PAUL ERLICH'S AMAZING 11-LIMIT GENERATOR

Whew! I had to find an alternative subject heading for this thread, quick!
That other one was getting too embarrassing. With Paul setting me up like
that, someone would just _have_ to shoot me down. :-)

Sorry I managed to paste the same 19-tone lattice twice in one of those
posts. Yikes!.

Paul, I hope you've stopped flaggellating yourself.

Back in 27-Dec-99 I posted the same 31-tone periodicity block.

/tuning/topicId_7341.html#7341

It was microtempered to give vastly more consonances than it would have in
JI (which is the whole point). But I didn't temper it all the way to 72-EDO
(although, thanks to Paul Erlich, I knew this was possible), because I was
trying to convince otherwise strict-JI types that they were crazy not to
take advantage of well-distributed 224:225's and 384:385's. I used a
meantone naming scheme.

Since the 27-Dec-99 version, which minimised the maximum error to 2.7c, I
came up with a better distribution that minimised beat rates in the
4:5:6:7:9:11. That's the distribution I described again yesterday. I
haven't yet looked at tempering the octave in this context.

As Paul pointed out, you get a lot more consonances if you are willing to
temper this 31-toner all the way to 72-EDO. This is still the case, even
when you take a subset from it in the right way, like Paul's very
interesting 21-tone "blackjack" scale (great name Paul).

No Paul, I never discovered that generator. You're thinking of the 125c one
(better at 126c) that makes 9 and 10 tone MOS's and works in 19,29,48,67-EDOs.

The 116.7c generator is your discovery. When I did the computer search for
good 7-limit generators, this one would have been rejected because the
width of a tetrad on its chain is 13 generators (6 up and 7 down from the
root), more than meantone's 10 generators. It is only as an 11-limit
generator that it comes into its own. I think it will be hard to beat,
particularly with such small errors.

In just terms, this generator is probably best thought of as a minor seventh.
1200-116.7 = 1083.3c
ln(15/8)/ln(2)*1200 = 1088.3c
At least that's how I tend to think of it when I see it on the lattice.

As Dan Stearns showed (indirectly), if we plug 116.7c into the old MOS
calculator (i.e. continued fraction calculator) we get strictly proper
scales with 10, 31, 41 or 72 notes and improper ones with 11 or 21 notes.
Of course such a scale with only 10 or 11 notes will have no 5-limit
triads, let alone any tetrads, pentads or hexads. It will however have 4 or
5 NEUTRAL TRIADS, since these only span 6 generators. And 3 or 4 4:5:7
chords. Melodically the 10 note MOS is just sssssssssM where M/s ~= 1.3;
not anything that would rate terribly highly by any of Rothenberg's
information-theoretic criteria. One would need to skip some steps and make
3-stepsize scales, e.g. where L = s+s. In steps of 72-EDO, s=7, M=9, L=14.

This generator is a goldmine! Are you gonna build a 21-tone guitar now Paul?

It's a pity 21-tones doesn't have any hexads. The 5:11 spanning 22
generators is the killer there. But it has so much else. Yes, it has
disproportionately more 11-limit consonances than any of the 19's we were
looking at.

116.7c generator

Intvl No. generators
making it up
---------------------
4:6 6
4:5 -7
5:6 13
4:7 -2
5:7 5
6:7 -8
4:9 12
5:9 19
7:9 14
4:11 15
5:11 22
6:11 9
7:11 17
9:11 3

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

I've attached a 5-limit map of the 31-tone scale with Paul's 21 tone scale
shown in darker colour.

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

Of course I had to dust off the spreadsheet I used to search for 5 and 7
limit generators and modify it to look for good 11-limit ones.

I asked it for those with 11-limit errors (RMS and max absolute) no worse
than a chain of meantone fourths and having a hexad width of no more than
22 generators (the hexad width using Paul's generator).

Surprisingly, there are 13. (Unless there are bugs in my search, quite
possible).

Gener- Hexad RMS MA
ator width err err
(cents) (gens) (cents)
-----------------------------
78.1 21 5.2 8.5
87.7 20 6.3 10.1 Morrison
116.7 22 2.3 3.4 Erlich
154.5 22 7.3 10.5
232.1 20 7.9 10.8
271.3 17 6.6 9.3
321.8 21 6.3 8.9
348.3 20 8.4 10.8
380.7 20 5.6 8.7
387.3 20 6.0 9.5
495.6 21 6.6 8.6
503.0 18 7.9 11.2 meantone
580.3 22 6.2 10.4

The first thing to notice is that the 116.7c minor seventh (8:15) generator
gives errors way lower than any of the others. Less than half the errors of
the nearest competitor, which is the 78.1c generator whose hexads do
however span one less generator.

The second thing to notice is that there is one and only one generator
whose hexads span fewer generators than do those of (extended) meantone.
That's the 271.3c subminor third (6:7) generator, and it has slightly lower
errors than meantone.

There's plenty to investigate here. Like what size MOS's they generate and
what EDOs they work in.
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

Dave Keenan wrote:

> The first thing to notice is that the 116.7c minor seventh (8:15)
generator
> gives errors way lower than any of the others. Less than half the
errors of
> the nearest competitor, which is the 78.1c generator whose hexads do
> however span one less generator.

Looks like we're on to a winner. But why call it 8:15? It's closest
to the 6th root of 3:2 or 117.0 cents. 16:15 would be 111.7 cents.
So we have a family of scales where the fifth is divided into 6 equal
steps. Or equivalently the neutral third is divided into 3. 10-equal
is the simplest case, where a fifth is exactly 6 steps.

> There's plenty to investigate here. Like what size MOS's they
generate and
> what EDOs they work in.

Here's a graph of the 21 from 31 scale:

C-----Ev----G-----Bv----D-----F^----A
/ \ / \ / \ / / \
/ \ / \ / \ / / \
D^----F#----A^----C#----E^----G#----B^ \
\/ \/ \/ \/ / \/ / \ / \ \
/\ /\ /\ /\ / /\ / \ / \ \
Ab----Cv----Eb----Gv----Bb \ C-----Ev
\/ \/ \/ \/ / \ \ / \
/\ /\ /\ /\ / \ \ / \
D-----F^----A \ D^----F#----A^
/ \ / \ \ / \/ / \ \/
/ \ / \ \ / /\ / \ /\
G#----B^ \ Db----Fv-/--Ab-/--Cv-\--Eb
\ \ / \ / \/ / \/ \/ / \/
\ \ / \ / /\ / /\ /\ / /\
\ C-----Ev----G-----Bv----D-----F^----A
\ / \ / \/ / \/ \/ / \/ / \ /
\ / \ / /\ / /\ /\ / /\ / \ /
D^-/--F#-\--A^-/--C#-\--E^-/--G#----B^
/ \/ / \/ \/ / \/ / \ / \
/ /\ / /\ /\ / /\ / \ / \
Db----Fv----Ab----Cv----Eb----Gv----Bb \
\ / \/ / \/ \/ \/ \/ / \ \
\ / /\ / /\ /\ /\ /\ / \ \
G--\--Bv-/--D--\--F^-/--A \ D^
\/ \/ / \/ / \ / \ \ /
/\ /\ / /\ / \ / \ \ /
C#----E^----G#----B^ \ Db----Fv----Ab
/ \ / \ \ / \ \/ /
/ \ / \ \ / \ /\ /
Gv----Bb \ C-----Ev-/--G--\--Bb-/--D
\ / \ / \
\ / \ / \
D^----F#----A^----C#----E^----G#

That's on a 7-limit/neutral third lattice. It happens that the
7-limit notes in the middle of the triangles are a neutral third apart
in 31-equal, so it works both ways. This is because the interval
540:539 is eliminated along with the 243:242 for neutral thirds.
Looks like both these commas are eliminated in 72= as well. Ooh! And
41=. So this kind of latticing works with 31, 41 or 72-equal, exactly
the family we're looking at.

For this graph, the 5-limit connections are only valid for 31-equal.
There are no complete 11-limit chords. The 21 notes are made up of
three 7-note neutral-third MOS scales.

Graham

Dave Keenan wrote:

> Gener- Hexad RMS MA
> ator width err err
> (cents) (gens) (cents)
> -----------------------------
> 78.1 21 5.2 8.5
> 87.7 20 6.3 10.1 Morrison
> 116.7 22 2.3 3.4 Erlich
> 154.5 22 7.3 10.5
> 232.1 20 7.9 10.8
> 271.3 17 6.6 9.3
> 321.8 21 6.3 8.9
> 348.3 20 8.4 10.8
> 380.7 20 5.6 8.7
> 387.3 20 6.0 9.5
> 495.6 21 6.6 8.6
> 503.0 18 7.9 11.2 meantone
> 580.3 22 6.2 10.4

Playing with these numbers, seeing how they come out in my favourite
11-limit tunings, gives some close matches.

116.7 = 7.002 * 72 /1200 2.3

380.7 = 13.007 * 41 /1200 5.6
321.8 = 10.995 * 41 /1200 6.3
87.7 = 2.996 * 41 /1200 6.3

78.1 = 2.994 * 46 /1200 5.2
495.6 = 18.998 * 46 /1200 6.6

387.3 = 10.005 * 31 /1200 6.0
580.3 = 14.991 * 31 /1200 6.2
271.3 = 7.009 * 31 /1200 6.6
154.5 = 3.991 * 31 /1200 7.3
503.0 = 12.994 * 31 /1200 7.9
232.1 = 5.996 * 31 /1200 7.9
348.3 = 8.998 * 31 /1200 8.4

Note how Dave's RMS error is unsurprisingly correlated with good
approximations to small ETs. I think each of Dave's generators is
represented exactly once, and I didn't ignore any equally good matches
to get that to work :) (The next best is actually 3.987*41/1200 for
the generator we started with. There's also 380.7=6.98*22/1200 and
271.3=4.97*22/1200)

Graham

Dave Keenan wrote:

> The second thing to notice is that there is one and only one generator
> whose hexads span fewer generators than do those of (extended) meantone.
> That's the 271.3c subminor third (6:7) generator, and it has slightly lower
> errors than meantone.

Right away we can know that a chain of three 7:6s will yield a good approximation of 8:5, which means
some near just 5:4s, octave equivalence assumed.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--