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First of all, these bars do not need to be pedaled, and the first draft does not have a puja (only a few bars have, and it is not long), and there are many skips on the left hand, so it is not recommended that you pedal. If you have to step on it, you can change it every two beats (half a measure) from measures 145 to 156, noting that there is a left-hand skip in the middle. There are too many skips in the left hand in other measures, so it's better not to step on it, and keep it crisp.
Beethoven himself doesn't seem to like to mark the score with skips, and the fingering is not marked, and the fingering should be c. in the current Chinese versionHansen annotated. So I think Beethoven probably wanted the players to play with their own feelings!
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Beethoven himself did not mark the pedals, and it was up to the player to decide.
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The pedal is really flexible, everyone uses a different pedal method, and if you listen to this sonata played by different pianists, you will never find two pianists using the same pedal method. The reason why a lot of music scores don't have pedals on them is because it depends mainly on your understanding of the work. If you really understand the piece deeply, you can use the pedals by feeling.
In addition, your teacher can also teach you how to make your own decisions about the use of pedals. I'm attaching to you here four different versions of the score, all of which have pedal markings, but all of them are different because they have to do with the aesthetics of the compiler of the score itself. Please see the four links below.
Pedal Marker Version 1
Pedal Marker Version 2
Pedal Marker Version 3
Pedal Marker Version 4
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You don't make it clear whether the fourth movement of Beethoven's Sonata in F minor refers to Op. 2-1 or Op. 26.
However, I can answer you with certainty: Beethoven's piano works must use pedals. Because the piano performance was already perfecting at that time, pedals were widely used.
According to Beethoven's student Carl Czerny, Beethoven himself used pedals in many places on the score that were not marked when he played the piano.
The pedal is known as the "soul" of the piano. It greatly enhances the artistic atmosphere of piano playing, and can create a warm, shocking, or soft, ethereal sound effect.
There are some basic principles and methods for the use of piano pedals, by adjusting the depth or shallowness of the pedal, to achieve different sound effects: such as creating a gentle and sweet, dreamlike artistic conception in the lyrical **, and then through its subtle finger touch technology, the piano can make the timbre of the piano colorful, intoxicating. On the contrary, in the passionate and agitated **, the pedal can increase the volume of the piano, create a rich and thick harmonic sound, resonate with the listener's heart, and create a thunderous and shocking sound effect (this is also described in Czerny).
However, the marking and use of the pedals will vary from person to person. It depends mainly on the composer and the compiler of the score. For scores without pedal markers, it is necessary to rely on the performer's (and his instructor's) deep understanding of the ** work and rich artistic practice experience to conceive and design.
Since you didn't specify the work number. It is difficult to explain how the pedals are used in detail for a specific piece.
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