Yahoo Tuning Groups Ultimate Backup tuning-math Kees tuning

I mentioned on tuning that it might be interesting to look at the Kees
metric in the same way as Tenney. If you do that, you can get a
definition of optimal tuning which is analogous to TOP tuning, one
feature of which is that octaves are pure. What you need to do is
minimize the tuning distance to the Kees version of the JI point ("JIP")
now using

p3|q3 - p3| + p5|q5 - p5| + ... + pn|qn - pn|

as the thing to be minimized, where pm is the log base whatever of the
odd prime m, and q3, q5 etc are appromimations, possible in the given
tempering, to p3, p5 etc.

For example, for 5-limit meantone we would want to minimize

cents(3)|x - cents(3/2)| + cents(5)|4x - cents(5)|,

with the result that x is approximately 696.5784. This would be the
Kees 5-limit meantone. In this case it is the same as minimax
meantone, which is 1/4 comma.

🔗Paul Erlich <perlich@aya.yale.edu>

Isn't this the same thing Graham posted about like a year and a half
ago?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I mentioned on tuning that it might be interesting to look at the
Kees
> metric in the same way as Tenney. If you do that, you can get a
> definition of optimal tuning which is analogous to TOP tuning, one
> feature of which is that octaves are pure. What you need to do is
> minimize the tuning distance to the Kees version of the JI point
("JIP")
> now using
>
> p3|q3 - p3| + p5|q5 - p5| + ... + pn|qn - pn|
>
> as the thing to be minimized, where pm is the log base whatever of
the
> odd prime m, and q3, q5 etc are appromimations, possible in the
given
> tempering, to p3, p5 etc.
>
> For example, for 5-limit meantone we would want to minimize
>
> cents(3)|x - cents(3/2)| + cents(5)|4x - cents(5)|,
>
> with the result that x is approximately 696.5784. This would be the
> Kees 5-limit meantone. In this case it is the same as minimax
> meantone, which is 1/4 comma.

🔗Paul Erlich <perlich@aya.yale.edu>

P.S. If you look at this diagram from my 'Middle Path' paper:

/tuning-math/files/hex2.gif

you can 'see' the JIP in the middle, and the regular hexagonal
contours can be taken to represent maximum Kees error . . . right?
(Not a rhetorical question)

The diagram was actually derived from TOP tunings (and the points
were plotted by treating them as what you might call projective 5-
limit 'vals' in trilinear coordinates) but seems to be saying
something about Kees tunings as well, if you assume pure octaves
throughout . . .

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> p3|q3 - p3| + p5|q5 - p5| + ... + pn|qn - pn|
>
> as the thing to be minimized ...

Sorry, this should be

|q3 - p3|/p3 + |q3 - p5|/p5 + ... + |qn - pn|/pn

It weights smaller primes more than larger ones.

Fans of 72-et may be happy to learn that the 7-limit Kees tuning
of miracle is very close to 72. The 11-limit tuning is closer to 103,
between 72 and 103. For meantone, the 5 limit meantone is 1/4 comma,
and the 7-limit meantone rather flat, between 50 and 69. The meanpop
Kees tuning is close to 112, which shows it must be good for
something. The 31&43, or "cage" 11-limit meantone is very close to 43.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

At 01:51 PM 7/26/2005, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>> >P.S. If you look at this diagram from my 'Middle Path' paper:
>> >
>> >/tuning-math/files/hex2.gif
>>
>> Heya Paul,
>>
>> Did you scan this from a printout? Or compress it with jpeg
>> at some point? It looks a little blurry.
>>
>> -Carl
>
>Yeah, Matlab created this thing as a jpeg, and then I changed it to a
>gif. Matlab's jpegs are bigger and allow you to see more detail, but I
>didn't think it was worth all the KB (vs. a little blurriness) to leave
>it as a jpeg.

Can you save as gif (or bmp) directly from Matlab? As you've
discovered, gif is more appropriate for this type of image than jpeg.
Not only will the quality be better if you don't jpeg it first, the
size will be even lower because the gif compression is bad at encoding
the blurry jpeg artifacts.

A good rule for this, by the way, is the just level of sharpness in
the image. jpeg is DCT-based, and so hates sharp edges. gif is LZW-
based, and works great almost all text and diagram stuff (sometimes
not true if your text has been anti-aliased).

>Note the date when that file was posted.

Anything special about it?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Graham Breed <gbreed@gmail.com>

On 26 Jul 2005, at 21:40, Gene Ward Smith wrote:

> Graham, are you around?

Hiya! Yes, I read everything, but I only have an illicit dialup connection so I can't keep up with the back and forth.

I did do a TOP-analog for the Kees metric. What it comes down to is taking the larger odd number in the comma ratio and doing the TOP optimization on that number alone. For meantone, you take 81:80, throw away 80 because it's an even number and so let 3 take all the tempering, which of course gives quarter-comma meantone. That gives the same result as you, but without involving 5 at all, so there may be a difference somewhere...

A variation is that for commas with a ratio of two odd numbers (I think 245:243 was given as an example) you do the very same TOP optimization instead of taking the larger number. In this case, the result is always the same as TOP, but with the whole scale stretched so that octaves are pure again. I think I proved that for the case of a single comma, but I expect it's true in general. You could think about it.

The other thing I did was a weighted RMS optimization. This is so obvious it would probably be easier to work it out yourself than have me explain it. One advantage over TOP is that it should be blindingly fast to calculate (I haven't actually benchmarked it). It's necessarily octave specific.

The optimization routines are a bit less obvious, so here's the Python code.

Equal Temperaments:

def getPORMSWE(self):
"""Return the prime, optimum, RMS, weighted error.

This is the RMS of the prime intervals where octave stretching
is allowed, with each prime interval weighted according to its size.
"""
avgStretches, avgSquares = self.getPrimeStretching()
return sqrt(1.0 - (avgStretches**2 / avgSquares))

def getPORMSWEStep(self):
"""Return the stretched step size
for the prime, optimum, RMS, weighted error.
"""
avgStretches, avgSquares = self.getPrimeStretching()
return avgStretches / avgSquares

def getPrimeStretching(self):
"""Used by getPORMSWE() and getPORMSWEStretch().
Not likely to be much use on its own.
"""
sumStretches = sumSquares = 0.0
primes = [1.0] + self.primes
size = len(primes)
for i in range(size):
stretch = self.basis[i] / primes[i]
sumStretches = sumStretches + stretch
sumSquares = sumSquares + stretch**2
return sumStretches/size, sumSquares/size

Linear temperaments:

def optimizePORMSWE(self):
"""Set the prime, optimum, RMS, weighted errors optimum"""
sx0 = sx1 = sx02 = sx12 = sx01 = 0.0

primes = [1.0]+self.primes
for i in range(len(primes)):
m = self.mapping[i]
x0 = m[0]/primes[i]
x1 = m[1]/primes[i]
sx0 = sx0 + x0
sx1 = sx1 + x1
sx02 = sx02 + x0**2
sx12 = sx12 + x1**2
sx01 = sx01 + x0*x1
#dedent

denom = sx02*sx12 - sx01**2
self.basis = ((sx0*sx12 - sx1*sx01)/denom,
(sx1*sx02 - sx0*sx01)/denom)

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> At 01:51 PM 7/26/2005, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >P.S. If you look at this diagram from my 'Middle Path' paper:
> >> >
> >> >/tuning-math/files/hex2.gif
> >>
> >> Heya Paul,
> >>
> >> Did you scan this from a printout? Or compress it with jpeg
> >> at some point? It looks a little blurry.
> >>
> >> -Carl
> >
> >Yeah, Matlab created this thing as a jpeg, and then I changed it
to a
> >gif. Matlab's jpegs are bigger and allow you to see more detail,
but I
> >didn't think it was worth all the KB (vs. a little blurriness) to
leave
> >it as a jpeg.
>
> Can you save as gif (or bmp) directly from Matlab?

Yes, but they're smaller which is what I meant above.

> As you've
> discovered, gif is more appropriate for this type of image than
>jpeg.

But the smaller size obscured some of the details in this case.

> Not only will the quality be better if you don't jpeg it first, the
> size will be even lower because the gif compression is bad at
encoding
> the blurry jpeg artifacts.

I'm well aware of that but in this case it was the superior tradeoff.

> >Note the date when that file was posted.
>
> Anything special about it?

It was a long time ago.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> >
> > can 'see' the JIP in the middle, and the regular hexagonal
> > > contours can be taken to represent maximum Kees error . . . right?
> > > (Not a rhetorical question)
> >
> > I think for Kees tuning, plotting 5-limit vals, you'd want rhombus
> > contours--multiples of the rhombus with x intercepts log(3) and -log
(3),
> > and y intercepts log(5) and -log(5).
>
> Ah, I see what may be the problem. I think I screwed up; I took the
> dual of the wrong metric.

Kees's contours in tone-space are clearly hexagonal, as he shows. (If
you don't see it, just look at a big JI lattice and draw the contours
for various odd-limits.) So, if I learned anything about duality from
you, the dual contours in tuning-space would have to be hexagonal
too . . . and in particular the ones shown in my plot(?) . . .

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> I did do a TOP-analog for the Kees metric. What it comes down to is
> taking the larger odd number in the comma ratio and doing the TOP
> optimization on that number alone. For meantone, you take 81:80,
> throw away 80 because it's an even number and so let 3 take all the
> tempering, which of course gives quarter-comma meantone. That gives
> the same result as you, but without involving 5 at all, so there may
> be a difference somewhere...

I screwed up my computation of Kees tuning, but it seems to me the
correct Kees tuning would be 697.1383 cents. The reason for that is
that the error in 3/2, divided by log 3, equals the error in 6/5,
divided by log 5.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
wrote:
>
> > I did do a TOP-analog for the Kees metric. What it comes down to
is
> > taking the larger odd number in the comma ratio and doing the
TOP
> > optimization on that number alone. For meantone, you take
81:80,
> > throw away 80 because it's an even number and so let 3 take all
the
> > tempering, which of course gives quarter-comma meantone. That
gives
> > the same result as you, but without involving 5 at all, so there
may
> > be a difference somewhere...
>
> I screwed up my computation of Kees tuning, but it seems to me the
> correct Kees tuning would be 697.1383 cents.

Nope.

> The reason for that is
> that the error in 3/2, divided by log 3, equals the error in 6/5,
> divided by log 5.

I'll use natural logs.

'3/2' = 697.1383 cents, error = 4.8167 cents, 4.8167/log(3) = 4.3843

'6/5' = 308.59 cents, error = 7.0562 cents, 7.0562/log(5) = 4.3843

'9/5' = 1005.7 cents, error = 11.873 cents, 11.873/log(9) = 5.4036

So no, this isn't the (minimax) Kees meantone tuning.

🔗Paul Erlich <perlich@aya.yale.edu>

Graham, if he was indeed talking about the same thing (minimax "Kees
tuning"), claimed that:

If the corresponding TOP tuning has impure octaves, the "Kees" tuning
is simply a uniform stretch/compression of the TOP tuning so as to
acheive pure octaves.

If the corresponding TOP tuning has pure octaves, the "Kees" tuning
will actually be different! This is not too surprising, though, since
any TOP tuning with pure octaves remains Tenney-optimal, striclty
speaking, if the octaves are detuned slightly, as long as that's not
enough to raise the *maximum* damage over all the primes (and hence
over all ratios). So the original TOP tuning would have been somewhat
arbitrary in this case anyway.

I think Graham is right. And after he sees the correction below, I
think Gene can prove it.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
> wrote:
> >
> > > I did do a TOP-analog for the Kees metric. What it comes down
to
> is
> > > taking the larger odd number in the comma ratio and doing the
> TOP
> > > optimization on that number alone. For meantone, you take
> 81:80,
> > > throw away 80 because it's an even number and so let 3 take all
> the
> > > tempering, which of course gives quarter-comma meantone. That
> gives
> > > the same result as you, but without involving 5 at all, so
there
> may
> > > be a difference somewhere...
> >
> > I screwed up my computation of Kees tuning, but it seems to me the
> > correct Kees tuning would be 697.1383 cents.
>
> Nope.
>
> > The reason for that is
> > that the error in 3/2, divided by log 3, equals the error in 6/5,
> > divided by log 5.
>
> I'll use natural logs.
>
> '3/2' = 697.1383 cents, error = 4.8167 cents, 4.8167/log(3) = 4.3843
>
> '6/5' = 308.59 cents, error = 7.0562 cents, 7.0562/log(5) = 4.3843
>
> '9/5' = 1005.7 cents, error = 11.873 cents, 11.873/log(9) = 5.4036
>
> So no, this isn't the (minimax) Kees meantone tuning.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> I think Graham is right. And after he sees the correction below, I
> think Gene can prove it.

I think I see the problem. I can't compare the relative error of
3/5 to 3, I have to compare the relative error of the conceptual
comma 3^log(5)/5^log(3) to 3. This should make Monz happy, as it is a
|0 log(5) -log(3)> vector we are talking about. Of course, I can come
arbitraily close, using actual 5-limit intervals, by rounding off.

Anyway, the result of all that is 1/4 comma meantone, which happens to
be what you get by shrinking the TOP tuned octave down to to a just 2.
I think I'd better write up some code; I was on the right track, I
think, but had a couple of brain farts.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > I think Graham is right. And after he sees the correction below,
I
> > think Gene can prove it.
>
> If you take a ratio of odd integers p/q, then you can cook up a
> corresponding quasi-comma 2^log2(q/p) * p/q.

Hmm? 2^log2(q/p) = q/p, and (q/p) * p/q = 1. So I don't get this.

> Taking the Tenney height
> of this gives the square of the Kees expressibility, so the logs are
> proportional. Hence Kees and Tenney are closely related.
>
> If we take the closest approach to the JIP under the above
contraint,
> we should get the Kees tuning. I'll think about this some more.

What constraint?

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > If you take a ratio of odd integers p/q, then you can cook up a
> > > corresponding quasi-comma 2^log2(q/p) * p/q.
> >
> > Hmm? 2^log2(q/p) = q/p, and (q/p) * p/q = 1. So I don't get this.
>
> It's not a number, it lives in Monzo land; it's a vector.

Clearly you need to use different notation or something. You
said "Taking the Tenney height of this gives the square of the Kees
expressibility" (I think you meant the square of the exponential of the
Kees expressibility) -- so you need to give people some way of possibly
performing this calculation and seeing it for themselves. Right now
your notation is giving me, at least, no clue (and there isn't exactly
a chorus of other people following you).

🔗Paul Erlich <perlich@aya.yale.edu>

Note the unanswered question at the bottom.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > I think Graham is right. And after he sees the correction
below,
> I
> > > think Gene can prove it.
> >
> > If you take a ratio of odd integers p/q, then you can cook up a
> > corresponding quasi-comma 2^log2(q/p) * p/q.
>
> Hmm? 2^log2(q/p) = q/p, and (q/p) * p/q = 1. So I don't get this.
>
> > Taking the Tenney height
> > of this gives the square of the Kees expressibility, so the logs
are
> > proportional. Hence Kees and Tenney are closely related.
> >
> > If we take the closest approach to the JIP under the above
> contraint,
> > we should get the Kees tuning. I'll think about this some more.
>
> What constraint?

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Paul Erlich wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > I think Graham is right. And after he sees the correction below, I
> > > think Gene can prove it.
> >
> > I think I see the problem. I can't compare the relative error of
> > 3/5 to 3, I have to compare the relative error of the conceptual
> > comma 3^log(5)/5^log(3) to 3.
>
> I think I know what you mean (despite the unfortunate notation), but I
> think in general there are *three* different things you need to compare
> the error of, not just two. Thus the hexagons.

Well, Paul, I'm glad you're understanding what Gene means,
because although I'm trying to follow the thread, it's not
making much sense.

Gene, is there some other way you could notate this stuff
to make important distinctions - like that betwen vectors
and scalars - a little clearer?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
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🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Clearly you need to use different notation or something. You
> said "Taking the Tenney height of this gives the square of the Kees
> expressibility" (I think you meant the square of the exponential of the
> Kees expressibility) -- so you need to give people some way of possibly
> performing this calculation and seeing it for themselves. Right now
> your notation is giving me, at least, no clue (and there isn't exactly
> a chorus of other people following you).

It's easier if you take the idea and convert it into an alternative
formula for expressibility. If |e2 e3 ... ep> is a monzo, then take
|u2 u3 ... up> = |log(2)e2 log(3)e3 ... log(p)ep> where log is to any
fixed base. Then expressibility is

|| |u2 u3 ... up> || = (|u2|+|u3|+...+|u2+u3+...+up|)/2

Dividing by 2 takes care of the square business, but you don't really
need to bother if you start from this point of view.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> It's easier if you take the idea and convert it into an alternative
> formula for expressibility. If |e2 e3 ... ep> is a monzo, then take
> |u2 u3 ... up> = |log(2)e2 log(3)e3 ... log(p)ep> where log is to any
> fixed base. Then expressibility is
>
> || |u2 u3 ... up> || = (|u2|+|u3|+...+|u2+u3+...+up|)/2

Sorry, I put u2 in this formula, which is nonsense. It should be

|| |* u3 u5 ... up> || = (|u3|+|u5|+...+|u3+u5+...+up|)/2

The Kees metric being a metric on pitch *classes*, which gives us a
(non-Euclidean) Kees lattice of pitch classes in Kees space.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
>
> Paul Erlich wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> > >
> > > > I think Graham is right. And after he sees the correction
below, I
> > > > think Gene can prove it.
> > >
> > > I think I see the problem. I can't compare the relative error of
> > > 3/5 to 3, I have to compare the relative error of the conceptual
> > > comma 3^log(5)/5^log(3) to 3.
> >
> > I think I know what you mean (despite the unfortunate notation),
but I
> > think in general there are *three* different things you need to
compare
> > the error of, not just two. Thus the hexagons.
>
> Well, Paul, I'm glad you're understanding what Gene means,
> because although I'm trying to follow the thread, it's not
> making much sense.

Not to me either. But the one thing I think I did manage to
understand above was what Gene meant by 3^log(5)/5^log(3) above.
Taken at face value, this quantity equals 1. But I think the first 3
and second 5 are meant to be tempered, while the first 5 and second 3
are not. So I might write this as

"3"^log(5)/"5"^log(3)

If you take the log, which in practice you always do, it's

log(5)*log("3") - log(3)*log("5")

So this is a quantity which equals zero when there's no tempering,
and also equals zero if 3 is tempered in one direction and 5 is
tempered somewhat more in the *same* direction.

I still think this diagram:

/tuning-math/files/hex2.gif

shows contours of Kees error if you take the tunings to have pure
octaves, and it would be nice if Gene could relate what he's saying
to these hexagonal contours (or show that they're wrong). For
example, 22-equal is hitting a contour on the side of the hexagon
that corresponds to what Gene calls "the error of the conceptual
comma 3^log(5)/5^log(3)", since as my legend indicates, the size of
the hexagon 22-equal is hitting corresponds to a tempering of 5 that
is narrow *relative* to the tempering of 3. As a result, intervals
like 5/3 and 9/5 suffer more than other intervals in 22-equal.

> Gene, is there some other way you could notate this stuff
> to make important distinctions - like that betwen vectors
> and scalars - a little clearer?

I wholeheartedly agree with this sentiment, Yahya.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> If you take the log, which in practice you always do, it's
>
> log(5)*log("3") - log(3)*log("5")

And if you take the Kees norm, you get

|log(5)*log("3") + log(3)*log("5")|

> I still think this diagram:
>
> /tuning-math/files/hex2.gif
>
> shows contours of Kees error if you take the tunings to have pure
> octaves, and it would be nice if Gene could relate what he's saying
> to these hexagonal contours (or show that they're wrong).

Why don't you tell me what you are plotting? What are the verticies of
the hexagon?

For
> example, 22-equal is hitting a contour on the side of the hexagon
> that corresponds to what Gene calls "the error of the conceptual
> comma 3^log(5)/5^log(3)", since as my legend indicates, the size of
> the hexagon 22-equal is hitting corresponds to a tempering of 5 that
> is narrow *relative* to the tempering of 3.

Shouldn't that be hitting a vertex, not a side?

We have

35/p3 - 22 = 0.0825
51/p5 - 22 = -0.0355
35/p3 - 51/p5 = 0.1180

The third number is largest in absolute value, and so this corner of
the hexagon should be the one the 22-equal line hits.

> > Gene, is there some other way you could notate this stuff
> > to make important distinctions - like that betwen vectors
> > and scalars - a little clearer?
>
> I wholeheartedly agree with this sentiment, Yahya.

I was thinking of a companion web page to TOP and Tenney space, but
haven't gotten feedback on my question re existing web pages.

🔗Kees van Prooijen <lists@kees.cc>

Wow, amazing to find this huge thread with my name on it.

The only thing I can think of to help you Gene, is that I think I
wasn't to clear with the links from my page.
Did you see these?

Paul's observations:
http://www.kees.cc/tuning/erl_perbl.html
And my reaction to that (especially regarding the hexagons):
http://www.kees.cc/tuning/lat_perbl.html

Other than that, I'm afraid I'm totally out of it by now. I'm glad I
could always rely on Paul to be an excellent advocate for what it was
I was trying to 'express'.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > If you take the log, which in practice you always do, it's
> >
> > log(5)*log("3") - log(3)*log("5")
>
> And if you take the Kees norm, you get
>
> |log(5)*log("3") + log(3)*log("5")|
>
> > I still think this diagram:
> >
> > /tuning-math/files/hex2.gif
> >
> > shows contours of Kees error if you take the tunings to have pure
> > octaves, and it would be nice if Gene could relate what he's
saying
> > to these hexagonal contours (or show that they're wrong).
>
> Why don't you tell me what you are plotting? What are the verticies
of
> the hexagon?
>
> For
> > example, 22-equal is hitting a contour on the side of the hexagon
> > that corresponds to what Gene calls "the error of the conceptual
> > comma 3^log(5)/5^log(3)", since as my legend indicates, the size
of
> > the hexagon 22-equal is hitting corresponds to a tempering of 5
that
> > is narrow *relative* to the tempering of 3.
>
> Shouldn't that be hitting a vertex, not a side?
>
> We have
>
> 35/p3 - 22 = 0.0825
> 51/p5 - 22 = -0.0355
> 35/p3 - 51/p5 = 0.1180
>
> The third number is largest in absolute value, and so this corner of
> the hexagon should be the one the 22-equal line hits.
>
> > > Gene, is there some other way you could notate this stuff
> > > to make important distinctions - like that betwen vectors
> > > and scalars - a little clearer?
> >
> > I wholeheartedly agree with this sentiment, Yahya.
>
> I was thinking of a companion web page to TOP and Tenney space, but
> haven't gotten feedback on my question re existing web pages.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

I don't get it. Why would you do that?

> > I still think this diagram:
> >
> > /tuning-math/files/hex2.gif
> >
> > shows contours of Kees error if you take the tunings to have pure
> > octaves, and it would be nice if Gene could relate what he's
saying
> > to these hexagonal contours (or show that they're wrong).
>
> Why don't you tell me what you are plotting?

I'm plotting "vals" in a 2D projective space, using tricentric
coordinates. The three "axes" (at 120-degree angles from one another)
correspond to log("3"/"2"), log("2"/"5"), and log("5"/"3") -- these
always add up to zero for any tuning, thus there aren't three but
only two degrees of freedom, thus the 2D plot suffices for all 5-
limit (projective) vals.

> What are the verticies of
> the hexagon?

The hexagons were plotted by Matlab -- they're contours of TOP error,
where each projective val is to represent the TOP tuning that
corresponds to it. Note that they're regular hexagons.

> For
> > example, 22-equal is hitting a contour on the side of the hexagon
> > that corresponds to what Gene calls "the error of the conceptual
> > comma 3^log(5)/5^log(3)", since as my legend indicates, the size
of
> > the hexagon 22-equal is hitting corresponds to a tempering of 5
that
> > is narrow *relative* to the tempering of 3.
>
> Shouldn't that be hitting a vertex, not a side?

No, clearly it's not hitting a vertex, as you can see from the graph.

> We have
>
> 35/p3 - 22 = 0.0825
> 51/p5 - 22 = -0.0355
> 35/p3 - 51/p5 = 0.1180

I think I know what you mean, but can you explain this notation? I
really don't get this notation.

> The third number is largest in absolute value, and so this corner of
> the hexagon should be the one the 22-equal line hits.

22-equal is represented by a point, not a line. If two of the three
numbers were equal in magnitude, you'd then be on a vertex. Since
they're all different, you're not an a vertex, you're on a side (the
side corresponding to the third number on your list, I believe).

> > > Gene, is there some other way you could notate this stuff
> > > to make important distinctions - like that betwen vectors
> > > and scalars - a little clearer?
> >
> > I wholeheartedly agree with this sentiment, Yahya.
>
> I was thinking of a companion web page to TOP and Tenney space, but
> haven't gotten feedback on my question re existing web pages.

What question was that? I've found them largely impenetrable, but I'd
be happy to look again.

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Gene,

And thank you for this!

Yahya

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > If you take the log, which in practice you always do, it's
> >
> > log(5)*log("3") - log(3)*log("5")
>
> And if you take the Kees norm, you get
>
> |log(5)*log("3") + log(3)*log("5")|

OK ... are you happy with the notation "3" for "a tempered 3",
or can we do better? I suggested 3' to Paul in my previous
message.

... <snip!>

I'm sorry to say I've forgotten the question ... Please refresh
my memory if you think I might be able to contribute anything
useful.

Earlier this week, I refreshed my copies of your xenharmony
pages on theory, and plan to reread them again this weekend.
This may clear some cobwebs from my mind - or perhaps spawn
a spate of other questions :-( !

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Paul,

Thank you for this.

Yahya

Paul Erlich wrote:
> --- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> > Paul Erlich wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > > <gwsmith@s...> wrote:
> > > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
> > > > <perlich@a...> wrote:
> > > >
> > > > > I think Graham is right. And after he sees the correction
> > > > > below, I think Gene can prove it.
> > > >
> > > > I think I see the problem. I can't compare the relative error of
> > > > 3/5 to 3, I have to compare the relative error of the conceptual
> > > > comma 3^log(5)/5^log(3) to 3.
> > >
> > > I think I know what you mean (despite the unfortunate notation),
> > > but I think in general there are *three* different things you
> > > need to compare the error of, not just two. Thus the hexagons.
> >
> > Well, Paul, I'm glad you're understanding what Gene means,
> > because although I'm trying to follow the thread, it's not
> > making much sense.
>
> Not to me either. But the one thing I think I did manage to
> understand above was what Gene meant by 3^log(5)/5^log(3) above.
> Taken at face value, this quantity equals 1. But I think the first 3
> and second 5 are meant to be tempered, while the first 5 and second 3
> are not. So I might write this as
>
> "3"^log(5)/"5"^log(3)

Thank you! Much better! Writing your "tempered 3" as "3" is
one way to go about it; how about writing t(3), or even 3', instead?
Thus: 3'^log(5)/5'^log(3)

If our character set weren't so limited, we might write 3 bar, to
convey the notion of averaging.

> If you take the log, which in practice you always do, it's
>
> log(5)*log("3") - log(3)*log("5")

or: log(5)*log(3') - log(3)*log(5')

This has the form of the determinant of the matrix:
log(3') log(5')
log(3) log(5)

which suggests we're comparing vectors:
log(3') log(5')
and:
log(3) log(5).

Is this coincidental?

>
> So this is a quantity which equals zero when there's no tempering,
> and also equals zero if 3 is tempered in one direction and 5 is
> tempered somewhat more in the *same* direction.
>
> I still think this diagram:
>
> /tuning-math/files/hex2.gif
>
> shows contours of Kees error ...

Remind me what you mean by this; is it the determinant-like
expression above?

> ... if you take the tunings to have pure
> octaves, and it would be nice if Gene could relate what he's saying
> to these hexagonal contours (or show that they're wrong). For
> example, 22-equal is hitting a contour on the side of the hexagon
> that corresponds to what Gene calls "the error of the conceptual
> comma 3^log(5)/5^log(3)", since as my legend indicates, the size of
> the hexagon 22-equal is hitting corresponds to a tempering of 5 that
> is narrow *relative* to the tempering of 3. As a result, intervals
> like 5/3 and 9/5 suffer more than other intervals in 22-equal.
>
> > Gene, is there some other way you could notate this stuff
> > to make important distinctions - like that betwen vectors
> > and scalars - a little clearer?
>
> I wholeheartedly agree with this sentiment, Yahya.
>

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Kees van Prooijen wrote:
> Wow, amazing to find this huge thread with my name on it.

Kees,
Just shows you what happens when you think out loud ...! :-)

> The only thing I can think of to help you Gene, is that I think I
> wasn't to clear with the links from my page.
> Did you see these?
>
> Paul's observations:
> http://www.kees.cc/tuning/erl_perbl.html
> And my reaction to that (especially regarding the hexagons):
> http://www.kees.cc/tuning/lat_perbl.html

I never saw these, so will explore, in "Kees" they help reduce
my confusion.

> Other than that, I'm afraid I'm totally out of it by now. I'm glad I
> could always rely on Paul to be an excellent advocate for what it was
> I was trying to 'express'.

Always helps to have another thinker to clarify your thoughts.

Regards,
Yahya

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🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:

> > shows contours of Kees error ...
>
> Remind me what you mean by this; is it the determinant-like
> expression above?

No. What I mean by Kees error (hey, I didn't start calling it that) is
similar to TOP damage in my paper. The difference is that octave-
equivalence is assumed; instead of Tenney Harmonic Distance, we use
Kees van Prooijen 'expressibility', otherwise known as the log of the
smallest odd limit the ratio falls into. I gave some examples in a
recent post, where I showed that Gene's guess as to the specification
of the Kees meantone tuning was incorrect.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > No. What I mean by Kees error (hey, I didn't start calling it
that) is
> > similar to TOP damage in my paper.
>
> Is there any point in my finishing the web page I started on this?

I would think so, but I think it's a good idea to talk the stuff over
for a while. Why haven't you responded to any of the latest posts
here on this? I've been eagerly anticipating your responses!

Anyway, what's the link again?

> I'd
> like to know where and why my TOP page becomes incomprehensible.

I'd love to collaborate with you on something that might be
comprehensible to both musicians and mathematicians. But shouldn't we
first talk this stuff through enough to know that we're on the same
page first?

🔗Paul Erlich <perlich@aya.yale.edu>

Hi Gene,

> I'd
> like to know where and why my TOP page becomes incomprehensible.

Clicking your name on the home page of this group, in an attempt to
find this old page of yours, doesn't work. But when it reappears, I
think the best thing, if Yahya's willing, would be for Yahya, a
relative newcomer, to work with you on increasing its general
comprehensibility to 'outsiders'. The depth in which Yahya read
my 'Middle Path' paper, and the quantity and quality of his questions
and suggestions for clarification, was very impressive and simply
unmatched by any of the other dozens of readers. He helped me to
realize a whole lot of things that you and I (and a couple of others
around here) take for granted because we've been talking to each other
for so long, but that the rest of the world would have no clue about.
Yahya, you're the best. How about it?

🔗Carl Lumma <ekin@lumma.org>

>> I'd like to know where and why my TOP page becomes incomprehensible.
>
>Clicking your name on the home page of this group, in an attempt to
>find this old page of yours, doesn't work. But when it reappears, I
>think the best thing, if Yahya's willing, would be for Yahya, a
>relative newcomer, to work with you on increasing its general
>comprehensibility to 'outsiders'. The depth in which Yahya read
>my 'Middle Path' paper, and the quantity and quality of his questions
>and suggestions for clarification, was very impressive and simply
>unmatched by any of the other dozens of readers. He helped me to
>realize a whole lot of things that you and I (and a couple of others
>around here) take for granted because we've been talking to each other
>for so long, but that the rest of the world would have no clue about.
>Yahya, you're the best. How about it?

Gene, DNS from xenharmony.org is apparently not working.

But Paul, a Google search quickly reveals...

http://66.98.148.43/~xenharmo/

...from which "Theory" and "TOP and Tenney space" are obvious
choices.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Paul Erlich wrote:
> Hi Gene,
>
> > I'd
> > like to know where and why my TOP page becomes incomprehensible.
>
> Clicking your name on the home page of this group, in an attempt to
> find this old page of yours, doesn't work. But when it reappears, I
> think the best thing, if Yahya's willing, would be for Yahya, a
> relative newcomer, to work with you on increasing its general
> comprehensibility to 'outsiders'. The depth in which Yahya read
> my 'Middle Path' paper, and the quantity and quality of his questions
> and suggestions for clarification, was very impressive and simply
> unmatched by any of the other dozens of readers. He helped me to
> realize a whole lot of things that you and I (and a couple of others
> around here) take for granted because we've been talking to each other
> for so long, but that the rest of the world would have no clue about.
> Yahya, you're the best. How about it?

Paul,
Thank you kindly - flattery will get you everywhere! :-)

But I doubt Gene needs my help to express himself. I find his stuff
on xenharmony perfectly lucid. Of course, it does read a lot like a
mathematical text, but then I'm used to reading those ...

Still, putting the maths to musical use is a challenge. I don't think
Gene even attempts to deal with that level - all he does, in effect,
is to lay out some interesting facts for our inspection, which we may
use however we see fit.

Gene,
Please excuse me if I misread your intent on your published pages.
And if you do want to explicate the possible musical significance
of your maths, please feel free to call on me if you think I might
be of any service.

Regards,
Yahya

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🔗monz <monz@tonalsoft.com>

hi Yayha, Gene, and Paul,

I, for one, really want to have a good understanding
of Gene's tuning theory work. It's virtually incomprehensible
to me, yet i feel intuitively from what i can glean from
it that it's very important and useful. Yahya, i encourage
you to work with Gene on popularizing his work since you
get it.

-monz

--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
>
> Paul Erlich wrote:
> >
> > Yahya, you're the best. How about it?
>
>
> Paul,
> Thank you kindly - flattery will get you everywhere! :-)
>
> But I doubt Gene needs my help to express himself.
> I find his stuff on xenharmony perfectly lucid.
> Of course, it does read a lot like a mathematical
> text, but then I'm used to reading those ...
>
> Still, putting the maths to musical use is a challenge.
> I don't think Gene even attempts to deal with that level
> - all he does, in effect, is to lay out some interesting
> facts for our inspection, which we may use however we
> see fit.
>
>
> Gene,
> Please excuse me if I misread your intent on your
> published pages. And if you do want to explicate the
> possible musical significance of your maths, please
> feel free to call on me if you think I might
> be of any service.