What's with 14

Joseph asked on MMM:

"What's with 14, though... it scores pretty badly on the famed Paul
Erlich accuracy chart... :)"

The tuning that the Zeta function likes for 14 has a flat octave, and
corresponds to <14 22 33 39 48|. It has the following TM bases:

5-limit: [27/25, 2048/1875]
7-limit: [21/20, 27/25, 2048/1875]
11-limit: [21/20, 27/25, 33/32, 242/225]

27/25 in the 5-limit, 21/20 and 27/25 together in the 7-limit, and
21/20,27/25 and 33/32 in the 11-limit give the beep temperament, so
this 14-et val is closely associated to beep. The top tuning has
octaves around four cents flat.

Another val regards 14 as a contorted version of 7 in the 5-limit; in
the 11-limit it is <14 22 32 39 48|. TM bases are

5-limit: [25/24, 81/80]
7-limit: [25/24, 49/48, 81/80]
11-limit: [25/24, 33/32, 45/44, 49/48]

This involves decimal, meantone and jamesbond, and the TOP tuning of
the octave is now quite sharp, not flat; 1209.43 cents.

Other vals are possible; for instance a father version is <14 23 33 40
49|. TM bases for this are

5-limit: [16/15, 15625/13122];
7-limit: [16/15, 50/49, 175/162];
11-limit: [16/15, 22/21, 50/49, 175/162];

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

/tuning-math/message/10323

wrote:
>
> Joseph asked on MMM:
>
> "What's with 14, though... it scores pretty badly on the famed Paul
> Erlich accuracy chart... :)"
>
> The tuning that the Zeta function likes for 14 has a flat octave,
and
> corresponds to <14 22 33 39 48|. It has the following TM bases:
>
> 5-limit: [27/25, 2048/1875]
> 7-limit: [21/20, 27/25, 2048/1875]
> 11-limit: [21/20, 27/25, 33/32, 242/225]
>
> 27/25 in the 5-limit, 21/20 and 27/25 together in the 7-limit, and
> 21/20,27/25 and 33/32 in the 11-limit give the beep temperament, so
> this 14-et val is closely associated to beep. The top tuning has
> octaves around four cents flat.
>
> Another val regards 14 as a contorted version of 7 in the 5-limit;
in
> the 11-limit it is <14 22 32 39 48|. TM bases are
>
> 5-limit: [25/24, 81/80]
> 7-limit: [25/24, 49/48, 81/80]
> 11-limit: [25/24, 33/32, 45/44, 49/48]
>
> This involves decimal, meantone and jamesbond, and the TOP tuning of
> the octave is now quite sharp, not flat; 1209.43 cents.
>
> Other vals are possible; for instance a father version is <14 23 33
40
> 49|. TM bases for this are
>
> 5-limit: [16/15, 15625/13122];
> 7-limit: [16/15, 50/49, 175/162];
> 11-limit: [16/15, 22/21, 50/49, 175/162];

***Well, most of this is, admittedly, a bit over my head... but I
believe I saw that the 3-limit is reflected in 14-tET, at least
according to the Erlich chart, and I don't see it listed in the
above... (??)

Thanks!

JP

--- In tuning-math@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:

> ***Well, most of this is, admittedly, a bit over my head... but I
> believe I saw that the 3-limit is reflected in 14-tET, at least
> according to the Erlich chart, and I don't see it listed in the
> above... (??)

You just chop the vals I list down. The first two give <14 22|, which
gives the same fifth as 7-et ("contorsion".) The last is <14 23|; you
could also try <14 25| I suppose. None of this is very good.

Hi Joseph.

As far as 5-limit goes, I have a suggestion.

Remember the big ET chart on Monz's equal temperament page:

http://www.tonalsoft.com/enc/eqtemp.htm

It's the first chart there . . .

Now mouse over "zoom: 1" above the chart. If you can't see the yellow
triangular grid, mouse over "zoom: 1" under "negatives".

You'll see 14 occuring three times on that chart . . . once
overlapping 7.

These are three ways of "using" 14-equal in the 5-limit.

Look at how large the errors are of the basic 5-limit consonances.

Two of the instances of 14, it is true, are fairly close to the "just
perfect fifths - just perfect fourths" line.

But they both have thirds that are off by around 30-80 cents.

The other instance of 14 (the one at the top) doesn't fare much
better, and the "perfect fifth" is some 70 cents sharp!!

Am I making any sense?

-Paul

--- In tuning-math@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
>
> /tuning-math/message/10323
>
> wrote:
> >
> > Joseph asked on MMM:
> >
> > "What's with 14, though... it scores pretty badly on the famed
Paul
> > Erlich accuracy chart... :)"
> >
> > The tuning that the Zeta function likes for 14 has a flat octave,
> and
> > corresponds to <14 22 33 39 48|. It has the following TM bases:
> >
> > 5-limit: [27/25, 2048/1875]
> > 7-limit: [21/20, 27/25, 2048/1875]
> > 11-limit: [21/20, 27/25, 33/32, 242/225]
> >
> > 27/25 in the 5-limit, 21/20 and 27/25 together in the 7-limit, and
> > 21/20,27/25 and 33/32 in the 11-limit give the beep temperament,
so
> > this 14-et val is closely associated to beep. The top tuning has
> > octaves around four cents flat.
> >
> > Another val regards 14 as a contorted version of 7 in the 5-
limit;
> in
> > the 11-limit it is <14 22 32 39 48|. TM bases are
> >
> > 5-limit: [25/24, 81/80]
> > 7-limit: [25/24, 49/48, 81/80]
> > 11-limit: [25/24, 33/32, 45/44, 49/48]
> >
> > This involves decimal, meantone and jamesbond, and the TOP tuning
of
> > the octave is now quite sharp, not flat; 1209.43 cents.
> >
> > Other vals are possible; for instance a father version is <14 23
33
> 40
> > 49|. TM bases for this are
> >
> > 5-limit: [16/15, 15625/13122];
> > 7-limit: [16/15, 50/49, 175/162];
> > 11-limit: [16/15, 22/21, 50/49, 175/162];
>
>
> ***Well, most of this is, admittedly, a bit over my head... but I
> believe I saw that the 3-limit is reflected in 14-tET, at least
> according to the Erlich chart, and I don't see it listed in the
> above... (??)
>
> Thanks!
>
> JP

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Now mouse over "zoom: 1" above the chart. If you can't see the
yellow
> triangular grid, mouse over "zoom: 1" under "negatives".

I'm getting a "not found" for these. Monz?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>

/tuning-math/message/10326

wrote:
> Hi Joseph.
>
> As far as 5-limit goes, I have a suggestion.
>
> Remember the big ET chart on Monz's equal temperament page:
>
> http://www.tonalsoft.com/enc/eqtemp.htm
>
> It's the first chart there . . .
>
> Now mouse over "zoom: 1" above the chart. If you can't see the
yellow
> triangular grid, mouse over "zoom: 1" under "negatives".
>
> You'll see 14 occuring three times on that chart . . . once
> overlapping 7.
>
> These are three ways of "using" 14-equal in the 5-limit.
>
> Look at how large the errors are of the basic 5-limit consonances.
>
> Two of the instances of 14, it is true, are fairly close to
the "just
> perfect fifths - just perfect fourths" line.
>
> But they both have thirds that are off by around 30-80 cents.
>
> The other instance of 14 (the one at the top) doesn't fare much
> better, and the "perfect fifth" is some 70 cents sharp!!
>
> Am I making any sense?
>
> -Paul
>

***Yes, I can see that a line drawn through the two 14s is
practically parallel to the just 3:2 line, the thirds, however, being
way off...

I'd forgotten how nice these charts look in "negative" mode... I
think certain features come out better that way, too...

JP

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

/tuning-math/message/10327

wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Now mouse over "zoom: 1" above the chart. If you can't see the
> yellow
> > triangular grid, mouse over "zoom: 1" under "negatives".
>
> I'm getting a "not found" for these. Monz?

***Possibly you did what I first did, Gene, and actually *clicked on*
the links, rather than just "mousing over" without clicking... I got
that error message first, too...

JP