here's what
you get
> by combining the two:

Combining the two what? How?

Paul wrote:

> > I did notice 72 came up a lot in the higher limits. Here's what
> you get
> > by combining the two:
>
> Combining the two what? How?

That's how the program works. It takes two equal temperaments, and finds
the linear temperament consistent with both of them. It comes from all
linear temperaments being describable in terms of large and small scale
steps, as Dan Stearns mentions every now and then.

Graham

--- In tuning-math@y..., graham@m... wrote:
> Paul wrote:
>
> > > I did notice 72 came up a lot in the higher limits. Here's what
> > you get
> > > by combining the two:
> >
> > Combining the two what? How?
>
> That's how the program works. It takes two equal temperaments, and finds
> the linear temperament consistent with both of them. It comes from all
> linear temperaments being describable in terms of large and small scale
> steps, as Dan Stearns mentions every now and then.

Hmm . . . please elaborate. I'd tend to think of this approach in terms of each equal
temperament being consistent with its own set of unison vectors; those UVs that are consistent
with both become the commatic unison vectors of the resulting linear temperament. I'd have
trouble immediately seeing why a linear temperament would necessarily be the result, but in
2D, Herman Miller's graph makes it very clear: each ET is a point, and combining two ETs
means drawing a line between those two points, and the slope of the line is the sole unison
vector of the resulting unison vector . . . it's cool to be able to glimpse into this thicket from so
many different directions.