simple comparison -- JI error in 72-tET

I was just wondering exactly how far off 72-tET actually *is* from
just intonation.

How exactly would that be calculated, again??

I can make a simple chart that would compare the Ptolemy Syntonic
Diatonic with 72-tET for simple intervals starting from 1/1:

INTERVAL JI 72-tET ERROR IN CENTS
9/8 204 200 -4
5/4 386 383 -3
4/3 498 500 +2
3/2 702 700 -2
5/3 884 883 -1
15/8 1088 1083 -5

Now if I am to do a simple average of the error for this, I get
-13/6 or -2 cents, correct??

However, this only takes in those specific cases where intervals are
starting on 1/1.

What is the mathematical method for finding the error between more
intervals in a more sophisticated way?

I would put this on the "math" list... but I'm sure that since *I'm*
asking it, it can't be all that complex! :)

_________ _________ ______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_24536.html#24536

> I was just wondering exactly how far off 72-tET actually *is* from
> just intonation.
>
> How exactly would that be calculated, again??
>
> I can make a simple chart that would compare the Ptolemy Syntonic
> Diatonic with 72-tET for simple intervals starting from 1/1:
>
> INTERVAL JI 72-tET ERROR IN CENTS
> 9/8 204 200 -4
> 5/4 386 383 -3
> 4/3 498 500 +2
> 3/2 702 700 -2
> 5/3 884 883 -1
> 15/8 1088 1083 -5
>
> Now if I am to do a simple average of the error for this, I get
> -13/6 or -2 cents, correct??
>
> However, this only takes in those specific cases where intervals
are starting on 1/1.
>

Whoopsie... I just realized since we are dealing with an ET, the
error is the same for any given interval across the range... (forgot
about that "nicety" for a moment)

But still, to find an "overall" error estimation for 72-tET, how many
intervals would we have to take??

Am I to assume, also, that with "blackjack," "canasta" or any of
the "subsets" of the overall 72-tET the SAME error pertains??

Or is that wrong, since there are a different total number of
intervals in each of these scales? It probably is wrong.

If so, then what is the average error in cents of 72-tET, and the
various "Miracle" families, "blackjack" and "canasta" in particular,
and how would that be calculated again??

_________ _______ ____
Joseph Pehrson

Joseph wrote:
> But still, to find an "overall" error estimation for 72-tET, how many
> intervals would we have to take??

You should read this article:
http://www.xs4all.nl/~huygensf/doc/realm.html

Fokker went to the 13th harmonic in his comparison of ETs.

> I was just wondering exactly how far off 72-tET actually *is* from
> just intonation.

There are different ways to evaluate this. Fokker defined his misfit
and deficiency measures. Mandelbaum created another combined error
factor. I myself have defined a relative error percentage.
You can find these measures at the beginning of the EQUAL/DATA
output. There's a Scala tip in tips.par that explains them.

Manuel

In-Reply-To: <9fqt6r+tsc6@eGroups.com>
Joseph Pehrson wrote:

> Whoopsie... I just realized since we are dealing with an ET, the
> error is the same for any given interval across the range... (forgot
> about that "nicety" for a moment)
>
> But still, to find an "overall" error estimation for 72-tET, how many
> intervals would we have to take??

Joe, let's take a step back here. 72-tET has no errors. It is what it
is. The errors only come in when you compare it to something else. So
it's up to *you* how many intervals you take.

For the case of tonality diamonds, though, you only need to take the
ratios with a power of two on the bottom, so long as you do the
calculation the right way. Let's take the 7-limit, because that's nice
and easy:

Ratio JI 72-tET error

1/1 0.0 0.0 0.0
3/2 702.0 700.0 -2.0
5/4 386.3 383.3 -3.0
7/4 968.8 966.7 -2.1

The worst 7-limit interval can be found by subtracting the lowest error
from the highest. So 0.0--3.0 = 3.0 cents. So 72-tET is accurate to
within 3 cents in the 7-limit. Remember to preserve the signs of the
errors for this calculation.

> Am I to assume, also, that with "blackjack," "canasta" or any of
> the "subsets" of the overall 72-tET the SAME error pertains??
>
> Or is that wrong, since there are a different total number of
> intervals in each of these scales? It probably is wrong.

A 7:4 is always 58 steps of 72 whether it's used in blackjack, canasta or
any other subset. So the error will always be the same. This happens to
be true for any particular Miracle temperament.

> If so, then what is the average error in cents of 72-tET, and the
> various "Miracle" families, "blackjack" and "canasta" in particular,
> and how would that be calculated again??

Averages are trickier, you do need to consider all intervals then. The
most popular is the root mean squared (RMS). So you take the errors in
all intervals, square them all, add them together and return the square
root. But I stay with the worst error, which tells me what I need to
know. The best Miracle temperament (near enough 72-tET) is accurate to
3.3 cents of 11-limit JI.

Graham

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:

/tuning/topicId_24536.html#24540

>
> Joseph wrote:
> > But still, to find an "overall" error estimation for 72-tET, how
many
> > intervals would we have to take??
>
> You should read this article:
> http://www.xs4all.nl/~huygensf/doc/realm.html
>
> Fokker went to the 13th harmonic in his comparison of ETs.
>
> > I was just wondering exactly how far off 72-tET actually *is* from
> > just intonation.
>
> There are different ways to evaluate this. Fokker defined his misfit
> and deficiency measures. Mandelbaum created another combined error
> factor. I myself have defined a relative error percentage.
> You can find these measures at the beginning of the EQUAL/DATA
> output. There's a Scala tip in tips.par that explains them.
>
> Manuel

Thanks, Manuel! I got that to work in Scala... I guess this was a
bit more complicated process, with different answers, than I had
imagined! I guess "blackjack" and "canasta" have the same combined
errors as 72-tET then?? They can't, obviously, be plugged in to
EQUAL/DATA since they are not equal scales...

_________ ________ _____
Joseph Pehrson

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <9fqt6r+tsc6@e...>

/tuning/topicId_24536.html#24541

Hi Graham!

This was an EXTREMELY helpful message!

> > For the case of tonality diamonds, though, you only need to take
the ratios with a power of two on the bottom, so long as you do the
> calculation the right way. Let's take the 7-limit, because that's
nice and easy:
>
> Ratio JI 72-tET error
>
> 1/1 0.0 0.0 0.0
> 3/2 702.0 700.0 -2.0
> 5/4 386.3 383.3 -3.0
> 7/4 968.8 966.7 -2.1
>
> The worst 7-limit interval can be found by subtracting the lowest
error from the highest. So 0.0--3.0 = 3.0 cents. So 72-tET is
accurate to within 3 cents in the 7-limit. Remember to preserve the
signs of the errors for this calculation.

This is patently obvious, now that you've shown it to me! Makes lots
of cents, too!

> > > If so, then what is the average error in cents of 72-tET, and
the various "Miracle" families, "blackjack" and "canasta" in
particular, and how would that be calculated again??
>
> Averages are trickier, you do need to consider all intervals then.

That's what I was afraid of!

>The most popular is the root mean squared (RMS). So you take the
errors in all intervals, square them all, add them together and
return the square root.

That's "mean" stuff.... I wonder if Paul has ever done this...

Thanks so much for the help!

Joe

_________ ________ ____
Joseph Pehrson

Joe, I hate to keep harping on the limitations of 72-tET (oh, maybe I
don't mind ;-> ), but the numbers you're generating show 72-tET in its
best light, "statically", if you will.

Where it would be much worse, with a full 16.67 cents "error", is when
you have to make an adjustment of a comma. Or accept drift. As we've
already discussed, though, that problem applies only to some types of
music!

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

/tuning/topicId_24536.html#24546

> Joe, I hate to keep harping on the limitations of 72-tET (oh, maybe
I don't mind ;-> ), but the numbers you're generating show 72-tET in
its best light, "statically", if you will.
>
> Where it would be much worse, with a full 16.67 cents "error", is
when you have to make an adjustment of a comma. Or accept drift. As
we've already discussed, though, that problem applies only to some
types of music!
>
> JdL

Thanks, John, for the update! I can see where that would happen in
certain kinds of progressions and "expectations."

It's obviously clear that your *OWN* methods of "adaptive just" are
far superior to 72-tET, since it's a kind of "moving target" so to
speak!

However, I don't believe your computer has to play music from a score
on a music stand, but can do it's calculations at its "leisure" so to
speak...

At least, that's what I *think* it's like. I suppose it could also
be some kind of robot that you have that brings your the newspapers
and your slippers (coffee, too) as well as plays various
instruments... but I don't believe that is the case...

_________ _______ _______
Joseph Pehrson

[I wrote:]
>>Joe, I hate to keep harping on the limitations of 72-tET (oh, maybe
>>I don't mind ;-> ), but the numbers you're generating show 72-tET in
>>its best light, "statically", if you will.

>>Where it would be much worse, with a full 16.67 cents "error", is
>>when you have to make an adjustment of a comma. Or accept drift. As
>>we've already discussed, though, that problem applies only to some
>>types of music!

[Joseph Pehrson:]
>Thanks, John, for the update! I can see where that would happen in
>certain kinds of progressions and "expectations."

>It's obviously clear that your *OWN* methods of "adaptive just" are
>far superior to 72-tET, since it's a kind of "moving target" so to
>speak!

>However, I don't believe your computer has to play music from a score
>on a music stand, but can do it's calculations at its "leisure" so to
>speak...

>At least, that's what I *think* it's like. I suppose it could also
>be some kind of robot that you have that brings your the newspapers
>and your slippers (coffee, too) as well as plays various
>instruments... but I don't believe that is the case...

Right on every count, Joe! But, that's why I favor "Reinhard" notation:
so that it would be possible to distribute out commas smoothly, and
also convey the idea to real musicians.

JdL

--- In tuning@y..., jpehrson@r... wrote:

> That's "mean" stuff.... I wonder if Paul has ever done this...

Of course, lots of times. And in according to my personal listenting
experience, the measures Graham referred to are more important than
the measures Manuel referred to. But even simpler is just looking at
the _maximum_ error -- the _worst_ "goof" that the ET makes.

For 3-limit consonances, 72-tET has a maximum error of 2 cents.
For 5-limit consonances, 72-tET has a maximum error of 3 cents.
For 7-limit consonances, 72-tET has a maximum error of 3 cents.
For 9-limit consonances, 72-tET has a maximum error of 4 cents.
For 11-limit consonances, 72-tET has a maximum error of 4 cents.
For 13-limit consonances, 72-tET has a maximum error of 7 cents.
For 15-limit consonances, 72-tET has a maximum error of 7 cents.
For 17-limit consonances, 72-tET has a maximum error of 7 cents.
72-tET is inconsistent for 19 and higher odd limits.

That's all you really need to know, in my opinion. Blackjack and
Canasta can be used for either 7-, 9-, or 11-limit harmony (though
there are no 11:10s in blackjack). If you're just using 7-limit
tetrads, like the ones you've color-coded on your keyboard, no
interval will ever be more than 3 cents off. And remember, it's the
_intervals_, not the _pitches_, that you should be comparing against
JI.

By comparison, 12-tET:

For 3-limit consonances, 12-tET has a maximum error of 2 cents.
For 5-limit consonances, 12-tET has a maximum error of 16 cents.
For 7-limit consonances, 12-tET has a maximum error of 33 cents.
For 9-limit consonances, 12-tET has a maximum error of 35 cents.
12-tET is inconsistent for 11 and higher odd limits.

19-tET:

For 3-limit consonances, 12-tET has a maximum error of 7 cents.
For 5-limit consonances, 12-tET has a maximum error of 7 cents.
For 7-limit consonances, 12-tET has a maximum error of 21 cents.
For 9-limit consonances, 12-tET has a maximum error of 21 cents.
19-tET is inconsistent for 11 and higher odd limits.

22-tET:

For 3-limit consonances, 22-tET has a maximum error of 7 cents.
For 5-limit consonances, 22-tET has a maximum error of 12 cents.
For 7-limit consonances, 22-tET has a maximum error of 17 cents.
For 9-limit consonances, 22-tET has a maximum error of 19 cents.
For 11-limit consonances, 22-tET has a maximum error of 20 cents.
22-tET is inconsistent for 13 and higher odd limits.

31-tET:

For 3-limit consonances, 31-tET has a maximum error of 5 cents.
For 5-limit consonances, 31-tET has a maximum error of 6 cents.
For 7-limit consonances, 31-tET has a maximum error of 6 cents.
For 9-limit consonances, 31-tET has a maximum error of 11 cents.
For 11-limit consonances, 31-tET has a maximum error of 11 cents.
31-tET is inconsistent for 13 and higher odd limits.

41-tET:

For 3-limit consonances, 41-tET has a maximum error of 0 cents.
For 5-limit consonances, 41-tET has a maximum error of 6 cents.
For 7-limit consonances, 41-tET has a maximum error of 6 cents.
For 9-limit consonances, 41-tET has a maximum error of 7 cents.
For 11-limit consonances, 41-tET has a maximum error of 11 cents.
For 13-limit consonances, 41-tET has a maximum error of 14 cents.
For 15-limit consonances, 41-tET has a maximum error of 14 cents.
41-tET is inconsistent for 17 and higher odd limits.

46-tET:

For 3-limit consonances, 46-tET has a maximum error of 2 cents.
For 5-limit consonances, 46-tET has a maximum error of 5 cents.
For 7-limit consonances, 46-tET has a maximum error of 9 cents.
For 9-limit consonances, 46-tET has a maximum error of 9 cents.
For 11-limit consonances, 46-tET has a maximum error of 9 cents.
For 13-limit consonances, 46-tET has a maximum error of 11 cents.
46-tET is inconsistent for 15 and higher odd limits.

53-tET:

For 3-limit consonances, 46-tET has a maximum error of 0 cents.
For 5-limit consonances, 46-tET has a maximum error of 1 cents.
For 7-limit consonances, 46-tET has a maximum error of 6 cents.
For 9-limit consonances, 46-tET has a maximum error of 6 cents.
46-tET is inconsistent for 11 and higher odd limits.

58-tET:

For 3-limit consonances, 58-tET has a maximum error of 1 cents.
For 5-limit consonances, 58-tET has a maximum error of 7 cents.
For 7-limit consonances, 58-tET has a maximum error of 7 cents.
For 9-limit consonances, 58-tET has a maximum error of 7 cents.
For 11-limit consonances, 58-tET has a maximum error of 7 cents.
For 13-limit consonances, 58-tET has a maximum error of 8 cents.
For 15-limit consonances, 58-tET has a maximum error of 8 cents.
For 17-limit consonances, 58-tET has a maximum error of 10 cents.
58-tET is inconsistent for 19 and higher odd limits.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> Joe, I hate to keep harping on the limitations of 72-tET (oh, maybe
I
> don't mind ;-> ), but the numbers you're generating show 72-tET in
its
> best light, "statically", if you will.
>
> Where it would be much worse, with a full 16.67 cents "error", is
when
> you have to make an adjustment of a comma. Or accept drift. As
we've
> already discussed, though, that problem applies only to some types
of
> music!
>
> JdL

Yes, music where the 81:80 vanishes, such as the vast majority of
Western common practice and Renaissance music. This is of course
because the diatonic scale, as you and I understand it, 'hides' the
81:80. That's OK, John . . . Joseph is currently writing music using
a scale (the blackjack scale) where the 81:80 never even occurs. All
the chains of fifths are either two or three fifths long -- never
longer -- and the 300-cent 'minor third' resulting from three fifth
is treated as dissonant, unlike the consonant 316-cent minor third
you find more often in this scale . . . the blackjack scale 'hides'
the 224:225 and the 2400:2401 . . . Joseph will be writing
progressions that 'pump' these commas, not the 81:80 . . . so I think
you can stop worrying now.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> Right on every count, Joe! But, that's why I favor "Reinhard"
notation:
> so that it would be possible to distribute out commas smoothly, and
> also convey the idea to real musicians.
>
I don't think anyone's opposing "Reinhard" notation in this context,
John. On the way to infinity, we simply think 72-tET is a
particularly nice stopping point, with great 21- and 31-tone subsets
that can be mapped to a composer's keyboard, and notational
inflections that can be mastered by a good conservatory-trained
performer in a reasonable amount of time. But no one claims it can do
everything. As you may have seen, I got very upset when Monz tried to
quantize Mozart's tuning to 72-tET -- the E-B fifth was 19 cents flat!

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

/tuning/topicId_24536.html#24564

> --- In tuning@y..., jpehrson@r... wrote:
>
> > That's "mean" stuff.... I wonder if Paul has ever done this...
>
> Of course, lots of times. And in according to my personal
listenting experience, the measures Graham referred to are more
important than the measures Manuel referred to. But even simpler is
just looking at the _maximum_ error -- the _worst_ "goof" that the
ET makes.
>

Thanks so much for this, Paul! This was a great post!

_________ ________ ______
Joseph Pehrson

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

/tuning/topicId_24536.html#24566

. . . the blackjack scale 'hides'
> the 224:225 and the 2400:2401 . . . Joseph will be writing
> progressions that 'pump' these commas, not the 81:80

It's great to know that the commas in blackjack are so comparatively
tiny! Everything I learn about blackjack makes me like it more and
more!

________ ______ _____
Joseph Pehrson

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

/tuning/topicId_24536.html#24568

On the way to infinity, we simply think 72-tET is a
> particularly nice stopping point, with great 21- and 31-tone
subsets that can be mapped to a composer's keyboard, and notational
> inflections that can be mastered by a good conservatory-trained
> performer in a reasonable amount of time.

That's a good summary of the arguments that "sold" me!

_______ ______ ________
Joseph Pehrson