More temperaments (was: white key scales (Dan, Miracle folk))

In-Reply-To: <9ekihc+rtsn@eGroups.com>
Dave Keenan wrote:

> --- In tuning@y..., graham@m... wrote:
> > This must be similar to what Dave Keenan's generator-finding program
> does.
>
> No. It works by brute force. It's an Excel spreadsheet. It simply does
> the same calculation for every generator between 0 to 600c in 0.1c
> increments. It calculates how many generators in a chain (up to some
> maximum) give the best (octave equivalent) approximation for each of
> 1:3, 1:5, 1:7, 1:11 and what the errors are. From those it calculates
> the hexad-width and the MA and RMS errors over all 11-limit intervals
> (consistency is guaranteed). Then it flags those that meet the set
> criteria for hexad-width and errors.

Oh, is that the thing I downloaded? I might have another look if I ever
get Excel installed on this machine. I thought you mentioned it could be
getting only local minima once, which would suggest a larger granularity
and optimising from there.

My program's working now. I rank temperaments by the product of the
number of steps for a complete chord and the minimax error, then show the
10 best ones. Miracle actually comes out as third best for the 11-limit.
I make it a 3.3 cent minimax error. The two better ones require more
notes to get that all-important hexad. You can get to 1.3 cents accuracy
with 49 notes or 2.4 cents with 31 notes.

The simplest 15-limit temperament in the top 10 is defined like this, by
octaves and generators:

( 1 0)
( 2 -1)
(-1 8)
(-3 14)
(13 -23)
(12 -20)

That means a 3:1 is two octaves minus a generator, so the generator is a
fourth. A 5:1 is 8 generators minus an octave. That looks like schismic
to me. It's consistent with 41 and 94-equal. You need 37 notes from a
chain of fourths to get a 15-limit chord, and it's accurate to 4.9 cents.

The number one 15-limit temperament is consistent with 87 and 72, defined
like this:

( 3 0)
( 6 -6)
( 8 -5)
( 8 2)
(11 -3)
(14 -14)

It needs 16 generators for a complete chord, but because the octave
divides into three I think that means 49 notes to the octave. So it's
fairly complex, but accurate to 2.8 cents. I don't plan on tuning it up
any time soon, but if anybody has a huge keyboard and wants to play with
15-limit harmony, well, now you know an accurate temperament exists.

Graham

--- In tuning@y..., graham@m... wrote:

> The number one 15-limit temperament is consistent with 87 and 72,
defined
> like this:
>
> ( 3 0)
> ( 6 -6)
> ( 8 -5)
> ( 8 2)
> (11 -3)
> (14 -14)
>
> It needs 16 generators for a complete chord, but because the octave
> divides into three

Ha! I was trying to persuade Dave to do a multiple-chain search, and
he wouldn't! (I'm understanding you to mean that the "interval of
equivalence" is 1/3 octave here?)

So _what is_ this generator, in 72-tET?

In-Reply-To: <9em516+3q18@eGroups.com>
Paul wrote:

> Ha! I was trying to persuade Dave to do a multiple-chain search, and
> he wouldn't! (I'm understanding you to mean that the "interval of
> equivalence" is 1/3 octave here?)

Yes, the 11- and 15-limits are riddled with them. There's even one that
divides the octave into 29 equal parts! 4.7 cents for the 15-limit, only
needs 1 generator, needing 30 for a complete chord? Mighty strange, but
it does mean you can use it with two manuals tuned to 29-equal a generator
apart. As I've shown before that 29-equal works quite well mapped to
Halberstadt, this is far from arbitrary.

That generator is 13.3 moct, or 15.9 cents.

> So _what is_ this generator, in 72-tET?

For the 15-limit winner, I make it 5 steps (or 4.98 for the minimax).

BTW, have you noticed the magical connection between Miracle and
diaschismic temperaments?

This is blue-skies stuff, so should we continue on tuning-math?

Graham

--- In tuning@y..., graham@m... wrote:

> BTW, have you noticed the magical connection between Miracle and
> diaschismic temperaments?

No.

> This is blue-skies stuff, so should we continue on tuning-math?

Yes!

--- In tuning@y..., graham@m... wrote:

> Yes, the 11- and 15-limits are riddled with them. There's even one
that
> divides the octave into 29 equal parts! 4.7 cents for the 15-
limit, only
> needs 1 generator, needing 30 for a complete chord? Mighty
strange, but
> it does mean you can use it with two manuals tuned to 29-equal a
generator
> apart. As I've shown before that 29-equal works quite well mapped
to
> Halberstadt, this is far from arbitrary.

My proposal of two 12-equal keyboards 15 cents apart must look pretty
good in the 5-limit, yes?

--- In tuning@y..., graham@m... wrote:
> Oh, is that the thing I downloaded?

No but it is an extension of that. Let me know if you ever want the
full thing.

>I thought you mentioned it
could be
> getting only local minima once, which would suggest a larger
granularity
> and optimising from there.

Maybe what you're think of is the fact that it only looks for the 1:3,
1:5, 1:7, 1:11 and enforces consistency from there. I still have this
nagging fear that if it looked independently for 3:5, 3:7, 5:7 etc. it
might find other good generators.

> My program's working now. I rank temperaments by the product of the
> number of steps for a complete chord and the minimax error, then
show the
> 10 best ones. Miracle actually comes out as third best for the
11-limit.
> I make it a 3.3 cent minimax error.

That's right. And 23 notes for the hexad.

> The two better ones require
more
> notes to get that all-important hexad. You can get to 1.3 cents
accuracy
> with 49 notes or 2.4 cents with 31 notes.

How does Miracle compare on this metric as a 7 or 9 limit generator?
Also as 7 ,9 or 11-limit with no 5's?

How does it compare if we give the number of notes more weight (say by
squaring it)? Do generators with greater errors than Miracle come out
better?

The 15 limit ones are interesting, but maybe not real practical. What
if you ignore ratios of 13?

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
.
>
> Maybe what you're think of is the fact that it only looks for the
1:3,
> 1:5, 1:7, 1:11 and enforces consistency from there. I still have
this
> nagging fear that if it looked independently for 3:5, 3:7, 5:7 etc.
it
> might find other good generators.

Uh-oh . . . better dispell that fear. Or else I'll have to go write
my own program!

> How does it compare if we give the number of notes more weight (say
by
> squaring it)?
> Do generators with greater errors than Miracle come out
> better?

Yes, it looks like approximations of 7-, 14-, and 17-tET with 30 to
44 cents maximum errors will then look better.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > Maybe what you're think of is the fact that it only looks for the
> 1:3,
> > 1:5, 1:7, 1:11 and enforces consistency from there. I still have
> this
> > nagging fear that if it looked independently for 3:5, 3:7, 5:7
etc.
> it
> > might find other good generators.
>
> Uh-oh . . . better dispell that fear. Or else I'll have to go write
> my own program!

The more the merrier.

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <9em516+3q18@e...>
> Paul wrote:
>
> > Ha! I was trying to persuade Dave to do a multiple-chain search,
and
> > he wouldn't!

I'm sure glad I didn't. It seems Graham came up with a much easier way
to find them than what I would have done.

This is exciting stuff Graham.

Just to recap for non-math readers. Graham Breed has found some scale
generators that give lower 11-limit errors than Miracle. But they
require more notes before an approximate 11-limit hexad appears. This
is not at all surprising. If you are willing to have more notes you
can always get better accuracy, ad infinitum.

But if, as an overall figure-of-demerit, you multiply their maximum
error in cents by the number of notes required to give a hexad, two of
them come out looking slightly better than Miracle. These are (best
first)
(a) eight chains of 16.20 c generators an eighth of an octave apart,
(b) two chains of 183.21 c generators a half octave apart.

Miracle is
(c) a single chain of 116.72 c generators.

All three embed consistenly into 72-EDO. But of course when you do
that they all have exactly the same errors and the figure-of-demerit
becomes purely the number of notes at which the first 11-limit hexad
appears.

(a) 56 (not 49 as first reported)
(b) 32 (not 31 as first reported)
(c) 23

So as a way of choosing good 11-limit subsets of 72-EDO, the Miracle
generator is by far the best.

In the next paragraph I'll write N hexads or pentads etc. When I mean
N otonal and N utonal.

If you are willing to step outside 72-EDO and have 32 notes then
(a) has 1.3 c error, has no hexads but (somewhat remarkably) has 8
3-5-7-9-11 pentads (no 1's).
(b) has 2.4 c error and has 2 hexads, 4 9-limit pentads, 6 7-limit
tetrads.
(c) has 3.3 c error and has 11 hexads.

The number of notes in MOS scales using these generators are (improper
in parenthesis)

(a) (8 16 24 32 40 48 56 64) 72 (80 152)
(b) 6 (8 14) 20 26 46 72 (118)
(c) (5 6 7 8 9) 10 (11 21) 31 41 72 (113)

So the smallest proper MOS's containing a hexad are respectively 72,
46, 31.

-- Dave Keenan

Dave Keenan wrote:

> Just to recap for non-math readers. Graham Breed has found some scale
> generators that give lower 11-limit errors than Miracle. But they
> require more notes before an approximate 11-limit hexad appears. This
> is not at all surprising. If you are willing to have more notes you
> can always get better accuracy, ad infinitum.
>
> But if, as an overall figure-of-demerit, you multiply their maximum
> error in cents by the number of notes required to give a hexad, two of
> them come out looking slightly better than Miracle. These are (best
> first)
> (a) eight chains of 16.20 c generators an eighth of an octave apart,
> (b) two chains of 183.21 c generators a half octave apart.

I've changed the figure-of-demerit so that, as Paul noticed on another
list, Miracle is now number 1. The difference is I'm squaring the number
of notes now. As per Dave's suggestion, although I'd already done it by
the time he made it.

The code and data are at <http://x31eq.com/temper.html>. If
the data don't answer your questions, you can get a Python interpreter
(<http://www.python.org/>) and hack the code.

> Miracle is
> (c) a single chain of 116.72 c generators.
>
> All three embed consistenly into 72-EDO. But of course when you do
> that they all have exactly the same errors and the figure-of-demerit
> becomes purely the number of notes at which the first 11-limit hexad
> appears.
>
> (a) 56 (not 49 as first reported)
> (b) 32 (not 31 as first reported)
> (c) 23

The number of notes depends on how you build the scales. That isn't
really defined where the generator isn't an octave. I think your numbers
are making such temperaments look worse then they are, but we'll have to
discuss it on the tuning-math list.

Graham

I wrote:
> If you are willing to step outside 72-EDO and have 32 notes then
> (a) has 1.3 c error, has no hexads but (somewhat remarkably) has 8
> 3-5-7-9-11 pentads (no 1's).
> (b) has 2.4 c error and has 2 hexads, 4 9-limit pentads, 6 7-limit
> tetrads.
> (c) has 3.3 c error and has 11 hexads.

That should have been

(c) has 3.3 c error and has 10 hexads.

(c) is Miracle.

-- Dave Keenan