More microtemperaments

I've altered my temperament finding program to accept only temperaments
with a worst error of less than 2.8 cents. I think this is the cutoff for
a microtemperament. Results are at

<http://x31eq.com/limit5.micro>
<http://x31eq.com/limit7.micro>
<http://x31eq.com/limit9.micro>
<http://x31eq.com/limit11.micro>
<http://x31eq.com/limit13.micro>
<http://x31eq.com/limit15.micro>

Some of them don't have as many as 10 results.

Graham

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

> I've altered my temperament finding program to accept only
temperaments
> with a worst error of less than 2.8 cents. I think this is the
cutoff for
> a microtemperament.

Is there somewhere where the meaning of this results is documented?
It's hard to tell what to think of an error less than 2.8 cents when
one doesn't know what the error is a departure from, for instance.

In-Reply-To: <9mfr8i+di6b@eGroups.com>
In article <9mfr8i+di6b@eGroups.com>, genewardsmith@juno.com () wrote:

> Is there somewhere where the meaning of this results is documented?
> It's hard to tell what to think of an error less than 2.8 cents when
> one doesn't know what the error is a departure from, for instance.

The departure is from a JI odd limit. That is, all odd numbers up to the
one you pick are involved in ratios, and then you octave reduce. The
"mapping by steps" is your homomorphism.

Graham

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

> The departure is from a JI odd limit. That is, all odd numbers up
to the
> one you pick are involved in ratios, and then you octave reduce.
The
> "mapping by steps" is your homomorphism.

I think the best way to answer my questions may be to do some
detective work and then ask about anything which remains unclear. I'm
going therefore to take a look at the first 5-limit example, and add
commentary.

My understanding is that we want 3, 5 and 5/3 within 2.8 cents. If a
is how sharp our "3" is, and b is how sharp the "5" is, then we want
a and b in the hexagonal region determined by

|a| <= 2.8, |b| <= 2.8, |a-b| <= 2.8

We need further conditions to determine the tuning, so let's look at
what Graham does.

"3 4 5 6 7 8 9 10 12 15 16 18 19 22 23 24 25 26 27 28"

I don't know what these are.

"5/19, 317.0 cent generator

basis: (1.0, 0.26416041678685936)"

If we set r = 0.26416041678685936, then 5/19 is a convergent for r.
It's not clear why it is singled out; convergents for r are 1/3, 1/4,
4/15, 5/19, 9/34,14/53, ...

"mapping by period and generator:
[(1, 0), (0, 6), (1, 5)]"

This seems to explain where r came from: if we send (a, b) to a +
b*r, then (1,0) goes to log_2(2) = 1, (0,6) goes to 6*r which turns
out to be log_2(3), and (1,5) goes to 1+5*r which is the
approximation of log_2(5) we get when both octaves and fifths are
exact and [-6,-5,6] is in the kernel. Hence, r = 3^(1/6). Is Graham's
basic condition that all primes up to the last will be exactly
represented?

"mapping by steps:
[(15, 4), (24, 6), (35, 9)]"

It seems as if this may have something to do with the convergents to
r. We have the 4-et [4, 6, 9] from the convergent 1/4 and the 15-et
[15, 24, 35] from the convergent 4/15. We may then proceed to the
others:

[19, 30, 44] = [ 4, 6, 9] + [15, 24, 35]
[34, 54, 79] = [15, 24, 35] + [19, 30, 44]
[53, 84, 123] = [19, 30, 44] + [34, 54, 79]

after which a slew of semiconvergents come in. Graham says this
is "my homomorphism", but I'm getting a whole collection.

"unison vectors:
[[-6, -5, 6]]"

2^(-6)*3^(-5)*6^5 = 15625/15552 is the unison vector for anything
using the matrix M =

[0 1]
[6 5]

to approximate log_2(3) and log_2(5) using 1 and r' as a basis, so
that [1, r']M = [6r', 1+5r'] and so
6 r' approximates log_2(3) and 1 + 5 r' approximates log_2(5)--the
column vector V =

[ 1 ]
[ 6 r']
[1+5r']

has unison vectors generated by [-6, -5, 6].

"highest interval width: 6"

How did we get to intervals and scales?

"complexity measure: 6 (7 for smallest MOS)"

How is this defined?

"highest error: 0.001126 (1.351 cents)"

5/3 is off by this amount.

"unique"

What is unique?

This system is so close to the 53-et that it would seem to make sense
to adjust the fifth, the octave or both and make it exactly the 53-et.

genewardsmith@juno.com () wrote:

> My understanding is that we want 3, 5 and 5/3 within 2.8 cents. If a
> is how sharp our "3" is, and b is how sharp the "5" is, then we want
> a and b in the hexagonal region determined by
>
> |a| <= 2.8, |b| <= 2.8, |a-b| <= 2.8

Yes.

> We need further conditions to determine the tuning, so let's look at
> what Graham does.
>
> "3 4 5 6 7 8 9 10 12 15 16 18 19 22 23 24 25 26 27 28"
>
> I don't know what these are.

They're 5-limit consistent equal temperaments, being used to calculate the
linear temperaments.

> "5/19, 317.0 cent generator
>
> basis: (1.0, 0.26416041678685936)"
>
> If we set r = 0.26416041678685936, then 5/19 is a convergent for r.
> It's not clear why it is singled out; convergents for r are 1/3, 1/4,
> 4/15, 5/19, 9/34,14/53, ...

Because 19=4+15 is the simplest sum of numbers from the above list that
fits this temperament.

> "mapping by period and generator:
> [(1, 0), (0, 6), (1, 5)]"
>
> This seems to explain where r came from: if we send (a, b) to a +
> b*r, then (1,0) goes to log_2(2) = 1, (0,6) goes to 6*r which turns
> out to be log_2(3), and (1,5) goes to 1+5*r which is the
> approximation of log_2(5) we get when both octaves and fifths are
> exact and [-6,-5,6] is in the kernel. Hence, r = 3^(1/6). Is Graham's
> basic condition that all primes up to the last will be exactly
> represented?

The condition is that the worst error is as low as possible.

> "mapping by steps:
> [(15, 4), (24, 6), (35, 9)]"
>
> It seems as if this may have something to do with the convergents to
> r. We have the 4-et [4, 6, 9] from the convergent 1/4 and the 15-et
> [15, 24, 35] from the convergent 4/15. We may then proceed to the
> others:
>
> [19, 30, 44] = [ 4, 6, 9] + [15, 24, 35]
> [34, 54, 79] = [15, 24, 35] + [19, 30, 44]
> [53, 84, 123] = [19, 30, 44] + [34, 54, 79]
>
> after which a slew of semiconvergents come in. Graham says this
> is "my homomorphism", but I'm getting a whole collection.

Then this is a subtlety of "homomorphism" I wasn't aware of. I remember
you showing how a linear (2-D) temperament can be described using the
mappings of two equal temperaments.

> "unison vectors:
> [[-6, -5, 6]]"
>
> 2^(-6)*3^(-5)*6^5 = 15625/15552 is the unison vector for anything
> using the matrix M =
>
> [0 1]
> [6 5]
>
> to approximate log_2(3) and log_2(5) using 1 and r' as a basis, so
> that [1, r']M = [6r', 1+5r'] and so
> 6 r' approximates log_2(3) and 1 + 5 r' approximates log_2(5)--the
> column vector V =
>
> [ 1 ]
> [ 6 r']
> [1+5r']
>
> has unison vectors generated by [-6, -5, 6].

Not sure about this bit.

> "highest interval width: 6"
>
> How did we get to intervals and scales?

From the set of 5-limit intervals: 1:1, 5:4, 6:5, 3:2 and equivalents and
inversions. The highest number of generators you need to describe all
these intervals is 6.

> "complexity measure: 6 (7 for smallest MOS)"
>
> How is this defined?

The number before times the number of periods to an octave. The number of
complete otonalities you can play is the number of notes in the generated
scale minus this.

> "highest error: 0.001126 (1.351 cents)"
>
> 5/3 is off by this amount.

That'll be it then.

> "unique"
>
> What is unique?

It means each interval being approximated has a unique mapping to the
temperament. For example, meantone fails to be unique in the 9-limit
because 9:8 and 10:9 map the same way.

> This system is so close to the 53-et that it would seem to make sense
> to adjust the fifth, the octave or both and make it exactly the 53-et.

Maybe, but it could be a useful way of choosing subsets of 53-et.

Graham

--- In tuning-math@y..., graham@m... wrote:
> I've altered my temperament finding program to accept only
temperaments
> with a worst error of less than 2.8 cents. I think this is the
cutoff for
> a microtemperament. Results are at
>
> <http://x31eq.com/limit5.micro>
> <http://x31eq.com/limit7.micro>
> <http://x31eq.com/limit9.micro>
> <http://x31eq.com/limit11.micro>
> <http://x31eq.com/limit13.micro>
> <http://x31eq.com/limit15.micro>
>
> Some of them don't have as many as 10 results.

Oh Graham, you're wonderful!

-- Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> I've altered my temperament finding program to accept only
temperaments
> with a worst error of less than 2.8 cents. I think this is the
cutoff for
> a microtemperament. Results are at
>
> <http://x31eq.com/limit5.micro>
> <http://x31eq.com/limit7.micro>
> <http://x31eq.com/limit9.micro>
> <http://x31eq.com/limit11.micro>
> <http://x31eq.com/limit13.micro>
> <http://x31eq.com/limit15.micro>
>
> Some of them don't have as many as 10 results.

Something must be wrong. How come schismic didn't make it into
5-limit? Couldn't you be missing some by not taking your consistent
ET's out far enough. But there's definitely no need to go past 215-tET
(within 2.8 cents of anything).

-- Dave Keenan

In-Reply-To: <9mhgt6+avlv@eGroups.com>
Dave Keenan wrote:

> Something must be wrong. How come schismic didn't make it into
> 5-limit? Couldn't you be missing some by not taking your consistent
> ET's out far enough. But there's definitely no need to go past 215-tET
> (within 2.8 cents of anything).

Yes, schismic comes from 12 and 29. I was taking the first 20 consistent
ETs, which only got as far as 28. So I've fudged it and am now taking the
first 21 instead. Schismic should now be top of
<http://x31eq.com/limit5.micro>. I used to take all consistent
ETs with fewer than 100 notes, but this meant a lot more were considered
for 15- than 5-limit.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9mhgt6+avlv@e...>
> Dave Keenan wrote:
>
> > Something must be wrong. How come schismic didn't make it into
> > 5-limit? Couldn't you be missing some by not taking your
consistent
> > ET's out far enough. But there's definitely no need to go past
215-tET
> > (within 2.8 cents of anything).
>
> Yes, schismic comes from 12 and 29. I was taking the first 20
consistent
> ETs, which only got as far as 28. So I've fudged it and am now
taking the
> first 21 instead. Schismic should now be top of
> <http://x31eq.com/limit5.micro>. I used to take all
consistent
> ETs with fewer than 100 notes, but this meant a lot more were
considered
> for 15- than 5-limit.
>
>
> Graham

SO how do you know you're still not missing any?