Circulating temperament with 19:15 and 19:16

Following the discussion of some weeks ago on the practical tunability
of 19:15 relative to 81:64 (i.e. possible and possibly better), here
are a couple of pseudo-historical circulating tunings with some
19-limit ratios.

To be exact:
19:16 at B-D#, F#-Bb, C#-F
16:15 at B-C and E-F
therefore 19:16 at C-Eb and C#-E

In the first temperament I chose C-E to be 5/4*1216/1215 = 304/243 and
E-G# to be 5/4*2048/2025 = 512/405, which leaves Ab-C also just a
little under 81/64.

If we approximate the schisma as 2 cents and 1216/1215 as 1.5c, we
have 23.5 cents of tempering distributed round the circle as

C -5 G -5 D -5 A -5 E -1.5 B -1 F# -1 C# +1.5 G# +2 Eb -1 Bb -1 F -1.5 C

which can be seen as a 19-ic adaptation of Kirnberger, and improves
keys with a moderate number of accidentals.

Alternatively if we allow C-E to widen a tiny bit to the benefit of
E-G# and Ab-C, we can preserve the 19:15 and 19:16 just ratios and
achieve four pure 3:2 as follows:

C -4.5 G -4.5 D -4.5 A -4.5 E -3.5 B 0 F# 0 C# +0.5 G# +1 Eb 0 Bb 0 F
-3.5 C

which is clearly a 19-ic adaptation of Vallotti with somewhat more
interesting key colours.

If one can tune 19:15 by ear then the following instruction applies.

Start at F, Bb and Eb are pure fifths; next tune C# below via 19:15.
F# and B are again pure fifths. Next tune E via 15:16 from F (use a
temporary pure third F-A), E-B should now be 'tempered' by 512/513. C
can likewise be tuned via 16:15 from B (use a temporary pure third
B-G) then F-C is also 'tempered' by 512/513.

Now D is placed to temper G-D-A regularly, then G and A are themselves
tempered to fill out C-G-D-A-E as a regular sequence. Finally G# is
tempered between Eb and C#.

Try it for Mozart sonatas...

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> Following the discussion of some weeks ago on the practical
> tunability of 19:15 relative to 81:64 (i.e. possible and
> possibly better), here are a couple of pseudo-historical
> circulating tunings with some 19-limit ratios.

Have you made Scala files for these? If so, could you post
them? With Scala, it's easy to check if they've been invented
before.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> > Following the discussion of some weeks ago on the practical
> > tunability of 19:15 relative to 81:64 (i.e. possible and
> > possibly better), here are a couple of pseudo-historical
> > circulating tunings with some 19-limit ratios.
>
> Have you made Scala files for these? If so, could you post
> them? With Scala, it's easy to check if they've been invented
> before.
>
> -Carl
>

Good point, but I'm not sure how exact I should be. It would be easier
to take the approximation of a pure fifth being 702 and the (S. and
P.) commas 22 and 24 - but what level of error does this introduce in
the 19:15's if I call them as Pythagorean plus 1.5c?

Answer 408 + 1.5 = 409.5, vs. the true value 409.24... so slightly
'bright' 19:15's, probably acceptable.

So I might do the sums tomorrow.
~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Good point, but I'm not sure how exact I should be.

I suggest either rational intervals, or at least three decimals of
accuracy in terms of cents.

> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> > > Following the discussion of some weeks ago on the practical
> > > tunability of 19:15 relative to 81:64 (i.e. possible and
> > > possibly better), here are a couple of pseudo-historical
> > > circulating tunings with some 19-limit ratios.
> >
> > Have you made Scala files for these? If so, could you post
> > them? With Scala, it's easy to check if they've been invented
> > before.
>
>
> Good point, but I'm not sure how exact I should be.

I believe Scala lets you specify a fudge factor in the
comparison, so not to worry.

> It would be easier
> to take the approximation of a pure fifth being 702 and
> the (S. and P.) commas 22 and 24 - but what level of error
> does this introduce in the 19:15's if I call them as
> Pythagorean plus 1.5c?
>
> Answer 408 + 1.5 = 409.5, vs. the true value 409.24... so
> slightly 'bright' 19:15's, probably acceptable.
>
> So I might do the sums tomorrow.

The more accurate you can be the better. Usually keeping
things as fractions of the comma(s) you're working with
until the last minute is the easiest approach.

-Carl