Because classical music has greatly affected our life and culture in its long history, it has attracted extensive attention from researchers to understand laws behind it. Based on statistical physics, here we use a different method to investigate classical music, namely, by analyzing cumulative distribution functions (CDFs) and autocorrelation functions of pitch fluctuations in compositions. We analyze 1,876 compositions of five representative classical music composers across 164 years from Bach, to Mozart, to Beethoven, to Mendelsohn, and to Chopin. We report that the biggest pitch fluctuations of a composer gradually increase as time evolves from Bach time to Mendelsohn/Chopin time. In particular, for the compositions of a composer, the positive and negative tails of a CDF of pitch fluctuations are distributed not only in power laws (with the scale-free property), but also in symmetry (namely, the probability of a treble following a bass and that of a bass following a treble are basically the same for each composer). The power-law exponent decreases as time elapses. Further, we also calculate the autocorrelation function of the pitch fluctuation. The autocorrelation function shows a power-law distribution for each composer. Especially, the power-law exponents vary with the composers, indicating their different levels of long-range correlation of notes. This work not only suggests a way to understand and develop music from a viewpoint of statistical physics, but also enriches the realm of traditional statistical physics by analyzing music.

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BackgroundTo accurately predict the movement of stock prices is always of both academic importance and practical value. So far, a lot of research has been reported to help understand the behavior of stock prices. However, some of the existing theories tend to render us the belief that the time series of stock prices are unpredictable on a long-term timescale. The question arises whether the long-term predictability exists in stock price dynamics.Methodology/Principal FindingsIn this work, we analyze the price reversals in the US stock market and the Chinese stock market on the basis of a renormalization method. The price reversals are divided into two types: retracements (the downward trends after upward trends) and rebounds (the upward trends after downward trends), of which the intensities are described by dimensionless quantities, and , respectively. We reveal that for both mature and emerging markets, the distribution of either retracements or rebounds shows two characteristic values, 0.335 and 0.665, both of which are robust over the long term.Conclusions/SignificanceThe methodology presented here provides a way to quantify the stock price reversals. Our findings strongly support the existence of the long-term predictability in stock price dynamics, and may offer a hint on how to predict the long-term movement of stock prices.

BackgroundBecause the movement of stock prices is not only ubiquitous in financial markets but also crucial for investors, extensive studies have been done to understand the law behind it. In particular, since the financial crisis in 2008, researchers have a more interest in investigating large market volatilities in order to grasp changing market trends.Methodology/Principal FindingsIn this work, we analyze the breakouts and breakdowns of both the Standard & Poor’s 500 Index in the US stock market and the Shanghai Composite Index in the Chinese stock market. The breakout usually represents an ongoing upward trend in technical analysis while the breakdown represents an ongoing downward trend. Based on the renormalization method, we introduce two parameters to quantize breakouts and breakdowns, respectively. We discover scaling behavior, characterized by power-law distributions for both the breakouts and breakdowns in the two financial markets with different power-law exponents, which reflect different market volatilities. In detail, the market volatility for breakdowns is usually larger than that for breakouts. Moreover, as an emerging market, the Chinese stock market has larger market volatilities for both the breakouts and breakdowns than the US stock market (a mature market). Further, the short-term volatilities show similar features for both the US stock market and the Chinese stock market. However, the medium-term volatilities in the US stock market are almost symmetrical for the breakouts and breakdowns, whereas those in the Chinese stock market appear to be asymmetrical for the breakouts and breakdowns.Conclusions/SignicanceThe methodology presented here provides a way to understand scaling and hence volatilities of breakouts and breakdowns in stock price dynamics. Our findings not only reveal the features of market volatilities but also make a comparison between mature and emerging financial markets.