Magic lattices

I've discovered that "Multiple Approximations Generated Iteratively and
Consistently" is an acronym for "MAGIC". What a coincidence!

Here are some lattices for the 19 note MOS with meantone naming

/ / \ \ / / \ /\ /\ /\ / \ B--/------F#
Eb-/---\--Bb-/------F---------C \ / \/ \/ /
\/ \/ \/ / \ \ / / \ / /\ /\ /
/\ /\ /\ / \ D#-/------A#--------E# \ /
A---------E \ / \/ \/ / \ / / \ \ /
\ \ / / \ / /\ /\ / \ G--/---\--D------
--\--Gb-/------Db--------Ab \ / \/ \/ \/ \ /
\/ \/ / \ / / \ / /\ /\ /\ \ Cb
/\ /\ / \ B--/------F#--------C#-------G# \ \
-----C \ / \/ \/ / \ / / \ \ / / \ /
/ / \ / /\ /\ / \ Eb-/---\--Bb-/------F-
D#-/------A#--------E# \ / \/ \/ \/ \/ / \
\/ / \ / / \ \ / /\ /\ /\ /\ /
/\ / \ G--/---\--D---------A---------E \ /
Ab \ / \/ \/ \/ \ / / \ \ / / \ /
\ \ / /\ /\ /\ \ Cb-/---\--Gb-/------Db-----
--\--F#--------C#-------G# \ \/ \/ \/ / \ /
\/ \ / / \ \ / / \ \ /\ /\ /\ / \ B-
/\ \ Eb-/---\--Bb-/---\--F---------C \ / \/ \

/ / \ \ / / \ /\ /\ /\ / \ B--/---\--F#
Eb-/---\--Bb-/------F---------C \ / \/ \/ \/
\/ \/ \/ / \ \ / / \ / /\ /\ /\
/\ /\ /\ / \ D#-/------A#--------E#--------Cb
A---------E \ / \/ \/ / \ / / \ \ / /
\ \ / / \ / /\ /\ / \ G--/---\--D--/---
\ Gb-/------Db--------Ab \ / \/ \/ \/ \/
\/ \/ / \ / / \ \ / /\ /\ /\ /\
/\ /\ / \ B--/---\--F#--------C#-------G# \
-----C \ / \/ \/ \/ \ / / \ \ / / \ /
/ / \ / /\ /\ /\ \ Eb-/---\--Bb-/------F-
D#-/------A#--------E#--------Cb \ \/ \/ \/ / \
\/ / \ / / \ \ / / \ /\ /\ /\ /
/\ / \ G--/---\--D--/------A---------E \ /
Ab \ / \/ \/ \/ \/ / \ \ / / \ /
\ \ / /\ /\ /\ /\ / \ Gb-/------Db-----
--\--F#--------C#-------G# \ / \/ \/ / \ /
\/ \ / / \ \ / / \ \ / /\ /\ / \ B-
/\ \ Eb-/---\--Bb-/---\--F---------C \ / \/ \

This is the 22 note MOS

/ / \ \ / / /\ /\ /\ /\ / \ B--/---\--F#
Eb-/---\--Bb-/------F---------C \ / \/ \/ \/
\/ \/ \/ / / \ \ / / \ \ / /\ /\ /\
/\ /\ /\ G^-/---\- D#-/---\--A#--------E#--------Cb
A---------E \ \/ \/ \/ \/ \ / / \ \ / /
\ \ / / \ /\ /\ /\ /\ \ G--/---\--D--/---
--\--Gb-/------Db--------Ab--------Ev \/ \/ \/ \/ /
\/ \/ / \ \ / / \ \ / / /\ /\ /\ /\ B#
/\ /\ / \ B--/---\--F#-/------C#-------G# \/ \
-----C \ / \/ \/ \/ \/ / / \ \ / / /\ /
/ / \ \ / /\ /\ /\ /\ Eb-/---\--Bb-/------F-
D#-/---\--A#--------E#--------Cb \/ \/ \/ \/ / / \
\/ \/ \ / / \ \ / / /\ /\ /\ /\ G^-/---
/\ /\ \ G--/---\--D--/------A---------E \ \/
Ab--------Ev \/ \/ \/ \/ / / \ \ / / \ /\
\ \ / / /\ /\ /\ /\ B#-/---\--Gb-/------Db-----
--\--F#-/------C#-------G# \ \/ \/ \/ / \ \ /
\/ \/ / / \ \ / / \ \ /\ /\ /\ / \ B-
/\ /\ Eb-/---\--Bb-/---\--F---------C \ / \/ \

Now Dave Keenan's found an alternative simplified Miracle lattice, let's
see if he can make anything of this.

This is my 24 note keyboard mapping, as used in
<http://x31eq.com/magicpump.mp3> with decimal names

/ / \ \ / / /\ /\ /\ /\ 5--/---\--1--/---\--7v
5v-/---\--0^-/------6^--------2^ \/ \/ \/ \/ / \/
\/ \/ \/ / / \ \ / / /\ /\ /\ /\ 9--/\--
/\ /\ /\ 8--/---\--4^-/------0---------6---------2v
0v--------5^ \/ \/ \/ \/ / / \ \ / / \ \ / /
\ \ / / /\ /\ /\ /\ 2--/---\--8v-/---\--4--/---
--\--7--/------3^--------9^ \/ \/ \/ \/ \/ \/ /
\/ \/ / / \ \ / / \ /\ /\ /\ /\ /\ /\ 1^
/\ /\ 5--/---\--1--/---\--7v--------3---------9v \/ \
-----2^ \/ \/ \/ \/ / \/ \ \ / / \ \ / / /\ /
/ / /\ /\ /\ /\ 9--/\--\--5v-/---\--0^-/------6^
4^-/------0---------6---------2v \/ \/ \/ \/ / / \
\/ / / \ \ / / \ \ /\ / /\ /\ /\ /\ 8--/---
/\ 2--/---\--7^-/---\--4--/------0v--------5^ \/ \/
9^ \/ \/ \/ \/ \/ \/ \ / / \ \ / / /\ /\
\ /\ /\ /\ /\ /\ /\ 1^-/---\--7--/------3^-----
--\--7v--------3---------9v \/ \/ \/ \/ / / \ \ /
/ \/ \ \ / / \ \ / / \ /\ /\ /\ /\ 5--/---\--1-
9--/\--\--5v-/---\--0^-/---\--6^--------2^ \/ \/ \/ \

C C# D Eb E F F# G G# A Bb B C C# D Eb
r r r r q p q r r r r r r q r
0 1 2 3 4 5 6

Eb E F F# G G# A Bb B C
r r r r r q p q r
6 7 8 9

Cents for the minimax tuning are

0.0
58.8
117.7
176.5
235.3
262.7
294.1
321.6
380.4
439.2
498.0
556.9
615.7
674.5
702.0
760.8
819.6
878.4
937.3
105.4
1082.3
1113.7
1141.2
1200.0

As scale steps:

(0 1 2 3 4 4 5 5 6 7 8 9 10 11 11 12 13 14 15 16 17 17 18 18 19)
(0 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 20 21 22)

Graham

> 380.4

I'm guessing this is the generator?

Paul wrote:

> > 380.4
>
> I'm guessing this is the generator?

Yes. It's at the top of <http://x31eq.com/limit9.txt>.

I wrote:

> Now Dave Keenan's found an alternative simplified Miracle lattice,
> let's see if he can make anything of this.

I came up with something overnight:

Eb----Bb----F-----C
B#/ \ Gb/ \ Db/ \ Ab/ Ev
G^ / D#\ / A#\ / E#\ / Cb
B-----F#----C#----G#
G / \ D / \ A / \ E /
Eb / Bb\ / F \ / C \ /
B#----Gb----Db----Ab----Ev
G^/ \ D#/ \ A#/ \ E#/ Cb
/ B \ / F#\ / C#\ / G#
G-----D-----A-----E
Eb/ \ Bb/ \ F / \ C /
B# / Gb\ / Db\ / Ab\ /
G^----D#----A#----E#----Cb
/ \ B / \ F#/ \ C#/ G#
/ G \ / D \ / E \ / E
Eb----Bb----F-----C
B#/ \ Gb/ \ Db/ \ Ab/ Ev
G^ / D#\ / A#\ / E#\ / Cb
B-----F#----C#----G#
G / \ D / \ A / \ E /
Eb / Bb\ / F \ / C \ /
B#----Gb----Db----Ab----Ev
G^/ \ D#/ \ A#/ \ E#/ Cb
/ B \ / F#\ / C#\ / G#
G-----D-----A-----E

The template is

5
1-----3-----9
\ /
\ /
7

Like Dave's new septimal-kleismic lattice, but unlike a normal 7-limit
lattice, pitch increases left-right for a 4:5:6:7:9 chord. That'd make it
good as a mapping for a hexagonal keyboard.

I also found this:

G#--B+--D#--F#+-A#
/ \ / \
G+/ B \ D+/ F#\ A+
/ \ / \
G---Bt--D---F+--A
\ / \ /
Gt\ Bb/ Dt\ Ft/ At
\ / \ /
Gb--Bbt-Db--Fbt-Ab

With the template

5

1-------3-------9---11
\ /
\ /
\ /
7

It works with 31-equal, but I haven't found any other consistent
temperaments for it. Although it does work with the meantone-like
neutral-third family 7, 24, 31, 38, ... with the mapping (2 8 -11 5), a
complexity measure of 20 and an approximation of the 11-limit to within
10.8 cents (not much of an improvement on 31-equal). It looks like a good
mapping for a rectangular keyboard, and might work with adaptive tuning.

Unison vectors are 176:175 or (4 0 -2 -1 1)H and 243:242 or (-1 5 0 0
-2)H. These combine to give 31104:30625 or (7 5 -4 -2)H. Dieses are
36:35, 45:44 and 55:54. Two dieses of 36:35 make a 25:24.

And a unified neutral-second/neutral-third lattice

B--D+-F#-A+-C#
|\ |\ |
| \ | \ |
A C+ E G+ B
| \ | \ |
| \| \|
G--Bt-D--F+-A
|\ |\ |
| \ | \ |
F At C Et G
| \ | \ |
| \| \|
Eb-Gt-Bb-Dt-F

It's like Dave Keenan's new Miracle lattice, but with extra rows.
Template

7
|
|
|
|
|
5 |
| |
| |
| 11 |
| |
| |
1-----3-----9--11

I don't know if there's a simpler position for the 7 ...

Graham

> From: <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, July 29, 2001 3:56 AM
> Subject: [tuning-math] Re: Magic lattices
>
>
> I wrote:
>
> > Now Dave Keenan's found an alternative simplified Miracle lattice,
> > let's see if he can make anything of this.
>
> I came up with something overnight:
> <etc.>

Hi Graham,

*Please* provide a legend for your notation.
There are so many different ones being used now
that I'm not sure what you mean by "Bt" etc.

(Yes, I openly admit my guilt in being one of
the advocates of a "non-standard" notation, and
thus one of the reason why legends are required...)

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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monz wrote:

> *Please* provide a legend for your notation.
> There are so many different ones being used now
> that I'm not sure what you mean by "Bt" etc.

t is a half-flat, + is a half-sharp.

I'm now looking at lattices where the 3-direction is reversed, and the
primary 5-limit chord is 3:4:5 rather than 4:5:6. This gives a 7-limit
template of

7
\
---5
/ \ /
3---1

using the septimal kleisma.

Ct--Ft--A#--D#--G#
/ \ / \ / \ / \ / \
G#--C#--F#--B---E---A
\ / \ / \ / \ / \ /
A---D---G---C---F
/ \ / \ / \ / \ /
F---Bb--Eb--Ab--Db
\ / \ / \ / \ /
Gb--B+--E+--A+

The clever thing is that one diagonal is the miracle generator, and the
other is the magic generator. The horizontal is obviously the meantone or
schismic generator.

You could bring the 11-limit in using neutral seconds, but it
breaks the melodic pattern. Neutral thirds wouldn't fit. There is
another 7-limit mapping:

5
/
7 /
3-----1

C#----F#
/ \ Eb/ \
/ Ct\ / Ft\
A-----D-----G
\ B+/ \ E+/
\ / C$\ /
Bb----Eb

You can set Ct==B+ and E+==Ft by splitting the fourth into equal parts.
That means 7:6 and 8:7 become equal. This is the famous interval class
with no name. It works in 29= and others. I think 26. Looks like this
would make a good ZTar mapping for such temperaments.

Graham

--- In tuning-math@y..., graham@m... wrote:
>
> I've discovered that "Multiple Approximations Generated Iteratively
and
> Consistently" is an acronym for "MAGIC". What a coincidence!

Tee hee! Yes I _had_ noticed that.

By the way, you can delete the second ocurrence of it in your catalog. The
5-limit one. That was my fault.

...
> Now Dave Keenan's found an alternative simplified Miracle lattice,
let's
> see if he can make anything of this.

The lattice I gave for Miracle works just as well for this temperament
(without the 11s of course), because the 224:225 is distributed in this
temperament too.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

In-Reply-To: <3.0.6.32.20010729153925.00a89630@uq.net.au>
Dave Keenan wrote:

> > I've discovered that "Multiple Approximations Generated Iteratively
> and
> > Consistently" is an acronym for "MAGIC". What a coincidence!
>
> Tee hee! Yes I _had_ noticed that.
>
> By the way, you can delete the second ocurrence of it in your catalog.
> The
> 5-limit one. That was my fault.

Oops.

I've also updated the Bohlen-Pierce entry.

> > Now Dave Keenan's found an alternative simplified Miracle lattice,
> let's
> > see if he can make anything of this.
>
> The lattice I gave for Miracle works just as well for this temperament
> (without the 11s of course), because the 224:225 is distributed in this
> temperament too.

Yes, it does! I also ended up with my own lattice that removes the
224:225.

Graham

--- In tuning-math@y..., graham@m... wrote:
> > Now Dave Keenan's found an alternative simplified Miracle lattice,
> > let's see if he can make anything of this.
>
> I came up with something overnight:
[ lattice deleted ]
>
> The template is
>
> 5
> 1-----3-----9
> \ /
> \ /
> 7
>
> Like Dave's new septimal-kleismic lattice, but unlike a normal
7-limit
> lattice, pitch increases left-right for a 4:5:6:7:9 chord.

Mine doesn't strictly-increase from left to right, but yours does.

> That'd make it
> good as a mapping for a hexagonal keyboard.

Yes indeed. In fact it is _THE_ mapping for such a keyboard. Try
putting the period, generator and number of notes into my keyboard
mapper spreadsheet (I think you have to wind the aspect-ratio up to
max to get it to look hexagonal).
http://dkeenan.com/Music/KeyboardMapper.xls

> I also found this:
>
>
> G#--B+--D#--F#+-A#
> / \ / \
> G+/ B \ D+/ F#\ A+
> / \ / \
> G---Bt--D---F+--A
> \ / \ /
> Gt\ Bb/ Dt\ Ft/ At
> \ / \ /
> Gb--Bbt-Db--Fbt-Ab
>
> With the template
>
> 5
>
> 1-------3-------9---11
> \ /
> \ /
> \ /
> 7
>
>
...
> And a unified neutral-second/neutral-third lattice
>
>
> B--D+-F#-A+-C#
> |\ |\ |
> | \ | \ |
> A C+ E G+ B
> | \ | \ |
> | \| \|
> G--Bt-D--F+-A
> |\ |\ |
> | \ | \ |
> F At C Et G
> | \ | \ |
> | \| \|
> Eb-Gt-Bb-Dt-F
>
>
> It's like Dave Keenan's new Miracle lattice, but with extra rows.
> Template
>
> 7
> |
> |
> |
> |
> |
> 5 |
> | |
> | |
> | 11 |
> | |
> | |
> 1-----3-----9--11
>
>
> I don't know if there's a simpler position for the 7 ...

Nor do I.

All these tempered lattices coming out of the woodwork! After trying
the full colour version of the 2D tempered lattice for Blackjack, I'm
thinking that 3D may be easier to deal with in some ways.

It's also useful to note that the keyboard mapping can always be
treated as a 2D tempered lattice.

I'm thinking now that one way to find the best 3D lattice for an
11-limit scale may be to put several periods of it into a spreadsheet
as a 5D triangular lattice and look at a 3D projection of it in the
same way I did with the dekany. Then rotate it until it is as thin as
possible in the into-the-screen dimension, followed by minor
adjustments so no note is obscured. But I don't have time.

-- Dave Keenan