Polyphonic music files were analyzed using the set of symbols that produced the Minimal Entropy Description, which we call the Fundamental Scale. This allowed us to create a novel space to represent music pieces by developing: (a) a method to adjust a textual description from its original scale of observation to an arbitrarily selected scale, (b) a method to model the structure of any textual description based on the shape of the symbol frequency profiles, and (c) the concept of higher order entropy as the entropy associated with the deviations of a frequency-ranked symbol profile from a perfect Zipfian profile. We call this diversity index the ‘2nd Order Entropy’. Applying these methods to a variety of musical pieces showed how the space of ‘symbolic specific diversity-entropy’ and that of ‘2nd order entropy’ captures characteristics that are unique to each music type, style, composer and genre. Some clustering of these properties around each musical category is shown. These methods allow us to visualize a historic trajectory of academic music across this space, from medieval to contemporary academic music. We show that the description of musical structures using entropy, symbol frequency profiles and specific symbolic diversity allows us to characterize traditional and popular expressions of music. These classification techniques promise to be useful in other disciplines for pattern recognition and machine learning.

We measured entropy and symbolic diversity of texts written in English and Spanish. We included texts by Literature Nobel laureates and other famous authors. We formed four groups of texts according to the combinations of language used and the author’s Literature Nobel Prize condition. Entropy, symbol diversity and symbol frequency profiles were compared for these four groups. We also built a scale sensitive to the quality of writing and evaluated its relationship with the Flesch’s readability index for English and the Szigriszt’s perspicuity index for Spanish. Results suggest a correlation between entropy and word diversity with quality of writing. Text genre also influences the resulting entropy and diversity of the text. Results suggest the plausibility of automated quality assessment of texts.

The complexity of the description of a system is a function of the entropy of its symbolic description. Prior to computing the entropy of the system’s description, an observation scale has to be assumed. In texts written in artificial and natural languages, typical scales are binary, characters, and words. However, considering languages as structures built around certain preconceived set of symbols, like words or characters, limits the level of complexity that can be revealed analytically. This study introduces the notion of the fundamental description scale to analyze the essence of the structure of a language. The concept of Fundamental Scale is tested for English and musical instrument digital interface (MIDI) music texts using an algorithm developed to split a text in a collection of sets of symbols that minimizes the observed entropy of the system. This Fundamental Scale reflects more details of the complexity of the language than using bits, characters or words. Results show that this Fundamental Scale allows to compare completely different languages, such as English and MIDI coded music regarding its structural entropy. This comparative power facilitates the study of the complexity of the structure of different communication systems.