Yahoo Tuning Groups Ultimate Backup tuning-math non-1200: Tenney/heursitic meantone temperament

No one commented on the graphs I posted around Christmas, but I'll
keep going, if only for myself . . .

The syntonic comma = 81/80 (21.506 cents) has a length of
log(81*80) = 4*log(2) + 4*log(3) + log(5)
in the Tenney lattice, as it comprises 4 rungs along the 2-axis, 4
rungs along the 3-axis, and 1 rung along the 5-axis.

If the anomaly is distributed uniformly and efficiently along this
length, the rungs along the 2-axis and 5-axis will be tempered wide,
and the rung along the 3-axis will be tempered narrow.

So the 2-axis rungs should be tempered wide by log(2)/log(81*80) of
the syntonic comma, or 1.6985 cents -- so each rung represents
1201.6985 cents. The 3-axis rungs should be tempered narrow by log
(3)/log(81*80) of the syntonic comma, or 2.6921 cents -- so each rung
represents 1899.2629 cents. The 5-axis rungs should be tempered wide
by log(5)/log(81*80) of the syntonic comma, or 3.9438 cents -- so
each rung represents 2790.2575 cents.

Check that the syntonic comma vanishes:

4*1899.2629 - 4*1201.6985 - 2790.2575 = 0

What would you call this kind of meantone? The 3:2 is flattened by
about 10/49-comma, while the octave is widened by about 3/38-comma.

Looks like this approach optimizes according to a weighted minimax
over the prime intervals. Comments?

🔗Paul Erlich <perlich@aya.yale.edu>

Looks like this approach optimizes according to (Tenney) weighted
minimax over ALL intervals.

IS THIS RIGHT??

No need to specify a consonance limit? -- wow that's hot.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> No one commented on the graphs I posted around Christmas, but I'll
> keep going, if only for myself . . .
>
> The syntonic comma = 81/80 (21.506 cents) has a length of
> log(81*80) = 4*log(2) + 4*log(3) + log(5)
> in the Tenney lattice, as it comprises 4 rungs along the 2-axis, 4
> rungs along the 3-axis, and 1 rung along the 5-axis.
>
> If the anomaly is distributed uniformly and efficiently along this
> length, the rungs along the 2-axis and 5-axis will be tempered
wide,
> and the rung along the 3-axis will be tempered narrow.
>
> So the 2-axis rungs should be tempered wide by log(2)/log(81*80) of
> the syntonic comma, or 1.6985 cents -- so each rung represents
> 1201.6985 cents. The 3-axis rungs should be tempered narrow by log
> (3)/log(81*80) of the syntonic comma, or 2.6921 cents -- so each
rung
> represents 1899.2629 cents. The 5-axis rungs should be tempered
wide
> by log(5)/log(81*80) of the syntonic comma, or 3.9438 cents -- so
> each rung represents 2790.2575 cents.
>
> Check that the syntonic comma vanishes:
>
> 4*1899.2629 - 4*1201.6985 - 2790.2575 = 0
>
> What would you call this kind of meantone? The 3:2 is flattened by
> about 10/49-comma, while the octave is widened by about 3/38-comma.
>
> Looks like this approach optimizes according to a weighted minimax
> over the prime intervals. Comments?

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Gene and you had an exchange. Gene suggested what I was
thinking,
> >> you said that each band represented only a denominator, not a
comma.
> >> Or are you talking about something else?
> >
> >Yes, a later pair of graphs.
>
> D'oh! Link? (I even went to tuning-math to find it myself, but
> this thread is not connected to anything).

/tuning-math/files/Paul/com5monz.gif

and

/tuning-math/files/Paul/com5rat.gif

> >> >No need to specify a consonance limit? -- wow that's hot.
> >>
> >> How do you mean? Isn't this implicit in the dimensionality of
the
> >> Tenney lattice you're using?
> >
> >No, no consonance limit (aka odd-limit) is implicity in the
> >dimensionality of the Tenney lattice you're using -- and even the
> >latter, aka prime-limit, doesn't need to be specified -- for
example
> >the results for the pythagorean comma will be valid in both lower
and
> >higher prime limits.
>
> I must not be tracking you -- the pythagorean comma is of course
> a 3-limit comma.

See the last paragraph of

/tuning/topicId_50628.html#50964

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

>> >> Gene and you had an exchange. Gene suggested what I was
>> >> thinking, you said that each band represented only a
>> >> denominator, not a comma.
>> >> Or are you talking about something else?
>> >
>> >Yes, a later pair of graphs.
>>
>> D'oh! Link? (I even went to tuning-math to find it myself, but
>> this thread is not connected to anything).
>
>/tuning-math/files/Paul/com5monz.gif
>
>and
>
>/tuning-math/files/Paul/com5rat.gif

These are the graphs I thought Gene replied about -- do they look
very similar to those?

>> >> >No need to specify a consonance limit? -- wow that's hot.
>> >>
>> >> How do you mean? Isn't this implicit in the dimensionality of
>> >> the Tenney lattice you're using?
>> >
>> >No, no consonance limit (aka odd-limit) is implicity in the
>> >dimensionality of the Tenney lattice you're using -- and even the
>> >latter, aka prime-limit, doesn't need to be specified -- for
>> >example the results for the pythagorean comma will be valid in
>> >both lower and higher prime limits.
>>
>> I must not be tracking you -- the pythagorean comma is of course
>> a 3-limit comma.
>
>See the last paragraph of
>
>/tuning/topicId_50628.html#50964

I don't see anything new in any of this, I'm afraid.

Let's back up. We're talking about the badness of commas? Typically
that's been a function of their size in cents and the number of notes
you can be expected to search to find them (which depends on their
distance on the lattice and the dimensionality of the lattice).
That's Gene's logflat badness, which I've implemented in Scheme.
Now, what exactly are you up to here?

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> No one commented on the graphs I posted around Christmas, but
I'll
> >> keep going, if only for myself . . .
>
> Gene and you had an exchange. Gene suggested what I was thinking,
> you said that each band represented only a denominator, not a comma.
> Or are you talking about something else?
>
> >No need to specify a consonance limit? -- wow that's hot.
>
> How do you mean? Isn't this implicit in the dimensionality of the
> Tenney lattice you're using?
>
> It would indeed be hot.
>
> -Carl

Let's try to get a better grasp of what happens in this particular
meantone, for a start. I could also do this for 7-limit intervals,
treating the 81/80 temperament as a 'planar' temperament, but
hopefully it's clear that the extra intervals won't have enough error
to exceed the bound. Same for any higher limit.

Interval...Approx....|Error|....Comp=log2(n*d)...|Error|/Comp
2:1........1201.70....1.70...........1..............1.70
3:1........1899.26....2.69..........1.58............1.70
4:1........2403.40....3.40...........2..............1.70
5:1........2790.26....3.94..........2.32............1.70
3:2.........697.56....4.39..........2.58............1.70
6:1........3100.96....0.99..........2.58............0.38
8:1........3605.10....5.10...........3..............1.70
9:1........3798.53....5.38..........3.17............1.70
10:1.......3991.96....5.64..........3.32............1.70
4:3.........504.13....6.09..........3.58............1.70
12:1.......4302.66....0.70..........3.58............0.20
5:3.........890.99....6.64..........3.91............1.70
15:1.......4689.52....1.25..........3.91............0.32
16:1.......4806.79....6.79...........4..............1.70
9:2........2596.83....7.08..........4.17............1.70
18:1.......5000.22....3.69..........4.17............0.88
5:4.........386.86....0.55..........4.32............0.13
20:1.......5193.65....7.34..........4.32............1.70
8:3........1705.83....7.79..........4.58............1.70
24:1.......5504.36....2.40..........4.58............0.52
25:1.......5580.52....7.89..........4.64............1.70
6:5.........310.70....4.94..........4.91............1.01
10:3.......2092.69....8.33..........4.91............1.70
30:1.......5891.22....2.95..........4.91............0.60
32:1.......6008.49....8.49...........5..............1.70
36:1.......6201.92....1.99..........5.17............0.38
8:5.........814.84....1.15..........5.32............0.22
40:1.......6395.35....9.04..........5.32............1.70
9:5........1008.27....9.33..........5.49............1.70
45:1.......6588.78....1.44..........5.49............0.26
16:3.......2907.53....9.49..........5.58............1.70
48:1.......6706.06....4.10..........5.58............0.73
25:2.......4378.82....6.19..........5.64............1.10
50:1.......6782.21....9.59..........5.64............1.70
27:2.......4496.09....9.77..........5.75............1.70
54:1.......6899.49....6.38..........5.75............1.11
12:5.......1512.40....3.24..........5.91............0.55
15:4.......2286.12....2.15..........5.91............0.36
20:3.......3294.39...10.03..........5.91............1.70
60:1.......7092.92....4.65..........5.91............0.79
1296:5.....
and so on. Thinking about a few of these example spacially should
help you see that the weighted error can never exceed

cents(81/80)/log2(81*80) = 1.70

for ANY interval.

Is there a just (RI) interval in this meantone? The idea of duality
leads me to guess 81*80:1 = 6480:1 . . .

6480:1....15194.10....0.03

almost, but no cigar.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> Gene and you had an exchange. Gene suggested what I was
> >> >> thinking, you said that each band represented only a
> >> >> denominator, not a comma.
> >> >> Or are you talking about something else?
> >> >
> >> >Yes, a later pair of graphs.
> >>
> >> D'oh! Link? (I even went to tuning-math to find it myself, but
> >> this thread is not connected to anything).
> >
> >/tuning-math/files/Paul/com5monz.gif
> >
> >and
> >
> >/tuning-math/files/Paul/com5rat.gif
>
> These are the graphs I thought Gene replied about -- do they look
> very similar to those?

Pretty similar.

> >> >> >No need to specify a consonance limit? -- wow that's hot.
> >> >>
> >> >> How do you mean? Isn't this implicit in the dimensionality
of
> >> >> the Tenney lattice you're using?
> >> >
> >> >No, no consonance limit (aka odd-limit) is implicity in the
> >> >dimensionality of the Tenney lattice you're using -- and even
the
> >> >latter, aka prime-limit, doesn't need to be specified -- for
> >> >example the results for the pythagorean comma will be valid in
> >> >both lower and higher prime limits.
> >>
> >> I must not be tracking you -- the pythagorean comma is of course
> >> a 3-limit comma.
> >
> >See the last paragraph of
> >
> >/tuning/topicId_50628.html#50964
>
> I don't see anything new in any of this, I'm afraid.
>
> Let's back up. We're talking about the badness of commas?
>Typically
> that's been a function of their size in cents

No, not really. Rather, it's the error they induce when tempered out.
I seem to have figured out a simple formula to get this when the
tempering is Tenney-weighted-minimax optimal over ALL intervals. And
a simple way to do such tempering.

> and the number of notes
> you can be expected to search to find them (which depends on their
> distance on the lattice and the dimensionality of the lattice).

I'm just looking at distance on the lattice, and dimensionality is
irrelevant -- it can be assumed to be infinite.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> No, not really. Rather, it's the error they induce when tempered out. > I seem to have figured out a simple formula to get this when the > tempering is Tenney-weighted-minimax optimal over ALL intervals. And > a simple way to do such tempering.

The odd-limit rule simplifies to

interval size / complexity

where "complexity" is the smallest number of intervals in the relevant limit that make up the comma. The result is the optimum minimax error by tempering out only this comma. So any temperament involving this comma must be at least this well tuned.

What you've done is plugged in the Tenney complexity, and got a result that'll have something to do with the weighted minimax. That's not too surprising, and should generalize to any weighted complexity measure. At least if it gives a result in terms of octaves.

I thought this was all assumed by your hypothesis anyway. We know that the Tenney complexity and odd limits are linked, depending on whether or not you enforce octave equialence. As geometric complexity looks like being an octave-specific weighted complexity measure, this may be the way to progress.

The problem remains knowing how best to combine these commas to get a temperament of a specific dimension. For that we need a straightness measure, as always. And if you're not enforcing a prime limit to start with, you'll need to take a variable number of commas.

Oh, BTW, I've been looking at commas of the form (n*n):(n+1)(n-1). Do they have a name? They always come from setting two n-integer limit intervals to be equivalent. They're also more likely than arbitrary superparticulars to belong to a low prime limit.. The 11-prime limit subset gives us

4:3, 9:8, 16:15, 25:24, 36:35, 49:48, 64:63, 81:80,
100:99, 225:224, 441:440, 2401:2400

Graham

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> Oh, BTW, I've been looking at commas of the form (n*n):(n+1)(n-1). Do
> they have a name?

In post 849 on this list I called them B(1, n), which isn't much of a
name, nor is Square(n) which I think I've also used. What about
squarejack as a name? Then we could also have trianglejacks and so
forth, if anyone feels inspired.

They always come from setting two n-integer limit
> intervals to be equivalent. They're also more likely than arbitrary
> superparticulars to belong to a low prime limit.. The 11-prime limit
> subset gives us
>
> 4:3, 9:8, 16:15, 25:24, 36:35, 49:48, 64:63, 81:80,
> 100:99, 225:224, 441:440, 2401:2400

Note 16/15, 81/80 and 2401/2400 have fourth powers in the numerator
and 81/80 = (9/8)/(10/9), 2401/2400 = (49/48)/(50/49) are jumping
jacks. We aslo have 225/224, where the numerator is a square of a
triangular number and is a jumping jack by 225/224 = (15/14)/(16/15).
36/35 has a numerator which is both square and triangular, making it a
high jack.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >No, not really. Rather, it's the error they induce when tempered
out.
> >I seem to have figured out a simple formula to get this when the
> >tempering is Tenney-weighted-minimax optimal over ALL intervals.
And
> >a simple way to do such tempering.
>
> What is that formula a way? Sorry, can we start at the top?

For the comma p/q, p>q, the number of cents you need to temper out is
cents(p/q) = log2(p/q)*1200.

The distance in the Tenney lattice (taxicab, by definition) is log2
(p*q).

So the tempering per unit length in the direction of the comma is
cents(p/q)/log2(p*q). This was (aside from a factor of 1200) the
vertical axis in my graphs.

Now, for all primes r,

If p contains any factors of r, the r-rungs in the lattice (which
have length log2(r)) are shrunk from
cents(r)
to
cents(r) - log2(r)*cents(p/q)/log2(p*q).
If q contains any factors of 2, they are instead stretched to
cents(r) + log2(r)*cents(p/q)/log2(p*q).

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > No, not really. Rather, it's the error they induce when tempered
out.
> > I seem to have figured out a simple formula to get this when the
> > tempering is Tenney-weighted-minimax optimal over ALL intervals.
And
> > a simple way to do such tempering.
>
> The odd-limit rule simplifies to
>
> interval size / complexity
>
> where "complexity" is the smallest number of intervals in the
relevant
> limit that make up the comma. The result is the optimum minimax
error
> by tempering out only this comma. So any temperament involving
this
> comma must be at least this well tuned.
>
> What you've done is plugged in the Tenney complexity, and got a
result
> that'll have something to do with the weighted minimax. That's not
too
> surprising, and should generalize to any weighted complexity
measure.
> At least if it gives a result in terms of octaves.
>
> I thought this was all assumed by your hypothesis anyway.

I don't see the relationship.

> We know that
> the Tenney complexity and odd limits are linked, depending on
whether or
> not you enforce octave equialence.

Yes, but for octave equivalence (pegged to 1200 cent octaves), I'd
like to eventually be able to use Kees's expressibility measure
instead of Tenney harmonic distance. Just as there was no
finitistic 'limit' assumed for my 'optimization' in the Tenney
lattice, no odd limit will have to be specified in the octave-
equivalent case (if it can work).

> As geometric complexity looks like
> being an octave-specific weighted complexity measure, this may be
the
> way to progress.

What do you mean?

> The problem remains knowing how best to combine these commas to get
a
> temperament of a specific dimension. For that we need a
straightness
> measure, as always.

That's why I was asking about heron's formula, etc. But if we have
some way of acheiving this Tenney-weighted minimax for the relevant
temperaments, we may be able to skip this step.

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >For the comma p/q, p>q, the number of cents you need to temper out
is
> >cents(p/q) = log2(p/q)*1200.
> >
> >The distance in the Tenney lattice (taxicab, by definition) is log2
> >(p*q).
> >
> >So the tempering per unit length in the direction of the comma is
> >cents(p/q)/log2(p*q). This was (aside from a factor of 1200) the
> >vertical axis in my graphs.
> >
> >Now, for all primes r,
> >
> >If p contains any factors of r, the r-rungs in the lattice (which
> >have length log2(r)) are shrunk from
> >cents(r)
> >to
> >cents(r) - log2(r)*cents(p/q)/log2(p*q).
> >If q contains any factors of 2, they are instead stretched to
> >cents(r) + log2(r)*cents(p/q)/log2(p*q).
>
> Thanks. I understand this 100%. But I don't understand what's
> new.

Where have you seen this before?

> Perhaps it has something to do with using this to get
> optimum generators for a linear temperament?

Well, that's exactly what this does (when the dimensionality is
right), as I've illustrated already in a few cases.

Here's something new -- Top meantone is, it seems, exactly 1/4-comma
meantone (I get 0.24999999999997, but that's probably just rounding
error) except a uniform (in cents, or log Hz) stretch of
1.00141543374547 is applied to all intervals . . .

> And I don't understand your 'limitless' claim -- since p/q contains
> the factors it does and no others, one wouldn't expect its vanishing
> to effect

affect?

> intervals different factors.

intervals with different factors? Well, 5:4 and 5:3 have *some*
factors differing from those in the Pythagorean comma, yet both
intervals are affected by its vanishing, in this scheme.

🔗Carl Lumma <ekin@lumma.org>

>> Perhaps it has something to do with using this to get
>> optimum generators for a linear temperament?
>
>Well, that's exactly what this does (when the dimensionality is
>right), as I've illustrated already in a few cases.
>
>Here's something new -- Top meantone is, it seems, exactly 1/4-comma
>meantone (I get 0.24999999999997, but that's probably just rounding
>error) except a uniform (in cents, or log Hz) stretch of
>1.00141543374547 is applied to all intervals . . .

Is the formula for that particularly hard?

>> And I don't understand your 'limitless' claim -- since p/q contains
>> the factors it does and no others, one wouldn't expect its vanishing
>> to effect
>
>affect?

Yes, I think so... :)

>> intervals different factors.
>
>intervals with different factors? Well, 5:4 and 5:3 have *some*
>factors differing from those in the Pythagorean comma, yet both
>intervals are affected by its vanishing, in this scheme.

But not 7:5, right?

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Perhaps it has something to do with using this to get
> >> optimum generators for a linear temperament?
> >
> >Well, that's exactly what this does (when the dimensionality is
> >right), as I've illustrated already in a few cases.
> >
> >Here's something new -- Top meantone is, it seems, exactly 1/4-
comma
> >meantone (I get 0.24999999999997, but that's probably just
rounding
> >error) except a uniform (in cents, or log Hz) stretch of
> >1.00141543374547 is applied to all intervals . . .
>
> Is the formula for that particularly hard?

Maybe someone can derive this 0.24999999999997 as a 1/4 symbolically.
I'd be very happy to see it.

> >> And I don't understand your 'limitless' claim -- since p/q
contains
> >> the factors it does and no others, one wouldn't expect its
vanishing
> >> to effect
> >
> >affect?
>
> Yes, I think so... :)
>
> >> intervals different factors.
> >
> >intervals with different factors? Well, 5:4 and 5:3 have *some*
> >factors differing from those in the Pythagorean comma, yet both
> >intervals are affected by its vanishing, in this scheme.
>
> But not 7:5, right?

Right. Meanwhile, it seems that 81:80 vanishing leaves 6480:1 within
a dust mite's excrement (which I'm allergic to, by the way) of
vanishing . . .

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Graham Breed <graham@microtonal.co.uk>

Me:
>>I thought this was all assumed by your hypothesis anyway.

Paul:
> I don't see the relationship.

From

error = comma size / complexity

using Tenny complexity, for a comma n:d:

error = log(n/d) / log(n*d)
= log[1 + (n-d)/d] / log[(n/d)*d*d]

The base of the logarithms doesn't matter, so we can use log(1+x) ~ x as (n-d)/d will be small for a comma. so this is much the same as

error = [(n-d)/d]/[log(d*d) + log(n/d)]
= (n-d)/d/2/log(d) + 1/d

as it happens. The 1/d term is small and can be neglected, giving

error = (n-d)/[2d*log(d)]

The heuristic error given here:

http://www.tonalsoft.com/enc/heuristic-error.htm

is |n-d|/(R*log(R))

for commas as usually written, the numerator is larger than the denominator, so the |n-d| is the same as (n-d). Depending on whether n or d is even, R may be the same as D. If it isn't exactly the same, it's going to be close because commas tend to be small (and have to be for the simplification to work) which means the numerator and denominator are of roughly equal size. The factor of 2 is disposable, as this isn't measuring anything in particular.

So they look pretty similar to me.

> Yes, but for octave equivalence (pegged to 1200 cent octaves), I'd > like to eventually be able to use Kees's expressibility measure > instead of Tenney harmonic distance. Just as there was no > finitistic 'limit' assumed for my 'optimization' in the Tenney > lattice, no odd limit will have to be specified in the octave-
> equivalent case (if it can work).

I don't see why it shouldn't work mathematically. Whether it has any musical meaning is a different matter. But why shouldn't it work? You temper each factor in whichever of n and d is odd, or the larger if they both are.

>>As geometric complexity looks like >>being an octave-specific weighted complexity measure, this may be > the >>way to progress.
> > What do you mean?

Odd limits are a simplification so that we always get whole numbers, and can think octave equivalently. The geometric complexity Gene gave, as far as I could understand it, was naturally continuous and octave specific. So if it makes it easier to work that way, we can, and go back to odd limits for the fine tuning.

>>The problem remains knowing how best to combine these commas to get > a >>temperament of a specific dimension. For that we need a > straightness >>measure, as always.
> > > That's why I was asking about heron's formula, etc. But if we have > some way of acheiving this Tenney-weighted minimax for the relevant > temperaments, we may be able to skip this step.

I don't see how we can skip the step of combining commas. How could it make sense to do so?

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Me:
> >>I thought this was all assumed by your hypothesis anyway.
>
> Paul:
> > I don't see the relationship.
>
> From
>
> error = comma size / complexity
>
> using Tenny complexity, for a comma n:d:
>
> error = log(n/d) / log(n*d)
> = log[1 + (n-d)/d] / log[(n/d)*d*d]

Oh, you mean "heuristic", not "hypothesis" (the latter concerns PBs
and DEs).

> > Yes, but for octave equivalence (pegged to 1200 cent octaves),
I'd
> > like to eventually be able to use Kees's expressibility measure
> > instead of Tenney harmonic distance. Just as there was no
> > finitistic 'limit' assumed for my 'optimization' in the Tenney
> > lattice, no odd limit will have to be specified in the octave-
> > equivalent case (if it can work).
>
> I don't see why it shouldn't work mathematically. Whether it has
any
> musical meaning is a different matter. But why shouldn't it work?
You
> temper each factor in whichever of n and d is odd, or the larger if
they
> both are.

Hmm . . . I'm a bit busy right now, so can you work out an example,
and show all the errors like I did with Top meantone here?

> >>As geometric complexity looks like
> >>being an octave-specific weighted complexity measure, this may be
> > the
> >>way to progress.
> >
> > What do you mean?
>
> Odd limits are a simplification so that we always get whole
numbers, and
> can think octave equivalently. The geometric complexity Gene gave,
as
> far as I could understand it, was naturally continuous and octave
> specific. So if it makes it easier to work that way, we can, and
go
> back to odd limits for the fine tuning.

I'll have to put this aside for later digestion . . .

> >>The problem remains knowing how best to combine these commas to
get
> > a
> >>temperament of a specific dimension. For that we need a
> > straightness
> >>measure, as always.
> >
> >
> > That's why I was asking about heron's formula, etc. But if we
have
> > some way of acheiving this Tenney-weighted minimax for the
relevant
> > temperaments, we may be able to skip this step.
>
> I don't see how we can skip the step of combining commas. How
could it
> make sense to do so?

I meant skip the step of getting a straightness measure. If we get
the right straightness measure, it won't matter which kernel basis we
pick for a given temperament. In which case we may be able to proceed
directly from the kernel to the relevant quantities. But I wouldn't
stress it at the moment . . .

Thanks for being you!

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:
> Maybe someone can derive this 0.24999999999997 as a 1/4 symbolically. > I'd be very happy to see it.

1.0014154337454717 you say? Yes, I've derived it symbolically -- which means I must have duplicated your method for tempering. That magic number is 2*log(81)/log(81*80). The calculation goes something this:

The 2 generator is tempered as cents(2)[1+k]

The 3 generator is tempered as cents(3)[1-k]

The 5 generator is tempered as cents(5)[1+k]

where k is log(81/80)/log(81*80). The + or - depends on whether factors of this prime occur in the denominator or numerator respectively.

So as 2 and 5 are just in quarter comma meantone, they must be stretched by 1+k here.

In quarter comma meantone, the 3 generator is tempered as

cents(3) - cents(81/80)/4

= cents(3) - cents(81)/4 + cents(80)/4
= cents(3) - cents(3) + cents(80)/4
= cents(80)/4

so the stretch from that to its new tempered value is

cents(3)[1-k]/[cents(80)/4]

= 4*cents(3)([1-k]/cents(80)
= [1-k]cents(81)/cents(80)

Now we need to substitute in k, so that stretch becomes

[1-cents(81/80)/cents(81*80)]*cents(81)/cents(80)

(cents are a special case of log)

= [(cents(81*80) - cents(81/80))/cents(81*80)]*cents(81)/cents(80)
= [2*cents(80)/cents(81*80)]*cents(81)/cents(80)
= 2*cents(81)/cents(81*80)
= 2*log(81)/log(81*80)

which is the same as 1+k, and you can work through that if you like.

Graham

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> Oh, you mean "heuristic", not "hypothesis" (the latter concerns PBs > and DEs).

Yes, well, I typed the right thing into Google anyway :-)

> Hmm . . . I'm a bit busy right now, so can you work out an example, > and show all the errors like I did with Top meantone here?

Meantone's the easy one. 81 is the odd part of 81/80. So we need to share the error of 81/80 amongst the factors of 81 according to their respective weights. As 3 is the only prime factor of 81, it takes all the tempering, and each 3 gets a quarter of it, hence quarter comma meantone.

> I meant skip the step of getting a straightness measure. If we get > the right straightness measure, it won't matter which kernel basis we > pick for a given temperament. In which case we may be able to proceed > directly from the kernel to the relevant quantities. But I wouldn't > stress it at the moment . . .

We can already get at everything from the wedgie, by finding out what temperament it leads to and looking at those quantities. If we can get "goodness" straight from the wedgie, that would save these calculations. This is what geometric complexity might do. And as it also gives us the straightness then, yes, it would mean we could skip straightness as an independent quantity.

The main thing is to get goodness of an incomplete wedgie. Like we could find out that 2401:2400 and 3025:3024 work well together, and so keep looking for the next comma. But maybe looking at the planar temperament would tell us that. Which is like what I'm assuming about pairs of good equal temperaments giving good linear temperaments.

> Thanks for being you!

Well, sure, I do it all the time.

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> > Maybe someone can derive this 0.24999999999997 as a 1/4
symbolically.
> > I'd be very happy to see it.
>
> 1.0014154337454717 you say? Yes, I've derived it symbolically --
which
> means I must have duplicated your method for tempering. That magic
> number is 2*log(81)/log(81*80). The calculation goes something
this:
>
> The 2 generator is tempered as cents(2)[1+k]
>
> The 3 generator is tempered as cents(3)[1-k]
>
> The 5 generator is tempered as cents(5)[1+k]
>
> where k is log(81/80)/log(81*80). The + or - depends on whether
factors
> of this prime occur in the denominator or numerator respectively.

Yes, this is the method, as I recently explained here to Carl. But I
didn't factor out [1+k] or [1-k] as multipliers -- that's a neat
trick.

> So as 2 and 5 are just in quarter comma meantone, they must be
stretched
> by 1+k here.

Ah -- that's the secret. Good going!

> In quarter comma meantone, the 3 generator is tempered as
>
> cents(3) - cents(81/80)/4
>
> = cents(3) - cents(81)/4 + cents(80)/4
> = cents(3) - cents(3) + cents(80)/4
> = cents(80)/4
>
> so the stretch from that to its new tempered value is
>
> cents(3)[1-k]/[cents(80)/4]
>
> = 4*cents(3)([1-k]/cents(80)
> = [1-k]cents(81)/cents(80)
>
> Now we need to substitute in k, so that stretch becomes
>
> [1-cents(81/80)/cents(81*80)]*cents(81)/cents(80)
>
> (cents are a special case of log)
>
> = [(cents(81*80) - cents(81/80))/cents(81*80)]*cents(81)/cents(80)
> = [2*cents(80)/cents(81*80)]*cents(81)/cents(80)
> = 2*cents(81)/cents(81*80)
> = 2*log(81)/log(81*80)
>
> which is the same as 1+k, and you can work through that if you like.
>
>
> Graham

Nice work, Graham!

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>> >Now, for all primes r,
>> >
>> >If p contains any factors of r, the r-rungs in the lattice (which
>> >have length log2(r)) are shrunk from
>> >cents(r)
>> >to
>> >cents(r) - log2(r)*cents(p/q)/log2(p*q).
>> >If q contains any factors of 2, they are instead stretched to
>> >cents(r) + log2(r)*cents(p/q)/log2(p*q).
>>
>> Thanks. I understand this 100%. But I don't understand what's
>> new.
>
>Where have you seen this before?

I guess in my head.

What if the generator isn't a just interval? Then isn't it still
the same kind of multivariable optimization that you guys have been
using all along?

>> And I don't understand your 'limitless' claim -- since p/q contains
>> the factors it does and no others, one wouldn't expect its vanishing
>> to effect

What I mean is, when extending meantone to a 7-limit mapping, it
will naturally implicate different commas and change the optimal
generator a bit, same as before. Well, we can't be sure until we
see how to combine commas. But to claim it doesn't require a limit
when it's currently limited to linear temperaments in the 5-limit...
Or am I all wet?

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >Now, for all primes r,
> >> >
> >> >If p contains any factors of r, the r-rungs in the lattice
(which
> >> >have length log2(r)) are shrunk from
> >> >cents(r)
> >> >to
> >> >cents(r) - log2(r)*cents(p/q)/log2(p*q).
> >> >If q contains any factors of 2, they are instead stretched to
> >> >cents(r) + log2(r)*cents(p/q)/log2(p*q).
> >>
> >> Thanks. I understand this 100%. But I don't understand what's
> >> new.
> >
> >Where have you seen this before?
>
> I guess in my head.
>
> >> Perhaps it has something to do with using this to get
> >> optimum generators for a linear temperament?
> >
> >Well, that's exactly what this does (when the dimensionality is
> >right), as I've illustrated already in a few cases.
>
> What if the generator isn't a just interval? Then isn't it still
> the same kind of multivariable optimization that you guys have been
> using all along?

I didn't make any assumptions about what the generator was above. The
same formula works for any generator, and even when there is no
generator, as is the case for 7-limit and above.

> >> And I don't understand your 'limitless' claim -- since p/q
contains
> >> the factors it does and no others, one wouldn't expect its
vanishing
> >> to effect
>
> What I mean is, when extending meantone to a 7-limit mapping, it
> will naturally implicate different commas and change the optimal
> generator a bit, same as before.

Yes, and there are different choices as to which commas to use to
extend meantone to a 7-limit linear temperament. But I wasn't talking
about that. I was talking about tempering out a single comma, which
would lead to a planar temperament in the 7-limit, etc.

> But to claim it doesn't require a limit
> when it's currently limited to linear temperaments in the 5-limit...

No, it's simply limited to temperaments of codimension 1. Though I've
only charted the 5-limit commas, the exact same method works for any
commas, and I'll be producing a 7-limit comma chart for Herman soon.

🔗Carl Lumma <ekin@lumma.org>

>> What if the generator isn't a just interval? Then isn't it still
>> the same kind of multivariable optimization that you guys have been
>> using all along?
>
>I didn't make any assumptions about what the generator was above. The
>same formula works for any generator,

You also didn't give the method for finding it, but I was assuming the
TOP 3:2 and 2:1 are the meantone generators, for example.

>and even when there is no
>generator, as is the case for 7-limit and above.

With a single comma there are two 5-limit generators, three 7-limit,
and so on. Or so I suppose.

>> >> And I don't understand your 'limitless' claim -- since p/q
>> >> contains the factors it does and no others,
//
>> What I mean is, when extending meantone to a 7-limit mapping, it
>> will naturally implicate different commas and change the optimal
>> generator a bit, same as before.
>
>Yes, and there are different choices as to which commas to use to
>extend meantone to a 7-limit linear temperament. But I wasn't talking
>about that. I was talking about tempering out a single comma, which
>would lead to a planar temperament in the 7-limit, etc.

And does the old method give different results when going from
5-limit linear to 7-limit planar? Or are you claiming the answer
is "no" when "old method" was minimax, and "yes" when it was
anything else?

-Carl

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> Hmm . . . by *all* the errors, I meant for lots and lots of > intervals, like I did.

Oh, well, here's the 9-limit with a few bonuses:

3:1 0.002827
5:1 0.000000
5:3 0.001930
7:1 0.000903
7:3 0.000693
7:5 0.000903
9:1 0.002827
9:5 0.002827
9:7 0.002027
15:1 0.001147
27:1 0.002827
27:5 0.002827

The method for 7-limit temperaments is to find the worst possible comma that's in the basis, and temper that out. For meantone, it's the 81:80, so quarter comma meantone is still the 7-prime limit optimum (like it is for the unweighted 7-odd limit). I don't know if it always works, but it can.

Also in miracle the worst 7-limit comma is the 5-limit one, [-25 7 6>. The comma size is 31.567 cents and the complexity 30032 cents (the size of [0 7 6>). The size of a fifth is then 1200*log2(3)*(1-31.567/30032)-1200 = 699.96 cents. A secor is a sixth or this, giving 116.66 cents. Here are the weighted 9-limit errors:

3:1 0.001051
5:1 0.001051
5:3 0.000334
7:1 0.000637
7:3 0.000043
7:5 0.000233
9:1 0.001051
9:5 0.000281
9:7 0.000487

The same tuning happens to work in the 11-limit as well, but it's no panacea because it fails beyond that. So here are some more numbers:

11:1 0.000344
11:3 0.000137
11:5 0.000361
11:7 0.000172
11:9 0.000619
13:1 0.002138
13:3 0.002588
13:5 0.002798
13:7 0.002621
13:9 0.003039
13:11 0.002460
15:1 0.001051
15:7 0.000594
15:11 0.000746
15:13 0.003076
17:1 0.002386
17:3 0.002794
17:5 0.002983
17:7 0.002823
17:9 0.003201
17:11 0.002677
17:13 0.000450
17:15 0.003391
19:1 0.000514
19:3 0.000906
19:5 0.001088
19:7 0.000935
19:9 0.001298
19:11 0.000794
19:13 0.001349
19:15 0.001481
19:17 0.001782
21:1 0.000786
21:5 0.000231
21:11 0.000515
21:13 0.002588
21:17 0.003007
21:19 0.001283

Oh yes, and this octave-equivalent method always gives the same result as your octave-specific one after you unstretch the scale so that octaves are just. Unless both the numerator and denominator of the comma are odd numbers, in which case you already get a just octave, and so only taking the larger one will give a different result.

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What if the generator isn't a just interval? Then isn't it still
> >> the same kind of multivariable optimization that you guys have
been
> >> using all along?
> >
> >I didn't make any assumptions about what the generator was above.
The
> >same formula works for any generator,
>
> You also didn't give the method for finding it, but I was assuming
the
> TOP 3:2 and 2:1 are the meantone generators, for example.

In truth, tempering a single comma (such as 81:80) from the 5-limit
lattice yields a 2-dimensional tuning system, with no unique choice
of generators. But if we assume *octave-repetition*, then we're back
to the usual period-generator mappings for the primes, which you can
invert to find the generator.

> >and even when there is no
> >generator, as is the case for 7-limit and above.
>
> With a single comma there are two 5-limit generators, three 7-limit,
> and so on. Or so I suppose.

Yes, but then there's even less uniqueness to the choice. Gene has
proposed "hermite reduction", perhaps the issue is worth another look.

> >> >> And I don't understand your 'limitless' claim -- since p/q
> >> >> contains the factors it does and no others,
> //
> >> What I mean is, when extending meantone to a 7-limit mapping, it
> >> will naturally implicate different commas and change the optimal
> >> generator a bit, same as before.
> >
> >Yes, and there are different choices as to which commas to use to
> >extend meantone to a 7-limit linear temperament. But I wasn't
talking
> >about that. I was talking about tempering out a single comma,
which
> >would lead to a planar temperament in the 7-limit, etc.
>
> And does the old method give different results when going from
> 5-limit linear to 7-limit planar?

I believe so, though I can't remember the specifics.

> Or are you claiming the answer
> is "no" when "old method" was minimax, and "yes" when it was
> anything else?

If you mean Tenney-weighted minimax over all intervals, then this
could very well be, though I don't think that was actually one of
the "old" methods that were tried around here.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > Hmm . . . by *all* the errors, I meant for lots and lots of
> > intervals, like I did.
>
> Oh, well, here's the 9-limit with a few bonuses:
>
> 3:1 0.002827
> 5:1 0.000000
> 5:3 0.001930
> 7:1 0.000903
> 7:3 0.000693
> 7:5 0.000903
> 9:1 0.002827
> 9:5 0.002827
> 9:7 0.002027
> 15:1 0.001147
> 27:1 0.002827
> 27:5 0.002827

So you're dividing by expressibility here? Interesting . . . !

> The method for 7-limit temperaments is to find the worst possible
comma
> that's in the basis, and temper that out. For meantone, it's the
81:80,
> so quarter comma meantone is still the 7-prime limit optimum (like
it is
> for the unweighted 7-odd limit). I don't know if it always works,
but
> it can.

Hmm . . . are you talking about "septimal meantone", or are you
talking about a 7-limit planar temperament?

>
> Also in miracle the worst 7-limit comma is the 5-limit one, [-25 7
6>.
> The comma size is 31.567 cents and the complexity 30032 cents (the
size
> of [0 7 6>). The size of a fifth is then
> 1200*log2(3)*(1-31.567/30032)-1200 = 699.96 cents. A secor is a
sixth
> or this, giving 116.66 cents. Here are the weighted 9-limit errors:
>
> 3:1 0.001051
> 5:1 0.001051
> 5:3 0.000334
> 7:1 0.000637
> 7:3 0.000043
> 7:5 0.000233
> 9:1 0.001051
> 9:5 0.000281
> 9:7 0.000487
>
> The same tuning happens to work in the 11-limit as well, but it's
no
> panacea because it fails beyond that. So here are some more
numbers:
>
> 11:1 0.000344
> 11:3 0.000137
> 11:5 0.000361
> 11:7 0.000172
> 11:9 0.000619
> 13:1 0.002138

Hold on. Are you talking about some 13-limit linear extension of
miracle, or a planar temperament that tempers out the same commas as
miracle?

> 13:3 0.002588
> 13:5 0.002798
> 13:7 0.002621
> 13:9 0.003039
> 13:11 0.002460
> 15:1 0.001051
> 15:7 0.000594
> 15:11 0.000746
> 15:13 0.003076
> 17:1 0.002386
> 17:3 0.002794
> 17:5 0.002983
> 17:7 0.002823
> 17:9 0.003201
> 17:11 0.002677
> 17:13 0.000450
> 17:15 0.003391
> 19:1 0.000514
> 19:3 0.000906
> 19:5 0.001088
> 19:7 0.000935
> 19:9 0.001298
> 19:11 0.000794
> 19:13 0.001349
> 19:15 0.001481
> 19:17 0.001782
> 21:1 0.000786
> 21:5 0.000231
> 21:11 0.000515
> 21:13 0.002588
> 21:17 0.003007
> 21:19 0.001283
>
> Oh yes, and this octave-equivalent method always gives the same
result
> as your octave-specific one after you unstretch the scale so that
> octaves are just. Unless both the numerator and denominator of the
> comma are odd numbers, in which case you already get a just octave,
and
> so only taking the larger one will give a different result.

Only taking the larger one?

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> So you're dividing by expressibility here? Interesting . . . !

I'm dividing by the complexity. What's expressiblility?

> Hmm . . . are you talking about "septimal meantone", or are you > talking about a 7-limit planar temperament?

The usual 7-limit meantone.

> Hold on. Are you talking about some 13-limit linear extension of > miracle, or a planar temperament that tempers out the same commas as > miracle?

A 21-limit extension of miracle, based on the best approximations in 31- and 41-equal, or whatever else my program is doing.

> Only taking the larger one?

I don't now of any realistic commas with only odd numbers. So say you were tempering out 15/13. Using your method will work fine, it won't be very accurate, but you'll get a temperament out. And, because there are no factors of 2 in the commma, there's nothing to say the octaves should be stretched. So you already have an octave equivalent temperament! My method will take the complexity as log(15) rather than log(13). So it'll give some other result, and it may have to be like this for full generality.

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > So you're dividing by expressibility here? Interesting . . . !
>
> I'm dividing by the complexity.

Which complexity measure?

> What's expressiblility?

Kees's metric -- the log of the lowest odd limit the ratio belongs to.

> > Hmm . . . are you talking about "septimal meantone", or are you
> > talking about a 7-limit planar temperament?
>
> The usual 7-limit meantone.

So how do you define and find "the worst comma"?

> > Hold on. Are you talking about some 13-limit linear extension of
> > miracle, or a planar temperament that tempers out the same commas
as
> > miracle?
>
> A 21-limit extension of miracle, based on the best approximations
in 31-
> and 41-equal, or whatever else my program is doing.

Same question.

> > Only taking the larger one?
>
> I don't now of any realistic commas with only odd numbers.

27/25.

> So say you
> were tempering out 15/13. Using your method will work fine, it
won't be
> very accurate, but you'll get a temperament out. And, because
there are
> no factors of 2 in the commma, there's nothing to say the octaves
should
> be stretched. So you already have an octave equivalent temperament!

Yes.

> My
> method will take the complexity as log(15) rather than log(13).

Who would use log(13)?

> So
> it'll give some other result,

???

🔗Carl Lumma <ekin@lumma.org>

>In truth, tempering a single comma (such as 81:80) from the 5-limit
>lattice yields a 2-dimensional tuning system, with no unique choice
>of generators.

Aren't the units on either dimension the generators?

>But if we assume *octave-repetition*, then we're back
>to the usual period-generator mappings for the primes, which you can
>invert to find the generator.

I never understood this process, or what differentiates a period
from a generator.

>> With a single comma there are two 5-limit generators, three 7-limit,
>> and so on. Or so I suppose.
>
>Yes, but then there's even less uniqueness to the choice. Gene has
>proposed "hermite reduction", perhaps the issue is worth another look.

Searching my new tuning-math archive for "hermite" yields 37 matches,
most from March of 2002.

The most recent bit (excluding the present thread) is from Gene
from November of last year...

>> You form a matrix with the octave (1 0 0 ...) at the top, then a
>> chromatic unison vector (it doesn't matter which) and below them
>> the commas. Take the adjoint (the inverse multiplied by the
>> determinant).
>
>This proceedure only works if you have a linear temperament.
>Something else you might try is finding a basis for the nullspace of
>the matrix formed from the commas alone, without your additions, and
>using this to obtain a reduced set of vals (which could involve some
>extra work.) From there, one can put the vals in the form you like; I
>am partial to Hermite reduction unless we are dealing with linear
>temperaments, in which case we do period-generator and make the the
>generator as small, greater than one, as possible, to get a standard
>reduced form.
>
>Sometimes it suffices to simply find all standard vals which make all
>of the commas zero and use this to start with. Finding the wedgie
>from the commas, and the matrix from the wedgie, will also work; that
>is how I would do this but I use Maple's Hermite reduction function
>for it.

From October 2003...

>> I really hate to ask, but what do wedgies have to do with mapping
>> generators to primes?
>
>I take the wedgie, and from it generate what I call the subgroup vals.
>Then I hermite-reduce these, and apply a further reduction to make
>the generators of the generator/period pair as small, greater than
>one, as possible. This gives a standarized period/generator for the
>temperament in question.

As usual, however, nowhere on this list can I find any explanation
of hermite reduction. Not what it is, not why we'd care, and not
how to calculate it.

>> And does the old method give different results when going from
>> 5-limit linear to 7-limit planar?
>
>I believe so, though I can't remember the specifics.
>
>> Or are you claiming the answer
>> is "no" when "old method" was minimax, and "yes" when it was
>> anything else?
>
>If you mean Tenney-weighted minimax over all intervals, then this
>could very well be, though I don't think that was actually one of
>the "old" methods that were tried around here.

I'm still partial to rms over all the intervals, but somehow I
think those doing rms around here were not including the 2s.

-Carl

🔗Graham Breed <graham@microtonal.co.uk>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> Which complexity measure?

The logarithm of the (larger) odd number in the comma ratio.

>>What's expressiblility?
> > Kees's metric -- the log of the lowest odd limit the ratio belongs to.

I think that's the same, yes.

> So how do you define and find "the worst comma"?

It's defined as having the largest value for size/complexity. How you find it as an interesting question. I'm searching a range of combinations of the defining pair of commas and hoping I don't miss any. In general, it's probably easier to find the minimax by finding the temperament mapping first.

>>A 21-limit extension of miracle, based on the best approximations > > in 31- > >>and 41-equal, or whatever else my program is doing.
> > > Same question.

I only know it's the worst comma in the 7-limit case, but as the 11-limit values are all within the weighted minimax it looks like it works there as well.

>>My >>method will take the complexity as log(15) rather than log(13).
> > Who would use log(13)?

Sorry, log(15*13)

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >In truth, tempering a single comma (such as 81:80) from the 5-
limit
> >lattice yields a 2-dimensional tuning system, with no unique
choice
> >of generators.
>
> Aren't the units on either dimension the generators?

The generators amount to a choice of basis for the tuning system.
There is no unique choice of basis. I don't know what you mean
by "units".

> >But if we assume *octave-repetition*, then we're back
> >to the usual period-generator mappings for the primes, which you
can
> >invert to find the generator.
>
> I never understood this process,

Solving a system of linear equations?

> or what differentiates a period
> from a generator.

In our parlance, when we assume *octave-repetition*, the 'period'
will be the generator that generates the octave all by itself, while
the 'generator' (usually the smallest possible is chosen, such as
fourths for meantone) will produce all the other notes in the tuning
in conjunction with the period -- they form a basis.

> >> And does the old method give different results when going from
> >> 5-limit linear to 7-limit planar?
> >
> >I believe so, though I can't remember the specifics.
> >
> >> Or are you claiming the answer
> >> is "no" when "old method" was minimax, and "yes" when it was
> >> anything else?
> >
> >If you mean Tenney-weighted minimax over all intervals, then this
> >could very well be, though I don't think that was actually one of
> >the "old" methods that were tried around here.
>
> I'm still partial to rms over all the intervals,

How can you do that? Does it even converge? Or do you not really
mean "all the intervals"?

> but somehow I
> think those doing rms around here were not including the 2s.

If you don't include all the intervals, but don't want to assume
octave-equivalence, you can use an integer limit, and I've posted
some integer-limit rms results on the tuning list and elsewhere. But
I don't like integer limit in comparison with Tenney limit,
especially a Tenney limit that you don't even have to specify (as
long as it's large enough to include all the primes you care about)!

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

>> I never understood this process,
>
>Solving a system of linear equations?

Uh-huh.

>> or what differentiates a period
>> from a generator.
>
>In our parlance, when we assume *octave-repetition*, the 'period'
>will be the generator that generates the octave all by itself, while
>the 'generator' (usually the smallest possible is chosen, such as
>fourths for meantone) will produce all the other notes in the tuning
>in conjunction with the period -- they form a basis.

Why are you assuming octave repetition, what does this assumption
amount to?

If 2 is in the map, one of the generators had better well generate
it. If it isn't in the map, assuming octave repetition seems like
a bad idea to me.

>> >> And does the old method give different results when going from
>> >> 5-limit linear to 7-limit planar?
>> >
>> >I believe so, though I can't remember the specifics.
>> >
>> >> Or are you claiming the answer
>> >> is "no" when "old method" was minimax, and "yes" when it was
>> >> anything else?
>> >
>> >If you mean Tenney-weighted minimax over all intervals, then this
>> >could very well be, though I don't think that was actually one of
>> >the "old" methods that were tried around here.
>>
>> I'm still partial to rms over all the intervals,
>
>How can you do that? Does it even converge? Or do you not really
>mean "all the intervals"?

I can't, and I mean all the odd-limit intervals including 2s, though
I suppose there may be difficulties in then allowing the size of the
2s to be a variable.

>> but somehow I
>> think those doing rms around here were not including the 2s.
>
>If you don't include all the intervals, but don't want to assume
>octave-equivalence, you can use an integer limit, and I've posted
>some integer-limit rms results on the tuning list and elsewhere.

Ok, that makes sense. Odd-limit with tempered 2s means an infinite
number of intervals to optimize. So I guess I've been asking for
integer limit all along.

>But
>I don't like integer limit in comparison with Tenney limit,
>especially a Tenney limit that you don't even have to specify (as
>long as it's large enough to include all the primes you care about)!

What's a Tenney limit?

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I never understood this process,
> >
> >Solving a system of linear equations?
>
> Uh-huh.

Well, the easiest way to understand is to solve one equation for one
variable, plug that solution into the other variables so that you've
eliminated one variable entirely, and repeat until you're done.

> >> or what differentiates a period
> >> from a generator.
> >
> >In our parlance, when we assume *octave-repetition*, the 'period'
> >will be the generator that generates the octave all by itself,
while
> >the 'generator' (usually the smallest possible is chosen, such as
> >fourths for meantone) will produce all the other notes in the
tuning
> >in conjunction with the period -- they form a basis.
>
> Why are you assuming octave repetition, what does this assumption
> amount to?

That you'll have the same pitches in each (possibly tempered) octave.

> If 2 is in the map, one of the generators had better well generate
> it. If it isn't in the map, assuming octave repetition seems like
> a bad idea to me.

Any recent cases where you'd prefer not to see 2 in the map?

> >> >> And does the old method give different results when going from
> >> >> 5-limit linear to 7-limit planar?
> >> >
> >> >I believe so, though I can't remember the specifics.
> >> >
> >> >> Or are you claiming the answer
> >> >> is "no" when "old method" was minimax, and "yes" when it was
> >> >> anything else?
> >> >
> >> >If you mean Tenney-weighted minimax over all intervals, then
this
> >> >could very well be, though I don't think that was actually one
of
> >> >the "old" methods that were tried around here.
> >>
> >> I'm still partial to rms over all the intervals,
> >
> >How can you do that? Does it even converge? Or do you not really
> >mean "all the intervals"?
>
> I can't, and I mean all the odd-limit intervals including 2s,

There are an infinite number of those, but if the octave is fixed at
1200 cents, you only need one member of each class, and then you have
a finite list of intervals, so you do get convergence.

> though
> I suppose there may be difficulties in then allowing the size of the
> 2s to be a variable.

Correct.

> What's a Tenney limit?

If the limit is L, it's all reduced ratios n/d such that n*d<=L (or
log(n*d)<=L, whatever).

🔗Graham Breed <graham@microtonal.co.uk>

Carl Lumma wrote:

> Hiya Graham! Let me rephrase the above. Say I'm using unweighted
> rms error over all the intervals in a given prime limit. I want to
> find the 5-limit linear temperament that minimizes this error, call
> it Alex, and then I want to find the 7-limit planar temperament
> that does the same, call it Ben. Now, are the 5-limit intervals in
> Ben going to be different sizes than they are in Alex? In TOP
> temperament, the answer is no (I think).

Hello!

How would an unweighted, unbounded RMS error work? The advantage of the weighting is that more complex intervals get lower weights, and so the weighted error stays roughly constant. Hence you can impose a weighted minimax over all intervals within a given prime limit. The interesting, and slightly unexpected, thing about TOP is that it goes straight to this weighted minimax.

Oh, you meant odd limit. Well, I'll leave that paragraph in anyway.

Still to the question. Yes, I think the answer is no. If you define a 7-limit planar temperament by a 5-limit comma, the TOP method will give you a 5-limit linear temperament along with a just 7:4. And that should also be the same temperament you'd get by finding the weighted minimax for that planar temperament by any other method.

Graham

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Carl Lumma wrote:
>
> > Hiya Graham! Let me rephrase the above. Say I'm using unweighted
> > rms error over all the intervals in a given prime limit. I want
to
> > find the 5-limit linear temperament that minimizes this error,
call
> > it Alex, and then I want to find the 7-limit planar temperament
> > that does the same, call it Ben. Now, are the 5-limit intervals
in
> > Ben going to be different sizes than they are in Alex? In TOP
> > temperament, the answer is no (I think).
>
> Hello!
>
> How would an unweighted, unbounded RMS error work? The advantage
of the
> weighting is that more complex intervals get lower weights, and so
the
> weighted error stays roughly constant. Hence you can impose a
weighted
> minimax over all intervals within a given prime limit.

And, in fact, over *all* intervals, if you're simply talking about
the effect of tempering out a given comma in general -- or, if we
really can generalize this, the effect of tempering out some "multi-
comma" so that the codimension is specified but the dimension can be
anything.

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> Interesting! And is that truly the only one that matters?

The size/complexity tells you the best value for the suitably weighted minimax in any temperament in which this comma vanishes. If a comma exists such that its size/complexity is equal to the optimim minimax error in a given linear temperament, and the comma is in the linear temperament's kernel, then the two temperaments must be identical.

I'm not sure if such a comma will always exist, but provided it does it's the only one you need for TOPS. It doesn't even have to be made up of integers, so long as it's a linear combination of commas that are.

Naturally, if such a comma does exist, there can be no other comma in the temperament that has a higher value of size/complexity.

> So imposing octave-equivalence amounts to a uniform stretch/squish > of "Top", unless the octaves are already just? Bizarre!

It's a generalization of the proof of the TOP meantone being stretched quarter-comma. All the factors that get tempered the same way as 2 will be stretched by the same amount. But if the octaves don't get tempered at all, some factors will be tempered in one method but not the other.

Graham

🔗Carl Lumma <ekin@lumma.org>

>How would an unweighted, unbounded RMS error work?

Hopefully by now you've seen the messages where I've seen the
"error" of my ways, and settled on integer limit, or odd-limit with
just octaves.

>The advantage of the
>weighting is that more complex intervals get lower weights, and so
>the weighted error stays roughly constant.

In fact, I once asked if the weighting couldn't be so steep that
additionally specifying a limit would be unnecessary (since the
contribution from the errors of higher-limit ratios would be
vanishingly small). IOW, could the concept of choosing a "limit" be
replaced entirely by that of choosing a steepness of the weighting
function?

>Hence you can impose a weighted
>minimax over all intervals within a given prime limit.

Aha! So why then isn't the prime limit also superfluous?

>Still to the question. Yes, I think the answer is no.

Cool, then you and Paul agree on that.

-Carl

🔗Carl Lumma <ekin@lumma.org>

>> >> I never understood this process,
>> >
>> >Solving a system of linear equations?
>>
>> Uh-huh.
>
>Well, the easiest way to understand is to solve one equation for one
>variable, plug that solution into the other variables so that you've
>eliminated one variable entirely, and repeat until you're done.

I remember this technique from Algebra, but I didn't think it would
be applicable here, since I assumed the variables wouldn't be
independent in that way. What do these equations look like?

>> Why are you assuming octave repetition, what does this assumption
>> amount to?
>
>That you'll have the same pitches in each (possibly tempered) octave.
>
>> If 2 is in the map, one of the generators had better well generate
>> it. If it isn't in the map, assuming octave repetition seems like
>> a bad idea to me.
>
>Any recent cases where you'd prefer not to see 2 in the map?

I'd always prefer to see it, but why assume?

>> What's a Tenney limit?
>
>If the limit is L, it's all reduced ratios n/d such that n*d<=L (or
>log(n*d)<=L, whatever).

Ok.

-Carl

🔗Graham Breed <graham@microtonal.co.uk>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > Interesting! And is that truly the only one that matters?
>
> The size/complexity tells you the best value for the suitably
weighted
> minimax in any temperament in which this comma vanishes. If a
comma
> exists such that its size/complexity is equal to the optimim
minimax
> error in a given linear temperament, and the comma is in the linear
> temperament's kernel, then the two temperaments must be identical.

I can't follow that right now.

> I'm not sure if such a comma will always exist,

Wow -- now that's an interesting question to consider.

> but provided it does
> it's the only one you need for TOPS.

My intuition says it doesn't exist.

> It doesn't even have to be made up
> of integers, so long as it's a linear combination of commas that
>are.

???

> It's a generalization of the proof of the TOP meantone being
stretched
> quarter-comma. All the factors that get tempered the same way as 2
will
> be stretched by the same amount. But if the octaves don't get
tempered
> at all, some factors will be tempered in one method but not the
other.

There's something that seems strange about your octave-equivalent
method. The comma is supposed to be distributed uniformly (per unit
length, taxicabwise) among its constituent "rungs" in the lattice.
But it seems that 81:80 = 81:5 would involve 1, not 0, rungs of 5 in
the octave-equivalent lattice. But the octave-equivalent lattice
can't be embedded in euclidean space, so this completely falls apart??

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> I never understood this process,
> >> >
> >> >Solving a system of linear equations?
> >>
> >> Uh-huh.
> >
> >Well, the easiest way to understand is to solve one equation for
one
> >variable, plug that solution into the other variables so that
you've
> >eliminated one variable entirely, and repeat until you're done.
>
> I remember this technique from Algebra, but I didn't think it would
> be applicable here, since I assumed the variables wouldn't be
> independent in that way.

They're not, you actually have an extra equation.

> What do these equations look like?

For meantone,

prime2 = period;
prime3 = period + generator;
prime5 = 4*generator.

You can throw out any equation -- say the first.

so generator = .25*prime5,
prime3 = period + .25*prime5,
period = prime3 - .25*prime5.

> >> Why are you assuming octave repetition, what does this assumption
> >> amount to?
> >
> >That you'll have the same pitches in each (possibly tempered)
octave.
> >
> >> If 2 is in the map, one of the generators had better well
generate
> >> it. If it isn't in the map, assuming octave repetition seems
like
> >> a bad idea to me.
> >
> >Any recent cases where you'd prefer not to see 2 in the map?
>
> I'd always prefer to see it, but why assume?

Agreed. But it's a "default", a "convention", that many would assume.

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> What do these equations look like?
> >> >
> >> >For meantone,
> >> >
> >> >prime2 = period;
> >> >prime3 = period + generator;
> >> >prime5 = 4*generator.
> >> >
> >> >You can throw out any equation -- say the first.
> >> >
> >> >so generator = .25*prime5,
> >> >prime3 = period + .25*prime5,
> >> >period = prime3 - .25*prime5.
> >>
> >> Sure, I've done these hundreds of times. But this is
> >> just the map -- where are all the errors of all the
> >> intervals?
> >>
> >> -Carl
> >
> >Just add up the primes that make them up!
>
> That sounds like TOP. I'm talking about the old way.

Oh. The old way, you start with the mapping, and then solve for the
period and generator, and then minimize your error function (which is
over some finite set of intervals) by varying the period & generator,
or in octave-equivalent cases, just the generator. You can use
calculus and express the error function in terms of the generator
size, take the derivative, set that equal to zero, and solve -- works
great for sum-squared error (p=2), weighted or unweighted. Other
methods work better for "harder" error functions like max-abs-error
(p=inf) or sum-abs-error (p=1).

🔗Carl Lumma <ekin@lumma.org>

>> That sounds like TOP. I'm talking about the old way.
>
>Oh. The old way, you start with the mapping, and then solve for the
>period and generator, and then minimize your error function (which is
>over some finite set of intervals) by varying the period & generator,
>or in octave-equivalent cases, just the generator. You can use
>calculus

Aha, I knew it! Calculus! :)

>and express the error function in terms of the generator
>size, take the derivative,

ok...

>set that equal to zero, and solve

Lost me here. The derivative itself is a curve, unless the
error/generator function is a straight line or something.

Wait -- are you saying that once the error fuctio starts
going up it'll never go down again?

Oh, and if we're doing integer limit don't we need two
generators?

>-- works great for sum-squared error (p=2), weighted or unweighted.

Good, that's all I want. I've got enough software to put my eye
out with, I ought to be able to set this up. By the way, this now
includes Matlab, if you'd prefer to illustrate with code.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> That sounds like TOP. I'm talking about the old way.
> >
> >Oh. The old way, you start with the mapping, and then solve for
the
> >period and generator, and then minimize your error function (which
is
> >over some finite set of intervals) by varying the period &
generator,
> >or in octave-equivalent cases, just the generator. You can use
> >calculus
>
> Aha, I knew it! Calculus! :)
>
> >and express the error function in terms of the generator
> >size, take the derivative,
>
> ok...
>
> >set that equal to zero, and solve
>
> Lost me here. The derivative itself is a curve,

Right, and where it meets the x-axis is where it equals zero.

> Oh, and if we're doing integer limit don't we need two
> generators?

We need two generators if we're talking about a 2D temperament --
either a planar temperament with octave-equivalence assumed, or a
linear temperament where we can vary the octave (or period) as well
as the generator.

> >-- works great for sum-squared error (p=2), weighted or unweighted.
>
> Good, that's all I want. I've got enough software to put my eye
> out with, I ought to be able to set this up. By the way, this now
> includes Matlab, if you'd prefer to illustrate with code.

Wow. Do you have the optimization toolbox?

🔗Carl Lumma <ekin@lumma.org>

>> >and express the error function in terms of the generator
>> >size, take the derivative,
>>
>> ok...
>>
>> >set that equal to zero, and solve
>>
>> Lost me here. The derivative itself is a curve,
>
>Right, and where it meets the x-axis is where it equals zero.

That's where the original function is flat, but how do we
know the original function isn't flat at multiple places?

>> Oh, and if we're doing integer limit don't we need two
>> generators?
>
>We need two generators if we're talking about a 2D temperament --
>either a planar temperament with octave-equivalence assumed, or a
>linear temperament where we can vary the octave (or period) as well
>as the generator.

I'm talking about linear temperaments now, strictly. And by
"integer limit", I mean variable octave, and I've been calling
the period a generator for, oh, over a year.

>> >-- works great for sum-squared error (p=2), weighted or unweighted.
>>
>> Good, that's all I want. I've got enough software to put my eye
>> out with, I ought to be able to set this up. By the way, this now
>> includes Matlab, if you'd prefer to illustrate with code.
>
>Wow. Do you have the optimization toolbox?

It looks like it. I just ran a "Large-scale unconstrained nonlinear
minimization" demo.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >and express the error function in terms of the generator
> >> >size, take the derivative,
> >>
> >> ok...
> >>
> >> >set that equal to zero, and solve
> >>
> >> Lost me here. The derivative itself is a curve,
> >
> >Right, and where it meets the x-axis is where it equals zero.
>
> That's where the original function is flat, but how do we
> know the original function isn't flat at multiple places?

If you're using least squares, the function is just a parabola, since
there are no terms involving higher powers (than 2) of the generator.

> >> Oh, and if we're doing integer limit don't we need two
> >> generators?
> >
> >We need two generators if we're talking about a 2D temperament --
> >either a planar temperament with octave-equivalence assumed, or a
> >linear temperament where we can vary the octave (or period) as
well
> >as the generator.
>
> I'm talking about linear temperaments now, strictly. And by
> "integer limit", I mean variable octave,

I didn't know you meant to use integer limit here. Then you have a
function of two variables, so the calculus is a little harder, but
the optimization toolbox has no problem -- and it's basically a
paraboloid anyway.

> and I've been calling
> the period a generator for, oh, over a year.

You were right.

> >> >-- works great for sum-squared error (p=2), weighted or
unweighted.
> >>
> >> Good, that's all I want. I've got enough software to put my eye
> >> out with, I ought to be able to set this up. By the way, this
now
> >> includes Matlab, if you'd prefer to illustrate with code.
> >
> >Wow. Do you have the optimization toolbox?
>
> It looks like it. I just ran a "Large-scale unconstrained nonlinear
> minimization" demo.

How did you get so lucky? Anyway, for any temperament, write a
function that computes your error function for a given choice of
period and generator, then minimize it using fmin or fmins or
whatever.

🔗Carl Lumma <ekin@lumma.org>

>> >> >and express the error function in terms of the generator
>> >> >size, take the derivative,
>> >>
>> >> ok...
>> >>
>> >> >set that equal to zero, and solve
>> >>
>> >> Lost me here. The derivative itself is a curve,
>> >
>> >Right, and where it meets the x-axis is where it equals zero.
>>
>> That's where the original function is flat, but how do we
>> know the original function isn't flat at multiple places?
>
>If you're using least squares, the function is just a parabola, since
>there are no terms involving higher powers (than 2) of the generator.

I understand that functions of the type f(x) -> x^2 + c are shaped
like parabolas, but x isn't a generator size here, it's the sum of
errors resulting from a generator size. If I took out the ^2 it
might be shaped like anything; are you saying the ^2 will turn it
into a parabola? Oh man I'm braindead, I used to know this stuff. :(

>> >> Oh, and if we're doing *integer limit* don't we need two
>> >> generators?
>> >
>> >We need two generators if we're talking about a 2D temperament --
>> >either a planar temperament with octave-equivalence assumed, or a
>> >linear temperament where we can vary the octave (or period) as
>> >well as the generator.
>>
>> I'm talking about linear temperaments now, strictly. And by
>> "integer limit", I mean variable octave,
>
>I didn't know you meant to use integer limit here.

Emphasis added.

>Then you have a
>function of two variables, so the calculus is a little harder, but
>the optimization toolbox has no problem -- and it's basically a
>paraboloid anyway.

Well I'm happy to learn the fixed-octave case first... or not.
However.

>> >> Good, that's all I want. I've got enough software to put my eye
>> >> out with, I ought to be able to set this up. By the way, this
>> >> now includes Matlab, if you'd prefer to illustrate with code.
>> >
>> >Wow. Do you have the optimization toolbox?
>>
>> It looks like it. I just ran a "Large-scale unconstrained
>> nonlinear minimization" demo.
>
>How did you get so lucky?

One of my best friends studies rat brains (vision alas), and we're
using matlab to model a potential business idea at the moment.

>Anyway, for any temperament, write a
>function that computes your error function for a given choice of
>period and generator, then minimize it using fmin or fmins or
>whatever.

Tx.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I understand that functions of the type f(x) -> x^2 + c are
shaped
> >> like parabolas, but x isn't a generator size here, it's the sum
of
> >> errors resulting from a generator size. If I took out the ^2 it
> >> might be shaped like anything;
> >
> >Huh? x + c is shaped like anything?
>
> Traditionally a line, but in this case x is actually this other
> function, the summed errors from these arbitrary external just
> ratio things.

No, silly goose :), the squaring is done *before* the summing! If
it's done *after* the summing, it has no effect (since the location
of the lowest value of a positive function is also the lowest
possible value of the function squared, and if you don't start with a
positive function, you're doing something wrong).

> As I move the generator size from 0-600 cents and
> pump it through say the meantone map, the errors could go up and
> down several times for all I know.

The error of each interval will be a straight line. The errors
squared will be parabolas. The sum of a set of parabolas is a
parabola, since the sum of any number of functions of order 2 is a
function of order 2 -- since you're squaring some linear functions,
then adding, you'll have quadratic terms, linear terms, and constant
terms to add, and that's all.

🔗Carl Lumma <ekin@lumma.org>

🔗Graham Breed <graham@microtonal.co.uk>

Me:
>>I'm not sure if such a comma will always exist,

Paul:
> Wow -- now that's an interesting question to consider.

Me:
>>but provided it does >>it's the only one you need for TOPS.

Paul:
> My intuition says it doesn't exist.

If it doesn't, the worst one you get should get you close to the optimum result.

Me:
>>It doesn't even have to be made up >>of integers, so long as it's a linear combination of commas that >>are.

Paul:
> ???

81**2:80**2 or 6561:6400 has the double the size and double the complexity of 81:80. That means its size/complexity ratio is the same, and so tempering it out gives exactly the same result as tempering out 81:80. So why stop there? The cube of 81:80 also gives the same result. Any power will do, and it doesn't have to be an integer. The square root of 81:80 works. Even the pith root works (provided you express it as a monzo -- otherwise, it's an arbitrary number).

That means instead of having two integers to define the resultant comma of a 7-limit linear temperament, you only need one real number. For example, the miracle kernel is (225:224)***i * (2401:2400)**j or i*[-5 2 2 -1> + j*[-5 -1 -2 4>. You can simplify that to x*[-5 2 2 -1> + (1-x)*[-5 -1 -2 4> so that you only have 1 variable instead of 2.

> There's something that seems strange about your octave-equivalent > method. The comma is supposed to be distributed uniformly (per unit > length, taxicabwise) among its constituent "rungs" in the lattice. > But it seems that 81:80 = 81:5 would involve 1, not 0, rungs of 5 in > the octave-equivalent lattice. But the octave-equivalent lattice > can't be embedded in euclidean space, so this completely falls apart??

81:5 involves three rungs of 3:1 and one rung of 3:5. For the 5-odd limit, these rungs are of equal length, and so that error has to be shared between them. That leaves 3:1 and 3:5 having an equal amount of temperament, and so 1:5 must be untempered.

This is how Dave did it on his original web page:

http://dkeenan.com/Music/DistributingCommas.htm

and we worked out at the time that it's correct. I didn't remember all that stuff about octave-specific optimization. It looks like you've cleared up the problems. But anyway, for the weighted, octave-equivalent case:

The complexity of 81:80 is entirely determined by the 81, and not at all by the 80. The error can only be shared amongst those intervals that contribute to the complexity. I don't see how this can be drawn on a lattice, but it doesn't need to be.

You can improve 3:1 at the expense of 3:5 and, because 3:5 has less weight, the worst weighted error for 5 odd-limit ratios goes down. However, the weighted error for 9:5 exceeds this, because it has the same complexity as 9:8 but is more poorly tuned. The true minimax is found by only tempering the 81.

You certainly can embed an octave-equivalent lattice in Euclidian space. The triangular lattice is an example, but there's no reason to connect up the 5:3s if you're measuring Euclidian distances. In general, for each prime interval you need to assign a weight (the unit lattice distance) and also an angle. And you can probably do size/complexity tempering with such a measure, but it'd be more complex, and I haven't tried it. I'm guessing it wouldn't give the true weighted minimax, because for that to work it looks like the complexity of a comma has to be the sum of the complexities of its factors (one way or another).

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Me:
> >>I'm not sure if such a comma will always exist,
>
> Paul:
> > Wow -- now that's an interesting question to consider.
>
> Me:
> >>but provided it does
> >>it's the only one you need for TOPS.
>
> Paul:
> > My intuition says it doesn't exist.
>
> If it doesn't, the worst one you get should get you close to the
optimum
> result.

I thought "the worst one" is what you weren't sure existed.

>
> Me:
> >>It doesn't even have to be made up
> >>of integers, so long as it's a linear combination of commas that
> >>are.
>
> Paul:
> > ???
>
> 81**2:80**2 or 6561:6400 has the double the size and double the
> complexity of 81:80. That means its size/complexity ratio is the
same,
> and so tempering it out gives exactly the same result as tempering
out
> 81:80.

Right, that's why I excluded such ratios from my charts.

> So why stop there? The cube of 81:80 also gives the same
> result. Any power will do, and it doesn't have to be an integer.
The
> square root of 81:80 works. Even the pith root works (provided you
> express it as a monzo -- otherwise, it's an arbitrary number).
>
> That means instead of having two integers to define the resultant
comma
> of a 7-limit linear temperament, you only need one real number.
For
> example, the miracle kernel is (225:224)***i * (2401:2400)**j or i*
[-5 2
> 2 -1> + j*[-5 -1 -2 4>. You can simplify that to x*[-5 2 2 -1> +
> (1-x)*[-5 -1 -2 4> so that you only have 1 variable instead of 2.

Hmm . . .

> > There's something that seems strange about your octave-equivalent
> > method. The comma is supposed to be distributed uniformly (per
unit
> > length, taxicabwise) among its constituent "rungs" in the
lattice.
> > But it seems that 81:80 = 81:5 would involve 1, not 0, rungs of 5
in
> > the octave-equivalent lattice. But the octave-equivalent lattice
> > can't be embedded in euclidean space, so this completely falls
apart??
>
> 81:5 involves three rungs of 3:1 and one rung of 3:5.

Oh yeah.

> For the 5-odd
> limit, these rungs are of equal length, and so that error has to be
> shared between them. That leaves 3:1 and 3:5 having an equal
amount of
> temperament, and so 1:5 must be untempered.

Ha!

> This is how Dave did it on his original web page:
>
> http://dkeenan.com/Music/DistributingCommas.htm
>
> and we worked out at the time that it's correct. I didn't remember
all
> that stuff about octave-specific optimization. It looks like
you've
> cleared up the problems. But anyway, for the weighted,
> octave-equivalent case:
>
> The complexity of 81:80 is entirely determined by the 81, and not
at all
> by the 80. The error can only be shared amongst those intervals
that
> contribute to the complexity. I don't see how this can be drawn on
a
> lattice, but it doesn't need to be.

But you already cleared up the lattice situation above.

> You can improve 3:1 at the expense of 3:5 and, because 3:5 has less
> weight, the worst weighted error for 5 odd-limit ratios goes down.
> However, the weighted error for 9:5 exceeds this, because it has
the
> same complexity as 9:8 but is more poorly tuned. The true minimax
is
> found by only tempering the 81.

OK . . .

> You certainly can embed an octave-equivalent lattice in Euclidian
>space.

Nope -- you need 'wormholes', i.e., non-euclidean features.

> The triangular lattice is an example,

It doesn't work -- 9:1 looks shorter than other ratios of 9.

> but there's no reason to connect
> up the 5:3s if you're measuring Euclidian distances.

I certainly hope we're not measuring Euclidean distances!

> In general, for
> each prime interval you need to assign a weight (the unit lattice
> distance) and also an angle.

There should be no angles defined here, just as there are none in the
Tenney lattice.

> And you can probably do size/complexity
> tempering with such a measure, but it'd be more complex, and I
haven't
> tried it. I'm guessing it wouldn't give the true weighted minimax,
> because for that to work it looks like the complexity of a comma
has to
> be the sum of the complexities of its factors (one way or another).

Right, but a non-euclidean version of the triangular lattice can
respect this, can't it? Looks like that's similar to what Kees was
geting at at the bottom of this page:

http://www.kees.cc/tuning/lat_perbl.html

except it doesn't look like he was thinking taxicab . . .

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> I thought "the worst one" is what you weren't sure existed.

There'll always be a worst comma, but it might not be as bad as the linear temperament it belongs to.

>>You certainly can embed an octave-equivalent lattice in Euclidian >>space. > > > Nope -- you need 'wormholes', i.e., non-euclidean features.

Which kind of lattice are you talking about? I meant the crystallographic one.

> It doesn't work -- 9:1 looks shorter than other ratios of 9.

Or longer, indeed. But so what?

> I certainly hope we're not measuring Euclidean distances!

Yes, Euclidian distances. That's how it comes to be in Euclidian space.

> There should be no angles defined here, just as there are none in the > Tenney lattice.

Why "should"? They have to be, or it won't work.

> Right, but a non-euclidean version of the triangular lattice can > respect this, can't it? Looks like that's similar to what Kees was > geting at at the bottom of this page:
> > http://www.kees.cc/tuning/lat_perbl.html
> > except it doesn't look like he was thinking taxicab . . .

I don't know, I don't see what the point is. I don't think the odd limit counts as a norm, which I think is a problem, although I don't remember the details.

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > I thought "the worst one" is what you weren't sure existed.
>
> There'll always be a worst comma, but it might not be as bad as the
> linear temperament it belongs to.

What's the worst comma for 12-equal in the 5-limit?

> >>You certainly can embed an octave-equivalent lattice in Euclidian
> >>space.
> >
> >
> > Nope -- you need 'wormholes', i.e., non-euclidean features.
>
> Which kind of lattice are you talking about? I meant the
> crystallographic one.

Me too -- but the lengths aren't compatible in a Euclidean space.
Remember the whole big "wormholes" discussion from years ago?

> > It doesn't work -- 9:1 looks shorter than other ratios of 9.
>
> Or longer, indeed. But so what?

See above.

> > I certainly hope we're not measuring Euclidean distances!
>
> Yes, Euclidian distances. That's how it comes to be in Euclidian
space.

We can either embed a lattice, with a taxicab distance, into
Euclidean space, or we can't. But just because we can, doesn't mean
we should use Euclidean distance! NONONONONONO!

> > There should be no angles defined here, just as there are none in
the
> > Tenney lattice.
>
> Why "should"? They have to be, or it won't work.

Why won't it? My Tenney, non-octave-equivalent way doesn't need
angles defined. You can choose any set of angles you want, and still
embed the result in Euclidean space, but that doesn't even matter --
what matters are the taxicab distances ONLY.

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> What's the worst comma for 12-equal in the 5-limit?

[0 28 -19> or 22876792454961:19073486328125

TOPping it gives a narrow octave of 0.99806 2:1 octaves.

> Me too -- but the lengths aren't compatible in a Euclidean space. > Remember the whole big "wormholes" discussion from years ago?

Yes, I remember all about the wormholes, and they don't have anything to do with this. You only need them for odd limits.

> We can either embed a lattice, with a taxicab distance, into > Euclidean space, or we can't. But just because we can, doesn't mean > we should use Euclidean distance! NONONONONONO!

You could try taxicab distance, I'm not sure it'd work right. But you can also use Euclidian distance, and it looks like a more straightforward way to me.

> Why won't it? My Tenney, non-octave-equivalent way doesn't need > angles defined. You can choose any set of angles you want, and still > embed the result in Euclidean space, but that doesn't even matter -- > what matters are the taxicab distances ONLY.

When did it become *your* way? The problem that either triangular or angular lattices solve doesn't arise in octave specific lattices, as we've always known. But Euclidian metrics can still be useful. From what I remember/understood, geometric complexity was one.

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

Gene's proposed canonical meantone:

5-limit: [1200., 1896.578429, 2786.313713]

Let's evaluate:

Interval...Approx....|Error|....Comp=log2(n*d)...|Error|/Comp
2:1........1200.00.....0.............1...............0
3:1........1896.58....5.38..........1.58............3.41

So already you've exceeded the maximum weighted error of my proposal
by a factor of 2!

> Interval...Approx....|Error|....Comp=log2(n*d)...|Error|/Comp
> 2:1........1201.70....1.70...........1..............1.70
> 3:1........1899.26....2.69..........1.58............1.70
> 4:1........2403.40....3.40...........2..............1.70
> 5:1........2790.26....3.94..........2.32............1.70
> 3:2.........697.56....4.39..........2.58............1.70
> 6:1........3100.96....0.99..........2.58............0.38
> 8:1........3605.10....5.10...........3..............1.70
> 9:1........3798.53....5.38..........3.17............1.70
> 10:1.......3991.96....5.64..........3.32............1.70
> 4:3.........504.13....6.09..........3.58............1.70
> 12:1.......4302.66....0.70..........3.58............0.20
> 5:3.........890.99....6.64..........3.91............1.70
> 15:1.......4689.52....1.25..........3.91............0.32
> 16:1.......4806.79....6.79...........4..............1.70
> 9:2........2596.83....7.08..........4.17............1.70
> 18:1.......5000.22....3.69..........4.17............0.88
> 5:4.........386.86....0.55..........4.32............0.13
> 20:1.......5193.65....7.34..........4.32............1.70
> 8:3........1705.83....7.79..........4.58............1.70
> 24:1.......5504.36....2.40..........4.58............0.52
> 25:1.......5580.52....7.89..........4.64............1.70
> 6:5.........310.70....4.94..........4.91............1.01
> 10:3.......2092.69....8.33..........4.91............1.70
> 30:1.......5891.22....2.95..........4.91............0.60
> 32:1.......6008.49....8.49...........5..............1.70
> 36:1.......6201.92....1.99..........5.17............0.38
> 8:5.........814.84....1.15..........5.32............0.22
> 40:1.......6395.35....9.04..........5.32............1.70
> 9:5........1008.27....9.33..........5.49............1.70
> 45:1.......6588.78....1.44..........5.49............0.26
> 16:3.......2907.53....9.49..........5.58............1.70
> 48:1.......6706.06....4.10..........5.58............0.73
> 25:2.......4378.82....6.19..........5.64............1.10
> 50:1.......6782.21....9.59..........5.64............1.70
> 27:2.......4496.09....9.77..........5.75............1.70
> 54:1.......6899.49....6.38..........5.75............1.11
> 12:5.......1512.40....3.24..........5.91............0.55
> 15:4.......2286.12....2.15..........5.91............0.36
> 20:3.......3294.39...10.03..........5.91............1.70
> 60:1.......7092.92....4.65..........5.91............0.79
> 1296:5.....
> and so on. Thinking about a few of these example spacially should
> help you see that the weighted error can never exceed
>
> cents(81/80)/log2(81*80) = 1.70
>
> for ANY interval.
>
> Is there a just (RI) interval in this meantone? The idea of duality
> leads me to guess 81*80:1 = 6480:1 . . .
>
> 6480:1....15194.10....0.03
>
> almost, but no cigar.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > What's the worst comma for 12-equal in the 5-limit?
>
> [0 28 -19> or 22876792454961:19073486328125

Wow. How did you find that?

> TOPping it gives a narrow octave of 0.99806 2:1 octaves.

Shall I proceed to calculate Tenney-weighted errors for all (well, a
bunch of) intervals? I hope you're onto something!

> > Me too -- but the lengths aren't compatible in a Euclidean space.
> > Remember the whole big "wormholes" discussion from years ago?
>
> Yes, I remember all about the wormholes, and they don't have
anything to
> do with this. You only need them for odd limits.

I thought that's what you were talking about in the thread where I
brought them up! Odd limit, right?

> > We can either embed a lattice, with a taxicab distance, into
> > Euclidean space, or we can't. But just because we can, doesn't
mean
> > we should use Euclidean distance! NONONONONONO!
>
> You could try taxicab distance, I'm not sure it'd work right. But
you
> can also use Euclidian distance, and it looks like a more
> straightforward way to me.
>
> > Why won't it? My Tenney, non-octave-equivalent way doesn't need
> > angles defined. You can choose any set of angles you want, and
still
> > embed the result in Euclidean space, but that doesn't even
matter --
> > what matters are the taxicab distances ONLY.
>
> When did it become *your* way?

Did someone publish it before? It's currently not Gene's way, anyway.

> The problem that either triangular or
> angular lattices solve doesn't arise in octave specific lattices,
as
> we've always known. But Euclidian metrics can still be useful.
From
> what I remember/understood, geometric complexity was one.

I've been trying to convince Gene otherwise, and he said something
about minor thirds being shorter than major thirds there . . .

🔗Graham Breed <graham@microtonal.co.uk>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> Wow. How did you find that?

Briefly (use the Reply thing so that indentation works),

Python 2.2.2 (#2, Feb 5 2003, 10:40:08)
[GCC 3.2.1 (Mandrake Linux 9.1 3.2.1-5mdk)] on linux-i386
Type "copyright", "credits" or "license" for more information.
IDLE 0.8 -- press F1 for help
>>> import temper
>>> h12 = temper.BestET(12,temper.limit5)
>>> ~temper.Wedgie(h12)^temper.WedgableRatio((81,80))
{}
>>> ~temper.Wedgie(h12)^temper.WedgableRatio((128,125))
{}
>>> temper.factorizeRatio(128,125)
(7, 0, -3, 0, 0, 0, 0, 0)
>>> def badness(a,b,c):
return (a + b*temper.log2(3) + c*temper.log2(5)) / (
abs(a) + abs(b)*temper.log2(3) + abs(c)*temper.log2(5))
>>> for i in range(20):
for j in range(-20,20):
if i or j:
print "%i %i %f" %(
i, j, badness(
7*j - 4*i,
4*i,
-i - 3*j))
0 -20 -0.002450
0 -19 -0.002450
0 -18 -0.002450
0 -17 -0.002450
0 -16 -0.002450
0 -15 -0.002450
0 -14 -0.002450
...
7 -2 0.000643
7 -1 0.001029
7 0 0.001415
7 1 0.001802
7 2 0.002189
7 3 0.002577
7 4 0.002964
7 5 0.002894
7 6 0.002841
...
19 15 0.002864
19 16 0.002846
19 17 0.002829
19 18 0.002813
19 19 0.002799
>>> badness(0,28,-19)
0.0029641729677381511
>>> psize = 28*temper.log2(3) - 19 * temper.log2(5)
>>> worstbad = 0.0
>>> for i in range(100):
for j in range(-100,100):
if i or j:
worstbad=max(worstbad,
abs(badness(
7*j - 4*i,
4*i,
-i - 3*j)))

>>> worstbad
0.0029641729677381628
>>> worstbad/badness(0,28,-19)
1.000000000000004
>>> (1+worstbad) * temper.log2(5)/28*12
0.99806172487682532
>>> (1-worstbad) * temper.log2(3)/19*12
0.99806172487682532
>>> print '%i:%i'%(3**28, 5**19)
22876792454961:19073486328125

>>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> > > Shall I proceed to calculate Tenney-weighted errors for all (well, a > bunch of) intervals? I hope you're onto something!

If you like.

> I thought that's what you were talking about in the thread where I > brought them up! Odd limit, right?

No, a lattice for octave-equivalent ratios.

> Did someone publish it before? It's currently not Gene's way, anyway.

How about Tenney?

Graham

🔗Graham Breed <graham@microtonal.co.uk>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > Wow. How did you find that?
>
> Briefly (use the Reply thing so that indentation works),

> 22876792454961:19073486328125

So it was a finite search? How do you know you won't keep finding
worse and worse examples if you go farther out? You might be
approaching a limit, but how do you know you'll ever reach it?

> >>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> >
> >
> > Shall I proceed to calculate Tenney-weighted errors for all
(well, a
> > bunch of) intervals? I hope you're onto something!
>
> If you like.

OK, later -- gotta go perform now.

> > I thought that's what you were talking about in the thread where
I
> > brought them up! Odd limit, right?
>
> No, a lattice for octave-equivalent ratios.

Yes, I meant that, with particular qualifications.

> > Did someone publish it before? It's currently not Gene's way,
anyway.
>
> How about Tenney?

No, I don't think he ever mentioned anything about comma-eating
systems with tempered octaves or anything like that.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

Gene, what do you get for the top system with the commas of 12-equal
(in other words, some stretching or squashing of 12-equal)? Graham
seems to gave gotten pretty close below, but no cigar . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
> > Paul Erlich wrote:
> >
> > > Wow. How did you find that?
> >
> > Briefly (use the Reply thing so that indentation works),
>
> > 22876792454961:19073486328125
>
> So it was a finite search? How do you know you won't keep finding
> worse and worse examples if you go farther out? You might be
> approaching a limit, but how do you know you'll ever reach it?
>
> > >>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> > >
> > >
> > > Shall I proceed to calculate Tenney-weighted errors for all
> (well, a
> > > bunch of) intervals? I hope you're onto something!
> >
> > If you like.
>
> OK, later -- gotta go perform now.

I'm back . . . Looks like you might be off in the last digit or two
(so maybe there is no worst comma?), but a lot of the Tenney-weighted
errors are in the 3.5549 - 3.5591 range, so you're probably pretty
close . . .

10 9 199.61 17.208 2.6508
9 8 199.61 4.298 0.69661
6 5 299.42 16.223 3.3062
5 4 399.22 12.91 2.9872
4 3 499.03 0.985 0.27476
3 2 698.64 3.313 1.2816
8 5 798.45 15.238 2.8633
5 3 898.25 13.895 3.5566
9 5 998.06 19.536 3.5573
2 1 1197.7 2.328 2.328
9 4 1397.3 6.626 1.2816
12 5 1497.1 18.551 3.1406
5 2 1596.9 10.582 3.1856
8 3 1696.7 1.343 0.29291
3 1 1896.3 5.641 3.5591
16 5 1996.1 17.566 2.7786
10 3 2095.9 11.567 2.3574
18 5 2195.7 21.864 3.368
15 4 2295.5 7.2693 1.2306
4 1 2395.3 4.656 2.328
9 2 2595 8.954 2.1473
5 1 2794.6 8.2543 3.5549
16 3 2894.4 3.671 0.6573
6 1 3094 7.969 3.0828
25 4 3193.8 21.165 3.1856
20 3 3293.6 9.2393 1.5642
15 2 3493.2 4.9413 1.007
8 1 3593 6.984 2.328
25 3 3692.8 22.15 3.556
9 1 3792.6 11.282 3.5591
10 1 3992.2 5.9263 1.784
32 3 4092 5.999 0.91101
12 1 4291.7 10.297 2.8723
25 2 4391.5 18.837 3.3375
27 2 4491.3 14.595 2.5361
15 1 4690.9 2.6133 0.66889
16 1 4790.7 9.312 2.328
18 1 4990.3 13.61 3.2638
20 1 5189.9 3.5983 0.83257
45 2 5389.5 0.69972 0.10778
24 1 5489.3 12.625 2.7536
25 1 5589.1 16.509 3.5549
27 1 5688.9 16.923 3.5591
30 1 5888.6 0.28529 0.05814
32 1 5988.4 11.64 2.328
36 1 6188 15.938 3.0828
40 1 6387.6 1.2703 0.23869
45 1 6587.2 3.0277 0.55131
48 1 6687 14.953 2.6774
50 1 6786.8 14.181 2.5126
54 1 6886.6 19.251 3.3452
60 1 7086.2 2.0427 0.34582
64 1 7186 13.968 2.328
72 1 7385.6 18.266 2.9605
75 1 7485.4 10.868 1.7447
80 1 7585.3 1.0577 0.16731
81 1 7585.3 22.564 3.5591
90 1 7784.9 5.3557 0.82499
96 1 7884.7 17.281 2.6243
100 1 7984.5 11.853 1.784

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

For an ET, just stretch so that the weighted errors of the most
upward-biased prime and most downward-biased prime are equal in
magnitude and opposite in sign. For 12-equal I take the mapping
[12 19 28]
divide (elementwise) by
[1 log2(3) log2(5)]
and get
[12.00000000000000 11.98766531785769 12.05894362605501]
Now we want to make the largest and smallest of these equidistant
from 12, so we divide [12 19 28] by their average
[12.05894362605501+11.98766531785769 ]/2
giving
0.99806172487683 1.58026439772164 2.32881069137926
So Graham had the all the digits right, I just needed more precision.
Multiply by 12, and we get
1197.67406985219 1896.31727726597 2794.57282965511
Here it's clear we're hitting the maximum, 3.557, with both 3 and 5.

10 9 199.61 17.209 2.6508
9 8 199.61 4.2977 0.69655
6 5 299.42 16.223 3.3061
5 4 399.22 12.911 2.9873
4 3 499.03 0.98586 0.275
3 2 698.64 3.3118 1.2812
8 5 798.45 15.237 2.863
5 3 898.26 13.897 3.557
9 5 998.06 19.535 3.557
2 1 1197.7 2.3259 2.3259
9 4 1397.3 6.6236 1.2812
12 5 1497.1 18.549 3.1402
5 2 1596.9 10.585 3.1864
8 3 1696.7 1.3401 0.29227
3 1 1896.3 5.6377 3.557
16 5 1996.1 17.563 2.7781
10 3 2095.9 11.571 2.3581
18 5 2195.7 21.86 3.3674
15 4 2295.5 7.2733 1.2313
4 1 2395.3 4.6519 2.3259
9 2 2595 8.9495 2.1462
5 1 2794.6 8.2591 3.557
16 3 2894.4 3.666 0.6564
6 1 3094 7.9637 3.0808
25 4 3193.8 21.17 3.1864
20 3 3293.6 9.245 1.5651
15 2 3493.2 4.9473 1.0082
8 1 3593 6.9778 2.3259
25 3 3692.8 22.156 3.557
9 1 3792.6 11.275 3.557
10 1 3992.2 5.9332 1.7861
32 3 4092.1 5.9919 0.90994
12 1 4291.7 10.29 2.8702
25 2 4391.5 18.844 3.3389
27 2 4491.3 14.587 2.5348
15 1 4690.9 2.6214 0.67097
16 1 4790.7 9.3037 2.3259
18 1 4990.3 13.601 3.2618
20 1 5189.9 3.6073 0.83464
45 2 5389.5 0.6904 0.10635
24 1 5489.3 12.616 2.7515
25 1 5589.1 16.518 3.557
27 1 5689 16.913 3.557
30 1 5888.6 0.29546 0.060214
32 1 5988.4 11.63 2.3259
36 1 6188 15.927 3.0808
40 1 6387.6 1.2813 0.24076
45 1 6587.2 3.0163 0.54924
48 1 6687 14.941 2.6753
50 1 6786.8 14.192 2.5146
54 1 6886.6 19.239 3.3431
60 1 7086.2 2.0305 0.34375
64 1 7186 13.956 2.3259
72 1 7385.7 18.253 2.9584
75 1 7485.5 10.881 1.7468
80 1 7585.3 1.0446 0.16524
81 1 7585.3 22.551 3.557
90 1 7784.9 5.3423 0.82292
96 1 7884.7 17.267 2.6222
100 1 7984.5 11.866 1.7861

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Gene, what do you get for the top system with the commas of 12-
equal
> (in other words, some stretching or squashing of 12-equal)? Graham
> seems to gave gotten pretty close below, but no cigar . . .
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
> wrote:
> > > Paul Erlich wrote:
> > >
> > > > Wow. How did you find that?
> > >
> > > Briefly (use the Reply thing so that indentation works),
> >
> > > 22876792454961:19073486328125
> >
> > So it was a finite search? How do you know you won't keep finding
> > worse and worse examples if you go farther out? You might be
> > approaching a limit, but how do you know you'll ever reach it?
> >
> > > >>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> > > >
> > > >
> > > > Shall I proceed to calculate Tenney-weighted errors for all
> > (well, a
> > > > bunch of) intervals? I hope you're onto something!
> > >
> > > If you like.
> >
> > OK, later -- gotta go perform now.
>
> I'm back . . . Looks like you might be off in the last digit or two
> (so maybe there is no worst comma?), but a lot of the Tenney-
weighted
> errors are in the 3.5549 - 3.5591 range, so you're probably pretty
> close . . .
>
> 10 9 199.61 17.208 2.6508
> 9 8 199.61 4.298 0.69661
> 6 5 299.42 16.223 3.3062
> 5 4 399.22 12.91 2.9872
> 4 3 499.03 0.985 0.27476
> 3 2 698.64 3.313 1.2816
> 8 5 798.45 15.238 2.8633
> 5 3 898.25 13.895 3.5566
> 9 5 998.06 19.536 3.5573
> 2 1 1197.7 2.328 2.328
> 9 4 1397.3 6.626 1.2816
> 12 5 1497.1 18.551 3.1406
> 5 2 1596.9 10.582 3.1856
> 8 3 1696.7 1.343 0.29291
> 3 1 1896.3 5.641 3.5591
> 16 5 1996.1 17.566 2.7786
> 10 3 2095.9 11.567 2.3574
> 18 5 2195.7 21.864 3.368
> 15 4 2295.5 7.2693 1.2306
> 4 1 2395.3 4.656 2.328
> 9 2 2595 8.954 2.1473
> 5 1 2794.6 8.2543 3.5549
> 16 3 2894.4 3.671 0.6573
> 6 1 3094 7.969 3.0828
> 25 4 3193.8 21.165 3.1856
> 20 3 3293.6 9.2393 1.5642
> 15 2 3493.2 4.9413 1.007
> 8 1 3593 6.984 2.328
> 25 3 3692.8 22.15 3.556
> 9 1 3792.6 11.282 3.5591
> 10 1 3992.2 5.9263 1.784
> 32 3 4092 5.999 0.91101
> 12 1 4291.7 10.297 2.8723
> 25 2 4391.5 18.837 3.3375
> 27 2 4491.3 14.595 2.5361
> 15 1 4690.9 2.6133 0.66889
> 16 1 4790.7 9.312 2.328
> 18 1 4990.3 13.61 3.2638
> 20 1 5189.9 3.5983 0.83257
> 45 2 5389.5 0.69972 0.10778
> 24 1 5489.3 12.625 2.7536
> 25 1 5589.1 16.509 3.5549
> 27 1 5688.9 16.923 3.5591
> 30 1 5888.6 0.28529 0.05814
> 32 1 5988.4 11.64 2.328
> 36 1 6188 15.938 3.0828
> 40 1 6387.6 1.2703 0.23869
> 45 1 6587.2 3.0277 0.55131
> 48 1 6687 14.953 2.6774
> 50 1 6786.8 14.181 2.5126
> 54 1 6886.6 19.251 3.3452
> 60 1 7086.2 2.0427 0.34582
> 64 1 7186 13.968 2.328
> 72 1 7385.6 18.266 2.9605
> 75 1 7485.4 10.868 1.7447
> 80 1 7585.3 1.0577 0.16731
> 81 1 7585.3 22.564 3.5591
> 90 1 7784.9 5.3557 0.82499
> 96 1 7884.7 17.281 2.6243
> 100 1 7984.5 11.853 1.784

🔗Paul Erlich <perlich@aya.yale.edu>

So the stretch factor is 24/(19/log2(3) + 28/log2(5)). This looks
related to the 'dual' of the comma Graham found, but I didn't have to
go looking for it . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> For an ET, just stretch so that the weighted errors of the most
> upward-biased prime and most downward-biased prime are equal in
> magnitude and opposite in sign. For 12-equal I take the mapping
> [12 19 28]
> divide (elementwise) by
> [1 log2(3) log2(5)]
> and get
> [12.00000000000000 11.98766531785769 12.05894362605501]
> Now we want to make the largest and smallest of these equidistant
> from 12, so we divide [12 19 28] by their average
> [12.05894362605501+11.98766531785769 ]/2
> giving
> 0.99806172487683 1.58026439772164 2.32881069137926
> So Graham had the all the digits right, I just needed more
precision.
> Multiply by 12, and we get
> 1197.67406985219 1896.31727726597 2794.57282965511
> Here it's clear we're hitting the maximum, 3.557, with both 3 and 5.
>
>
> 10 9 199.61 17.209 2.6508
> 9 8 199.61 4.2977 0.69655
> 6 5 299.42 16.223 3.3061
> 5 4 399.22 12.911 2.9873
> 4 3 499.03 0.98586 0.275
> 3 2 698.64 3.3118 1.2812
> 8 5 798.45 15.237 2.863
> 5 3 898.26 13.897 3.557
> 9 5 998.06 19.535 3.557
> 2 1 1197.7 2.3259 2.3259
> 9 4 1397.3 6.6236 1.2812
> 12 5 1497.1 18.549 3.1402
> 5 2 1596.9 10.585 3.1864
> 8 3 1696.7 1.3401 0.29227
> 3 1 1896.3 5.6377 3.557
> 16 5 1996.1 17.563 2.7781
> 10 3 2095.9 11.571 2.3581
> 18 5 2195.7 21.86 3.3674
> 15 4 2295.5 7.2733 1.2313
> 4 1 2395.3 4.6519 2.3259
> 9 2 2595 8.9495 2.1462
> 5 1 2794.6 8.2591 3.557
> 16 3 2894.4 3.666 0.6564
> 6 1 3094 7.9637 3.0808
> 25 4 3193.8 21.17 3.1864
> 20 3 3293.6 9.245 1.5651
> 15 2 3493.2 4.9473 1.0082
> 8 1 3593 6.9778 2.3259
> 25 3 3692.8 22.156 3.557
> 9 1 3792.6 11.275 3.557
> 10 1 3992.2 5.9332 1.7861
> 32 3 4092.1 5.9919 0.90994
> 12 1 4291.7 10.29 2.8702
> 25 2 4391.5 18.844 3.3389
> 27 2 4491.3 14.587 2.5348
> 15 1 4690.9 2.6214 0.67097
> 16 1 4790.7 9.3037 2.3259
> 18 1 4990.3 13.601 3.2618
> 20 1 5189.9 3.6073 0.83464
> 45 2 5389.5 0.6904 0.10635
> 24 1 5489.3 12.616 2.7515
> 25 1 5589.1 16.518 3.557
> 27 1 5689 16.913 3.557
> 30 1 5888.6 0.29546 0.060214
> 32 1 5988.4 11.63 2.3259
> 36 1 6188 15.927 3.0808
> 40 1 6387.6 1.2813 0.24076
> 45 1 6587.2 3.0163 0.54924
> 48 1 6687 14.941 2.6753
> 50 1 6786.8 14.192 2.5146
> 54 1 6886.6 19.239 3.3431
> 60 1 7086.2 2.0305 0.34375
> 64 1 7186 13.956 2.3259
> 72 1 7385.7 18.253 2.9584
> 75 1 7485.5 10.881 1.7468
> 80 1 7585.3 1.0446 0.16524
> 81 1 7585.3 22.551 3.557
> 90 1 7784.9 5.3423 0.82292
> 96 1 7884.7 17.267 2.6222
> 100 1 7984.5 11.866 1.7861
>
>
>
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > Gene, what do you get for the top system with the commas of 12-
> equal
> > (in other words, some stretching or squashing of 12-equal)?
Graham
> > seems to gave gotten pretty close below, but no cigar . . .
> >
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
> > wrote:
> > > > Paul Erlich wrote:
> > > >
> > > > > Wow. How did you find that?
> > > >
> > > > Briefly (use the Reply thing so that indentation works),
> > >
> > > > 22876792454961:19073486328125
> > >
> > > So it was a finite search? How do you know you won't keep
finding
> > > worse and worse examples if you go farther out? You might be
> > > approaching a limit, but how do you know you'll ever reach it?
> > >
> > > > >>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> > > > >
> > > > >
> > > > > Shall I proceed to calculate Tenney-weighted errors for all
> > > (well, a
> > > > > bunch of) intervals? I hope you're onto something!
> > > >
> > > > If you like.
> > >
> > > OK, later -- gotta go perform now.
> >
> > I'm back . . . Looks like you might be off in the last digit or
two
> > (so maybe there is no worst comma?), but a lot of the Tenney-
> weighted
> > errors are in the 3.5549 - 3.5591 range, so you're probably
pretty
> > close . . .
> >
> > 10 9 199.61 17.208 2.6508
> > 9 8 199.61 4.298 0.69661
> > 6 5 299.42 16.223 3.3062
> > 5 4 399.22 12.91 2.9872
> > 4 3 499.03 0.985 0.27476
> > 3 2 698.64 3.313 1.2816
> > 8 5 798.45 15.238 2.8633
> > 5 3 898.25 13.895 3.5566
> > 9 5 998.06 19.536 3.5573
> > 2 1 1197.7 2.328 2.328
> > 9 4 1397.3 6.626 1.2816
> > 12 5 1497.1 18.551 3.1406
> > 5 2 1596.9 10.582 3.1856
> > 8 3 1696.7 1.343 0.29291
> > 3 1 1896.3 5.641 3.5591
> > 16 5 1996.1 17.566 2.7786
> > 10 3 2095.9 11.567 2.3574
> > 18 5 2195.7 21.864 3.368
> > 15 4 2295.5 7.2693 1.2306
> > 4 1 2395.3 4.656 2.328
> > 9 2 2595 8.954 2.1473
> > 5 1 2794.6 8.2543 3.5549
> > 16 3 2894.4 3.671 0.6573
> > 6 1 3094 7.969 3.0828
> > 25 4 3193.8 21.165 3.1856
> > 20 3 3293.6 9.2393 1.5642
> > 15 2 3493.2 4.9413 1.007
> > 8 1 3593 6.984 2.328
> > 25 3 3692.8 22.15 3.556
> > 9 1 3792.6 11.282 3.5591
> > 10 1 3992.2 5.9263 1.784
> > 32 3 4092 5.999 0.91101
> > 12 1 4291.7 10.297 2.8723
> > 25 2 4391.5 18.837 3.3375
> > 27 2 4491.3 14.595 2.5361
> > 15 1 4690.9 2.6133 0.66889
> > 16 1 4790.7 9.312 2.328
> > 18 1 4990.3 13.61 3.2638
> > 20 1 5189.9 3.5983 0.83257
> > 45 2 5389.5 0.69972 0.10778
> > 24 1 5489.3 12.625 2.7536
> > 25 1 5589.1 16.509 3.5549
> > 27 1 5688.9 16.923 3.5591
> > 30 1 5888.6 0.28529 0.05814
> > 32 1 5988.4 11.64 2.328
> > 36 1 6188 15.938 3.0828
> > 40 1 6387.6 1.2703 0.23869
> > 45 1 6587.2 3.0277 0.55131
> > 48 1 6687 14.953 2.6774
> > 50 1 6786.8 14.181 2.5126
> > 54 1 6886.6 19.251 3.3452
> > 60 1 7086.2 2.0427 0.34582
> > 64 1 7186 13.968 2.328
> > 72 1 7385.6 18.266 2.9605
> > 75 1 7485.4 10.868 1.7447
> > 80 1 7585.3 1.0577 0.16731
> > 81 1 7585.3 22.564 3.5591
> > 90 1 7784.9 5.3557 0.82499
> > 96 1 7884.7 17.281 2.6243
> > 100 1 7984.5 11.853 1.784

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> right, right . . . just wanted to do a cross-check between you and
> graham . . . so what's the formula for top linear in 7-limit (for
> pajara and/or in general)?

I'm not using a formula, I'm finding points in a subpace of the val
space closest to JIP. This may give a simpilical region, in which
case we can find the centroid by taking the average of the points
(remember, we are in a vector space.) Another way to look at it is
that we could replace the Tenney norm with this:

|| |e2 e3 ... ep> ||_a =
(|e2|^a + |e3 log2(3)|^a ... + |ep log2(p)|^a)^(1/a)

If 1/a+1/b=1, the dual space for that norm would be

|| <f2 f3 ... fp> || =
(|f2|^b + |f3/log2(3)|^b + ... + |fp log2(p)}^b}^(1/b)

As a-->1, b--> infinity, and the point we want would be the limit of
this. I wouldn't recommend it as a computation method.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>For an ET, just stretch so that the weighted errors of the most
>upward-biased prime and most downward-biased prime are equal in
>magnitude and opposite in sign. For 12-equal I take the mapping
>[12 19 28]
>divide (elementwise) by
>[1 log2(3) log2(5)]
>and get
>[12.00000000000000 11.98766531785769 12.05894362605501]
>Now we want to make the largest and smallest of these equidistant
>from 12, so we divide [12 19 28] by their average
>[12.05894362605501+11.98766531785769 ]/2
>giving
>0.99806172487683 1.58026439772164 2.32881069137926

Awesome!

>So Graham had the all the digits right, I just needed more precision.
>Multiply by 12, and we get
>1197.67406985219 1896.31727726597 2794.57282965511
>Here it's clear we're hitting the maximum, 3.557, with both 3 and 5.
>
>
> 10 9 199.61 17.209 2.6508
> 9 8 199.61 4.2977 0.69655
> 6 5 299.42 16.223 3.3061
> 5 4 399.22 12.911 2.9873
> 4 3 499.03 0.98586 0.275
> 3 2 698.64 3.3118 1.2812
> 8 5 798.45 15.237 2.863
> 5 3 898.26 13.897 3.557
> 9 5 998.06 19.535 3.557
> 2 1 1197.7 2.3259 2.3259
> 9 4 1397.3 6.6236 1.2812
> 12 5 1497.1 18.549 3.1402
> 5 2 1596.9 10.585 3.1864
> 8 3 1696.7 1.3401 0.29227
> 3 1 1896.3 5.6377 3.557
> 16 5 1996.1 17.563 2.7781
> 10 3 2095.9 11.571 2.3581
> 18 5 2195.7 21.86 3.3674
> 15 4 2295.5 7.2733 1.2313
> 4 1 2395.3 4.6519 2.3259
> 9 2 2595 8.9495 2.1462
> 5 1 2794.6 8.2591 3.557
> 16 3 2894.4 3.666 0.6564
> 6 1 3094 7.9637 3.0808
> 25 4 3193.8 21.17 3.1864
> 20 3 3293.6 9.245 1.5651
> 15 2 3493.2 4.9473 1.0082
> 8 1 3593 6.9778 2.3259
> 25 3 3692.8 22.156 3.557
> 9 1 3792.6 11.275 3.557
> 10 1 3992.2 5.9332 1.7861
> 32 3 4092.1 5.9919 0.90994
> 12 1 4291.7 10.29 2.8702
> 25 2 4391.5 18.844 3.3389
> 27 2 4491.3 14.587 2.5348
> 15 1 4690.9 2.6214 0.67097
> 16 1 4790.7 9.3037 2.3259
> 18 1 4990.3 13.601 3.2618
> 20 1 5189.9 3.6073 0.83464
> 45 2 5389.5 0.6904 0.10635
> 24 1 5489.3 12.616 2.7515
> 25 1 5589.1 16.518 3.557
> 27 1 5689 16.913 3.557
> 30 1 5888.6 0.29546 0.060214
> 32 1 5988.4 11.63 2.3259
> 36 1 6188 15.927 3.0808
> 40 1 6387.6 1.2813 0.24076
> 45 1 6587.2 3.0163 0.54924
> 48 1 6687 14.941 2.6753
> 50 1 6786.8 14.192 2.5146
> 54 1 6886.6 19.239 3.3431
> 60 1 7086.2 2.0305 0.34375
> 64 1 7186 13.956 2.3259
> 72 1 7385.7 18.253 2.9584
> 75 1 7485.5 10.881 1.7468
> 80 1 7585.3 1.0446 0.16524
> 81 1 7585.3 22.551 3.557
> 90 1 7784.9 5.3423 0.82292
> 96 1 7884.7 17.267 2.6222
> 100 1 7984.5 11.866 1.7861

The alaska tunings are essentially circulating versions of this
tuning.

-Carl

🔗Carl Lumma <ekin@lumma.org>

>>So Graham had the all the digits right, I just needed more precision.
>>Multiply by 12, and we get
>>1197.67406985219 1896.31727726597 2794.57282965511
>>Here it's clear we're hitting the maximum, 3.557, with both 3 and 5.
>>
>>
>> 10 9 199.61 17.209 2.6508
>> 9 8 199.61 4.2977 0.69655
>> 6 5 299.42 16.223 3.3061
>> 5 4 399.22 12.911 2.9873
>> 4 3 499.03 0.98586 0.275
>> 3 2 698.64 3.3118 1.2812
>> 8 5 798.45 15.237 2.863
>> 5 3 898.26 13.897 3.557
>> 9 5 998.06 19.535 3.557
>> 2 1 1197.7 2.3259 2.3259
>> 9 4 1397.3 6.6236 1.2812
>> 12 5 1497.1 18.549 3.1402
>> 5 2 1596.9 10.585 3.1864
>> 8 3 1696.7 1.3401 0.29227
>> 3 1 1896.3 5.6377 3.557
>> 16 5 1996.1 17.563 2.7781
>> 10 3 2095.9 11.571 2.3581
>> 18 5 2195.7 21.86 3.3674
>> 15 4 2295.5 7.2733 1.2313
>> 4 1 2395.3 4.6519 2.3259
>> 9 2 2595 8.9495 2.1462
>> 5 1 2794.6 8.2591 3.557
>> 16 3 2894.4 3.666 0.6564
>> 6 1 3094 7.9637 3.0808
>> 25 4 3193.8 21.17 3.1864
>> 20 3 3293.6 9.245 1.5651
>> 15 2 3493.2 4.9473 1.0082
>> 8 1 3593 6.9778 2.3259
>> 25 3 3692.8 22.156 3.557
>> 9 1 3792.6 11.275 3.557
>> 10 1 3992.2 5.9332 1.7861
>> 32 3 4092.1 5.9919 0.90994
>> 12 1 4291.7 10.29 2.8702
>> 25 2 4391.5 18.844 3.3389
>> 27 2 4491.3 14.587 2.5348
>> 15 1 4690.9 2.6214 0.67097
>> 16 1 4790.7 9.3037 2.3259
>> 18 1 4990.3 13.601 3.2618
>> 20 1 5189.9 3.6073 0.83464
>> 45 2 5389.5 0.6904 0.10635
>> 24 1 5489.3 12.616 2.7515
>> 25 1 5589.1 16.518 3.557
>> 27 1 5689 16.913 3.557
>> 30 1 5888.6 0.29546 0.060214
>> 32 1 5988.4 11.63 2.3259
>> 36 1 6188 15.927 3.0808
>> 40 1 6387.6 1.2813 0.24076
>> 45 1 6587.2 3.0163 0.54924
>> 48 1 6687 14.941 2.6753
>> 50 1 6786.8 14.192 2.5146
>> 54 1 6886.6 19.239 3.3431
>> 60 1 7086.2 2.0305 0.34375
>> 64 1 7186 13.956 2.3259
>> 72 1 7385.7 18.253 2.9584
>> 75 1 7485.5 10.881 1.7468
>> 80 1 7585.3 1.0446 0.16524
>> 81 1 7585.3 22.551 3.557
>> 90 1 7784.9 5.3423 0.82292
>> 96 1 7884.7 17.267 2.6222
>> 100 1 7984.5 11.866 1.7861
>
>The alaska tunings are essentially circulating versions of this
>tuning.

Which was based on...

! zeta12.scl
!
12 equal zeta tuning
12
!
99.807
199.614
299.422
399.229
499.036
598.843
698.650
798.457
898.265
998.072
1097.879
1197.686

...Notice the similarity...

-Carl