## 4 Tests and Results  
In order to compare the differences obtained when observing a written message at the scales of characters, words and the fundamental scale, we designed an Example Text. Table 1 shows the symbols obtained after the analysis of the Example Text at the three observation scales used in this study. The entropies calculated at the scales of characters and words were 0.81 and 0.90 respectively, the entropy at the fundamental scale was 0.76; an important reduction of the information required to describe the same message. These results also get along with our intuition. Clearly, the selection of a certain character-string as a fundamental symbol, is favored by the frequency of appearance of the string of characters. As a result, the \"_space character_\" (represented as \(\omega\) in the table) is recognized as the most frequent fundamental symbol. It indeed is an important structural piece in any English text, since it defines the beginning and the end of natural words. The length of the string of characters also favors the survival of the symbol in its competence with other prospective symbols. The string \"_describ_\", for example, appears twice in the Example Text and the algorithm recognized it as a symbol. On the other hand, the 11-char long string \"_An adverb_\" also appears two times, but the algorithm found it more effective in reducing the overall entropy, to break that phrase apart and increase the appearances of other symbols. A similar case is that of the word \"_adverb_\", which appears in nine instances (not including those written with the first capital letter) on the Example Text. But the entropy minimization problem found a more important entropy reduction by splitting the word \"_adverb_\" in shorter and more frequent symbols as \"_dv_\" (10 times), or the characters as \"_e_\" (70 times), \"_a_\" (40 times), ), \"_r_\" (33 times), and \"_b_\"(12 times). In another experiment, we contrasted two different types of communication systems by performing tests over full real messages. The first test is based on a text description written in English and the second in test based on the text file associated to music coded using the MIDI format. The English text is a speech by [PERSON] given in 1950 during the Nobel Prize ceremony. The MIDI music is a version of the 4 th movement of [PERSON]'s ninth symphony. The sizes of these descriptions are near the limit of applicability of the algorithm. English descriptions of 1300 words or less can be processed in short times of less than a minute. Larger English texts have to be segmented using the control computational complexity criteria mentioned in Section 3.6 to reach reasonable working times. [PERSON]'s speech was fractioned in seven pieces. For MIDI music files, the processing times show an attitude of sharp increase starting for music pieces lasting about 3 min. The version of 4 th movement of [PERSON]'s ninth symphony used, is a 25 min long piece. It was necessary to process it by fractioning in 20 segments. To reveal the differences of descriptions when observed at different scales, symbol frequency distributions were produced. For the English text, characters, words and the fundamental scale were applied. For the MIDI music text distributions at character and fundamental scale were constructed. Words do not exist as scale for music. The corresponding detailed set of fundamental symbols can be seen in Appendix B. The frequency distributions were ordered upon the frequency rank of the symbols, thus the obtained were Zipf's profiles. Table 2 shows the length \(N\), the diversity \(D\) and the entropy \(h\) obtained for these two descriptions analyzed at several scales and Figure 3 shows the corresponding [PERSON]'s profiles for [PERSON]'s speech English speech and [PERSON]'s 9 th Symphony's 4 th movement. Both descriptions' profiles are presented at the scales they were analyzed: character-scale and the fundamental scale for both, English and music, and the word-scale only for English.