## 4 The Relationship between the Interpretation Process and Scale
### A Model to Quantify the Information of Different Interpretations  
Consider a description \(\mathcal{M}\) as the sequence of symbols \(A_{i}\) all of them contained in the alphabet \(\mathcal{A}=\{A_{1},\ A_{2},\ A_{3},\ \ldots,\ A_{n}\}\). Thus, \(\mathcal{M}\) can be seen as a specific sequence of the letters (or characters) of alphabet \(\mathcal{A}\). A descriptor of \(\mathcal{M}\) is the probability distribution \(P(A_{i})\) where \(A\) is the random variable associated with the frequency with which each letter of alphabet \(\mathcal{A}\) is encountered within message \(\mathcal{M}\). Then, using \(f_{A_{i}}\) to represent the number of times the letter \(A_{i}\) appears within the message, we can model message \(\mathcal{M}\) writing:  
\[\mathcal{M}:\ P(A)=\{P(A=A_{1}),P(A=A_{2}),\ldots,\ P(A=A_{i}),\ \ldots,\ P(A=A_{n})\}=\{f_{A_{1}},f_{A_{2}},\ \ldots,f_{A_{n}}\}. \tag{1}\]  
Then, the entropy \(h_{\mathcal{A}n}\) associated to the message \(\mathcal{M}\) when observed as the \(n\) different letters of alphabet \(\mathcal{A}\), equals the entropy of distribution \(P(A)\),  
\[h_{\mathcal{A}n}=\ -\sum_{i=1}^{n}f_{A_{i}}\ \log_{n}f_{A_{i}}. \tag{2}\]  
The figure computed in Expression (2) holds for the entropy obtained when reading the message one letter at a time; each letter would be considered as a symbol and no symbol would be allowed to \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Nature/Dimensions** & **Language Group** & **Examples** & **Alphabet Components** & **Enodings** \\ \hline \hline \multirow{8}{*}{\begin{tabular}{} \end{tabular} } & \multirow{8}{*}{Music} & \multirow{8}{*}{Music} & Chromatic scale & Pentagram Notation \\  & & Western music & Major scale, Minor scales & \\  & & & Pentantic scale & \\  & & Indu-raga music & Other Scales & \\  & & & Sound Volume & \\  & & & & Alteration signs & \\  & & Chinese music & Rhythms & \\  & & & Tempos & \\ \cline{3-5}  & & African Traditional & & \\  & & Music & Rhythms, Tempos & \\ \cline{2-5}  & & Alarms & Stridency of harmonies & Characters of a recording \\  & & & Ringing tones & Ascending or descending toral & Amplitude and Frequency of \\  & & Warning Calls & patterns & analog recording \\ \hline \hline \multirow{8}{*}{\begin{tabular}{} \end{tabular} } & \multirow{8}{*}{2D graphical expressions} & \multirow{8}{*}{2D graphical expressions} & Color spectrum & Pixel Size \\  & & & Brush Strokes & Pixel Color \\ \cline{1-1} \cline{3-5}  & & & & \\ \cline{1-1} \cline{3-5}  & & Charts \& Diagrams & Geometric Shapes & Shapes, colors, borders \\ \hline \multirow{8}{*}{\
\begin{tabular}{} \end{tabular} } & Numerical system, 1 dimension & Numbers & Digits and decimal point & 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, \(\cdot\cdot\) \\  & & & & Binary, any other base \\ \cline{1-1} \cline{3-5}  & & Equations, & Mathematical operators & Conventional Math \\ \cline{1-1} \cline{3-5}  & 1+ dimensions & Math Expressions & Char-strings declared as Symbols & ASCII code \\ \cline{1-1} \cline{3-5}  & & & & Binary \\ \hline \multirow{8}{*}{
\begin{tabular}{} \end{tabular} } & \multirow{8}{*}{Genomes} & ADN & Adenine, Thymine, Cytosine, & \multirow{8}{*}{A, T, C, G} \\  & & & Guanine & \\ \cline{1-1} \cline{3-5}  & & & & \\ \cline{1-1} \cline{3-5}  & & RNA & Adenine, Uracil, Cytosine, & \\ \cline{1-1} \cline{3-5}  & & & & \\ \cline{1-1} \cline{3-5}  & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._be formed by more than a single letter. This is what is called _'reading at the scale of letters'_. We know this would be a very ineffective way of reading. Fortunately, there are better ways of reading message \(\mathcal{M}\) by grouping neighbor characters \(A_{i}\) to form \(D\) different multi-character symbols \(Y_{j}\). This approach may be effective in the sense of reducing the associated entropy.