Surveying marvel diamonds

The complete list of marvel-tempered 5-limit diamonds of 19 notes or
less and containing 7 otonal tetrads or more consists of three scales,
generated from [1,3,5,45,75,225], [1,3,9,15,45,675], and
[1,5,15,25,75,1125]. They all have 19 notes and 7 otonal tetrads
exactly, and since the scales are diamonds they are self-inverse, and
hence also have 7 utonal tetrads each. The first scale has 10 triads
of each kind, and the other two 8. The "rather remarkable" first scale
still seems the most interesting of these, but they are all worth
looking at.

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

Here is a list of all marvel diamonds with six generators and of 21
notes or less and 8 otonal tetrads or more. They all have exactly this
many, and four are distinguished by having 12 major and 12 minor triads.

[1, 3, 5, 9, 15, 225]
21 8 12

[1, 3, 5, 9, 45, 225]
21 8 12

[1, 3, 5, 15, 25, 225]
21 8 12

[1, 3, 5, 25, 75, 225]
21 8 12

[1, 3, 5, 45, 225, 675]
21 8 10

[1, 3, 9, 15, 135, 675]
21 8 10

[1, 5, 15, 25, 225, 1125]
21 8 10

[1, 5, 15, 25, 375, 1125]
21 8 10

[1, 3, 9, 15, 225, 675]
21 8 10

[1, 3, 5, 75, 225, 1125]
21 8 10

[1, 3, 15, 45, 75, 3375]
21 8 8

[1, 15, 45, 75, 675, 3375]
21 8 8

[1, 5, 15, 45, 75, 3375]
21 8 8

[1, 15, 45, 75, 1125, 3375]
21 8 8

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> Here is a list of all marvel diamonds with six generators and of 21
> notes or less and 8 otonal tetrads or more.

The list has duplicates--for instance the first two are the same (and
the same as rostaim, the marvelized Dante Rosati scale) and the second
two are likewise the same.

> Here is a list of all marvel diamonds with six generators and of
> 21 notes or less and 8 otonal tetrads or more. They all have
> exactly this many, and four are distinguished by having 12 major
> and 12 minor triads.

Is one of these Blackjack?

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@...> wrote:
>
> > Here is a list of all marvel diamonds with six generators and of
> > 21 notes or less and 8 otonal tetrads or more. They all have
> > exactly this many, and four are distinguished by having 12 major
> > and 12 minor triads.
>
> Is one of these Blackjack?

Nope.

Do you have a Scala file for [1,3,5,45,75,225]?

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@...> wrote:
>
> Do you have a Scala file for [1,3,5,45,75,225]?

I discussed this scale some a while back:

! trab19.scl
transformed Euclidean ball, a^2-ab+b^2 metric diamond =
{1,3,5,15*{1,3,5},225}
! also projected A3 marvel ball, 11a^2-14ab+11b^2 < 45
! [1,3,5,45,75,225] diamond
19
!
16/15
9/8
256/225
75/64
6/5
5/4
32/25
4/3
45/32
64/45
3/2
25/16
8/5
5/3
128/75
225/128
16/9
15/8
2

! trab19marv.scl
1/4 kleismic tempered trab19
19
!
115.587047
200.054240
231.174094
268.798786
315.641287
384.385833
431.228334
499.972880
584.440073
615.559927
700.027120
768.771666
815.614167
884.358713
931.201214
968.825906
999.945760
1084.412953
1200.000000

Thanks. What does the 1/4-kleisma tempering add? And what
sort of temperament is that (quartminorthirds or something)?

-C.

>> Do you have a Scala file for [1,3,5,45,75,225]?
>
>! trab19.scl
>transformed Euclidean ball, a^2-ab+b^2 metric diamond =
>{1,3,5,15*{1,3,5},225}
>! also projected A3 marvel ball, 11a^2-14ab+11b^2 < 45
>! [1,3,5,45,75,225] diamond
>19
>!
>16/15
>9/8
>256/225
>75/64
>6/5
>5/4
>32/25
>4/3
>45/32
>64/45
>3/2
>25/16
>8/5
>5/3
>128/75
>225/128
>16/9
>15/8
>2
>
>! trab19marv.scl
>1/4 kleismic tempered trab19
>19
>!
>115.587047
>200.054240
>231.174094
>268.798786
>315.641287
>384.385833
>431.228334
>499.972880
>584.440073
>615.559927
>700.027120
>768.771666
>815.614167
>884.358713
>931.201214
>968.825906
>999.945760
>1084.412953
>1200.000000

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Thanks. What does the 1/4-kleisma tempering add? And what
> sort of temperament is that (quartminorthirds or something)?

It tempers out 225/224, so that the 7 otonal and 7 utonal tetrads have
better 7's at the expense of detuning 3 and 5 a little. The exact tuning
is one that makes the 7's exact, and makes the 3/2 and 5/4 flat by the
same amount, which is (225/224)^(1/4), or 1.93 cents. It's analogous
to 1/4-comma meantone as a meantone tuning.

There's plenty of other choices, of course--for instance, the
least-error patent val tempering out 225/224 is 228-et. That's 12*19,
which sounds like it must be good for something, and in fact it's
excellent for Waage temperament. The 3's are the same as 72-et, the
good old 12-et 3, but the 5's and 7's are better.