graphic depiction of 5-limit consistency of equal temperaments

no, this is not x-rated . . .

you all may recall

/tuning/files/perlich/equaltemp.jpg

or, more likely, the first graph at

/tuning/files/dict/eqtemp.htm

these graphs show how far the basic 5-limit consonant intervals
deviate from JI in various ETs.

when i created this, i was asked why certain ETs appear to be
missing . . .

there are three reasons:

1. certain low-cardinality ETs have errors so large that they fell
off the edges of the graph -- if you want to see where 7, 8, 9, and
10 lie, check out this zoom-out:
/tuning/files/perlich/equaltemp2.jpg
which also shows the directions of the errors for all these graphs.

2. certain ETs are just multiples of simpler ones, but do not provide
any better 5-limit consonances, so they would lie on the same point
as the simpler ones (for example, 24 would lie on the same point as
12). the simplest ET is the one shown for each point.

3. certain ETs are *inconsistent* -- see below.

if we remove the consistency requirement, there are three ways we can
recreate the graph. try to get all three in different windows so that
you may quickly switch between them:

/tuning/files/perlich/miller1.gif
/tuning/files/perlich/miller2.gif
/tuning/files/perlich/miller3.gif

each of them uses only 2/3 of the 5-limit consonances to determine
where each ET is plotted. hopefully this is self-explanatory from the
text on the graphs.

notice what happens to inconsistent ETs when you switch between the
different graphs. they move! 40-ET, for example, moves from the upper
right, to the lower right, to the far left. if the red lines of

/tuning/files/dict/eqtemp.htm

are superimposed, one sees that 40-ET is a wuerschmidt temperament in
the first case, a diminished ("octatonic") temperament in the second
case, and a meantone temperament in the third case.

the ETs that remain in place on all three graphs are the consistent
ones, and the ones that appear in the first two links given in this
message. consistency therefore means that the mapping to JI is done
the same way no matter which of the consonances' best approximations
are taken.

below one of these "consistent-only" graphs, namely the first graph on

/tuning/files/monzo/blackjack/blackjack.htm

carl lumma includes 40-ET among the ETs in the diminished
("octatonic") family, but not in either the wuerschmidt or meantone
families. i'm not sure why carl did that. since 40-ET does not appear
on the graph, it would probably make most sense to exclude it
altogether . . . or one could include it in all three categories, in
which case the graph should show it in all three locations, like this
graph (a superimposition of miller1.gif, miller2.gif, and
miller3.gif) does:

/tuning/files/perlich/miller4.gif

similar comments could be made for other 5-limit inconsistent
ETs . . .

>carl lumma includes 40-ET among the ETs in the diminished
>("octatonic") family, but not in either the wuerschmidt or
>meantone families. i'm not sure why carl did that.

Where do you see that? I made a list once by looking at
the graph... if I saw the in which it originally appeared
I could tell you what I did, if it matters.

-Carl

--- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:
> >carl lumma includes 40-ET among the ETs in the diminished
> >("octatonic") family, but not in either the wuerschmidt or
> >meantone families. i'm not sure why carl did that.
>
> Where do you see that? I made a list once by looking at
> the graph... if I saw the in which it originally appeared
> I could tell you what I did, if it matters.
>
> -Carl

if you saw the . . . in which it originally appeared?

the graph is there, right above your table . . . it never had 40 on
it . . . and yet you included 40 as diminished/octatonic, and not as
meantone or wuerschmidt.

> if you saw the . . . in which it originally appeared?

Post.

> the graph is there, right above your table . . .

Where's my table? I followed all the links in your last post
and didn't see it.

-Carl

i wrote,

> you all may recall
>
> /tuning/files/perlich/equaltemp.jpg
>
> or, more likely, the first graph at
>
> /tuning/files/dict/eqtemp.htm
>
> these graphs show how far the basic 5-limit consonant intervals
> deviate from JI in various ETs.

[. . .]

>upper
> right, to the lower right, to the far left. if the red lines of
>
> /tuning/files/dict/eqtemp.htm
>
> are superimposed, one sees that 40-ET is a wuerschmidt temperament
in
> the first case, a diminished ("octatonic") temperament in the
second
> case, and a meantone temperament in the third case.
>
>
> the ETs that remain in place on all three graphs are the consistent
> ones, and the ones that appear in the first two links given in this
> message. consistency therefore means that the mapping to JI is done
> the same way no matter which of the consonances' best
approximations
> are taken.
>
> below one of these "consistent-only" graphs, namely the first graph
on
>
>
/tuning/files/monzo/blackjack/blackjack.htm
>
> carl lumma includes 40-ET among the ETs in the diminished

sorry all, this link should have been the same as the two above it:

/tuning/files/dict/eqtemp.htm

hopefully you can find your table now, carl!

> sorry all, this link should have been the same as the two
> above it:
>
> /tuning/files/dict/eqtemp.htm
>
> hopefully you can find your table now, carl!

Monz heavily modified that table, but I wouldn't rule out
the possibility that it was my mistake. I think the
original was posted either here or to tuning-math.

-Carl

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39388.html#39388

>
***I really love these graphs! These are really fantastic, but I
have some questions about them!

They seem to be a basic visual plot of the "struggle" between the 3-
limit and 5-limit consonances, with the result graphically depicted.
It's almost like a struggle between the best approximations of the
major 6th/minor 3rd, major 3rd/minor 6th and the perfect fifths and
fourths resulting in these various ETs! It's amazing to have it
graphed like this!

I printed out the three "Miller" graphs, since it was *very*
difficult to get them all in my browser. (Anybody able to do
that?? :)

Anyway, I'm gathering by the text, Paul, that you're saying that you
only used 2/3 of all the possible just intervals for the axes that
say "approx. may not be best...)

Why was it done like that again??

Also, I'm completely mystified why certain of the "inconsistent"
scales jump around like that! That's amazing, but whyso??

Maybe I need to first off know what is meant by an "inconsistent"
scale! :)

Anyway, it looks as though having various "completeness" of data in
certain axes for these scales really throw them into a different
light...

I'd like to know a bit more how that happens.

And, how were these wonderful graphs made?? They are truly amazing!

Joseph Pehrson

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> /tuning/topicId_39388.html#39388
>
> >
> ***I really love these graphs! These are really fantastic, but I
> have some questions about them!
>
> They seem to be a basic visual plot of the "struggle" between the 3-
> limit and 5-limit consonances,

well, 3-limit consonances *are* 5-limit consonances!

> with the result graphically depicted.
> It's almost like a struggle between the best approximations of the
> major 6th/minor 3rd, major 3rd/minor 6th and the perfect fifths and
> fourths resulting in these various ETs! It's amazing to have it
> graphed like this!
>
> I printed out the three "Miller" graphs, since it was *very*
> difficult to get them all in my browser. (Anybody able to do
> that?? :)
>
> Anyway, I'm gathering by the text, Paul, that you're saying that
you
> only used 2/3 of all the possible just intervals

2/3 of the 5-limit consonances -- 2 of the 3 5-limit consonant
interval classes.

> for the axes that
> say "approx. may not be best...)

. . . that *don't* say "approx. may not be best" . . . only 1 axis in
each graph says "approx. may not be best".

> Why was it done like that again??

there are three axes, corresponding to the three consonant interval
classes. if you find an ET's best approximation to two of them,
you've already determined the ET's position on the graph -- it's only
a two-dimensional graph! the third interval class depends on the
other two -- for example, if you've decided what the major third is,
based on the best approximation to 5:4, and you've decided what the
minor third is, based on the best approximation to 6:5, then the
relevant fifth, the fifth shown in the graph, results from stacking
the two thirds -- even if a better fifth (closer approximation to
3:2) is available in the tuning.

> Also, I'm completely mystified why certain of the "inconsistent"
> scales jump around like that! That's amazing, but whyso??

continuing with the same example . . . if you now create a different
graph, plotting the ET based on its best approximation to the 3:2
fifth, it will have to appear in a different place in this graph than
in the graph above . . . clear?

> Maybe I need to first off know what is meant by an "inconsistent"
> scale! :)

hopefully it's obvious from the above . . . yes?

> Anyway, it looks as though having various "completeness" of data in
> certain axes for these scales really throw them into a different
> light...
>
> I'd like to know a bit more how that happens.

let me know if you still need any gaps filled.

> And, how were these wonderful graphs made?? They are truly amazing!

matlab . . . here's the matlab code for miller3:

****************************************************************
vt=0;
tt=0;
axis square
axis off
axis([317.5575 351.5575 491.7981 525.7981])
hold on
for a=1:99;
g=round(log(3/2)/log(2)*a)/a*1200;
t=round(log(6/5)/log(2)*a)/a*1200;
v=g-t;
vt(a)=v;
tt(a)=t;
if ~(any((t==tt(1:a-1))&(v==vt(1:a-1))))
plot(v*sqrt(3)/2,t+v/2,'g.')
h=text(v*sqrt(3)/2,t+v/2,num2str(a));
set(h,'FontSize',9)
end
end
line([(318.5575+350.5575)/2 (318.5575+350.5575)/2],[ 292.7981
724.7981])
line([(318.5575+350.5575)/2+(292.7981-724.7981)*sqrt(3)/2
(318.5575+350.5575)/2-(292.7981-724.7981)*sqrt(3)/2],[292.7981
724.7981])
line([(318.5575+350.5575)/2-(292.7981-724.7981)*sqrt(3)/2
(318.5575+350.5575)/2+(292.7981-724.7981)*sqrt(3)/2],[292.7981
724.7981])
title('deviations of 5-limit consonances from JI in equal
temperaments')
****************************************************************

those last three "line" statements simply draw the three axes.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39388.html#39616

> matlab . . . here's the matlab code for miller3:
>
> ****************************************************************
>

***Thanks so much for this interesting post, Paul. I've run out of
time, but I'll have a few more questions shortly...

Joseph

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> > Also, I'm completely mystified why certain of the "inconsistent"
> > scales jump around like that! That's amazing, but whyso??
>
> continuing with the same example . . . if you now create a different
> graph, plotting the ET based on its best approximation to the 3:2
> fifth, it will have to appear in a different place in this graph than
> in the graph above . . . clear?

Another way to explain it is that what is being plotted are vals, and the two vals in question are quite different.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39388.html#39616

> well, 3-limit consonances *are* 5-limit consonances!
>

***I guess I really messed up my terminology with this one! I meant,
of course the fifths/fourths as plotted against the major
thirds/sixths and the minor thirds/sixths!

>
> > for the axes that
> > say "approx. may not be best...)
>
> . . . that *don't* say "approx. may not be best" . . . only 1 axis
in each graph says "approx. may not be best".
>

***Oh, FINALLY I get it! *TWO* of the axes are the "best
approximations" and plot the ET. Therefore the *THIRD* one is
extraneous, so it's "not so good... :)"

It would make sense that there would be three graphs... :)

>
> there are three axes, corresponding to the three consonant interval
> classes. if you find an ET's best approximation to two of them,
> you've already determined the ET's position on the graph -- it's
only
> a two-dimensional graph! the third interval class depends on the
> other two -- for example, if you've decided what the major third
is,
> based on the best approximation to 5:4, and you've decided what the
> minor third is, based on the best approximation to 6:5, then the
> relevant fifth, the fifth shown in the graph, results from stacking
> the two thirds -- even if a better fifth (closer approximation to
> 3:2) is available in the tuning.
>

***You know, *now* I remember this business about "consistency..." I
remember the bit about adding up the intervals in the inconsistent
scales and not having them come up with a good approximation of the
*composite* interval...

Still, it seems like some of them, 40-tET, really "hops around" a
lot! I guess that would make sense, since once the "gravitational
pull" of one of the axes is invalidated, "all hell" could break loose
with the plotting...

> matlab . . . here's the matlab code for miller3:
>
> ****************************************************************
> vt=0;
> tt=0;
> axis square
> axis off
> axis([317.5575 351.5575 491.7981 525.7981])
> hold on
> for a=1:99;
> g=round(log(3/2)/log(2)*a)/a*1200;
> t=round(log(6/5)/log(2)*a)/a*1200;

***That looks like a translation into cents, right there...

Etc.

***Very cool!

Joseph