By Alfred Brendel
During this definitive assortment of a well known pianist's writings—some of them by no means ahead of released in English—Alfred Brendel brings the readability and originality of expression that signify his performances to the broadcast web page. even if discussing Bach, Mozart, Beethoven, Schubert, Liszt, Schoenberg, or different pianists, his reflections will turn out useful to severe piano gamers and listeners alike.
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Extra info for Alfred Brendel on Music: Collected Essays
The first step is to differentiate φm (θ) to find its local maxima and minima. We concentrate on the interval 0 ≤ θ ≤ π, since φm (2π − θ) = 7Josiah Willard Gibbs described this phenomenon in a series of letters to Nature in 1898 in correspondence with A. E. H. Love. He seems to have been unaware of the previous treatment of the subject by Henry Wilbraham in his article On a certain periodic function, Cambridge & Dublin Math. J. 3 (1848), 198–201. 5. THE GIBBS PHENOMENON −φm (θ). 8). So the zeros of φ′m (θ) occur at θ = (2k+1)π m+1 m−1 8 2kπ θ = m , 0 ≤ k ≤ ⌊ 2 ⌋.
Instead, the series is converging to a value which is equal to the average of the higher and the lower values of the function. 5. Finally, there is a very interesting phenomenon which is happening right near the discontinuity. There is an overshoot, which never seems to get any smaller. Does this mean that the series is not converging properly? Well, not quite. At each given value of θ, the series is converging just fine. It’s just when we look at values of θ closer and closer to the discontinuity that we find problems.
An argument similar to the above shows that the value at the kth critical point kπ of φm (θ) tends to 0 sinθ θ dθ as m tends to infinity. 9028233. 8588% of the size of the discontinuity. The Gibbs phenomenon can be interpreted in terms of the response of an amplifier as follows. No matter how good your amplifier is, if you feed it a square wave, the output will overshoot at the discontinuity by roughly 9%. This is because any amplifier has a frequency beyond which it does not respond. Improving the amplifier can only increase this frequency, but cannot get rid of the limitation altogether.