"oracle temperament" (11-limit 31&51p / miracle extension)

Hi,

I want to propose my first temperament, "oracle". :)
Here's the data:

Oracle

Commas: 121/120, 225/224, 1029/1024

POTE generator: ~11/8 = 541.668
Map: [<1 7 -4 1 3|, <0 -12 14 4 1|]
Wedgie: <<12 -14 -4 -1 -50 -40 -43 30 46 11||
EDOs: 11, 20, 31, 42b, 51, 82e, 113e, 144ee
Badness: ?

--------------------------

Motivation:

I really like the linear tuning with a generator of
14\31 (= 14 steps in 31-EDO) and an octave period.
In 31-EDO, this tuning has a strong relationship to
the tunings associated with orwell and miracle, since
14\31 equals two orwell generators, and two 17\31s
make up a miracle generator; the secor 3\31.

This is why I want to look at this tuning as a marvel
temperament related to orwell and miracle; hence the
name "oracle". An alternative interpretation for the
tuning would casablanca temperament, but I prefer the
former.

Oracle is mainly a "worse" 11-limit miracle extension:
7-limit miracle tempers out 225/224 and 1029/1024,
and adding 121/120 means the generator 16/15 (or 15/14)
plus an octave is divided into two 16/11s (or 22:15s),
the new generator. As a result, all even multiples of
the generator can be associated with miracle temperament,
while all odd multiples represent 11-limit ratios.
You can look at is as an oracle that predicts a miracle,
since oracle always implies 7-limit miracle. ;)

Orwell tempers out 99/98, 121/120 and 225:224. Removing
99/98 leads to the rank-3 11-limit temperament Artemis,
and adding 1029/1024 leads to oracle. There's a clearly
recognizable relationship between orwell and oracle,
though it's not as simple to describe as with oracle and
miracle (*I think*).

--------------------------

A modulation example: (orwell -> oracle -> miracle)

As I mentioned, in 31-EDO two orwell generators 7\31 make
up an oracle generator 14\31, and two oracle generators
17\31 (modulo octave) make up a marvel generator 3\31.

Here's a modulation example to demonstrate this relation:

0 3 7 10 14 17 21 24 28 31 - Orwell[9]
0 3 10 14 17 21 28 31 - 7-note Orwell[9] subscale
0 3 11 14 17 20 28 31 - Oracle[7]
0 3 6 11 14 17 20 25 28 31 - Oracle[9]
0 3 6 9 12 19 22 25 28 31 - Miracle[9]

...and here are the same modulations, using an Orwell[9]
notation:

0 1 2 3 4 5 6 7 8 9 - Orwell[9]
0 1 3 4 5 6 8 9 - 7-note Orwell[9] subscale
0 1 3^ 4 5 6v 8 9 - Oracle[7]
0 1 2v 3^ 4 5 6v 7^ 8 9 - Oracle[9]
0 1 2v 3v 4b 5# 6^ 7^ 8 9 - Miracle[9]

Here, v and ^ represent an Orwell[9] chroma, and b = vv
and # = ^^ represent an Oracle[9] chroma.

I hope you can see why I want to associate this tuning
with oracle temperament in such a situation, s.th.
the associated ratios don't change too much. ;)

If you're interested in modulations in 31-EDO (or other EDOs),
you can read more here:
http://xenguitarist.com/index.php?PHPSESSID=fGBU13rCHvSMAUJR7hziS0&topic=434.0

...or reply here:
/tuning/topicId_106136.html#106136

This is just the basic concept, and I extended the idea
by associating triads with linear tunings.
However, since none replied I assumed people were either not
interested, or I did a bad job at explaining / advertising my
idea, so I refrained from posting stuff that further builds on
those basics...

--------------------------

Finally, here's a list of stacked generators (left column)
and some associated ratios (right column) for oracle:
(no guarantee for correctness)

0 | 1:1, 225:224, 121:120, 1029:1024
1 | 11:8, 15:11
2 | 15:8, 28:15
3 | 14:11
4 | 7:4, 225:128
5 | 105:88
6 | 49:30, 105:64
7 | 49:44
8 | 32:21, 49:32
9 | 44:21
10| 10:7, 64:45
11| 64:33
12| 4:3
13| 11:6, 20:11
14| 5:4
15| 28:33, 55:32
16| 7:6, 75:64
17| 35:22
18| 35:32
19| ?
20| 64:63, 50:49
21| ?
22| 40:21, 256:135
23| 55:42
24| 16:9
25| 11:9
26| 5:3
27| 55:48
28| 14:9, 25:16
29| 77:72, 35:33
30| 35:24
31| 196:99
32| 49:36
33| ?
34| 80:63
35| 110:63
36| 32:27
37| 44:27, 160:99
38| 10:9
39| 50:33, 55:36
40| 25:24, 28:27
...
62| 160:81, 125:63

Best
- Gedankenwelt

On 6/9/2013 1:47 PM, gedankenwelt94 wrote:
> Hi,
>
> I want to propose my first temperament, "oracle". :)
> Here's the data:
>
> Oracle
>
> Commas: 121/120, 225/224, 1029/1024
>
> POTE generator: ~11/8 = 541.668
> Map: [<1 7 -4 1 3|,<0 -12 14 4 1|]
> Wedgie:<<12 -14 -4 -1 -50 -40 -43 30 46 11||
> EDOs: 11, 20, 31, 42b, 51, 82e, 113e, 144ee
> Badness: ?

I don't see a name for this, and I don't know another temperament called "oracle", so that seems like a reasonable name.

One thing that's interesting about this is how many intervals can be divided in half. It's got an equal division of 5/4 but it's not a meantone. There's an interval halfway between 8/7 and 5/4 that could act as a generic minor third.

Here's some missing ratios:

> 19| ?
49:33

> 21| ?
88:63

> 33| ?
245:132

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> I don't see a name for this, and I don't know another temperament called
> "oracle", so that seems like a reasonable name.

Thanks for the confirmation! I saw it's already included in the
temperament finder (thanks Graham!).
I added an entry in the xenharmonic wiki under gamelismic clan /
miracle:

http://xenharmonic.wikispaces.com/Gamelismic+clan#Miracle-Oracle

It's directly under hemimiracle, which looks very similar.

I think the following would be a good 13-limit extension:

13-limit
Commas: 105/104, 121/120, 196/195, 512/507

POTE generator: ~11/8 = 541.617
Map: [<1 7 -4 1 3 1|, <0 -12 14 4 1 6|]
EDOs: 11, 20, 31, 51

The accuracy for 13/8 is about as bad as for 11/8,
but the complexity for the 2.7.11.13-limit is very low.

Another possible extension would be this one:

Mirwell
Commas: 121/120, 225/224, 275/273, 1029/1024

POTE generator: ~11/8 = 541.596
Map: [<1 7 -4 1 3 -13|, <0 -12 14 4 1 37|]
EDOs: 31, 82e, 113e, 144ee

It's very accurate in the 2.3.5.7.13-limit, but the complexity
for 13/8 is quite high, so I don't know if it's worth proposing
it as a temperament.
The name is again a portmanteau of "miracle" and "orwell" with
an emphasis on "orwell", which should be a hint that all commas
except 1029/1024 are from 13-limit orwell.

This one is just an idea, not a proposal by me.

Btw., is there a simple method to determine the (TE simple)
badness? I could probably calculate it using the algorithm
described here if necessary, though:

http://xenharmonic.wikispaces.com/Tenney-Euclidean+temperament+measures#TE simple badness

> One thing that's interesting about this is how many intervals can be
> divided in half. It's got an equal division of 5/4 but it's not a
> meantone. There's an interval halfway between 8/7 and 5/4 that could act
> as a generic minor third.

Yeah, all odd multiples of the generator are 11-limit intervals
that divide a 7-limit miracle interval in half - just as with
hemimiracle.

Those 11-limit intervals are very similar in size to intervals
that differ by the chroma of the 31-tone MOS, which represents
99:98.
It is tempered out in 31-tet, and is only 1 step in 51, 82e,
113e and 144ee, since there 31 is a maximally even scale.
Applying this chroma once up or down to an 11-limit interval
will lead to a 7-limit miracle interval that is similar in size.

For example, 5/4 (14 generators) is divided into two 49/44
(7 gen.). Going a chroma up (-31 gen.) will lead to 9/8
(-24 gen.), and going a chroma down (+31 gen.) to 10/9
(38 gen.).

The interval close to a minor third is 105/88 (5 gen.), and
bisects 10/7 (10 gen.). Applying a chroma up and down will
lead to 6/5 (-26 gen.) and 32/27 (36 gen.), respectively.

You can take any of those 11-limit intervals, look which
7-limit miracle interval they bisect, and find two 7-limit
miracle intervals that are only slightly smaller/larger than
the 11-limit interval.

> Here's some missing ratios:
>
> > 19| ?
> 49:33
>
> > 21| ?
> 88:63
>
> > 33| ?
> 245:132

Thanks! I'm a little disappointed I didn't find the first two, they can be easily derived from 4:3.

On Thursday 13 June 2013 5:50:48 gedankenwelt94 wrote:
> Btw., is there a simple method to determine the (TE
> simple) badness? I could probably calculate it using the
> algorithm described here if necessary, though:
>
> http://xenharmonic.wikispaces.com/Tenney-Euclidean+temper
> ament+measures#TE simple badness

For a row matrix <J| and a matrix <M| that would be a row
matrix if it were for an equal temperament, and transposes |
J> and |M>, it's

sqrt[determ(<M|M>/<J|J> - <M|J><J|M>/<J|J><J|J>)]

Whether that's simpler than using wedge products depends
mainly on whether you have a library for doing wedge
products.

There's also a special formula for a rank two temperament
with weighted equal temperament mappings m1 and m2:

sqrt[var(m1) * var(m2) - cov(m1,m2) * cov(m1,m2)]

where var() is the variance and cov() is the covariance.

Graham

Thanks for your answer, and sorry it took me so long to reply!

Using wedgies seems to be simpler, though I have to learn some
basics first.

I wanted to reply earlier, but I wanted to figure some things
out by myself, first, and failed due to some apparently wrong
assumptions.

E.g., I read that TE simple badness = TE complexity * TE error,
and I assumed I could
- calculate TE complexity by weighting a wedgie with log2(2),
log2(3), log2(5) etc., and calculate the root mean square
- use the TE error determined by your temperament finder,
and divide it by 1200
- calculate the product of both, and assure that it is identical
with the badness specified on a Xenharmonic Wiki page about
the temperament in question.

However, the values didn't coincide at all.
I suppose the different sources / methods use different
weightings, or the badness specified for temperaments on
Xen Wiki isn't TE simple badness?

xenwolf asked a similar question here:
http://xenharmonic.wikispaces.com/share/view/62807180?replyId=63500848

I found some useful documents about the topic on your page, and
I'm gladly willing to read them, but this will take some time.

Best
- Gedankenwelt

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On Thursday 13 June 2013 5:50:48 gedankenwelt94 wrote:
> > Btw., is there a simple method to determine the (TE
> > simple) badness? I could probably calculate it using the
> > algorithm described here if necessary, though:
> >
> > http://xenharmonic.wikispaces.com/Tenney-Euclidean+temperament+measures#TE simple badness
>
> For a row matrix <J| and a matrix <M| that would be a row
> matrix if it were for an equal temperament, and transposes |
> J> and |M>, it's
>
> sqrt[determ(<M|M>/<J|J> - <M|J><J|M>/<J|J><J|J>)]
>
> Whether that's simpler than using wedge products depends
> mainly on whether you have a library for doing wedge
> products.
>
> There's also a special formula for a rank two temperament
> with weighted equal temperament mappings m1 and m2:
>
> sqrt[var(m1) * var(m2) - cov(m1,m2) * cov(m1,m2)]
>
> where var() is the variance and cov() is the covariance.
>
>
> Graham