## 5 Discussion  
The results clearly showed the calculus of the entropy content of a communication system varies in important ways, depending on the scale of analysis. Looking at a language at the scale of characters provides a different picture than examining it at the level of words, or at the here described fundamental scale. Thus, in order to compare different communication systems, we need to use a similar scale applicable to each communication system. We showed that the fundamental scale presented here is applicable to very different communication systems, such as music, computer programs, and natural languages. This allows us to perform comparative studies regarding the systems entropy and thus to infer about the relative complexity of different communication systems. In both examples analyzed, the profiles at the scale of characters and the fundamental scale run close to each other, within the range of the most frequent symbols to the symbols with a rank placed near the mid logarithmic scale. For points with lower ranking, the fundamental-scale profile extends its tail toward the region of low symbol frequencies. The closeness of fundamental and character scaled profiles in the high frequency region, indicates that the character-scaled language \(\mathbf{B_{1}}\) is a subset of the fundamental scale language. The language at fundamental scale, having a greater symbolic diversity and therefore more degrees of freedom, finds a way to generate a symbol frequency distribution with a lower entropy as compared to the minimal entropy distribution when the description is viewed at the scale of words. Focusing in the fundamental scale profiles, the symbols located in the lower rank region--the tail of the profile--tend to be longer symbols formed by more than one character. These multi-character symbols, which cannot exist at the character scale, are formed at the expense of instances of single character symbols typically located in the profile's head. This explains the nearly constant gap between the two profiles in the profiles' heads. The English description, observed at the scale of words, produces a symbol profile incapable of showing short symbols--fragments of a word--which would represent important aspects of a spoken language as syllabus and other typical fundamental language sounds. On the opposite extreme, by observing at the character scale, the profile forbids considering strings of characters as symbols, thus meaningful words or structures cannot appear at this scale, missing important information about the structure of the described system. The fundamental scale, on the other hand, appears as an intermediate scale capable of capturing the essence of the most elementary structure of a language, as its alphabet, as well as larger structures which represent the result of language evolution in its way to form more specialized and complex symbols. The same applies for music MIDI representation. There is no word scale for music, but clearly the character scale does not capture the richness that undoubtedly is present in this type of language. Another difference between the fundamental scale, and other scales is the sensitivity to the order of the symbols as they appear in the text. At the scale of words or the scale of characters, the symbol frequency profile does not vary as the symbol order. The profiles depend only on the number of appearances of each symbol, word or character, depending on the subject scale. The profile built at the fundamental scale does change as the symbol order is altered, not because of the symbol order itself, but because the symbol set recognized as fundamental, changes when the order or words or characters are modified. As a consequence, the character and word scales do not have any sense of grammar. The fundamental scale and its corresponding profile, on the other hand, is affected by the order in which words are organized--or disorganized--and is therefore sensitive to the rules of grammar.