Beethoven's Appassionata comma

On page 509 of _The Harmonic Experience_, W. A. Mathieu provides a
harmonic map of Beethoven's "Appassionata" sonata, which begins and
ends in F minor.

The first thing to notice is that Beethoven invokes enharmonic
equivalence at two points in the piece. If the notation is
kept "consistent" and enharmonic equivalence is not used, then the
piece begins in F minor and ends in Abbbb minor! But clearly
Beethoven wanted to return to the home key at the end. This makes it
clear that Beethoven, unlike Mozart and most other composers since
c.1480, was not assuming meantone temperament, but was instead
assuming a closed, cyclic system of 12 pitches per octave.

According to Mathieu, the music (if analysed in JI), through its
exploratory harmonic path, moves up a Great Diesis (128:125) to land
in the key of Ab major (which would actually be Gbbb major had
enharmonic equivalence not been employed). Then it moves up a
Diaschisma (2048:2025) to return to F minor (Abbbb minor without
enharmonic equivalence). (The second half of the piece spends a lot
of time in a harmonically static mode.) If these commas both vanish,
the tuning system must be 12-equal or some other closed, cyclic 12-
tone system.

If Mathieu's analysis is correct and the Didymic comma (syntonic
comma or 81:80, which vanishes in meantone) doesn't actually come
into play in this piece, a JI rendition of the piece would end up
(128:125)*(2048:2025) = 262144:253125 higher than it began. My new
paper (for those who have been looking at the draft) calls the
temperament where 262144:253125 vanishes "Subchrome". But I'm
changing this name to "Passion".

Correspondingly, I'd like to change the name of "Superchrome". The
first half of the alphabet is off-limits, since I've already done
that part of the paper. Any ideas? "Papaya"?

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> If Mathieu's analysis is correct and the Didymic comma doesn't
> actually come
> into play in this piece, a JI rendition of the piece would end up
> (128:125)*(2048:2025) = 262144:253125 higher than it began.

What is the ratio 262144:253125 in cents?

BTW, can I assume you didn't receive my question(s) last night
regarding the paper?

Cheers,
Jon

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> > If Mathieu's analysis is correct and the Didymic comma doesn't
> > actually come
> > into play in this piece, a JI rendition of the piece would end up
> > (128:125)*(2048:2025) = 262144:253125 higher than it began.
>
> What is the ratio 262144:253125 in cents?

It's 60.6 cents. For Monz and others, its "monzo" is [18 -4 -5>,
which means it's

2^18
---------
3^4 * 5^5

.

> BTW, can I assume you didn't receive my question(s) last night
> regarding the paper?

Sorry, Jon. The "stretch" e-mail server is useless, for the time
being, so if you sent them there, please send them to a different e-
mail. Or post them. Both the tuning-math list and this list have had
critiques of my paper posted -- and the former set are not
necessarily more technical than the latter!! You might be surprised.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> It's 60.6 cents. For Monz and others, its "monzo" is [18 -4 -5>,
> which means it's
>
> 2^18
> ---------
> 3^4 * 5^5

Thanks. The first answer certainly makes cents to me, the second is
once again in the language of the monks...

> Sorry, Jon. The "stretch" e-mail server is useless, for the time
> being, so if you sent them there, please send them...

Yeah, sorry, I just happened to be reading the paper and the list at
the same time, so I clicked on your name. I'll resend when traffic
slows down on this list, but I've also got a nice picture of my new
cat sleeping on your paper (the first diagram to be exact) - it
obviously causes Dr. Cricket no trouble! :)

Cheers,
Jon

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> > It's 60.6 cents. For Monz and others, its "monzo" is [18 -4 -5>,
> > which means it's
> >
> > 2^18
> > ---------
> > 3^4 * 5^5
>
> Thanks. The first answer certainly makes cents to me, the second is
> once again in the language of the monks...

Does it make sense to you if I say 2^18 = 262144?

What if I say 3^4 * 5^5 = 253125?

Have I lost you already?

> Yeah, sorry, I just happened to be reading the paper and the list at
> the same time, so I clicked on your name. I'll resend when traffic
> slows down on this list,

I hope you won't wait that long.

> but I've also got a nice picture of my new
> cat sleeping on your paper (the first diagram to be exact) - it
> obviously causes Dr. Cricket no trouble! :)

:x)

(smiling cat -- see the whiskers?)