That Scale

In-Reply-To: <940394585.9925@onelist.com>
Paul Erlich wrote:

>The interpretation of a neutral third as 11:9, a neutral seventh as 11:6,
>and a half-sharp fourth as 11:8, as well as the vanishing of the 243:242,
is
>equivalent to using your scale and "believing in 11-limit harmony", is it
not?

Partly, but if you write the scale in JI, the 243:242 doesn't vanish.
It's like with meantone: there are a number of ways to convert the scale
to JI, all of which will lose a consonance (and therefore a chord).
There's no reason why one of these should be singled out as "Graham's
scale".

There are also some major and minor thirds in there, which I'd prefer to
tune to 5:4 and 6:5. With the 7-note MOS, I have tried approximating to
JI preserving the 6:5. It fits quite well but, as the seconds are 12:11,
11:10, 10:9 and 9:8, there's no distinction between large and small
intervals. With 10 notes, the 5-limit intervals are more important as
there are more of them, including complete 5-limit chords.

Hey, I've even found my own lattice for the 10 note scale!

C---E---G---B---D
\ \ / \ / \
\ \ / \
\ / \ / \ \
Gb--Bb--Db--F---A---C

This isn't JI because

1) C is on there twice

2) All neutral thirds are equal.

Point (2) is shown by neutral thirds being directly between the perfect
fifths. This makes the lattice cluttered, but it's the best way I can
find of representing these scales. Let's get rid of a C.

C---E---G---B---D
\ \ / \ / \
\ \ / \
\ / \ / \ \
Gb--Bb--Db--F---A

I think transposing that scale to each of its degrees will give

*---*---*---*---*---*---*---*---*
\ \ / \ / \ / \ / \ / \ / \
\ \ \ \ \ \ \ \
\ / \ / \ / \ / \ / \ / \ / \
*---*---*---*---*---*---*---*---*
\ \ / \ / \ / \ / \ / \ / \
\ \ \ \ \ \ \ \
\ / \ / \ / \ / \ / \ / \ \
*---*---*---*---*---*---*---*---*

Which, as it's still a single string of neutral thirds, is also

*---*---*---*---*---*---*---*---*---*
\ / \ / \ / \ / \ / \ / \ / \ / \ / \
/ / \ \ \ \ \ \ \ \
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \
*---*---*---*---*---*---*---*---*---*---*---*---*---*
\ \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ \ \ \ \ \ \ \ / /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \
*---*---*---*---*---*---*---*---*---*

And can be extended even further to either side. If anybody's following
this, perhaps they could tell me if that's right.

Sorry, what was the question again?

Graham wrote,

>Partly, but if you write the scale in JI, the 243:242 doesn't vanish.
>It's like with meantone: there are a number of ways to convert the scale
>to JI, all of which will lose a consonance (and therefore a chord).
>There's no reason why one of these should be singled out as "Graham's
>scale".

That's why I drew the lattice diagram for the complete scale with all the
multiple meanings, and noted that the top and bottom rows were equivalent,
and that thus the diagram repeated infinitely. Did you see that? That
infinite diagram corresponds to what I called Graham's scale. The part of
the diagram that I drew contained all the consonances in the tempered
system. Correct?

>Which, as it's still a single string of neutral thirds, is also

>*---*---*---*---*---*---*---*---*---*
> \ / \ / \ / \ / \ / \ / \ / \ / \ / \
> / / \ \ \ \ \ \ \ \
> / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
>*---*---*---*---*---*---*---*---*---*---*---*---*---*
> \ \ / \ / \ / \ / \ / \ / \ / \ / \ /
> \ \ \ \ \ \ \ \ / /
> \ / \ / \ / \ / \ / \ / \ / \ / \ / \
> *---*---*---*---*---*---*---*---*---*

Wait a minute -- doesn't that middle row imply a chain of 7 fifths? Where is
that in your 10-note scale?

In-Reply-To: <940496063.2404@onelist.com>
Paul Erlich, digest 362.12, wrote:

> >Partly, but if you write the scale in JI, the 243:242 doesn't vanish.
> >It's like with meantone: there are a number of ways to convert the
> scale >to JI, all of which will lose a consonance (and therefore a
> chord). >There's no reason why one of these should be singled out as
> "Graham's >scale".
>
> That's why I drew the lattice diagram for the complete scale with all
> the
> multiple meanings, and noted that the top and bottom rows were
> equivalent,
> and that thus the diagram repeated infinitely. Did you see that? That
> infinite diagram corresponds to what I called Graham's scale. The part
> of
> the diagram that I drew contained all the consonances in the tempered
> system. Correct?

Yes, sorry. I wasn't following as closely as I should have. The first
posts where you talked about "Graham's scale" didn't include diagrams, so
I thought it was still JI. Otherwise, I saw diagrams with ratios, and
assumed they meant JI, without checking them properly.

I still prefer the 5-limit intervals to be there, but you aren't psychic,
so you couldn't have known that.

> >Which, as it's still a single string of neutral thirds, is also
>
> >*---*---*---*---*---*---*---*---*---*
> > \ / \ / \ / \ / \ / \ / \ / \ / \ / \
> > / / \ \ \ \ \ \ \ \
> > / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
> >*---*---*---*---*---*---*---*---*---*---*---*---*---*
> > \ \ / \ / \ / \ / \ / \ / \ / \ / \ /
> > \ \ \ \ \ \ \ \ / /
> > \ / \ / \ / \ / \ / \ / \ / \ / \ / \
> > *---*---*---*---*---*---*---*---*---*
>
> Wait a minute -- doesn't that middle row imply a chain of 7 fifths?
> Where is
> that in your 10-note scale?

This is the 19 note cross set(?) with all equivalences. That should be 19
neutral thirds, so chains of 9 and 10 fifths. You don't get them all,
because the diagram isn't complete. All rows are equivalent!

These are the two step size EDOs less than (an arbitrary) 72, that
work with this algorithm (when rounded to the nearest integer):

((LOG(1)-LOG(1))*(n/LOG(2))
((LOG(12)-LOG(11))*(n/LOG(2))
((LOG(9)-LOG(8))*(n/LOG(2))
((LOG(11)-LOG(9))*(n/LOG(2))
((LOG(4)-LOG(3))*(n/LOG(2))
((LOG(16)-LOG(11))*(n/LOG(2))
((LOG(3)-LOG(2))*(n/LOG(2))
((LOG(18)-LOG(11))*(n/LOG(2))
((LOG(16)-LOG(9))*(n/LOG(2))
((LOG(11)-LOG(6))*(n/LOG(2))
((LOG(2)-LOG(1))*(n/LOG(2))

17e (L=2 & s=1)
0 141 212 353 494 635 706 847 988 1059 1200

24e (L=3 & s=1)
0 150 200 350 500 650 700 850 1000 1050 1200

31e (L=4 & s=1)
0 155 194 348 503 658 697 852 1006 1045 1200

38e (L=5 & s=1)
0 158 189 347 505 663 695 853 1011 1042 1200

41e (L=5 & s=2)
0 146 205 351 498 644 702 849 995 1054 1200

55e (L=7 & s=2)
0 153 196 349 502 655 698 851 1004 1047 1200

58e (L=7 & s=3)
0 145 207 352 497 641 703 848 993 1055 1200

65e (L=8 & s=3)
0 148 203 351 498 646 702 849 997 1052 1200

This excludes reducible mappings (34 and 48e here, as they are
duplicates of 17 and 24e), as well as the EDOs (22, 29, 39, 46, 53,
56, 63 & 70) that use three step sizes to mimic the JI step structure
that distinguishes between the 12/11 and the 88/81 (for example, 22e
where L=3&2 and s=1, which could be said to be analogous to the 22e
mapping of 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, where L=4&3 and
s=2).

Dan

Graham wrote,

>This is the 19 note cross set(?) with all equivalences.

Why is it that you and Dan both started dealing with the cross set of the
scale with its inversion rather than the scale itself? Am I missing
something?

Dan,

I see you are looking for ET versions of Graham's scale. I would approach it
differently, but probably get the same results.

First, I would eliminate all ETs which are inconsistent with respect to (1,
3, 9, 11), or the 11-limit with 5 and 7 removed. That leaves the following
ETs: 2, 5, 7, 10, 14, 15, 17, 19, 22, 24, 26, 29, 31, 34, 38, 39, 41, 46,
48, 53, 55, 56, 58, 63, 65, 70, 72 . . . . Then I'd remove all ETs where the
243/242 doesn't vanish. That leaves 7, 10, 14, 17, 24, 29, 31, 34, 38, 41,
48, 55, 58, 65, 72, . . . . Now as far as the 7L+3s scale, in 7 s=0, in 10
L=s, 14 is the same as 7, 34 is the same as 17, and 48 and 72 are the same
as 24, leaving

17, 24, 31, 38, 41, 55, 58, and 65.

So I got the same results as you, but in principle I would argue that my
method is better, because there may be cases when simply looking at how the
ET approximates a version of the scale entirely expressed in JI ignores the
important structural characteristics that define the scale. For example, we
have seen that a full JI expression of the pentachordal decatonic scale can
be quite at odds with a 22-tET representation, but 22 not only presents the
relevant consonances and chords reasonably well, but also allows many more
that are not possible in JI.

[Paul H. Erlich:]
>So I got the same results as you, but in principle I would argue that
my method is better,

Different anyway... most all the algorithms I give are just my
attempts at quick and easy ways to specifically sieve through
(sometimes enormous) piles of options, and seldom is there any one
sort of (rigorous) overriding theoretical concern.

> First, I would eliminate all ETs which are inconsistent with respect
to (1, 3, 9, 11), or the 11-limit with 5 and 7 removed.

While I understand the concept, what exactly is the process that you
use here? (Steps 2 and 3 of the process you outlined would seem to be
taken care of by the algorithm I gave, though I guess for that to be
true, step 2 would have to assume only two step sizes.)

Dan

In-Reply-To: <940580858.17767@onelist.com>
Paul Erlich, digest 363.14 wrote:

> >This is the 19 note cross set(?) with all equivalences.
>
> Why is it that you and Dan both started dealing with the cross set of
> the
> scale with its inversion rather than the scale itself? Am I missing
> something?

I moved the scale to start at each of its degrees. I don't even know what
"inversion" means here: turning the scale upside-down gives a
transposition of itself. If this does make a difference, I didn't take
account of it.

One of us must me missing something.

I wrote:

>> First, I would eliminate all ETs which are inconsistent with respect
>> to (1, 3, 9, 11), or the 11-limit with 5 and 7 removed.

>While I understand the concept, what exactly is the process that you
>use here?

Paul Hahn's method is easiest. In each ET, calculate the closest approximation to
3/2, 9/8, and 11/8. Express each approximation's signed error in terms of steps
of the ET. These errors will range from -0.5 to 0.5. If (and only if) the
difference between the largest error and the smallest error is less than 0.5,
then the tuning is consistent with respect to (1, 3, 9, 11). In other words, all
the "consonant" ratios -- those found in the 1-3-9-11 diamond (the cross-set of
1/1 9/8 11/8 3/2 with its inverse) will be able to combine with one another to
form "consonant" triads and tetrads without ever having to use any approximation
to any ratio other than the best one. I used to evaluate all these ratios before
Paul Hahn pointed out this shortcut.

>(Steps 2 and 3 of the process you outlined would seem to be
>taken care of by the algorithm I gave, though I guess for that to be
>true, step 2 would have to assume only two step sizes.)

Well, consistency combined with the 243:242 vanishing automatically guarantees
that Graham's scale will have no more than two step sizes.

[Paul Erlich:]
> Express each approximation's signed error in terms of steps of the
ET.

Isn't this the same as saying the difference between
((LOG(N)-LOG(D))*(n/LOG(2)) and its value when rounded to the nearest
whole number (in other words, if N/D=3/2 and n=9, then
((LOG(3)-LOG(2))*(9/LOG(2)) = ~5.3, and ~5.3 - 5 = ~0.3)? If this is
off the mark, could you give me a specific example, or perhaps say the
same thing in a different way?

(thanks,)
Dan

> [Dan Stearns, TD 363.17]
>
> ... most all the algorithms I give are just my
> attempts at quick and easy ways to specifically sieve through
> (sometimes enormous) piles of options, and seldom is there any
> one sort of (rigorous) overriding theoretical concern.

Which is exactly what I love about this forum and about all
the great research into different avenues of the tuning world.

In fact, Dan, your mathematical postings are a model of a
straight-and-to-the-point expository style that my discursive
self can really appreciate. (but I enjoy reading your text
ramblings too...)

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Free software, free e-mail, and free Internet access for a month!
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Graham wrote,

>I moved the scale to start at each of its degrees. I don't even know what
>"inversion" means here: turning the scale upside-down gives a
>transposition of itself. If this does make a difference, I didn't take
>account of it.

The result of combining all the modes of a scale on one tonic is equivalent
to the cross-set of a scale with its own inversion (yes, the upside-down
version of the scale, in this case a transposition of the original scale).
That's what you were doing, right?

>One of us must me missing something.

Well, I was just wondering why boh you and Dan seem so interested in the big
19-tone scale rather than the original 10-tone scale, and why you both
seemed to slip from discussing the small one into discussing the big one
without any indication that you were doing so.

I wrote,

>>Express each approximation's signed error in terms of steps of the
>>ET.

Dan Stearns wrote,

>Isn't this the same as saying the difference between
>((LOG(N)-LOG(D))*(n/LOG(2)) and its value when rounded to the nearest
>whole number (in other words, if N/D=3/2 and n=9, then
>((LOG(3)-LOG(2))*(9/LOG(2)) = ~5.3, and ~5.3 - 5 = ~0.3)?

You got it!

[Paul H. Erlich:]
>Express each approximation's signed error in terms of steps of the
ET.

[Dan Stearns:]
>Isn't this the same as saying the difference between
((LOG(N)-LOG(D))*(n/LOG(2)) and its value when rounded to the nearest
whole number?

[Paul}
>You got it!

I thought so, and I even looked at this before... but I'm either still
missing something or carrying out the shortcut (of Paul Hahn's) that
you previously outlined:

"Paul Hahn's method is easiest. In each ET, calculate the closest
approximation to 3/2, 9/8, and 11/8. Express each approximation's
signed error in terms of steps of the ET. These errors will range
from -0.5 to 0.5. If (and only if) the difference between the largest
error and the smallest error is less than 0.5, then the tuning is
consistent with respect to (1, 3, 9, 11)."

with respect to any (odd numbered) harmonic sequence (as opposed to
just a 1, 3, 9, 11) will only give a rough approximation... For
instance, what would this method make 10e consistent through?

Dan

Dan Stearns wrote,

>For
>instance, what would this method make 10e consistent through?

10-equal is consistent through the 7-limit (meaning 1 3 5 7), but not the
9-limit. The harmonic with the largest positive rounding error is 5 (+0.22),
and the harmonic with the largest negative rounding error is 9 (-0.30). So
we see that the 9-limit "consonant" interval 9/5 will not be represented by
its closest approximation if we construct a chord like 4:5:9 and use the
best approximations of 5:4 and 9:4.

See Paul Hahn's page http://library.wustl.edu/~manynote/consist3.txt for the
the stepsize and the greatest error among the primary or "consonant"
intervals within the harmonic limit, both in cents, for all ETs up to
200TET, for all harmonic limits within which they are consistent (Thanks
Paul H.!). The columns to the right of the | symbol correspond to harmonic
limits 3, 5, 7, 9, etc.

-Paul E.

[Paul H. Erlich:]
>10-equal is consistent through the 7-limit (meaning 1 3 5 7), but not
the 9-limit.

Yes, that's what I thought... but before you wrote "if (and only if)
the difference between the largest error and the smallest error is
less than 0.5, then the tuning is consistent," and I took this to mean
the difference between the largest and smallest difference from
zero... But what I thought was going on is what you describe here:
"The harmonic with the largest positive rounding error is 5 (+0.22),
and the harmonic with the largest negative rounding error is 9
(-0.30)."

So anyway, I Got it now... and thanks for the link to Paul Hahn's
(http://library.wustl.edu/~manynote/consist3.txt) table.

Dan

In-Reply-To: <940926037.14272@onelist.com>
Paul Erlich (digest 367.8) wrote:

> Graham wrote,
>
> >I moved the scale to start at each of its degrees. I don't even know
> what >"inversion" means here: turning the scale upside-down gives a
> >transposition of itself. If this does make a difference, I didn't
> take >account of it.
>
> The result of combining all the modes of a scale on one tonic is
> equivalent
> to the cross-set of a scale with its own inversion (yes, the upside-down
> version of the scale, in this case a transposition of the original
> scale).
> That's what you were doing, right?

Yes. This is what I would expect "inversion" to mean, but it contradicts
the usual meaning wrt chord inversions.

> >One of us must me missing something.
>
> Well, I was just wondering why boh you and Dan seem so interested in
> the big
> 19-tone scale rather than the original 10-tone scale, and why you both
> seemed to slip from discussing the small one into discussing the big one
> without any indication that you were doing so.

I did it because Dan did it!

I introduced the diagram (in digest 362.6) by saying "I think transposing
that scale to each of its degrees will give" which I thought indicated
what I was doing.

As the discussion was about "my scale" circled around, I thought I'd show
what my scale would look like after that operation. It didn't look like
the diagrams you two were producing, and I've now gone back through the
digests to see why.

First of, this diamond thing of *my* scale on *my* lattice is

A---C/--E---G/--B---D/--F#
\ / \ / \ / \ / \ / \ / \
/ / / / / / \
/ \ / \ / \ / \ / \ / \ \
A\--C---E\--G---B\--D---F/--A---C/
\ \ / \ / \ / \ / \ / \ /
\ / / / / / /
\ / \ / \ / \ / \ / \ / \
Eb--G\--Bb--D\--F---A\--C

You can join the edges so it infinitely repeats. This isn't quite what I
posted before. I've put note names in to show where the neutral thirds
are. The base scale is normal meantone.

Here's the latest version of how Paul thinks the scale should look

*===*.����*.����*.����*.����*.����*.����*.����*.====*.
`._ \`._/ \`._/ \`._/ \`._/ \`._/ \`._/ \`._/ \`._
`._\ /`._\ /`._\ /`._\ /`._\ /`._\ /`._\ /`._\ `._
*.===`*.���`*.���`*.���`@.���`*����`*����`*====`*.
`._ \`._/ \`._/ \`._/ \`._/ \`._ \`._/ \`._/ \`._
`._\ /`._\ /`._\ /`._\ /`._\ /`._\ /`._\ /`._\ `._
`*====`*����`*����`*����`*����`*����`*����`*����`*===*

from digest 359.

As well as not having the 5-limit intervals, that's drawn on a different
lattice. On my lattice, the scale is a single string of neutral thirds,
which is trivial. On the lattice that Dan originally drew, it is

*---*---*---*---*---*---*---*---*---*
\ / \ / \ / \ / \ / \ / \ / \ / \ /
*---*---*---*---*---*---*---*---*
/ \ / \ / \ / \ / \ / \ / \ / \ / \
*---*---*---*---*---*---*---*---*---*

Where the primary consonance is

11
--
/ 9 \
/ \
/ \
1 / \ 3
- ----------- -
1 2

I've now discovered that digest 357 is where the diagram switched so the
triangle is this consonance:

11
--
/ 8 \
/ \
/ \
1 / \ 3
- ----------- -
1 2

and that does mean the two diagrams show the same scale. So we're all
happy. But I still think

Eb-G\-Bb-D\-F--A\-C--E\-G--B\-D--F/-A--C/-E--G/-B--D/-F#

is simplest.

For a more *interesting* scale, how about 17 notes, which is an MOS.

Bb--D\--F---A\--C---E\--G---B\
\
C/--E---G/--B---D/--F# \
\ / \ / \ / \ / \ / \ \
/ / / / / \ Bb
/ \ / \ / \ / \ / \ \
C---E\--G---B\--D---F/--A---C/--E
\ \ / \ / \ / \ / \ /
F# \ / / / / /
\ \ / \ / \ / \ / \ / \
\ Bb--D\--F---A\--C---E\
\
F/--A---C/--E---G/--B---D/--F#

I've added some 7-limit intervals there which are consistent with
31-equal. Here's the same thing with 24 notes

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

Try showing all that on your puny JI lattice!!!!!

Now we do have all the intervals from the 11-limit, it may be worth using
the other lattice explained here:

http://x31eq.com/lattice.htm#11limit

But that would mean work. As the scales are generated from a string of
neutral thirds, the neutral third-based lattice is easier.

>Yes. This is what I would expect "inversion" to mean, but it contradicts
>the usual meaning wrt chord inversions.

I know. I think I picked up this non-usual usage from Daniel Wolf. I should
try to drop it.

>I did it because Dan did it!

Oh!

Graham Breed wrote,

>Now we do have all the intervals from the 11-limit, it may be worth using
>the other lattice explained here:

>http://x31eq.com/lattice.htm#11limit

I see a section on 7-limit, but not on 11-limit, at that page.

In-Reply-To: <941014214.24137@onelist.com>
Paul Erlich wrote:

> I see a section on 7-limit, but not on 11-limit, at that page.

By 'eck, you're right! I added the 11-limit stuff nearly two weeks ago,
but obviously forgot to upload it. I have done so now, so let's try
again.

http://x31eq.com/lattice.htm#11limit