Yahoo Tuning Groups Ultimate Backup tuning-math 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"?

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> 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".

Sounds like an overrated movie by Mel Gibson, or a perfume. Why not
call it "appassionata"?

> 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"?

The single most striking thing about the comma remains how close it
is to 21/20; another factor to bear in mind is that it is a 12-equal
comma, but also a 23 and 35 et comma, so you might think it is in a
natural partnership with 36/35 for a 7-limit version ("Number 58".)

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > 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".
>
> Sounds like an overrated movie by Mel Gibson, or a perfume. Why not
> call it "appassionata"?

I've already cut out, pasted, and all that stuff that Carl couldn't
believe I was doing, for the first half of the alphabet. Moreover, I
want short names.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Herman Miller <hmiller@IO.COM>

🔗jjensen142000 <jjensen14@hotmail.com>

I *almost* got that book today from the music library... I had
the call number on a slip of paper in my pocket and everything,
but I was just too busy :(

How am I ever going to make it through 500+ pages though?

--Jeff

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Graham Breed <graham@microtonal.co.uk>

jjensen142000 wrote:

> I *almost* got that book today from the music library... I had
> the call number on a slip of paper in my pocket and everything,
> but I was just too busy :(
> > How am I ever going to make it through 500+ pages though?

My copy covers Appassionata on p.349, and not in the detail Paul describes. Could he have a different edition? Amazon doesn't mention it.

Oh, has everybody seen Eytan Agmon's Scarlatti analysis?

Graham

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Graham Breed <graham@microtonal.co.uk>

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> > >>Oh, has everybody seen Eytan Agmon's Scarlatti analysis?
> > > No, where is that?

In Theory Only, vol. 11, no. 5, pp.1-8.

It's not as interesting as it looks from the title (Equal Division of the Octave in a Scarlatti Sonata) because the equal divisions are only 12 to the octave :( But it's an interesting example of an octatonic scale and enharmonic modulation from the 18th Century.

Graham

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> jjensen142000 wrote:
>
> > I *almost* got that book today from the music library... I had
> > the call number on a slip of paper in my pocket and everything,
> > but I was just too busy :(
> >
> > How am I ever going to make it through 500+ pages though?
>
> My copy covers Appassionata on p.349, and not in the detail Paul
> describes. Could he have a different edition? Amazon doesn't
mention it.

This looks like the one I've seen two copies of, and was referring to:

http://www.amazon.com/exec/obidos/tg/detail/-/0892815604/103-5498842-
0174200?v=glance

563 pages. Yup.

🔗jjensen142000 <jjensen14@hotmail.com>

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

I still haven't actually seen this book, but upon thinking about
this posting a little, it seems that the relevant issue is the
particular path of modulations: Fm --> Ab --> Fm, not whether
they occurred in the Appassionata sonata or somewhere else.
Therefore, maybe you would want names that describe this function?
In other words, name (or classify) the temperments by what
modulations they enable...?

just a thought...
Jeff

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:

> I still haven't actually seen this book, but upon thinking about
> this posting a little, it seems that the relevant issue is the
> particular path of modulations: Fm --> Ab --> Fm, not whether
> they occurred in the Appassionata sonata or somewhere else.
> Therefore, maybe you would want names that describe this function?
> In other words, name (or classify) the temperments by what
> modulations they enable...?

That would probably be the comma method of naming; base the name on
sucessive commas for the 5, 7, 11 etc limits, where at each limit
they form a basis for the kernel, or alternatively if they determine
the temperament after reduction of the wedgie, and we choose the
smallest Tenney height which works. Each method leads to a unique
name for each linear temperament which makes the family relationships
clear. I suppose along with the TM basis for a temperament I could
give the one or both of the comma names sometimes:

7-limit pajara TM basis: {50/49, 64/63}

7-limit comma name, first method: (2048/2025, 64/63)

7-limit comma name, second method: (2048/2025, 50/49)

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > 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"?
>
>
>
> I still haven't actually seen this book, but upon thinking about
> this posting a little, it seems that the relevant issue is the
> particular path of modulations: Fm --> Ab --> Fm, not whether
> they occurred in the Appassionata sonata or somewhere else.
> Therefore, maybe you would want names that describe this function?
> In other words, name (or classify) the temperments by what
> modulations they enable...?
>
> just a thought...
> Jeff

Seems a bit far-fetched. There are a whole heck of a lot of different
paths one can follow to traverse a 262144:253125, and of course the
starting point is completely irrelevant. Meanwhile, "Fm --> Ab -->
Fm" alone doesn't tell you anything about temperament at all -- this
could just as well be a JI chord progression.