What is higher, A# or Bb?

An old question - and I thought I had answered it long ago...
Calculating the ratios based on pure fifth and pure major third
(5-limit) I got that Bb is higher - I even wrote an entry into my
Music and mathematics FAQ
(http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_A#_Bb).

But now I read a newsgroup posting stating the opposite: that A# is
higher than Bb, at least if it appears as a leading tone - because a
leading tone works the better the smaller the interval is.
Now I am quite a little confused - can the experts here clear that up?
I want to have correct data on my website...
--
Hans Straub

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

> I want to have correct data on my website...

A# and Bb are the same in 12-equal, from which you may immediately
conclude that it represents the boundry between two regions, one
where A# is sharper than Bb, and one where it is flatter. For justly
tuned fifths, as you discovered, A# is sharper than Bb, hence for
fifths flatter than the 1/11-comma meantone of 12-equal, A# will be
flatter than Bb. It all can be derived therefore from what you
already knew, with no math.

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

> An old question - and I thought I had answered it long ago...
> Calculating the ratios based on pure fifth and pure major third
> (5-limit) I got that Bb is higher -

Sorry, I am waiting for my new glasses to arrive and I see now you
were in the 5-limit, which really confuses things since now you've
got two sizes of whole tones, 9/8 and 10/9, plus 16/15 and 25/24
semitones. You have a Bb which is a minor third above G, and another
which is a fourth above F; and lots of ways of defining A#, etc etc.
It's confusing enough for your web page just with fifths, and I'd
stick to that. However if you absolutely must you can claim that C#
is a 25/24 above C, which it is if it is the major third above an A
of 5/6, and Db is a major third below an F of 4/3, and hence is (4/3)
(4/5) = 16/15; and 16/15 > 25/24.

> But now I read a newsgroup posting stating the opposite: that A# is
> higher than Bb, at least if it appears as a leading tone - because a
> leading tone works the better the smaller the interval is.

That just says some people play that way; it's confusing enough
without that issue added as well.

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:
> An old question - and I thought I had answered it long ago...
> Calculating the ratios based on pure fifth and pure major third
> (5-limit) I got that Bb is higher - I even wrote an entry into my
> Music and mathematics FAQ
> (http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_A#_Bb).

In 5-limit JI, every note name occurs at an infinite number of
different acoustical pitches, separated by 81:80 from one another. So
it's fairly nonsensical to consider either A# or Bb to be higher in
JI.

In meantone tuning, though, A# is lower than Bb, and this was the
accepted practice in the 1480-1780 era in Europe. Historical meantone
tunings feature either exactly pure or very nearly pure major thirds,
and perfect fifths flattened by only 4-7 cents from JI (2-5 cents
from 12-equal).

> But now I read a newsgroup posting stating the opposite: that A# is
> higher than Bb, at least if it appears as a leading tone - because a
> leading tone works the better the smaller the interval is.

This philosophy conforms with the medieval Pythagorean tuning, and
also agrees with the "expressive intonation" doctrine that has been
widely taught since 1800. Some here have argued differently, but my
read of current teaching and hearing of current practice is that pure
major thirds have largely fallen by the wayside as a musical
desideratum, major thirds 14 cents wide, or even wider (say F#-A#
if "expressive intonation" is followed) having become the entrenched
norm.

> Now I am quite a little confused - can the experts here clear that
up?
> I want to have correct data on my website...
> --
> Hans Straub

You seem to have a confusion between *adaptive JI* and *strict JI* on
your website. Even in those minority of cases where strings and
voices do lean toward just intonation, it is toward *adaptive JI* --
each chord being roughly in JI within itself -- and virtually never
toward *strict JI* -- where scale degrees and horizontal intervals
seek to approximate simple-integer ratios. Monz' dictionary has some
definitions for these terms that might be enlightening -- and
hopefully not too confusing -- to look at.

Anyway, your website looks like a promising beginning, I'm sure
others here will join me in offering to help you fix and expand this
website and make it more useful!

-Paul

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

>
> This philosophy conforms with the medieval Pythagorean tuning, and
> also agrees with the "expressive intonation" doctrine that has been
> widely taught since 1800. Some here have argued differently, but my
> read of current teaching and hearing of current practice is that
pure
> major thirds have largely fallen by the wayside as a musical
> desideratum, major thirds 14 cents wide, or even wider (say F#-A#
> if "expressive intonation" is followed) having become the
entrenched
> norm.

> -Paul

Fortunately, this space serves well for all who think this "musicalia
desiderata" justify somehow a reformulation of the current state of
things, at least at a personal level.

Max.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:
>
> > Now I am quite a little confused - can the experts here clear
that
> up?
> > I want to have correct data on my website...
>
> You seem to have a confusion between *adaptive JI* and *strict JI*
on
> your website. Even in those minority of cases where strings and
> voices do lean toward just intonation, it is toward *adaptive JI* -
-
> each chord being roughly in JI within itself -- and virtually
never
> toward *strict JI* -- where scale degrees and horizontal intervals
> seek to approximate simple-integer ratios. Monz' dictionary has
some
> definitions for these terms that might be enlightening -- and
> hopefully not too confusing -- to look at.
>

Yes - I think it is adaptive JI I wanted to address.
Now let's see, I'll have to look for a reformulation of that section
without going to much into details...

> Anyway, your website looks like a promising beginning, I'm sure
> others here will join me in offering to help you fix and expand
this
> website and make it more useful!
>

Thanks - I am always glad for input!
--
HAns Straub