Rational meantone fifths

Since 6144/6125 is very nearly 1/4 of a comma, lowering a fifth by
this comma produces a rational 7-limit meantone fifth which is close
to 1/4 comma.

Other similar such fifths may be obtained from other commas.

5120/5103 about 4/15 comma, or near the Golden Meantone

3136/3125 about 2/7 comma meantone

65625/65536 near 1/9 comma meantone

10976/10935 near 3/10 comma meantone

16875/16807 near 1/3 comma meantone

1600000/1594323 very near 2/7 comma; 5-limit

1224440064/1220703125 near 1/4 comma; 5-limit

540/539 near 1/7 comma

441/440 near 2/11 comma

385/384 near 1/5 comma

We can obtain rational meantone p-limit scales by using the above
meantone fifths and then reducing according to a microtemperament.
Of course, we can also play this game with any other temperament we
care to pick on.

I reduced the 6125/4096 fifth meantone via ennealimmal, and the result
according to Scala is actually closer to Woolhouse's 7/26-comma and
Golden Meantone, which is fine by me. Not so fine is the fact that the
numbers are large and ugly looking, but it is a 7-limit rational
meantone anyway.

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

> I reduced the 6125/4096 fifth meantone via ennealimmal, and the result
> according to Scala is actually closer to Woolhouse's 7/26-comma and
> Golden Meantone, which is fine by me. Not so fine is the fact that the
> numbers are large and ugly looking, but it is a 7-limit rational
> meantone anyway.

Since ennealimmal reduction didn't produce a very nice-looking result
(though in fact it would *sound* fine) I decided to reduce further, by
adding 32805/32768, which gives the complete comma set for 171-equal.
As expected this gave a much more reduced result, which turns out to
be a kind of mutant 19-equal meantone with nine exactly pure minor thirds.
So here it is, a 7-limit rational meantone.

! meanred.scl
171-et Hahn reduced rational Meantone[12]
12
!
672/625
125/112
6/5
56/45
75/56
25/18
112/75
45/28
5/3
224/125
625/336
2

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

> Since ennealimmal reduction didn't produce a very nice-looking result
> (though in fact it would *sound* fine) I decided to reduce further, by
> adding 32805/32768, which gives the complete comma set for 171-equal.
> As expected this gave a much more reduced result, which turns out to
> be a kind of mutant 19-equal meantone with nine exactly pure minor
thirds.
> So here it is, a 7-limit rational meantone.

270-et has a better meantone fifth that 171, so I tried Hahn reducing
to it; the result does not look as nice and does not have all those
pure minor thirds, but it is a good meantone.

! meanqr.scl
270-et Hahn reduced rational 6125/4096 Meantone[12]
12
!
16/15
28/25
448/375
784/625
8192/6125
6144/4375
6125/4096
625/392
375/224
25/14
15/8
2

From: "Gene Ward Smith" <gwsmith@s>
> Something else I've mentioned along these same lines is (224/9)^(1/8),
> which gives pure 9/7 supermajor thirds.
>
> supermajor thirds fifth: 695.61 cents
>
> This is actually more a 69-equal idea, but 69 is also an interesting
> equal temperament, with a fifth very close to the Wilson equal beating
> fifth.
>
> 69-et fifth: 695.62 cents
> Wilson fifth: 695.63 cents

So am I right in assuming there is a meantone tuning with the "Wilson"
fifth? Where does it come from? From the top of my head I'm now unable to
figure out the minor second. What is its size there?
Which intervals have the same beat rates, if you say it's an equal-beating
tuning? My favorite equal-beating meantone is the one with the same beat
rates in 4/5 and 3/5 (like E-C and E-G). It has a minor second of ~120.33
cents.

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@t...> wrote:

> > 69-et fifth: 695.62 cents
> > Wilson fifth: 695.63 cents
>
> So am I right in assuming there is a meantone tuning with the "Wilson"
> fifth? Where does it come from? From the top of my head I'm now
unable to
> figure out the minor second. What is its size there?

The Wilson fifth is a fifth f such that f^4 - 2 f - 2 = 0. Since this
is a meantone, the major third will be f^4/4, so a 15/8 will
approximate to
f^5/4; hence 16/15 is 8/f^5, which works out to 121.85 cents.

> Which intervals have the same beat rates, if you say it's an
equal-beating
> tuning?

The beats of the major thirds, minor thirds, and fifths within a given
major triad in close root position are the same. If f is the Wilson
fifth, and t=f^4/4 is the Wilson major third, then (6t-5f)/(4t-5)=-1,
(4t-5)/(2f-3)=1, and (2t-3)/(6t-5f)=-1. You might want to look at this
page:

http://66.98.148.43/~xenharmo/brat.html

From: "Gene Ward Smith" <gwsmith@s>

> You might want to look at this page:
>
> http://66.98.148.43/~xenharmo/brat.html

Something strange must have happened there. The main page loads OK but
whenever I try one of the links, I'm told the page can't be found. Nice
going guys.
Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@t...> wrote:

> Something strange must have happened there. The main page loads OK but
> whenever I try one of the links, I'm told the page can't be found.

It's my fault. I would guess there is some kind of website utility
which tells you if you have broken links; I'll look for it.

Strangely enough, I did add something to the site today:

http://66.98.148.43/~xenharmo/hahn.htm

On Sun, 01 Aug 2004 20:37:18 -0000 Gene wrote

>I would guess there is some kind of website utility
>which tells you if you have broken links; I'll look for it.

Hi Gene,

try this one from World Wide Web consortium

http://validator.w3.org/checklink

Peter
www.midicode.com

--- In tuning@yahoogroups.com, Peter Frazer <paf@e...> wrote:
> On Sun, 01 Aug 2004 20:37:18 -0000 Gene wrote
>
> >I would guess there is some kind of website utility
> >which tells you if you have broken links; I'll look for it.
>
>
> Hi Gene,
>
> try this one from World Wide Web consortium
>
> http://validator.w3.org/checklink

Thanks; I'll need to figure it out, as it doesn't seem to think I have
any broken links.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > Since ennealimmal reduction didn't produce a very nice-looking
result
> > (though in fact it would *sound* fine) I decided to reduce
further, by
> > adding 32805/32768, which gives the complete comma set for
171-equal.
> > As expected this gave a much more reduced result, which turns
out to
> > be a kind of mutant 19-equal meantone with nine exactly pure
minor
> thirds.
> > So here it is, a 7-limit rational meantone.
>
> 270-et has a better meantone fifth that 171, so I tried Hahn
reducing
> to it; the result does not look as nice and does not have all
those
> pure minor thirds, but it is a good meantone.
>
> ! meanqr.scl
> 270-et Hahn reduced rational 6125/4096 Meantone[12]
> 12
> !
> 16/15
> 28/25
> 448/375
> 784/625
> 8192/6125
> 6144/4375
> 6125/4096
> 625/392
> 375/224
> 25/14
> 15/8
> 2

A plus is that it also has the rough outline of a bell curve!

Here's another plug for Google: I found this very interesting thread
by chance. If this list were on Google, the topic I was interested
in would (Ratioonal meantone tunings) would have lead me to it, and
I wouldn't have had to wade through.

Perhaps we can convince Carl to start his Google group up? The
non-believers can slowly be weaned by starting cross-posting, and
then not cross-posting.

-A.

--- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:

> Here's another plug for Google: I found this very interesting thread
> by chance. If this list were on Google, the topic I was interested
> in would (Ratioonal meantone tunings) would have lead me to it, and
> I wouldn't have had to wade through.
>
> Perhaps we can convince Carl to start his Google group up? The
> non-believers can slowly be weaned by starting cross-posting, and
> then not cross-posting.

i thought that a Google group had already been created ...?

well, whether it already exists or still needs to be
created, its URL should be displayed prominently on
the homepage of *this* list, so that we know how to find it.

-monz

>Perhaps we can convince Carl to start his Google group up? The
>non-believers can slowly be weaned by starting cross-posting, and
>then not cross-posting.

Isn't it already started?

http://groups-beta.google.com/group/tuning

-C.