Yahoo Tuning Groups Ultimate Backup tuning the limits of justness [19-tET]

I have some questions concerning Just Intonation and it's
relationship to a couple of scales... notably the great Paul Erlich
just 19-tone scale #3 and it's relationship to 19-tET.

For those of you who might wish a "refresher" in this scale, please
refer to post #19050, scale #3

/tuning/topicId_19050.html#19050

I've been playing around with this scale and, so far, very much enjoy
some of the just sonorities that seem to come from it.

HOWEVER, naturally the thought occurred to me how many such just
intervals in TRANSPOSITION within the 19 tone octave I was actually
finding, and how far "off" such transpositions were from just
intervals...

I also wanted to compare the scale to 19-tET to see how much more
"unjust" that scale was for any given interval in transposition.

I took "basic" just intervals, since I figured those were the easiest
for me to aurally verify the "justness."

The following figures are the difference in cents between the various
intervals in transposition within the 19-tone octave:

5:4 = 386 just

386
386
379
386
365
386
379
379
386
365
386
379
386
386

Well... 5/4 doesn't do too badly in this scale. I see 8 just 5:4s
(386 cents), 4 are off by -7 cents (the "spooky" 7) and 2 are off by
-21 cents.

Compared to 19-equal it seems to be "better," since the number of
just intervals is more than half... and in 19-equal ALL the 5:4s are,
of course, 379 cents, all -7 low.

Of course, in a way it's kind of a "toss up," since -7 is pretty
close to just for each of the thirds.

Let's face it... it's a tie for the 5:4.

Let's be adventurous and try the 6:5 = 316 just:

In the 19-tone "just" scale:

316
337
316
316
302
316
316
330
316
316
302
316
316
337
316

OK... that's 10 just minor thirds, 2 are off by -14 cents and 2 are
off by +21 cents.

HOWEVER, as we know, ALL the 6:5s in 19-tET are JUST: 316 cents!

So 19-tET is the big winner in the minor third category.

Mark your scorecards!

Let's spring for the 3:2 = 702 just. That has to be useful somewhere
down the road...

702
716
695
702
666
702
695
716
702

OK... We've got 4 just "perfect" fifths, 2 are off by +14 cents, 2
are off by -7 cents and 1 is off by -35 cents...

In 19-tET all are 695, or off by -7 cents.

Pooh... looks like the just scale is a "loser" here... I can hardly
hear -7 cents!

Mark your scorecards!

I'm feeling very Medieval at the moment, so let's look at the 4:3=
498 just:

498
519
512
519
498
498
498
498
498
519
512
519
498

Hmmm. 6 just, 4 off +21 cents, 2 off +14 cents. In 19-tET all are
off -7 cents. None really pure...

I think it's a tie here...

Please mark your scorecards...

Let's look at the 7:6= 267 just:

267
267
267
253
232
253
246
267
267
246
253
232
253
267
267
267

Well, we have here 8 just 7:6s, 4 are off by -14 cents, 2 are off by
+35 cents and 2 are off by +21 cents.

In 19-tET, all are off by -14 cents.

I think we have a winner here with the just scale! Mark your
scorecards!

I'm feeling very 9:7= 435 just. Let's try that kind of third.

435
449
449
449
435
449
428
449
435
449
435
449
449
449
435

Hmmm. 4 just, 8 off by +14 and 1 off by -7.

In 19-tET all are off by -7 cents.

With only 4 just, I would say that 19-tET is the "winner" here. Mark
your cards, please.

Lets go for the 5:3 = 884 cents just.

Well, I don't even have to post the results, since we know that the
major sixth is JUST in 19-tET.

It's a clear "winner" in 19-tET....

Those are the most obvious intervals, but there are others:

52:18 = 568 just

and

36:25 = 632 just

For BOTH of these intervals, 19-tET differs from just NOT AT ALL, and
transposes throughout the entire scale!

Well, let's look at our "scorecard" and rate the 9 interval choices
we have made so far:

BETTER IN JUST SCALE??: 5:4 (a tie), 4:3 (a little debatable-- a
tie) and 7:6

3 classes of intervals-- with two "ties"

BETTER ("juster") in 19-tET:

6:5, 3:2, 9:7, 5:3, 25:18, 36:25...

6 intervals.

So, it seems that 19-tET produces MORE just intervals for 6 classes
of intervals to the 3 classes of intervals for the "just" scale among
the 9 basic classes selected...

Can it be that 19-tET is actually "juster" than just??

Of course, there is a "downside." The fact that, as a temperament
the MAJORITY of intervals are "off" (not the minor thirds and major
sixes, though, nor the 25:18 and 36:25 intervals!)

Also, there is the "cloying regularity" of an equal temperament.
That might, though, make it easy for performers to learn, but it
seems so "mechanical." Little boxes... ugh.

ALSO, it doesn't correspond to such nice visual lattices... or at
least corresponds only roughly.

HOWEVER, you've got to admit that, in some quarters, it could be
claimed that 19-tET is SUPERIOR from a JUST standpoint to this "just"
scale we have been discussing.

Yes, no??

What did I do wrong in this study...???

thanks!

Joseph Pehrson

🔗paul@stretch-music.com

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

Hi Paul!

Thanks so much for your help with this er, "comparison study..." I
really appreciate it! Thanks for finding the "mistakes!"

> >
> > Can it be that 19-tET is actually "juster" than just??
>

> First of all, you rigged the contest. 25:18 and 36:25 are such
large-number intervals that you only ever could have been interested
in because 19-tET approximates them so well (by stacking two of those
just 6:5s). So drop them.
>

You'r right... these are silly. They're just "anomalies..." Dropped.

> Secondly, 5:3 is just the inversion of 6:5. You counted that one
twice. Drop one of those.
>

Ok..

> Thirdly, you clearly left out 7:4 and 7:5.
>

How did you know that 7:4 and 7:5 would come out better for the just
scale?? Experience, I guess...

Anyway... here's the result and, you're right of course, they should
have been included:

7/4 = approx 969 cents

Well, the 12:7 is the closest interval in this scale, and in 19-tET
it is always -22 cents flat...

However, in the just scale I find (just within ONE octave for now...):

933
969
962
969
933

In the "just scale" I see now 2 just 7:4s, 2 are off by -36 cents and
1 is off by -7 cents...

BUT, the 19-tET is consistently off by -22 cents for that interval...

So, yes, the 7:4 is "BETTER" in the just scale!

Going on...

Let's take 7:5 as 582 cents

The closest interval in the just scale is the 25:18 at -14 cents
flat...

Within the octave:

568
583
583
583
547
568
547
583
583
583
568

Hmm.... Mighty impressive. I see 6 just 7:5s, 3 are off by -14
cents, and 2 are off by -35.

Well, with 19-tET off by -14 for ALL these intervals, I do have to
conclude that 7:5 is ALSO a big winner in this category...

> Fourthly, if you're going to include 9:7, you should include 9:4
and 9:5 as well.

Let's omit for now...

So our "scorecard tally" has changed drastically:

For the just scale:

SUPERIOR: 5:4, 7:4, 7:5, 7:6

This MUST be what you meant when you said 19-tET was "lousy" in the 7-
limit (!!)

DEBATABLE: 3:2 (we'll leave out the 4:3 this time, ahem :) )

INFERIOR: 6:5 (well, you can't beat those just minor thirds!)

So, for the 6 intervals we are currently considering:

4 are now superior, against 2 possibly inferior.

The just scale is now ahead. There is justice in this world!

Of course, this is really to my advantage as well, since my synth is
already set up to play the Erlich just scale #3 and I have already
started composing with it!

Talk about "rigging the results!"

Glad it's working out this way! :)

_______ _______ _ _____
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21931.html#21936

> >
> > Can it be that 19-tET is actually "juster" than just??
>

That makes sense. Those really are "anomalies"

> Secondly, 5:3 is just the inversion of 6:5. You counted that one
twice. Drop one of those.
>

OK makes sense.

> Thirdly, you clearly left out 7:4 and 7:5.
>

Ok so you must have known that these intervals would be better
in the just scale! That must be what you meant when you said that 19-
tET is so poor at the 7-limit!

If I am to take the 7:4 interval as 969 cents, the closest
approximation in the just scale is 12:7. For that, the 19-tET scale
is off -22 cents through the octave range.

Let's take the just 7:4 as 969 cents

I get, through the octave range in the just scale:

933
969
962
969
933

I see 2 just intervals, 2 are off by -36 cents, and 1 is off by
only -7 cents.

Since the 12:7 is off -22 for ALL the intervals in the 19-tET scale,
I think we would have to declare the just scale the "winner" in the
7:4 category.

Now on to the 7:5.

If we are to take the 7:5 as 582 cents, our closest interval in the
just scale is 25:18, which is -14 cents throughout for the 19-tET
scale.

Here are the cents value for 7:5 through the octave of the just scale:

568
583
583
583
547
568
547
568
547
583
583
583
568

Hmmm. I see SIX just 7:5s! Quite a few, 3 are off by -14 and 2
are off by -35. Since 19-tET is -14 flat throughout, I would say
that, again, the just scale is clearly the "winner" in this category!

> Fourthly, if you're going to include 9:7, you should include 9:4
and 9:5 as well.
>

We'll drop these for now

So, the final tally with the 7-limit intervals is, for the just scale:

BETTER:

5:4, 7:4, 7:5, 7:6 (clearly a winner here in the 7-limit!)

DEBATABLE:

3:2 (we'll leave out the 4:3, ahem, for the moment :) )

INFERIOR:

6:5. Well, it's hard to beat those just minor thirds in 19-tET!

So our scorecard is now in favor of the just scale. Of the 6
intervals chosen for comparison, now 4 are superior in the just scale.

This is fortunate, since I have already set up the synthesizer for
this scale, and have started composing in it!

Talk about "skewing" the results!

__________ _____ _ _____
Joseph Pehrson