# A Proposal about the Meaning of Scale, Scope and Resolution in the Context of the Information Interpretation Process

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## Abstract

**:**

## 1. Introduction

- Section 4. The interpretation process is hypothesized over the selection of symbols with the criterion of minimizing the description’s entropy, computed over a written version of the message.
- Section 5. The proposed concepts of scale, resolution and scope are used to evaluate descriptions of different nature. Five experiments are performed to evaluate the information content of each experiment’s description seen at different scales of observation.

## 2. Resolution, Scale, Scope and Other Properties of Descriptions

#### 2.1. Resolution

#### 2.2. Scale

#### 2.3. Scope

#### 2.4. Symbols, Alphabet and Encoding

## 3. More about the Meaning of Scale

#### 3.1. An Intuitive Notion of Scale

#### 3.2. The Alphabet as a ‘Coding’ Tool

## 4. The Relationship between the Interpretation Process and Scale

#### 4.1. A Model to Quantify the Information of Different Interpretations

**Example 1**

**(E1).**

**Example 2**

**(E2).**

**Example 3**

**(E3).**

#### 4.2. The Entropy Reduction Process

**Case 1. Symbol Splitting:**Split symbol ${Y}_{j}$ into two smaller symbols. In this case, the object symbol ${Y}_{j}$ is replaced by new symbols ${Y}_{j}^{\prime}$ and ${Y}_{j+1}^{\prime}$. The sub-indexes of all successive symbols will increase by one. The new symbolic diversity also increases from $D$ to $D+2$. One could think that symbol ${Y}_{j}$ as the only instance of that symbol in the entire text. If that were the case, by replacing it with ${Y}_{j}^{\prime}$ and ${Y}_{j+1}^{\prime}$, it would generate only one additional symbol. However, this seems very unlikely because, in order to be part of a minimal entropy set of symbols, ${Y}_{j}$ should appear more than once in the message. Another possibility is that the new symbols ${Y}_{j}^{\prime}$ and ${Y}_{j+1}^{\prime}$ already exist within the message. This is also unlikely because the minimal entropy criterion has already ‘decided’ to represent the sequence ${Y}_{j}^{\prime}{Y}_{j+1}^{\prime}$ with the ‘compacted’ form ${Y}_{j}$. Concluding, for Case 1 we can use $V=D+2$ as the symbolic diversity for this message interpretation. Example 4 (E4) illustrates this situation:

**Example 4**

**(E4).**

**Case 2. Symbols Boundary Drifting:**We refer to ‘boundary drifting’ as the effect produced when the end and start of neighbor symbols shifts, leaving constant the total length of both symbols but adding to a symbol the same number of characters diminished on the other. The boundary between two adjacent symbols ${Y}_{j}$ and ${Y}_{j+1}$ moves making one symbol larger and the other smaller and resulting in new symbols ${Y}_{j}^{\prime}$ and ${Y}_{j+1}^{\prime}$. The sub-indexes of symbols do not shift but the new symbolic diversity increases from $D$ to $D+2$ due to the birth of new symbols ${Y}_{j}^{\prime}$ and ${Y}_{j+1}^{\prime}$ which did not exist before. Again, as reasoned for Case 1, if the symbols ${Y}_{j}^{\prime}$ and ${Y}_{j+1}^{\prime}$ had existed before, they would likely appear as minimal entropy symbols ${Y}_{j}$ and ${Y}_{j+1}$, prior to the boundary drifting. Thus, for Case 2 we can use $V=D+2$ as the symbolic diversity for this message interpretation.

**Example 5**

**(E5).**

**Case 3. Symbols Joining:**Two adjacent symbols ${Y}_{j}$ and ${Y}_{j+1}$ are replaced by one larger symbol ${Y}_{j}^{\prime}$. Depending on the number of instances of symbols ${Y}_{j}$ and ${Y}_{j+1}$, the symbolic diversity may reduce by one or two symbols, or may not reduce at all. Thus, the new diversity will be in the interval $V=\left[D-2,D\right]$.

#### 4.3. The Interpretation Process

#### 4.4. The Scale Is a Choice

#### 4.5. The Fundamental Scale

#### 4.6. Scale Downgrading

## 5. Some Experiments with Different Language Expressions

#### 5.1. Natural Languages

#### 5.2. Same Symbolic Structure: Different Perceptions

#### 5.3. Partial Changes of Resolution and Scope

#### 5.4. The Impact of Reorganizing

#### 5.5. Music

## 6. Discussion

#### 6.1. Advantages of a Quantitative Notion of Scale

#### 6.2. The Fundamental Scale as a Language Descriptor

#### 6.3. Usefulness and Applications of the Concept of Scale

## 7. Conclusions

**Scale**: The set of different symbols used in a description. The scale can be numerically expressed as the symbolic diversity $D$ of the system’s description interpretation.

**Scope**: The total number of symbols used in a description.

**Resolution**: The density of symbols (alphabet-symbols or encoded symbols) used to create the symbols used in a description.

## Conflicts of Interest

## References

- Heylighen, F. Modelling Emergence, World Futur. J. Gen. Evol.
**1991**, 31, 89–104. [Google Scholar] [CrossRef] - Bar-Yam, Y. A mathematical theory of strong emergence using multiscale variety. Complexity
**2004**, 9, 15–24. [Google Scholar] [CrossRef] - Bar-Yam, Y. Multiscale Complexity/Entropy. Adv. Complex Syst.
**2004**, 7, 47–63. [Google Scholar] [CrossRef] - Ryan, A. Emergence is coupled to scope, not level. Complexity
**2007**, 13, 67–77. [Google Scholar] [CrossRef] - Prokopenko, M.; Boschetti, F.; Ryan, A.J. An information-theoretic primer on complexity, self-organisation and emergence. Complexity
**2008**, 15, 11–28. [Google Scholar] [CrossRef] - Fernandez, N.; Maldonado, C.; Gershenson, C. Information Measures of Complexity, Emergence, Self-Organization, Homeostasis and Autopoiesis; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Turner, M.G.; O’Neill, R.V.; Gardner, R.H.; Milne, B.T. Effects of changing spatial scale on the analysis of landscape pattern. Landsc. Ecol.
**1989**, 3, 153–162. [Google Scholar] [CrossRef] - Benz, U.C.; Hofmann, P.; Willhauck, G.; Lingenfelder, I.; Heynen, M. Multi-resolution, object-oriented fuzzy analysis of remote sensing data for GIS-ready information. ISPRS J. Photogramm. Remote Sens.
**2004**, 58, 239–258. [Google Scholar] [CrossRef] - Febres, G.; Jaffe, K. A Fundamental Scale of Descriptions for Analyzing Information Content of Communication Systems. Entropy
**2015**, 17, 1606–1633. [Google Scholar] [CrossRef] - Piasecki, R.; Plastino, A. Entropic descriptor of a complex behaviour. Physics A
**2010**, 389, 397–407. [Google Scholar] [CrossRef] - Zipf, G.K. Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology; Addison-Welesly: New York, NY, USA, 1949. [Google Scholar]
- Rissanen, J. Modelling by the shortest data description. Automatica
**1978**, 14, 465–471. [Google Scholar] [CrossRef] - Hansen, M.H.; Yu, B.; Mark, H.H.; Bin, Y. Model Selection and the Principle of Minimum Description Length. J. Am. Stat. Assoc.
**2001**, 96, 746–774. [Google Scholar] [CrossRef] - Febres, G.; Jaffe, K. Calculating entropy at different scales among diverse communication systems. Complexity
**2016**, 21, 330–353. [Google Scholar] [CrossRef]

**Figure 1.**The profile representing a Musical Instrument Data Interface (MIDI) version of Beethoven’s 9th Symphony 3rd Movement. (

**a**) shows the full fundamental scale description having 2828 different symbols; (

**b**,

**c**) represent the degraded versions of the same profile with 513 and 65 symbols respectively.

**Figure 2.**Entropy $h$ vs. description length $L$ in symbols. Graphs show the relationship between entropy $h$ and scope $L$ (symbol length) for descriptions expressed in: (

**a**) English; (

**b**) Spanish. The effect of the observation scale over the resulting observed entropy is shown for three observation scales: characters, words and the fundamental scale.

**Figure 3.**Two perceptions of a 2D mosaic with a resolution 56 × 56 pixels. Mosaic (

**a**) shows pixels with four different colors. Same color pixels are grouped and separated by white pixels forming triangles; Mosaic (

**b**) shows vertical and horizontal white lines dimmed to gray.

**Figure 4.**Effects of changes of resolution and scope over a 2D representation of polygons. Graphic representation of a language scale downgrading from scale $D$ to scale $S$ ($S<D$). The total number of symbols at scale $D$, representing $D$ different symbols on graphs (

**a**,

**b**) (at the top of the Figure), are transformed into $S$ different symbols when the language is represented on the scale $S$, as in the graphs (

**c**,

**d**) (at the bottom of the Figure). Also, graphs (

**b**,

**d**) (on the right side of the Figure) exhibit greater scope $L$ than graphs (

**a**,

**c**) (on the left side of the Figure).

**Figure 5.**Four interpretations of the same distribution of 30 squares colored with five different tones of blue. Numbers indicate each tone used. The lightest is represented by 1 and the darkest with 5. Each tone appears with the same frequencies in the four graphs. (

**a**) Shows the 30 squared randomly ordered; (

**b**) The squares are ordered according to the rule indicating that no darker square can appear below or at the right of another lighter square; (

**c**) Shows groups of symbols formed by a regular shaped lattice of 1 × 6 bricks; (

**d**) Shows with black borders the groups of squares forming symbols to reduce the entropy of this interpretation.

**Figure 6.**A tiny fraction of the text which constitutes the Beethoven’s 5th symphony 1st movement segment interpreted with orchestra. Extracted from an .MP3 sound recorded file.

**Table 1.**Examples of Languages of different nature and their corresponding writing and encoding systems.

Nature/Dimensions | Language Group | Examples | Alphabet Components | Encodings |
---|---|---|---|---|

English | Conventional writing | |||

Spanish | letters | Binary | ||

Alphabetic | Russian | Braille | ||

Natural Languages | French | Punctuation signs | Morse Code | |

1 Dimension | Arabic | ASCII code | ||

Chinese | ||||

Syllabic | Korean | Phonograms | ||

Japanese | ||||

Chromatic scale | Pentagram Notation | |||

Western music | Major scale, Minor scales | |||

Pentatonic scale | ||||

Indu-raga music | Other Scales | |||

Music | Sound Volume | |||

Alteration signs | Cypher | |||

Chinese music | Rhythms | |||

Sonic | Tempos | |||

1 Dimension | ||||

African Traditional Music | Rhythms, Tempos | |||

Alarms | Stridency of harmonies | Characters of a recording | ||

Alarm and Warning signals | Ringing tones | Ascending or descending tonal patterns | Amplitude and Frequency of analog recording | |

Warning Calls | ||||

Color spectrum | Pixel Size | |||

Painting | ||||

Graphic | Brush Strokes | Pixel Color | ||

2 Dimensions | 2D graphical expressions | |||

Charts & Diagrams | Geometric Shapes | Shapes, colors, borders | ||

Numerical system, 1 dimension | Numbers | Digits and decimal point | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ‘.’ Binary, any other base | |

Mathematics 1+ Dimensions | Equations, 1+ dimensions | Math Expressions | Mathematical operators Char-strings declared as Symbols | Conventional Math |

ASCII code | ||||

Binary | ||||

ADN | Adenine, Thymine, Cytosine, Guanine | A, T, C, G | ||

Biology | Genomes | |||

1 Dimension | RNA | Adenine, Uracil, Cytosine, Guanine | A, U, C, G |

**Table 2.**Properties of each interpretation of 2D patterns shown in Figure 3.

Figure | Figure 3a,b | Figure 3a | Figure 3b |
---|---|---|---|

Scale Name | Pixels | Symbols | Symbols |

Data Representation | Pixels | Triangles | Diagonal Bands |

Resolution Rhorz | 56 | 3 | 6 |

Resolution Rvert | 56 | 3 | 1 |

Resolution Rangle | - ^{1} | 4 | 1 ^{2} |

Resolution Rcolor | 4 | 4 ^{2} | 2 |

Scope (Length) L | 3136 | 36 | 6 |

Scale (Diversity) D | 4 | 4 ^{3} | 2 ^{†} |

Entropy h [0 to 1] | 1 | 1 | 1 |

Specific diversity d | 0.001 | 0.111 | 0.333 |

^{1}This degree of freedom does not exist for single pixels;

^{2}Only four angular positions are required;

^{3}Triangles: light blue, light green, dark blue, dark green;

^{†}Light band, dark band.

**Table 3.**Balance of information for the 2D example presented Figure 4.

Figure | Figure 4a | Figure 4b | Figure 4c | Figure 4d | ||||
---|---|---|---|---|---|---|---|---|

Scale Name | Pixels | Symbols | Pixels | Symbols | Pixels | Symbols | Pixels | Symbols |

Data Representation | 0’s & 1’s | Polygons | 0’s & 1’s | Polygons | 0’s & 1’s | Polygons | 0’s & 1’s | Polygons |

Resolution Rhorz | 28 | 28 | 46 | 46 | 13 | 13 | 23 | 23 |

Resolution Rvert | 28 | 28 | 28 | 28 | 13 | 13 | 13 | 13 |

Resolution Rangle | - ^{1} | 8 ^{2} | - ^{1} | 8 ^{2} | - ^{1} | 8 ^{2} | - | 8 |

Scope (Length) L | 784 | 8 ^{2} | 1288 | 10 | 169 | 8 | 299 | 8 |

Scale (Diversity) D | 2 | 5 ^{3} | 2 | 3 ^{†} | 3 | 4 ^{††} | 3 | 5 ^{†††} |

Entropy h [0 to 1] | 0.970 | 0.928 | 0.999 | 0.646 | 0.934 | 0.813 | 0.940 | 0.861 |

Specific diversity d | 0.003 | 0.625 | 0.002 | 0.300 | 0.018 | 0.500 | 0.010 | 0.625 |

^{1}This degree of freedom does not exist. The concept of single pixels’ angular position degenerates;

^{2}Only eight angular positions are required to describe symbols represented;

^{3}Square, triangle, trapezoids, large rectangle, small rectangle;

^{†}Square, triangle, trapezoids;

^{††}Large Triangle, rectangle and noise: trapezoids, stairs-like polygon;

^{†††}Large Triangle, small triangle, wedge and noise: trapezoids, stairs-like polygon.

**Table 4.**Properties of several interpretations of the 2D patterns shown in Figure 5.

Figure | Figure 5a | Figure 5b | Figure 5c | Figure 5d |
---|---|---|---|---|

Scale Name | Symbols | Symbols | Symbols | Symbols |

Data Representation | Single Squares | Single Squares | Organized Squares | Organized Squares |

Resolution Rhorz | 6 | 6 | 2 | 3 |

Resolution Rvert | 5 | 5 | 5 | 3 |

Resolution Rcolor | 5 ^{1} | 5 ^{1} | Varies ^{2} | Varies ^{3} |

Scope (Length) L | 30 ^{4} | 30 ^{4} | 10 | 16 |

Scale (Diversity) D | 5 ^{1} | 5 ^{1} | 7 ^{†} | 6 ^{††} |

Entropy h [0 to 1] | 0.943 | 0.943 | 0.970 | 0.812 |

Symb. Info. Yh [bits] | 0.167 | 0.167 | 0.700 | 0.375 |

^{1}Different colors for 1 × 1 array of squares;

^{2}Approximation of different combinations of ordered 3 × 1 squares;

^{3}Different combinations of 2 × 2, 2 × 1, 1 × 2 and 1 × 1 arrays of ordered squares;

^{4}30 = 1’s + 2’s + 3’s + 4’s + 5’s = 11 + 6 + 5 + 3 + 5;

^{†}5 5 4 | 5 4 3 | 3 3 2 | 3 2 2 | 3 2 1| 2 1 1 | 1 1 1;

^{††}5 5 − 5 5 | 4 3 | 3 2 − 2 1 | 5 | 4 | 1.

**Table 5.**Effects of different observation scales over the quantity of information of a segment of Beethoven’s 5th Symphony versioned by a full orchestra and piano solo.

Beethoven’s: | 5th Symphony. 1st Mov. Segment. Orch | 5th Symphony. 1st Mov. Segment. Piano | ||||
---|---|---|---|---|---|---|

Scale Name | Binary | Characters | Fundamental | Binary | Characters | Fundamental |

Data Representation | Zeroes and Ones | Letters, Punct. and Other Signs | Recognized Min. Entropy Symbols | Zeroes and Ones | Letters, Punct. and Other Signs | Recognized Min. Entropy Symbols |

Resolutn. R (symb./s) | 188,948 | 23,618 | 4517 | 192,669 | 24,084 | 4241 |

Scope LD | 5,668,432 | 708,554 | 135,519 | 7,514,080 | 939,260 | 165,387 |

Scope L2 | 5,668,432 | 5,668,432 | 1,084,152 | 7,514,080 | 7,514,080 | 1,323,096 |

Scale value Diversity D | 2 (0 & 1) | 252 | 4635 | 2 (0 & 1) | 257 | 13,808 |

Symbolic entropy h | near 1 | 0.990 | 0.893 | near 1 | 0.990 | 0.722 |

Specific Diversity d | 0.049 | 0.00036 | 0.03420 | 0.049 | 0.00027 | 0.08349 |

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Febres, G.L.
A Proposal about the Meaning of Scale, Scope and Resolution in the Context of the Information Interpretation Process. *Axioms* **2018**, *7*, 11.
https://doi.org/10.3390/axioms7010011

**AMA Style**

Febres GL.
A Proposal about the Meaning of Scale, Scope and Resolution in the Context of the Information Interpretation Process. *Axioms*. 2018; 7(1):11.
https://doi.org/10.3390/axioms7010011

**Chicago/Turabian Style**

Febres, Gerardo L.
2018. "A Proposal about the Meaning of Scale, Scope and Resolution in the Context of the Information Interpretation Process" *Axioms* 7, no. 1: 11.
https://doi.org/10.3390/axioms7010011