ET triads & lattice patterns

[Lumma:]
> When you hit 53, you just love it.
> It is pretty much indistinguishable from just,
> at least under the conditions I was using...

53-Eq has a step size almost midway between
the Pythagorean [3-limit] and Syntonic [5-limit]
commas.

Syntonic Comma 81:80 21.51 c
53-Eq step size 2^(1/53) 22.64 c
Pythagorean Comma (3^12):(2^19) 23.46 c

The mean interval between the commas would
have a value of 22.48 cents.

If one desires an ET which balances good 5-limit
rational approximation against complexity and size
of resources, 53-Eq's the winner.

[Lumma:]
> 41 sounds better than 31, but not as good as
> I thought it would.

An | 8 4 | Euler Genus (eight 3/2s wide by four 5/4s high
= 45 different 5-limit pitches) contains 4 schismas,
thus, assuming the schisma [= 2 cents: the difference
between the commas] to be imperceptible, and ignoring it,
there is audibly a system of 41 different pitches.
One might think that 41-Eq would represent this best,
but this is not the case.

For this Genus, the largest error of 53-Eq from JI
is only 3.09 cents, and the smallest is 0.068 of a cent!
(for the 3/2)

Because the Euler Genus is symmetrical, whatever
error value one Eq note has for a particular ratio,
there will be a complementary inverse error for
the nearest approximation to the complement ratio.

For comparison, some other ETs mentioned by Carl,
representing the same Euler Genus, would have the
following min/max errors (in cents), with the implied
ratios (and assuming their complements):

ERROR FROM 5-limit JI
ET minimum maximum
12 1.96 3/2 35.19 100/81
19 0.15 6/5 29.17 225/128
22 1.86 75/64 26.00 128/81
31 0.78 5/4 19.16 2025/1024
41 0.48 3/2 13.59 100/81
53 0.07 3/2 3.09 2025/1024

From the table one can see a dramatic increase in
closeness to the 3- and 5-limit ratios at 53-Eq,
as well as similarities between some of these temperaments
as far as how well they approximate 5-limit JI.

12-, 41-, and 53-Eq accumulate larger errors
going in both directions on the lattice. In other
words, the greatest errors are at the corners of the
lattice.

This is not the case for 31-Eq, because its error is
so small for 5/4 and so much greater for 3/2, that
going along the any of the 5-axes does not increase
the error by much, but along any of the 3-axes does.
The error for *all* intervals with 81 [= 3^4] as a
factor is near the maximum of 19 cents.

In fact it is exactly the opposite of 12-Eq, which
implies the 3/2s much better than the 5/4s.

[Lumma:]
> 19 and 22 sound more alike than any in the bunch.

With 19- and 22-Eq the errors are more evenly-distributed,
although also more chaotically, because there are
somewhat significant errors for both 3/2 and 5/4,
7 cents for both ratios in 19-Eq, and 7 and 4 cents
respectively for 22-Eq. Generally speaking,
viewed on the lattice, there is a dip in the amount
of error which reaches its low point along a diagonal
axis.

[Lumma:]
> 19 sounds a tiny bit better,
> maybe because of its one just interval.

With 19-Eq the dip travels diagonally as powers
of 5 increase and powers of 3 decrease, and vice-versa.
Because the amount of error is the same for both
axes, there is an extreme diagonal periodicity in the
implied 5-limit lattice of 19-Eq.

With 22-Eq the dip travels diagonally exactly the
opposite direction as 19-Eq, as the exponents of
both numbers get larger/smaller.

As Carl pointed out, for 53-Eq the error is virtually
negligible for the whole implied 5-limit lattice.

I could have put all this on a diagram . . .

- Monzo
http://www.ixpres.com/interval/monzo/homepage.html
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Two important things I left out of my last
post concerning lattice patterns of ETs:

1)
The ET degree which falls nearest an approximated
ratio may not necessarily be the one which would
be most likely to be used to represent that ratio.
Issues such as consistency may come into play

I haven't looked at it for these scales, but
24-Eq has major inconsistencies. One example:
(I'm using ^ with the letter-name to mean
"half-sharp"; where it appears with numbers
it indicates exponents, as usual.)

Assume C = 2^(0/24) [i.e., 1:1].

The best approximation of a 5:4 is E 2^(8/24),
an interval of 2^(8/24), or 8 24-Eq degrees.

However, the best approximation of the 5:4 above
that is G^ 2^(15/24), an interval of only 2^(7/12)
above 2^(8/24). Consistency would dictate that we
use G# 2^(16/24), but that is farther away from
the 25:16 ratio.

2)
Some of the ETs that aren't as good as 53 in 5-limit
are better than it for higher-limit ratios. 22-Eq,
for instance, gives approximations to 7- and 11-limit
ratios with no worse error than its approximations in
the 5-limit.

-Monzo
http://www.ixpres.com/interval/monzo/homepage.html
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Joe Monzo wrote,

>I haven't looked at it for these scales, but
>24-Eq has major inconsistencies. One example:
>(I'm using ^ with the letter-name to mean
>"half-sharp"; where it appears with numbers
>it indicates exponents, as usual.)

>Assume C = 2^(0/24) [i.e., 1:1].

>The best approximation of a 5:4 is E 2^(8/24),
>an interval of 2^(8/24), or 8 24-Eq degrees.

>However, the best approximation of the 5:4 above
>that is G^ 2^(15/24), an interval of only 2^(7/12)
>above 2^(8/24). Consistency would dictate that we
>use G# 2^(16/24), but that is farther away from
>the 25:16 ratio.

I wouldn't consider that a major inconsistency. 25:16 doesn't really
arise as a consonance on its own; it is normally constructed as two
5:4s, so as long as each of those is approximated well, you're golden.

Here's a major inconsistency in 24-Eq, and the one with which I
introduced the consistency issue here years ago:
The best approximation of a 5:4 is 2^(8/24); the best approximation of a
7:5 is 2^(12/24); but the best approximation of a 7:4 (the result of
stacking a 5:4 and a 7:5) is 2^(19/24), while the other two
approximations stack up to 2^(20/24).