3rd-best 11-limit temperament

While the top two temperaments in Graham's 11-limit list are
essentially 31-out-of-72 and 46-out-of-72, the third one has the
lowest complexity measure of all in this list. Can anyone discuss
this, in terms of unison vectors, etc.?

12/53, 271.1 cent generator

basis:
(1.0, 0.22594789337911292)

mapping by period and generator:
[(1, 0), (0, 7), (3, -3), (1, 8), (3, 2)]

mapping by steps:
[(31, 22), (49, 35), (72, 51), (87, 62), (107, 76)]

unison vectors:
[[7, 8, 0, -7, 0], [-21, 3, 7, 0, 0], [-21, -2, 0, 0, 7]]

highest interval width: 17
complexity measure: 17 (22 for smallest MOS)
highest error: 0.007764 (9.317 cents)

Why is this better than an ME 22-out-of-46, which has a maximum error
of 8.6 cents in the 11-limit, probably reducable further in a non-ET
setting?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> While the top two temperaments in Graham's 11-limit list are
> essentially 31-out-of-72 and 46-out-of-72, the third one has the
> lowest complexity measure of all in this list. Can anyone discuss
> this, in terms of unison vectors, etc.?

One thing you might note about it is that it is the 22-31 linear
temperament, with tunings calculated by Graham's minimax condition.
If we set B = 2^(0.2259478...), then his basis is 2^r B^s; we could
also have a basis U = 2^5 B^(-22), V = 2^(-7) B^31 and in this basis
we approximate rational numbers q in the 11-limit by
U^h31(q) V^h22(q).

--- In tuning-math@y..., genewardsmith@j... wrote:

> One thing you might note about it is that it is the 22-31 linear
> temperament, with tunings calculated by Graham's minimax condition.
> If we set B = 2^(0.2259478...), then his basis is 2^r B^s; we could
> also have a basis U = 2^5 B^(-22), V = 2^(-7) B^31 and in this
basis
> we approximate rational numbers q in the 11-limit by
> U^h31(q) V^h22(q).

Another thing to note is how easy it is to find the generators and
MOS scales if you don't proceed all the way down to the PB. In the
22-31 case, we have a continued fraction [1,2,2,4] for 31/22; the
penultimate convergent is 7/5, and our generator is U^7 V^5. For the
41-31 temperament, we have a continued fraction [1, 3, 10], which
leads to a penultimate continued fraction 4/3, and a generator
W^4 X^3 which is the miracle generator; here W and X are to be
selected by some optimality condition, such as minimax or least
squares on a tonality diamond (if I have the jargon right.)

This is easy enough that I've been meaning to suggest that Manuel
consider putting into Scala a routine to calculate Gen(m, n, p) and
Mos(n,m,p) for two ets m and n and a prime limit p; in case m and n
are not relatively prime this needs to be adjusted by working inside
of the interval of repetition. Of course one can also think of this
in terms of the ets generated by linear combinations of hm and hn, as
for instance h53 = h22 + h31 and h72 = h31 + h41.

In-Reply-To: <9pjjc9+5lu5@eGroups.com>
Gene wrote:

> This is easy enough that I've been meaning to suggest that Manuel
> consider putting into Scala a routine to calculate Gen(m, n, p) and
> Mos(n,m,p) for two ets m and n and a prime limit p; in case m and n
> are not relatively prime this needs to be adjusted by working inside
> of the interval of repetition. Of course one can also think of this
> in terms of the ets generated by linear combinations of hm and hn, as
> for instance h53 = h22 + h31 and h72 = h31 + h41.

That's roughly what my Python module does, and Manuel's welcome to take
that code as inspiration. <http://x31eq.com/temper.py> The
temperament's returned by

temper.Temperament(m, n, temper.primes[:q])

where q is the number of prime intervals you're using, not including the
2:1 which is the interval of equivalence. So the temperament in question
is

temper.Temperament(31, 22, temper.primes[:4])

Graham

In-Reply-To: <9pibm5+jcmt@eGroups.com>
Paul wrote:

> While the top two temperaments in Graham's 11-limit list are
> essentially 31-out-of-72 and 46-out-of-72, the third one has the
> lowest complexity measure of all in this list. Can anyone discuss
> this, in terms of unison vectors, etc.?
>
>
> 12/53, 271.1 cent generator
>
> basis:
> (1.0, 0.22594789337911292)
>
> mapping by period and generator:
> [(1, 0), (0, 7), (3, -3), (1, 8), (3, 2)]
>
> mapping by steps:
> [(31, 22), (49, 35), (72, 51), (87, 62), (107, 76)]
>
> unison vectors:
> [[7, 8, 0, -7, 0], [-21, 3, 7, 0, 0], [-21, -2, 0, 0, 7]]
>
> highest interval width: 17
> complexity measure: 17 (22 for smallest MOS)
> highest error: 0.007764 (9.317 cents)

This originally came out of Dave Keenan's spreadsheet. Note that it's
compatible because the period is an octave. It was noted then that it was
the simplest approximation -- better than meantone at 18 and schismic at
19.

The unison vectors are not in their simplest terms. Dan Stearns claimed
before to have an algorithm for finding unison vectors, so I'd still like
to see it.

[[7, 8, 0, -7, 0]
+ 2*[-21, 3, 7, 0, 0]
-------------------
[-35 14 14 -7 0]
= 7*[ -5 2 2 -1 0]

[-21, 3, 7, 0, 0]
- 2*[-21, -2, 0, 0, 7]]
---------------------
[21 7 7 0 -14]
= 7*[ 3 1 1 0 -2]

[[7, 8, 0, -7, 0]
- 3*[-21, -2, 0, 0, 7]]
---------------------
[ 70 14 0 -7 -21]
= 7*[ 10 2 0 -1 -3]

so we have new unison vectors

[-5 2 2 -1 0]
[ 3 1 1 0 -2]
[10 2 0 -1 -3]

I don't know what you were planning to do with them. To check the
determinants

|3 -1 0 0|
|2 2 -1 0| = -31
|1 1 0 -2|
|2 0 -1 -3|

|0 2 -2 0|
|2 2 -1 0| = 22
|1 1 0 -2|
|2 0 -1 -3|

and the adjoint of

| 1 0 2 -2 0|
|-4 4 -1 0 0|
|-5 2 2 -1 0|
| 3 1 1 0 -2|
|10 2 0 -1 -3|

is

| -31 -22 38 -36 24|
| -49 -35 60 -57 38|
| -72 -51 88 -84 56|
| -87 -62 107 -102 68|
|-107 -76 131 -124 83|

> Why is this better than an ME 22-out-of-46, which has a maximum error
> of 8.6 cents in the 11-limit, probably reducable further in a non-ET
> setting?

Presumably, you mean

>>> temper.Temperament(46, 22, temper.primes[:4])

3/34, 52.2 cent generator

basis:
(0.5, 0.043499613319368802)

mapping by period and generator:
[(2, 0), (3, 2), (5, -4), (5, 7), (7, -1)]

mapping by steps:
[(46, 22), (73, 35), (107, 51), (129, 62), (159, 76)]

unison vectors:
[[-3, -1, -1, 0, 2], [4, 0, -2, -1, 1], [0, -5, 1, 2, 0]]

highest interval width: 11
complexity measure: 22 (24 for smallest MOS)
highest error: 0.007005 (8.406 cents)

The increase in complexity is greater than the improved accuracy. It's
as complex as Miracle, but more than double the error! It doesn't make
limit11.key because it isn't unique. Ah! It is in limit11.mos. Note
that the diaschismic temperament including 46 and 58 does make the
13-limit list (right at the bottom of mine) as does the 94+41 schismic.

Graham

Graham wrote:
>That's roughly what my Python module does, and Manuel's welcome to take
>that code as inspiration. <http://x31eq.com/temper.py>

It would be a good addition, however it's quite a big piece of code.
So probably I'd be quicker to rethink the algorithm myself. I already
have the code for minimax temperament in Ada, although I'm not sure
yet if it can be applied. I need to let this stuff sink in too.
I'd also want to make the set of consonant p-limit intervals user
definable.

Manuel

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

> > Why is this better than an ME 22-out-of-46, which has a maximum
error
> > of 8.6 cents in the 11-limit, probably reducable further in a non-
ET
> > setting?
>
> Presumably, you mean
>
> >>> temper.Temperament(46, 22, temper.primes[:4])
>
> 3/34, 52.2 cent generator
>
> basis:
> (0.5, 0.043499613319368802)
>
> mapping by period and generator:
> [(2, 0), (3, 2), (5, -4), (5, 7), (7, -1)]
>
> mapping by steps:
> [(46, 22), (73, 35), (107, 51), (129, 62), (159, 76)]
>
> unison vectors:
> [[-3, -1, -1, 0, 2], [4, 0, -2, -1, 1], [0, -5, 1, 2, 0]]
>
> highest interval width: 11
> complexity measure: 22 (24 for smallest MOS)

24? What about the 22-tone MOS?

> highest error: 0.007005 (8.406 cents)
>
> The increase in complexity is greater than the improved accuracy.

OK.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
>
> > One thing you might note about it is that it is the 22-31 linear
> > temperament, with tunings calculated by Graham's minimax
condition.
> > If we set B = 2^(0.2259478...), then his basis is 2^r B^s; we
could
> > also have a basis U = 2^5 B^(-22), V = 2^(-7) B^31 and in this
> basis
> > we approximate rational numbers q in the 11-limit by
> > U^h31(q) V^h22(q).
>
> Another thing to note is how easy it is to find the generators and
> MOS scales if you don't proceed all the way down to the PB. In the
> 22-31 case, we have a continued fraction [1,2,2,4] for 31/22; the
> penultimate convergent is 7/5, and our generator is U^7 V^5. For
the
> 41-31 temperament, we have a continued fraction [1, 3, 10], which
> leads to a penultimate continued fraction 4/3, and a generator
> W^4 X^3 which is the miracle generator; here W and X are to be
> selected by some optimality condition, such as minimax or least
> squares on a tonality diamond (if I have the jargon right.)

Well, that's Graham's way of putting it . . . Anyway, the rest of
this stuff looks wonderful, but needs to be presented in a much more
user-friendly way for musicians. I think you could do a great service
by putting all these ideas in a "for dummies" kind of document, with
familiar examples and such. If you wrote such a paper in the next
couple of months, you could probably get it published in
Xenharmonikon 18 . . .

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

> Ah! It is in limit11.mos.

What's limit11.mos?

Paul Erlich wrote:

> > highest interval width: 11
> > complexity measure: 22 (24 for smallest MOS)
>
> 24? What about the 22-tone MOS?

The number of otonal (or utonal) complete chords is always the number of
notes in the scale minus the complexity measure. So 24 notes gives you 2
otonalities. 22 notes would give you no complete chords. That follows
from 22-22=0.

Graham

--- In tuning-math@y..., graham@m... wrote:
> Paul Erlich wrote:
>
> > > highest interval width: 11
> > > complexity measure: 22 (24 for smallest MOS)
> >
> > 24? What about the 22-tone MOS?
>
> The number of otonal (or utonal) complete chords is always the
number of
> notes in the scale minus the complexity measure. So 24 notes gives
you 2
> otonalities. 22 notes would give you no complete chords. That
follows
> from 22-22=0.

So 22-out-of-46 MOS gives you no hexads? That's odd, since the 22-out-
of-46 omnitetrachordal scale (which is very similar) does. This may
be the first time we're seeing an omnitetrachordal scale look
harmonically better than its MOS counterpart. Veddy veddy
interresteeng.

Paul wrote:

> What's limit11.mos?

It's one of a series of files on my website that use the smallest MOS as
the complexity measure in the figure of demerit. Check back through this
forum and you'll find the discussion. It happens that the 46+22
temperament, with a smallest MOS of 24, performs much better here than
with its standard complexity measure of 22.

Graham

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

If you wrote such a paper in the next
> couple of months, you could probably get it published in
> Xenharmonikon 18 . . .

Thanks for the suggestion; I've been meaning to ask about how much of
this stuff you think is publishable and how much might suffer from
the "too mathematical" virus. I hope to clean things up, put up a web
page, and then think about any further possibilities.

--- In tuning-math@y..., graham@m... wrote:
> The unison vectors are not in their simplest terms. Dan Stearns
claimed
> before to have an algorithm for finding unison vectors, so I'd
still like
> to see it.

There are such things as lattice basis reduction algorithms, but this
isn't even a lattice basis.

> > Why is this better than an ME 22-out-of-46, which has a maximum
error
> > of 8.6 cents in the 11-limit, probably reducable further in a non-
ET
> > setting?

> Presumably, you mean
>
> >>> temper.Temperament(46, 22, temper.primes[:4])
>
> 3/34, 52.2 cent generator
>
> basis:
> (0.5, 0.043499613319368802)

It seems to me he probably means the 22-24 system, with 22+24=46, and
not the 22-46 system, with 22+46=68. Paul?

--- In tuning-math@y..., genewardsmith@j... wrote:

> > 3/34, 52.2 cent generator
> >
> > basis:
> > (0.5, 0.043499613319368802)
>
> It seems to me he probably means the 22-24 system, with 22+24=46,
and
> not the 22-46 system, with 22+46=68. Paul?

I don't know. Graham got the right generator for the system I meant.
Does that mean you're wrong, Gene? I don't know. I do find it
interesting that though the 22-tone MOS has no 11-limit hexads, the
corresponding 22-tone omnitetrachordal scale has some.

Me:

> > > 3/34, 52.2 cent generator
> > >
> > > basis:
> > > (0.5, 0.043499613319368802)

Gene:

> > It seems to me he probably means the 22-24 system, with 22+24=46,
> and
> > not the 22-46 system, with 22+46=68. Paul?

Paul:

> I don't know. Graham got the right generator for the system I meant.
> Does that mean you're wrong, Gene? I don't know. I do find it
> interesting that though the 22-tone MOS has no 11-limit hexads, the
> corresponding 22-tone omnitetrachordal scale has some.

This is the system generated from the consistent mappings of 46- and
22-equal. There's also a system consistent with 46 -and 58-equal which I
called "diaschismic". It does have a 22 note MOS, but it's too complex to
be the 22 from 46 that Paul asked for.

Graham

--- In tuning-math@y..., graham@m... wrote:
> Me:
>
> > > > 3/34, 52.2 cent generator
> > > >
> > > > basis:
> > > > (0.5, 0.043499613319368802)
>
> Gene:
>
> > > It seems to me he probably means the 22-24 system, with 22+24=46,
> > and
> > > not the 22-46 system, with 22+46=68. Paul?
>
> Paul:
>
> > I don't know. Graham got the right generator for the system I meant.
> > Does that mean you're wrong, Gene? I don't know. I do find it
> > interesting that though the 22-tone MOS has no 11-limit hexads, the
> > corresponding 22-tone omnitetrachordal scale has some.
>
> This is the system generated from the consistent mappings of 46- and
> 22-equal. There's also a system consistent with 46 -and 58-equal which I
> called "diaschismic". It does have a 22 note MOS, but it's too complex to
> be the 22 from 46 that Paul asked for.
>
Are you sure? This is the shrutar system, which you once said was not consistent with 22-equal.
So maybe Gene was right?

Paul wrote:

> Are you sure? This is the shrutar system, which you once said was
not consistent with 22-equal.
> So maybe Gene was right?

The two systems are very different melodically. The diaschismic
shruti scale in 46-equal looks like:

3 1 3 1 3 1 3 1 3 1 3 1 3 3 1 3 1 3 1 3 1 3
S r R g G M m P d D n N S'

And the 22+46 temperament:

2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
S r R g G M m P d D n N S'

The former looks more like the canonical shruti scale, and so that's
what I'd have expected if you mentioned tuning shrutis to 46-equal.
The two are the same if you take only the named 12 notes. I half
remember looking at this before.

The straight diaschismic mapping is simpler in the 5- and 9-limits.
They're both as complex in the 7-limit, and the shrutar is simpler in
the 11-limit, but not so far as to be a compelling 11-limit
temperament.

Graham

--- In tuning-math@y..., "Graham Breed" <graham@m...> wrote:
> Paul wrote:
>
> > Are you sure? This is the shrutar system, which you once said was
> not consistent with 22-equal.
> > So maybe Gene was right?
>
> The two systems are very different melodically. The diaschismic
> shruti scale in 46-equal looks like:
>
> 3 1 3 1 3 1 3 1 3 1 3 1 3 3 1 3 1 3 1 3 1 3
> S r R g G M m P d D n N S'
>
> And the 22+46 temperament:
>
> 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
> S r R g G M m P d D n N S'
>
I meant the latter.
>
> The former looks more like the canonical shruti scale, and so
that's
> what I'd have expected if you mentioned tuning shrutis to 46-equal.

Right, but this is a 7- and 11-limit adaptation of the sruti idea,
keeping only the named 12 notes in their "canonical" tuning.

> The two are the same if you take only the named 12 notes. I half
> remember looking at this before.
>
> The straight diaschismic mapping is simpler in the 5- and 9-
limits.
> They're both as complex in the 7-limit, and the shrutar is simpler
in
> the 11-limit, but not so far as to be a compelling 11-limit
> temperament.

Well, I do get an 11-limit hexad that includes the open strings,
which is probably the only one I'd be able to play anyway.