Music as Science, The C Minor Series No. 9: The Mountain-Top—Beethoven's Op. 111

DAILY DOSE of BEETHOVEN (July 13, 2020)

Recently, some readers have communicated difficulties in comprehending some aspects of the C Minor Series. Even professional musicians are likely to encounter that difficulty, because few are taught to think about music as constructive geometry, but rather as chord progressions. Therefore, we appreciate our readers working through these musical and scientific challenges.

Today, we arrive at the apex of this C Minor idea with Beethoven's “Piano Sonata No. 32 in C Minor, Op. 111,” and we hope that the power of the music itself will help clarify the idea.

The greatest works of art have a paradoxical quality to them. If you’ve only heard the music of JS Bach, but had never heard Mozart and Beethoven, you might think that it just couldn't get any better; there is a quality of beauty and perfection in his masterpieces. Indeed, there were associates of Bach who made the claim that it was not possible to advance beyond him. Yet when we encounter the next great work in the series, we realize that it does get better!

We have heard several pieces that examine what we termed the "Lydian" intervals in music. (Ft 1) The usual approach to composing a piece of music is to start out with a theme, in a specific key, and then broaden your investigation, working your way up, by developing new themes and keys. We have already seen how Mozart in his “Fantasy”, started out from a more universal musical space, and worked his way down, deriving passing keys from that higher dimension.

In Op. 111, Beethoven begins completely from that higher universal space, in what would be known in physics as the “continuous domain”. He begins in no key, and without a theme.

Today instead of an audio explanation of these matters, we post short sections of the score, alongside a complete performance of the first movement. We encourage our readers to listen to it one step at a time.

Here is the first movement with score, as performed by Mituko Uchida: https://youtu.be/WGg9cE-ceso

Example 1: presents the first 2 measures. It begins with the left hand playing an octave of Eb, followed by a drop to an F# octave. That is the “diminished seventh” interval we have encountered in several of the C Minor compositions. It is followed by F# A C Eb in the right hand, the first of our three “double-lydians”. It resolves tentatively, on G.

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Then listen to Example 2: It repeats the same pattern but starts on the tones Ab-B, the “diminished seventh” from the opening of Bach's “Musical Offering”. The right hand follows with Ab D F Ab, the second of our “double-Lydians”, and comes to rest on an unstable C. Now we have covered two out of three of these configurations, and there should be only one left.

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If Beethoven repeated the same pattern, he would play Db-E in the left hand, followed by Db E G Bb in the right, coming to rest on F. Then we would have a neat little package of the three double-lydians, as they resolve to C F and G. He almost does that but...The left hand does give us Db-E, and then it moves up to F, while the right hand is a surprise—Db F Bb Db. (See Example 3).

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This a Bb minor chord! Why does he do that? We think likely to make the point that the double-lydians, because of their ambiguity, can resolve in many different ways to many keys. Any one of the four tones could be a "leading tone” that resolves upwards by a half-step. Beethoven has opened new pathways for composing!

Is Bach's “chromatic scale” present? Look at example 4 and you will see that Beethoven, like Mozart, has placed it in the bass line. You will also see it is an ascending chromatic scale, which is not so obvious at first. Up until 1:07 in this recording, there is no key established—a full keyless minute. After that, the note G (the dominant of C), grows until the piece actually resolves into a theme (Allegro Brio), and a key (C Minor) at 1:50.

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Tomorrow, we will discuss this miraculous transformation.

Ft 1. The term "Lydian interval" comes from the medieval Lydian mode. That mode differs from an F major scale by one note. Whereas F major has a Bb for its fourth term, F G A Bb C D E F, the Lydian mode has B, F G A B C D E F.
Instead of a perfect 4th between notes 1 and 4 of the scale, we have an augmented fourth, a “tritone”. Beethoven explored the properties of the Lydian mode in the slow movement of his String Quartet Op. 132.

Here again is the first movement, performed by Mituko Uchida: https://youtu.be/WGg9cE-ceso

https://youtu.be/WGg9cE-ceso

(Example one)

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