Yahoo Tuning Groups Ultimate Backup tuning [tuning] 22 + 34 =/= 56 (was: Re: Warped Canon page updated!)

> From: Herman Miller <hmiller@IO.COM>
> To: <tuning@yahoogroups.com>
> Sent: Friday, June 22, 2001 7:20 PM
> Subject: Re: [tuning] Re: Warped Canon page updated!
>
>
> ... 56-TET sounds very nice.
> Its minor sevenths aren't quite low enough to put them in the category of
> tunings that suggest 7-limit harmony, but they still sound pretty good. A
> bit of the 5-limit goodness of 34 and the 7-limit goodness of 22 (since 56
> = 22 + 34).

Umm... Herman, there's no real substance to this claim.
Just because 56 = 22 + 34, it doesn't mean that 56-EDO
will have pitches or intervals in common with 22- or
34-EDO. The logarithmic division of the 2:1 is different
for all three tunings, and not many of the degrees really
match up.

Here's a graph I made comparing all three:
/tuning/files/monz/22-34-56edo.jpg

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗David J. Finnamore <daeron@bellsouth.net>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: Herman Miller <hmiller@I...>
> > To: <tuning@y...>
> > Sent: Friday, June 22, 2001 7:20 PM
> > Subject: Re: [tuning] Re: Warped Canon page updated!
> >
> >
> > ... 56-TET sounds very nice.
> > Its minor sevenths aren't quite low enough to put them in the
category of
> > tunings that suggest 7-limit harmony, but they still sound pretty
good. A
> > bit of the 5-limit goodness of 34 and the 7-limit goodness of 22
(since 56
> > = 22 + 34).
>
>
> Umm... Herman, there's no real substance to this claim.
> Just because 56 = 22 + 34, it doesn't mean that 56-EDO
> will have pitches or intervals in common with 22- or
> 34-EDO. The logarithmic division of the 2:1 is different
> for all three tunings, and not many of the degrees really
> match up.
>
> Here's a graph I made comparing all three:
> /tuning/files/monz/22-34-56edo.jpg

Yeah, but look at the nearest 12-out-ofs and the nearest diatonic
sets.

David Finnamore

🔗Herman Miller <hmiller@IO.COM>

On Fri, 22 Jun 2001 23:33:48 -0700, "monz" <joemonz@yahoo.com> wrote:

>Umm... Herman, there's no real substance to this claim.
>Just because 56 = 22 + 34, it doesn't mean that 56-EDO
>will have pitches or intervals in common with 22- or
>34-EDO. The logarithmic division of the 2:1 is different
>for all three tunings, and not many of the degrees really
>match up.

Clearly they don't have many pitches "in common", but I never made that
claim! (Obviously, ET's will only have pitches in common if they have
common factors, in this case 2.)

To clarify what I intended: if two EDO's (say x-TET and y-TET) both have
good approximations of some interval, then the scale produced by adding
together the number of steps per octave in both tunings (x+y-TET) will have
an approximation that's somewhere in between. It might be better than both,
or it might only be not quite as bad as the least good approximation.

What's happening with 56-TET is that 34-TET's minor seventh (988.2c) isn't
a very good ~7/4 (968.8c), but 22-TET's is better (981.8c). Thus, 56-TET's
minor seventh has got to be better than 34-TET's, but since both 22-TET and
34-TET have sharp minor sevenths (if intended as a ~7/4 approximation),
56-TET's approximation to ~7/4 isn't as good as 22-TET's. It happens to be
985.71 cents, but you don't need to do the calculation to know that it's
going to be somewhere in between the 22-TET and 34-TET values.

This seems like magic, but it makes sense if you look at a permutation
lattice. I first saw this at http://www.bikexprt.com/tunings/tunings3.htm,
but I'd been making charts that looked similar to that before I saw it
online. There's a similar chart somewhere in the Erv Wilson archive. Note
that 56 fits between 22 and 34* on the diagram, and there are straight
lines connecting all three scales. Why this works is probably a topic for
the tuning-math list (and I don't fully understand it myself), but the fact
that it *does* work is useful in characterizing EDO's for their musical
properties. All meantone ET's, for instance, are various sums of 12 and 7
(12+7=19, 19+7=26, 19+12=31, 31+12=43, 31+19=50, etc.)

*actually, 34-TET is shown as 17-TET in the chart, but you can imagine
where the missing lines would be, or print out the chart and fill them in.

(In fact, both 34-TET and 56-TET have better ~7/4's, but they don't show up
in the retuning, since the 12-TET original doesn't distinguish between ~7/4
and ~16/9. Since the errors for the ~7/4 are opposite in 22-TET and 34-TET,
56-TET actually has a better ~7/4 than either of them.)

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

🔗monz <joemonz@yahoo.com>

> ----- Original Message -----
> From: Herman Miller <hmiller@IO.COM>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, June 23, 2001 2:39 PM
> Subject: Re: [tuning] 22 + 34 =/= 56 (was: Re: Warped Canon page updated!)
>
>
> To clarify what I intended: if two EDO's (say x-TET and y-TET) both have
> good approximations of some interval, then the scale produced by adding
> together the number of steps per octave in both tunings (x+y-TET) will
have
> an approximation that's somewhere in between. It might be better than
both,
> or it might only be not quite as bad as the least good approximation.

Ah... OK, now *that* makes perfectly good sense!

Thanks for taking the trouble to explain this further... the
rest of your post makes things very clear.

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗Paul Erlich <paul@stretch-music.com>

Yes there is, Monz. You must have forgotten that only a few short
weeks ago, we went over the "relative error theorem", which shows
that simply adding the cardinalities of two ETs will result in an ET
which has small errors for a given interval if the original ETs have
small errors for that interval.

🔗monz <joemonz@yahoo.com>

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, June 24, 2001 2:43 PM
> Subject: [tuning] 22 + 34 =/= 56 (was: Re: Warped Canon page updated!)
>
>
> > Umm... Herman, there's no real substance to this claim.
>
> Yes there is, Monz. You must have forgotten that only a few short
> weeks ago, we went over the "relative error theorem", which shows
> that simply adding the cardinalities of two ETs will result in an ET
> which has small errors for a given interval if the original ETs have
> small errors for that interval.

Yup, I missed that or forgot it. Well... uh... "a few short weeks
ago" we were also going over the amazing "new" MIRACLE temperaments,
the demise of the practicaltonality list in vivid brilliant flames,
your having been banned from it, the birthpangs of several new lists,
impersonations of McLaren, etc. etc., and I was gushing all over
the new-found applicability of 72-EDO to Starr Labs instruments.
So yes, I forgot it...

Thanks to Herman, Marc Jones, and you for setting me straight.

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., Herman Miller <hmiller@I...> wrote:
> Why this works is probably a topic for
> the tuning-math list (and I don't fully understand it myself),

It's actually very simple. The _relative_ (that is, in units of the
ET in question) signed errors simply add when you "add" ETs. We went
over this a few weeks ago, and I called it the "relative error
theorem".

> but the fact
> that it *does* work is useful in characterizing EDO's for their
musical
> properties.

Exactly.