Re: BP linear temperament

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Perhaps my view was too severe, but it definitely seems to contradict
> the _spirit_ of the near-just approximations to all simple ratios of
> odd numbers, since the approximations to simple ratios involving even
> numbers are so poor.

Oh no they're not. For sure they are bad in 13-ED3, but the 7-limit
(1,2,4,5,7) optimum version of the Bohlen-Pierce-Stearns temperament (1:3
period, 442.60 cent generator) has no error greater than 8.4 cents. The
octave itself is only 3.8c narrow.

A chain of 15 generators (16 notes) is required for a complete 7-limit
pentad (e.g. 3:4:5:6:7 or 4:5:6:7:9).

Remember that we have tritave equivalence instead of octave equivalence. So
for example 1:5, 3:5, 5:9 are all "tritave equivalent" so they all have the
same error magnitude in this temperament, and they do not have the same
error as 4:5, 5:6, or 9:10.

This 7-limit MA (max-absolute) optimum generator is a 7:9 widened by 2/15
of the following "BPS-comma".

1647086 2^1 * 7^7
------- = ------- ~= 56.37 c
1594323 3^13

And in fact the following 10-limit ratios come in under 8.4 cents error as
well. 1:10 (9:10), 7:8, 7:10. But not 1:8 (8:9) or 5:8, (11.3c and 12.1c
errors).

The 10-limit (1,2,4,5,7,8,10) MA optimum generator is 442.77 cents with max
error of 8.9c. This is 3/22 BPS-comma.

So we get denominators of convergents (and semiconvergents) for this
temperament being:
4 (5) (9) 13 (17) 30 43 (73) 116
These can be read as MOS cardinalities and the larger ones as possibly
useful ED3s.

--------
Keyboard
--------
I've attached a .gif) of the keyboard mapping (showing cents) for 30 notes
of this temperament, which also makes it clear that the "natural" notation
for it would have only 4 nominals to the tritave.
To see a 3:4:5:6:7 pattern, look at notes marked 0 491 885 1196 1459 (cents).

------
Guitar
------
This 30-note per tritave proper MOS will work incredibly well on a guitar!
Simply tune adjacent open strings one generator (a supermajor third) apart.
The smallest step is 48 cents. This is no worse than a 24-EDO guitar, but
notice that it has 19 steps per octave.

4:5:6:7:9 barre chords should be playable.

Here's the optimum rotation of the scale for fretting such a guitar. The
zero of the rotation shown on the keyboard map, corresponds to the 16th
fret (1017 cents) below. The keyboard scale is then playable everywhere on
the top string, and as far from the nut as possible on the other strings.

Fret Step
posn size (to the nearest cent)
-----------
0 48
48 84
132 48
179 84
263 48
311 84
395 48
443 84
526 48
574 48
622 84
706 48
754 84
837 48
885 84
969 48
1017 48
1065 84
1148 48
1196 84
1280 48
1328 84
1411 48
1459 48
1507 84
1591 48
1639 84
1722 48
1770 84
1854 48
1902

Good work Dan!

Regards,
-- Dave Keenan

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

--- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > Perhaps my view was too severe, but it definitely seems to
contradict
> > the _spirit_ of the near-just approximations to all simple ratios
of
> > odd numbers, since the approximations to simple ratios involving
even
> > numbers are so poor.
>
> Oh no they're not.

I meant those producable by a small number of generators. In any
case, I think it's worth pursuing this from _both_ angles.

How about the triple-BP scale I discovered
(http://members.aol.com/bpsite/scales.html#anchor417408)? Is there a
particular linear temperament that this cries out for?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote:
> > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > Perhaps my view was too severe, but it definitely seems to
> contradict
> > > the _spirit_ of the near-just approximations to all simple
ratios
> of
> > > odd numbers, since the approximations to simple ratios involving
> even
> > > numbers are so poor.
> >
> > Oh no they're not.
>
> I meant those producable by a small number of generators.

I think 7 is a small number. That's how many it needs for the octave
(and hence the fifth) with 3.8c errors.

Prime No. Generators
----- --------------
2 7
3 0
5 2
7 -1

> In any
> case, I think it's worth pursuing this from _both_ angles.

Sure. If 2s are omitted, the MA optimum generator is 439.82c and the
max error is 4.8c.

> How about the triple-BP scale I discovered
> (http://members.aol.com/bpsite/scales.html#anchor417408)? Is there a
> particular linear temperament that this cries out for?

Probably, but I don't have time. A suggested approach: Give the
simplest frequency ratio you can for each scale degree. Then list the
step sizes between these, classify them into two sizes, L and s, then
figure out what the generator is from that.
-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> >
> > I meant those producable by a small number of generators.
>
> I think 7 is a small number. That's how many it needs for the
octave
> (and hence the fifth) with 3.8c errors.
>
> Prime No. Generators
> ----- --------------
> 2 7
> 3 0
> 5 2
> 7 -1

By these standards I think 7 is a large number! It's much larger in
magnitude than 0, 2, or -1. And to get some of the simplest ratios
involving 2 (such as 4:3, 5:4, and 7:4) you need around 14
generators. Quite a qualitative difference, in my opinion, compared
with what you need for the basic BP consonances.

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> Probably, but I don't have time. A suggested approach: Give the
> simplest frequency ratio you can for each scale degree.

Hmm . . . this would be the Kees van Prooijen definition of a
periodicity block. Have you studied this page? Manuel, can we do this
in SCALA?

> Then list the
> step sizes between these, classify them into two sizes, L and s,
then
> figure out what the generator is from that.

The Hypothesis in action!

Paul wrote:

>> Probably, but I don't have time. A suggested approach: Give the
>> simplest frequency ratio you can for each scale degree.

>Hmm . . . this would be the Kees van Prooijen definition of a
>periodicity block. Have you studied this page? Manuel, can we do this
>in SCALA?

Yes, with APPROXIMATE while using one of the attribute functions
for your definition of "simple".

>> Then list the
>> step sizes between these, classify them into two sizes, L and s,
>> then figure out what the generator is from that.

> The Hypothesis in action!

In case of the Triple BP scale I don't see a generator for it.
If we omit one of 13/11 and 25/21 and do FIT/MODE, we see there
are two approximations that have two step sizes, a mode of 26th
root of 3 and one of 32nd root of 3, but neither is Myhill.
If there would be one, finding the generator is easy by doing
QUANTIZE for that ET and then SHOW DATA.

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
>
> >> Probably, but I don't have time. A suggested approach: Give the
> >> simplest frequency ratio you can for each scale degree.
>
> >Hmm . . . this would be the Kees van Prooijen definition of a
> >periodicity block. Have you studied this page? Manuel, can we do
this
> >in SCALA?
>
> Yes, with APPROXIMATE while using one of the attribute functions
> for your definition of "simple".
>
> >> Then list the
> >> step sizes between these, classify them into two sizes, L and s,
> >> then figure out what the generator is from that.
>
> > The Hypothesis in action!
>
> In case of the Triple BP scale I don't see a generator for it.

Are you including the full 39 pitches per tritave?

Hi Dan,

>I suppose I really should just get off my duff and install your
>wonderful program Scala (though I can do the above without it easy
>enough as well, and old habits sometimes die hard)!

Yes, I know. Well you could still decide not to use it after having
it tried.

>Does Scala have a function that can take and arbitrary set of notes
>and quantize it so that it can be rendered as a 2D lattice (i.e., in
>"triads" where the 1D plane is also the two stepsize MOS generator) --
>preferably in its simplest (nearest to the 1/1) configuration?

If I understand what you mean, i.e. can Scala calculate a multilinear
temperament that approximates an arbitrary scale in some optimal way,
no. It can however calculate a linear temperament, least squares and
minimax optimal. It can also approximate a scale by a JI scale for a
given set of primes.

Manuel

Paul wrote:
>Are you including the full 39 pitches per tritave?

No, I wasn't. But trying that, after filling the gaps I got the scale:
1/1 65/63 35/33 27/25 55/49 15/13 25/21 11/9 49/39 9/7 65/49 15/11 7/5
13/9 49/33 75/49 11/7 21/13 5/3 77/45 135/77 9/5 13/7 21/11 49/25
99/49 27/13 15/7 11/5 147/65 7/3 117/49 27/11 63/25 13/5 147/55 25/9
99/35 189/65 3/1
I saw there's no L+S approximation to it that is Myhill.
If you give me the set of unison vectors I can try to find some other
versions using the periodicity block shaping of Scala and see if any
of those can be approximated by a MOS, but the chance looks small that
it can be found like this by hand.

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
> >Are you including the full 39 pitches per tritave?
>
> No, I wasn't. But trying that, after filling the gaps I got the
scale:
> 1/1 65/63 35/33 27/25 55/49 15/13 25/21 11/9 49/39 9/7 65/49 15/11
7/5
> 13/9 49/33 75/49 11/7 21/13 5/3 77/45 135/77 9/5 13/7 21/11 49/25
> 99/49 27/13 15/7 11/5 147/65 7/3 117/49 27/11 63/25 13/5 147/55
25/9
> 99/35 189/65 3/1
> I saw there's no L+S approximation to it that is Myhill.
> If you give me the set of unison vectors

You can find the unison vectors by calculating the ratios of
(differences between) various pairs of generic seconds, pairs of
generic thirds, etc.

> I can try to find some other
> versions using the periodicity block shaping of Scala and see if any
> of those can be approximated by a MOS, but the chance looks small
that
> it can be found like this by hand.

Well at least you're not trying to do it entirely by hand!

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
>
> If you give me the set of unison vectors I can try to find some
other
> versions using the periodicity block shaping of Scala and see if any
> of those can be approximated by a MOS, but the chance looks small
that
> it can be found like this by hand.

P.S. if you temper out all but one of the unison vectors, you're
guaranteed to get an MOS, by my Hypothesis . . . right? I'd look for
a set of unison vectors that are all super-super-particular (n=d+2),
then temper out all but the largest one (the one with the smallest
numbers) . . .

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> You can find the unison vectors by calculating the ratios of
> (differences between) various pairs of generic seconds, pairs of
> generic thirds, etc.

Here are some I found, just by looking at the generic seconds:

243:245
273:275
1323:1331
1617:1625

That should be enough . . . in 4-D, (5,7,11,13) space, with 3 as the
interval of equivalence, these UVs become

(-1 -2 0 0)
(-2 1 -1 1)
( 0 2 -3 0)
(-3 2 1 -1)

and the determinant of this matrix is -39. Whew!

Now what's the generator of the MOS we get by tempering out all of
the above UVs, except for, say, 243:245?

In-Reply-To: <9k71vn+ll5m@eGroups.com>
Paul wrote:

> > You can find the unison vectors by calculating the ratios of
> > (differences between) various pairs of generic seconds, pairs of
> > generic thirds, etc.
>
> Here are some I found, just by looking at the generic seconds:
>
> 243:245
> 273:275
> 1323:1331
> 1617:1625
>
> That should be enough . . . in 4-D, (5,7,11,13) space, with 3 as the
> interval of equivalence, these UVs become
>
> (-1 -2 0 0)
> (-2 1 -1 1)
> ( 0 2 -3 0)
> (-3 2 1 -1)
>
> and the determinant of this matrix is -39. Whew!

Oh, you want 13-limit now, do you? My program gives

13/56, 279.0 cent generator

basis:
(1.0, 0.23248676035896343)

mapping by period and generator:
[(1, 0), (-1, 7), (1, 2), (2, -1), (-2, 18), (0, 10)]

mapping by steps:
[(43, 13), (27, 8), (63, 19), (76, 23), (94, 28), (100, 30)]

unison vectors:
[[-2, -1, -1, 1, 0, 1], [9, 2, -7, 0, 0, 0], [-13, 1, 0, 7, 0, 0], [4, 18,
0, 0, -7, 0]]

highest interval width: 19
complexity measure: 19 (30 for smallest MOS)
highest error: 0.009850 (11.820 cents)

(Caveat: cents are not cents, the consonance limit is arbitrary)

> Now what's the generator of the MOS we get by tempering out all of
> the above UVs, except for, say, 243:245?

(-1 -2 0 0)(2) (0)
(-2 1 -1 1)(-1) = (-13)
( 0 2 -3 0)(18) (-56)
(-3 2 1 -1)(10) (0)

So only two of your UVs work with my MOS.

Taking my UVs

[[-2, -1, -1, 1, 0, 1], [9, 2, -7, 0, 0, 0], [-13, 1, 0, 7, 0, 0], [4, 18,
0, 0, -7, 0]]

These are in (3,2,5,7,11,13) space. So they are, um, 91/90, 78732/78125,
1647086/1594323 and 21233664/19487171. I expect they can be simplified.

The real generator of my MOS is
0.23248676035896343*log2(3)*1200=442.2 cents. Not considering octaves, I
don't get a scale that looks right in the top 10. See
<http://x31eq.com/tritave.nonoct.py> and
<http://x31eq.com/tritave.nonoct.txt> as well as
<http://x31eq.com/tritave.py> and
<http://x31eq.com/tritave.txt>.

Graham

Paul wrote:

> Here are some I found, just by looking at the generic seconds:
>
> 243:245
> 273:275
> 1323:1331
> 1617:1625

Creating a periodicity block with these unison intervals was an
immediate result! This is the scale:
1/1 77/75 35/33 27/25 55/49 63/55 25/21 11/9 1701/1375 9/7 33/25
15/11 7/5 275/189 81/55 75/49 11/7 441/275 5/3 77/45 135/77 9/5
275/147 21/11 49/25 99/49 567/275 15/7 11/5 25/11 7/3 825/343 27/11
63/25 55/21 147/55 135/49 99/35 225/77 3/1
It is 11-limit. The 56th root of 3 quantisation is Myhill and the
generator is 11/7 (782.49 c.) or 21/11 (1119.46 c.).
The least squares best generator for the above scale is 780.8095
cents, s=39.924 cents (22) and L=60.213 cents (17). So it's very
close to the 95th root of 3.
1/1 39.9241 100.1373 140.0614 200.2747 240.1987 300.4120 340.3361
380.2601 440.4734 480.3975 540.6107 580.5348 640.7481 680.6721
740.8854 780.8095 820.7335 880.9468 920.8709 981.0841 1021.0082
1081.2215 1121.1455 1161.0696 1221.2829 1261.2069 1321.4202
1361.3443 1421.5575 1461.4816 1521.6949 1561.6189 1601.5430
1661.7563 1701.6803 1761.8936 1801.8177 1862.0309 3/1

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
>
> > Here are some I found, just by looking at the generic seconds:
> >
> > 243:245
> > 273:275
> > 1323:1331
> > 1617:1625
>
> Creating a periodicity block with these unison intervals was an
> immediate result! This is the scale:
> 1/1 77/75 35/33 27/25 55/49 63/55 25/21 11/9 1701/1375 9/7 33/25
> 15/11 7/5 275/189 81/55 75/49 11/7 441/275 5/3 77/45 135/77 9/5
> 275/147 21/11 49/25 99/49 567/275 15/7 11/5 25/11 7/3 825/343 27/11
> 63/25 55/21 147/55 135/49 99/35 225/77 3/1
> It is 11-limit. The 56th root of 3 quantisation is Myhill and the
> generator is 11/7 (782.49 c.) or 21/11 (1119.46 c.).
> The least squares best generator for the above scale is 780.8095
> cents, s=39.924 cents (22) and L=60.213 cents (17).

Your least squares optimization uses what mapping from generators to
primes? And it considers the errors for intervals 3:5, 3:7, 3:11,
3:13, 5:7, 5:11, 5:13, 7:11, 7:13, 11:13 . . . right?

> So it's very
> close to the 95th root of 3.
> 1/1 39.9241 100.1373 140.0614 200.2747 240.1987 300.4120 340.3361
> 380.2601 440.4734 480.3975 540.6107 580.5348 640.7481 680.6721
> 740.8854 780.8095 820.7335 880.9468 920.8709 981.0841 1021.0082
> 1081.2215 1121.1455 1161.0696 1221.2829 1261.2069 1321.4202
> 1361.3443 1421.5575 1461.4816 1521.6949 1561.6189 1601.5430
> 1661.7563 1701.6803 1761.8936 1801.8177 1862.0309 3/1
>
> Manuel

Paul wrote:
>Your least squares optimization uses what mapping from generators to
>primes? And it considers the errors for intervals 3:5, 3:7, 3:11,
>3:13, 5:7, 5:11, 5:13, 7:11, 7:13, 11:13 . . . right?

Not exactly, it was an approximation to all pitches of the "11-limit"
scale given. A bit more work, but can do that too. Then the result
is a virtually equal scale with a LS generator of 780.2702 cents.
Differences are
3:5 6.647 / 5.906
3:7 3.772 / 4.513
3:11 5.993 / 6.734
3:13 2.880 / 2.139
5:7 -2.876 / -2.135
5:11 -0.654 / 0.087
5:13 -3.768 / -4.509
7:11 2.222 / 1.481
7:13 -0.892 / -1.633
11:13 -3.114 / -3.855
The first one is the optimised value, the second one is the one
occurring fewer times in the scale.
The minimax optimal generator is 780.3520 cents giving a slightly
less equal scale with differences of
3:5 6.157 / 8.604
3:7 4.017 / 1.570
3:11 6.157 / 3.710
3:13 1.817 / 4.264
5:7 -2.140 / -4.587
5:11 0.0 / -2.447
5:13 -4.340 / -1.893
7:11 2.140 / 4.587
7:13 0.247 / -2.200
11:13 -4.340 / -1.893

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
> >Your least squares optimization uses what mapping from generators
to
> >primes? And it considers the errors for intervals 3:5, 3:7, 3:11,
> >3:13, 5:7, 5:11, 5:13, 7:11, 7:13, 11:13 . . . right?
>
> Not exactly, it was an approximation to all pitches of the "11-
limit"
> scale given.

Now why would anyone want that? It's the consonant intervals, not the
pitches, that we care about approximating well. (At least that's my
philosophy.)

> A bit more work, but can do that too. Then the result
> is a virtually equal scale

You mean virtually the 39th root of 3? That's more what I would have
expected.

> with a LS generator of 780.2702 cents.
> Differences are
> 3:5 6.647 / 5.906
> 3:7 3.772 / 4.513
> 3:11 5.993 / 6.734
> 3:13 2.880 / 2.139
> 5:7 -2.876 / -2.135
> 5:11 -0.654 / 0.087
> 5:13 -3.768 / -4.509
> 7:11 2.222 / 1.481
> 7:13 -0.892 / -1.633
> 11:13 -3.114 / -3.855
> The first one is the optimised value, the second one is the one
> occurring fewer times in the scale.

Can you explain what the latter means?

>> Not exactly, it was an approximation to all pitches of the "11-
>> limit" scale given.

>Now why would anyone want that? It's the consonant intervals, not the
>pitches, that we care about approximating well. (At least that's my
>philosophy.)

But the former doesn't exclude the latter. Approximating the
intervals everywhere in the scale smoothes out a just scale more than
somebody might want.
The posted "11-limit" periodicity block contains 13 chords 3:5:7:11.
The "approximation to pitches" scale contains 30 with a maximum
deviation of 7.072 cents and the "approximation to consonant
intervals" scale 39 with a 6.647 cents maximum deviation. The
L/s stepsize ratio of the former is 3:2 and 1.0153:1 of the latter.
So I would say it has merit if you're interested in a scale that is
more equal than just but not as equal as ET. By the way the Scala
command I used, BISTEP, approximates a scale by one with two step
sizes, but it doesn't require the step pattern to be Myhill.

>You mean virtually the 39th root of 3? That's more what I would have
>expected.

Yes.

>> The first one is the optimised value, the second one is the one
>> occurring fewer times in the scale.

>Can you explain what the latter means?

Because the scale is almost equal, both interval sizes of each
interval class are close. These are the deviations of the intervals
5/3 7/3 11/3 13/3 (the chord 3:5:7:11:13) in the minimax best scale,
so you can see it:
Locations of 5/3 7/3 11/3 13/3:
0 - 18 - 30 - 46 - 52 diff. 6.157, 1.570, 3.710, 1.817
1 - 19 - 31 - 47 - 53 diff. 6.157, 4.017, 6.157, 1.817
2 - 20 - 32 - 48 - 54 diff. 6.157, 4.017, 6.157, 1.817
3 - 21 - 33 - 49 - 55 diff. 6.157, 4.017, 6.157, 4.264
4 - 22 - 34 - 50 - 56 diff. 6.157, 4.017, 6.157, 1.817
5 - 23 - 35 - 51 - 57 diff. 6.157, 4.017, 6.157, 4.264
6 - 24 - 36 - 52 - 58 diff. 6.157, 4.017, 6.157, 1.817
7 - 25 - 37 - 53 - 59 diff. 8.604, 4.017, 6.157, 4.264
8 - 26 - 38 - 54 - 60 diff. 6.157, 4.017, 6.157, 1.817
9 - 27 - 39 - 55 - 61 diff. 6.157, 4.017, 6.157, 1.817
10 - 28 - 40 - 56 - 62 diff. 6.157, 4.017, 6.157, 1.817
11 - 29 - 41 - 57 - 63 diff. 6.157, 4.017, 6.157, 1.817
12 - 30 - 42 - 58 - 64 diff. 6.157, 4.017, 6.157, 4.264
13 - 31 - 43 - 59 - 65 diff. 6.157, 4.017, 6.157, 1.817
14 - 32 - 44 - 60 - 66 diff. 8.604, 4.017, 6.157, 4.264
15 - 33 - 45 - 61 - 67 diff. 6.157, 4.017, 6.157, 1.817
16 - 34 - 46 - 62 - 68 diff. 6.157, 1.570, 3.710, 1.817
17 - 35 - 47 - 63 - 69 diff. 6.157, 4.017, 6.157, 1.817
18 - 36 - 48 - 64 - 70 diff. 6.157, 4.017, 6.157, 1.817
19 - 37 - 49 - 65 - 71 diff. 6.157, 4.017, 6.157, 4.264
20 - 38 - 50 - 66 - 72 diff. 6.157, 4.017, 6.157, 1.817
21 - 39 - 51 - 67 - 73 diff. 8.604, 4.017, 6.157, 4.264
22 - 40 - 52 - 68 - 74 diff. 6.157, 4.017, 6.157, 1.817
23 - 41 - 53 - 69 - 75 diff. 8.604, 4.017, 6.157, 4.264
24 - 42 - 54 - 70 - 76 diff. 6.157, 4.017, 6.157, 1.817
25 - 43 - 55 - 71 - 77 diff. 6.157, 4.017, 6.157, 1.817
26 - 44 - 56 - 72 - 78 diff. 6.157, 4.017, 6.157, 4.264
27 - 45 - 57 - 73 - 79 diff. 6.157, 4.017, 6.157, 1.817
28 - 46 - 58 - 74 - 80 diff. 6.157, 4.017, 6.157, 4.264
29 - 47 - 59 - 75 - 81 diff. 6.157, 4.017, 6.157, 1.817
30 - 48 - 60 - 76 - 82 diff. 8.604, 4.017, 6.157, 4.264
31 - 49 - 61 - 77 - 83 diff. 6.157, 4.017, 6.157, 1.817
32 - 50 - 62 - 78 - 84 diff. 6.157, 1.570, 6.157, 1.817
33 - 51 - 63 - 79 - 85 diff. 6.157, 4.017, 6.157, 1.817
34 - 52 - 64 - 80 - 86 diff. 6.157, 4.017, 6.157, 1.817
35 - 53 - 65 - 81 - 87 diff. 6.157, 4.017, 6.157, 4.264
36 - 54 - 66 - 82 - 88 diff. 6.157, 4.017, 6.157, 1.817
37 - 55 - 67 - 83 - 89 diff. 8.604, 4.017, 6.157, 4.264
38 - 56 - 68 - 84 - 90 diff. 6.157, 4.017, 6.157, 1.817

Manuel