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Create complete phoneme databases of historical languages
With relative effort and simple means, it is possible to create databases for language research that expand understanding of languages that have been lost or are still researched.
In this topic I would like to explain step by step how, with appropriate (manual) effort, databases can be created, for example for phonemes, syllables, pronunciation, etc. created using conventional office software (complex spreadsheet documents).
The purpose of this project is to demonstrate one area of possibilities in which mathematics (combinatorics) can support language research. This is based on the assumption that a database with, for example, all possible linguistic and textual expression options can expand and enhance one's own engagement with a researched language. For reasons of effort, the scope of this project must be limited to texts or text modules with very few placeholders (e.g. phonemes or syllables). However, the principle can be expanded accordingly with appropriate technical know-how or human resources.
The connections to be described below are equally suitable for gaining an insight into the combinatorics that are useful for deciphering texts and is therefore necessary (cryptological basics). Since the overall topic is very complex, I needed several attempts here in the forum to find a suitable form for this context.

The Sumerian language and its associated basic phonemic inventory
(This topic cannot possibly provide an overview of the overall aspects of the Sumerian language. Please refer to the relevant sources listed in the bibliography, for example.)
The Sumerian language is well suited to provide a formal introduction to the topic of this post: the phoneme inventory of the Sumerian language has (to my knowledge and with reference to the sources I used) four vowels (/a e i u/) opposite 16 consonants, which makes the Sumerian language "quite simple" [gwiki1]. The advantage of this combination and number of vowels and consonants - from a mathematical point of view - is that a combination of 4 + 16 elements can be viewed in a very clear combinatorial manner - and therefore easy to represent. This helps to explain mathematical and combinatorial (cryptological) connections on the topic well and clearly.

Combinatorial basics for creating databases for e.g. phonemes or syllables of a language
From a combinatorial perspective, the (possible and special principle described here) of creating a database - e.g. for phonemes or syllables - of a language (explained here using the selected example of the Sumerian language) can be explained as follows. The procedure is split into various successive basic steps. These steps can all be done either by hand or in the form of computer table documents (please be sure to note the warnings in the disclaimer!):

Step 1:
In order to create the database, the analyzed language and - ideally a completely known phoneme inventory* - must first be determined. This definition also determines the number of elements to be mathematically (combinatorically) combined with one another as well as the resulting “chain combinations”.
Due to the scope required, I will only address the special case of (to our current knowledge) "completely known" language phoneme inventories in passing. For the sake of simplicity, at this point in the discussion I am assuming that the phoneme inventory of the Sumerian language known to us today - to our current knowledge - has been fully and completely deciphered (However, as an actual amateur and non-linguist, I could be wrong about this; my main focus here is on mathematical questions about the structure and deciphering of languages. In order to be able to explain the relevant procedures well, I need a database that is as clearly clarified as possible.).
If I continue to deal with the Sumerian language and its phoneme inventory, for the sake of simplicity (in order to be able to explain the entire procedure described here well) I will assume that the Sumerian language actually has a total of 20 phonemes (4 vowels and 16 associated ones).

Step2:
The specified number of elements is combined in chain combinations until the desired number of placeholders (more on this later) lined up one after the other is achieved: in order to get an introduction to the topic, In the following, I will initially use a smaller number of elements to explain the respective principle for combining elements that can be combined with each other (e.g. phonemes or e.g. also syllables) - there are different methods for this. In order to understand how the elements are combined with each other and what mathematical results result from this, it is necessary to deal with the mathematical concept of the Cartesian product (more on this later) and later with a principle - also in direct comparison with the principle of the Cartersian product - that I call the "telephone book principle".

Step3:
If all elements of a project (e.g. phonemes or syllables) - if technical; e.g. manually; possible - have been successfully combined with one another, the results can be combined into a specific database (more on the benefits and use of such a database later).

About the Cartesian product
The term Cartesian propuct is a term from the (so-called "naive") set theory (according to Cantor) [Reiss/Schmieder, 2007]; [gwiki2]. The Cartesian product describes the “crossed” (simple) combination of elements of two sets. The Cartesian product can be clearly explained using two equal sets (i.e. two sets that contain exactly the same elements). If we look at the sets A, B with A = {1, 2, 3} and B = {1, 2, 3}, we get a good basis for explaining the Cartesian product. At this point it´s necessary to explain that a mathematical set in the sense of set theory can contain an unordered specific number of elements. However a tuple is commonly spoken of in the mathematical sense when we consider an ordered number of elements: a set can therefore contain both an ordered and an unordered number of elements, a tuple always contains a number of ordered elements. For example, in the field of computer science, the ordered consideration of numbers of elements (tuples) is often used [Reiss/Schmieder, 2007].
For the combinatorial creation of phoneme databases for languages, for example, it makes sense, but is generally not important, whether and how we organize the phonemes of the phoneme inventory (here using the example of the Sumerian language with 4 vowels and 16 associated consonants): we can therefore use sets or tuples, just like we prefer it: In the context in which we define our own definition of order for a basic data base of elements, this order in the mathematical sense corresponds - in my opinion - not to a tuple (depending on the definition), but in any case to a specifically ordered set. However, due to these mathematical pitfalls when it comes to defining an "ordered number of elements", it is easier for me to choose mathematical ordering systems that are as close as possible - e.g. according to the order of numbers or letters in specific systems - so that I can call such ordered sets "tuples".
This step fulfills a purpose - which I will discuss later - but which does not change the overall result of "chain combinatorics": this is because in this topic I only explain the creation of complete element databases, i.e. all elements are combined with each other until all possible combinations have been exhausted. The orderly consideration of elements to be combined therefore fulfills its purpose more in a simplified mathematical consideration and in the subsequent possible assignment of elements in the form of an index (more on this later).
The purpose of such an index - as far as I know so far (which I must consciously formulate as a conjecture here) - is that "linguistic peculiarities", e.g. those of a specific author of historical documents, can be better assigned, among other things.
Elevating a defined number of elements to a Cartesian product means, so to speak, "crossing" every single element with every other element (e.g.) of a set (here this term is meant exclusively in the strictly mathematical sense).
We can therefore write in simplified terms, e.g.:

Cartesian product from A to B or A to A(1) or in the notation I use from A(0) to A(1) = AχB

Applied to a simple ("two-dimensional") tabular structure, this means that every element of the set/tuple A(0) is combined with every element of the set/tuple A(1). Important for "language research" in the sense of creating phonetic databases, for example, is to aim for A(0) = A(1) (the exact same elements should be contained in set/tuple A as in set/tuple A( 1). This condition does not necessarily have to be met in the sense of the topic described here, but it makes combinatorial work much easier. The result of applying this principle is therefore a very good, clear "data economy" (in the sense of "data economy"). This type the approach actually saves (experience has shown) time and concentration (when working), e.g.:

A(0) = {1, 2, 3} and A(1) = {1, 2, 3}

You can also write: "The sets A and B are equally powerful (because each of the two sets contains exactly the same elements)". Furthermore, when considering tuples it could be written: "Tuple A and tuple B have the same power and contain exactly the same elements with exactly the same number of elements.", e.g.:

(x1 ... x3)(0) = {1, 2, 3}
(x1 ... x3)(1) = {1, 2, 3}

This connection can be presented in a table, for example, as follows (here in notation with tuples) (see appendix 1).
(Note: The separation of elements with commas does not represent comma numbers in the sense of German spelling, but rather a list of elements that belong to one another.)
The elements ordered (here according to a certain scheme) arise from A(0)XA(1) as:

A(0)XA(1) = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}

which therefore represent the Cartesian product of A(0) and A(1).

Upon closer analysis of the Cartesian product of A(0)XA(1), two particularly noteworthy aspects emerge:
The table can be mathematically referred to as a square matrix or "matrix". Because the number space of natural numbers can be described with the properties of "multidimensionality", I also refer to such a matrix as "two-dimensional".
In the matrix, two reflecting areas (across the main diagonal of the matrix) can be identified, as well as the elements located on the main diagonal. If the matrix is broken down into cell contents in the form of a table and described, a matrix structure is created that can be divided into 3 main areas. I basically call these areas according to the areas: alpha (α), beta (β), gamma (γ) (see appendix 2). This classification and the names I have chosen here should in no way be viewed as fixed. Mathematical expressions are always open to discussion and this method of representation and naming is only an example. This classification and chosen name represents one of many different options and serves to better describe the matrix contents. In the following it is now possible to formally describe the matrix and its contents:

R(mxn) is a squarematrix if m = n (more on this later).

Applied to the possible combination of phonemes, for example, the principle of the Cartesian product with its specific results should not be confused with the following combinatorial notation, in which all elements of a set are crossed with all elements of the same set and represented in all possible notations ("dictionary notation" or "telephone book notation"), at:

{(a, b, c) ∈ A, (a, b, c) ∈ B}
A = {a, b, c}
B = {a, b, c}

linear combination options for A, B:

1, 1, 1 or a, a, a [line I]
1, 1, 2 or a, a, b [line II]
1, 1, 3 or a, a, c [line III]
1, 2, 1 or a, b, a [line IV]
1, 2, 2 or a, b, b [line V]
1, 2, 3 or a, b, c [line VI]
1, 3, 1 or a, c, a [line VII]
1, 3, 2 or a, c, b [line VIII]
1, 3, 3 or a, c, c [line IX]
2, 1, 1 or b, a, a [line X]
2, 1, 2 or b, a, b [line XI]
2, 1, 3 or b, a, c [line XII]
2, 2, 1 or b, b, a [line XIII]
2, 2, 2 or b, b, b [line XIV]
2, 2, 3 or b, b, c [line XV]
2, 3, 1 or b, c, a [line XVI]
2, 3, 2 or b, c, b [line XVII]
2, 3, 3 or b, c, c [line IXX]
3, 1, 1 or c, a, a [line XX]
3, 1, 2 or c, a, b [line XXI]
3, 1, 3 or c, a, c [line XXII]
3, 2, 1 or c, b, a [line XXIIII]
3, 2, 2 or c, b, b [line XXIV]
3, 2, 3 or c, b, c [line XXV]
3, 3, 1 or c, c, a [line XXVI]
3, 3, 2 or c, c, b [line XXVII]
3, 3, 3 or c, c, c [line XXVIII]

This notation is also of great importance for combintorics in the sense of "linguistic research" and e.g. deciphering historical texts with their mathematically resulting possibilities - and to say it in advance: it is the completely complete, but extremely complex method. With the small example of 3 elements to be combined, it produces a different number of combination possibilities, namely 3*3*3 = 3^3 = 27 possibilities, while elevating comparatively 4 elements to the Cartesian product to the Cartesian product (here "first stage", or 1st evolution) results in a number of 4*4 combinations = 16 combinations (when using 4 elements, basic base of elements that are raised to the Cartesian product).
The differences and effects to generate combinations between these two essential methods will be discussed later.
In short, what you may have already noticed: A Cartesian product (of two numbers of elements combined with each other always produces a square number of possible combinations, while the so-called “dictionary notation” always produces a number of possible combinations the size of a proportional number according to the principle of powers, e.g.:

Table of resulting combinations depending on the used principle:
comb = "combinations"
CP = Cartesian Product (here always "squared" Cartesian product)
TBP = Telephone Boon Principle
x = number of elements contained in the base of elements to be combined

[x] / [comb. CP] / [comb. TBP] // [line]
[1] / [1^2 = 1] / [1^1 = 1] // [line]
[2] / [2^2 = 4] / [2^2 = 4] // [line]
[3] / [3^2 =9] / [3^3 = 27] // [line]
[4] / [4^2 = 16] / [4^4 = 256] // [line]
[5] / [5^2 = 25] / [5^5 = 3125] // [line]
[6] / [6^2 = 36] / [6^6 = 46656] // [line]
[7] / [7^2 = 49] / [7^7 = 823543] // [line]
etc.etc.

As is clear from the specifically proportionally strong increase in results when using the (maximally complete, i.e. gap-free) method of the telephone book principle, one of the main objectives is to have databases that are as complete as possible for, for example, phonemes or syllables - or for the encoding or deciphering of texts, for example Number of possible combinations to analyze
as much as possible (and as far in advance of a project as possible). The application of the principle of the Cartesian product, among other essential methods, is suitable for exactly this purpose; set at a specifically sensible time in a project and of course used at the appropriate point.

The characteristic that a Cartesian product (with a single exception which will be discussed later - exclusively in a square matrix, i.e. with A = B) always generates a square number of possible combinations, is important in terms of data economy when creating complete tabular data lists (here databases with e.g. phonemes or syllabic databases) have a big advantage, which I will explain below.
First of all: both methods offer their own specific advantages and possible applications, which depend on what is to be achieved. Both methods can be combined with each other if this makes sense. However, I will discuss the different methods (and the possible combination of the two dirrefent methods in specific areas) one after the other and initially concentrate on the method of determining combinations using the Cartesian product principle.

Determine chain combinations by creating and combining Cartesian products
The big and special advantage for determining combinations (chain combinations) using the principle of the Cartesian product is that Cartesian products when applied to square matrices always output a square number of possible combinations, specifically sorted (please note the only exception here: I will explain further below). This connection can be used up to a certain resource-related limit when manually determining (in this case in tabular form) phoneme or syllable combinations (as well as letter combinations, for example). “Resources” here mean factors such as writing and computing time and the time that you are able to devote to concentrated work (which can of course vary greatly from person to person).
In order to better explain what this advantage is, a short excursion into the world of so-called figurable numbers and thus into the world of triangular numbers and square numbers as well as the Gaussian formula for triangular numbers is necessary.
As the young Gauss (1777 - 1855) [gWiki8], one of the most famous mathematicians in the world today and probably of all time, recognized at a young age, square numbers in the number space of natural numbers can always be formed according to the same scheme by combining two directly successive triangular numbers. Even if it is strongly assumed that the ancient Greeks, for example, were aware of this connection, Gauss is considered to be the first who verifiably published about this and summarized the principle in a formula [duSautoy,39].
The principle of forming square numbers can be explained relatively simply using two series of numbers (triangular numbers) placed next to each other. The formation of the triangular numbers in turn follows their very own - characteristic - recursivity, whereby the recursivity of both number series; of triangular numbers and square numbers; can be conclusively related to each other in a tabular overview of number series (this represents one of a few possible structural proofs, about which I cannot judge at this point, based on my current knowledge, to what extent they are generally known).

Squarenumbers and their formation:
at:

ℕ = {0, 1, 2, 3, 4, 5, 6, 7, ...}

x² = squarenumbers
x² ∈ ℕ = {0, 1, 4, 9, 16, 25, 36, 49 ...}

▲ = triangular numbers
▲ ∈ ℕ = {0, 1, 3, 6, 10, 15, 21, 28, 36 ...}

Evolution of triangular numbers in ℕ:
0+1 = 1
1+2 = 1
1+2+3 = 6
1+2+3+4 = 10
1+2+3+4+5 = 15
1+2+3+4+5+6 = 21
1+2+3+4+5+6+7 = 28
1+2+3+4+5+6+7+8 = 36

alternatively you can write e.g:
▲(1) = ∑(0...1) = 1
▲(2) = ∑(0...2) = 3
▲(3) = ∑(0...3) = 6
▲(4) = ∑(0...4) = 10
etc.etc.

Another way I use for triangular numbers is:
▲∑(1) = 1
▲∑(2) = 3
▲∑(3) = 6
▲∑(4) = 10
etc.etc.

Evolution of square numbers in ℕ:
0 + 1 = 1
1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25
15 + 21 = 36
etc.etc.

The developmental recursivity of triangular numbers and square numbers:
As a structural proof of the recursivity of triangular numbers and square numbers, the following connection essentially represents a modified excerpt from Pasqual's triangle (see Appendix 3).
The graphical representation is suitable for clarifying the structural connection between triangular numbers and square numbers (see Figure 4; slight distortions in the representation are possible due to screen technology and formatting technology, so that the figures shown are not 100% correctly displayed as square figures).

Developmental connection in the defined matrix between triangular numbers and square numbers
The graphical representation shows how triangular numbers and square numbers can be viewed and defined quantitatively in their mutually dependent, definable evolution: a quantitative part of the larger specific triangular number, which forms a correspondingly specific square number with a preceding specific triangular number, can be quantitatively assigned to the main diagonal of one square matrix and thus the elements lying on the main diagonal of a specific square matrix means: the elements located in the square matrix previously defined here with the specific designation (gamma) can be set theoretically 8depending on the point of view) either of the element set (alpha or the set of elements (beta) can be assigned in the matrix. This allows statements about the formation of specific sets to be described in a simplified manner. However, in view of the benefit of the defined matrix, set theory makes it easier to identify the elements located on the main diagonal of the defined specific matrix the fraction (gamma) should be viewed as an independent set.
With regard to the formation of square numbers from directly successive specific triangular numbers, it makes sense to introduce a further consideration of the square figure: a specific square figure with side length x can be designed with x² elements (e.g. square pieces of cardboard or mussel shells arranged in a square). This definition establishes the connection to the figured numbers, which was already intensively researched and discussed by the ancient Greeks. By referring to the figurable numbers, it can also be directly transferred to calculating with the matrix defined here:
Set-theoretically, of a number of x² elements (which are each specifically formed from the sets Alpha, Beta, Gamma), a number of to express how square numbers can be formed from triangular numbers. Furthermore, further fundamental statements about square numbers (and thus square figures) are possible, which can be directly transferred to the handling of the matrix defined here:

statement: the elements contained in the set Alpha+Beta+Gamma of a square matrix can always be divided according to the following key if we define and view the square matrix as a (square) Cartesian product: the elements contained in the Alpha and Beta fraction regions of the matrix reflect each other in a definable way, while the elements in the Gamma fraction region each have a specific unique selling point if we consider the elements of the Matrix as sets and their elements. This results in set theory:

This set-theoretic connection can be used to create, for example, phonemic or syllabic databases in order to achieve the most efficient data economy.

EXAMPLE:
If a number of 4 different elements are raised to form a (square) Cartesian product, the following possible combinations arise, e.g.:

A(0), A(1) = {1,,2 3, 4}
A(0) = A(1)

resulting combinations (in a specially defined sorting):
(see appendix 6)

{A(0)XA(1) ∈ ℕ / (a, b, c, d), (a = 1, b = 2, c = 3, d = 4), (a, b, c, d ∈ A(0), a, b, c, d ∈ A(1) / A(0)XA(1) = (Α,Β,Γ)}
A(0)XA(1) ∈ ℕ - beta = {(1,1), (2,1), (3,1), (4,1), (2,2), (3,2), (4,2), (3,3), (4,3), (4,4)}

When (manually) creating databases for, for example, phonemes and syllables of a specific language, a - specific - considerable amount of data (according to a fixed calculation key) can be saved if the "backward reading" option is used.
In this context, "reading backwards" is actually meant in the literal sense: with the appropriate disposition or experience and training, it is possible to recognize the meaning of text even in texts written backwards. Experienced readers can make use of this effect when creating, for example, phoneme databases based on the principle of using the Cartesian product of basic data sets.
Creating a database (here as an example) for phonemes or syllables, for example, using the principle of the Cartesian product works according to the method described below:

Create a database manually, e.g. for phonemes or syllables, using the principle of the Cartesian product:
In order to create a database manually, for example for phonemes or syllables, using the principle of the Cartesian product, it is necessary to implement the following steps for the principle of creating a database described here:

Step 1: Define the basic inventory of phonemes or syllables.
Step 2: Collect elements of the basic inventory to form a (square) Cartesian product.
Step 3: Eliminate unnecessary data when using the “read backwards” option or save it from the start.

(Note: Steps 2 and 3 are repeated frequently depending on the desired combination result in the sense of "chain combinations" - more on this later.)

To the individual steps:

Step 1: Define the basic inventory of phonemes or syllables
For the following application example, I define a selected (here fictitious) basic inventory of syllables. The individual syllables as elements of the set A(ß) with altogether 4 different elements are here:
1 = bo
2 = kun
3 = tun
4 = tos

A(0) = {bo, kun, tun, tos}

The individual syllables can optionally be represented as numbers in a square matrix. If A(0) and A(1) are now raised to the Cartesian product at A(0) = A(1), the corresponding (all) possible combinations of the syllables arise in the resulting square matrix (see appendix 7 and 8).

How useful this approach can be for creating databases with complete combination numbers of, for example, phonemes or syllables becomes clear when (also fictional) translations (here in the example into English) are assigned to the fantasy syllables, e.g.:

1 = bo = "boat"
2 = kun = "far"
3 = do = "wide water"
4 = tos = "move"

The (complete and gap-free combinations) created in the square matrix as a (square) Cartesian product can be compared to the reduced set of elements: it becomes clear here that the textual contexts of meaning can also be grasped (with appropriate skills or training), if they are recorded reading backwards:

Overall combinations determined in the example used here for four different (autonomous) syllables (here also “solitary”):
(listed in the specific sorting selected here)

Basic inventory of defined solitaires (here syllables) and their assignment:
1 = bo = "boat"
2 = kun = "far"
3 = tun = "wide water"
4 = tos = "move"

Resulting syllable constellations in A(0)XA(1) (squared Cartesian Product):
centered:
I: bo,bo
II: kun,kun
III: tun,tun
IV: tos,tos
- - -
{regular} (centered) [mirrored]
I: {kun,bo} [bo,kun]
II: {tun,bo} [bo,tun]
III: {tos,bo} {tun,kun} [kun,tun] [bo,tos]
IV: {tos,kun} [kun,tos]
V: {tos,tun} [tun,tos]

Resulting syllable constellations in A(0)XA(1) (squared Cartesian Product) "translated":
centered:
I: boat,boat
II: far,far
III: wide water,wide water
IV: move,move
- - -
{regular} (centered) [mirrored]
I: {far,boat} [boat,far]
II: {wide water,boat} [boat,wide water]
III: {move,boat} {wide water,far} [far,wide water] [boat,move]
IV: {move,far} [far,move]
V: {move,wide water} [wide water,move]

The meaning of this combinatorial and combination-eliminating approach becomes clear when we consider the data economic effect of this approach in chain combinations.

Chain combinatorial data economics in (square) Cartesian products
The previous explanations have shown that when determining the number of combinations in square matrices using the principle of (square) Cartesian products, elements (and thus data) can be eliminated if the "read backwards" option is used in the result output. This “data economy” or “data efficiency” can be precisely calculated in terms of an “ideally efficient data yield”. The corresponding calculation options for triangular numbers and square numbers are used for this purpose. These relationships can be clearly illustrated in a table overview:

Data economic yield when applying the read-back option in square Cartesian products:
generally for squarematrixes A(0)XA(1); "elements" = e.g. phonems or syllables

Tabloe-columns:
[C1] = [elements; x]
[C2] = [combinations; x²]
[C3] = [data saving (combinations; proportionate)]
[C4] = [data saving (combinations; fractional)]
[C5] = [data saving (combinations; percent)]

[C1] / [C2] / [C3] / [C4] / [C5] // [line]
[x] / [x²] / [prop.] / [fract.] / [%]
- - - - - - - - - - - - - - - - - - - - - - - - - -
[1] / [1] / [0] / [-] / [0.00%] // [line I]
[2] / [4] / [1] / [1/4] / [25.00%] // [line II]
[3] / [9] / [3] / [3/9] / [33.33p%] // [line III]
[4] / [16] / [6] / [6/16] / [37.50%] // [line IV]
[5] / [25] / [10] / [10/25] / [40.00%] // [line V]
[6] / [36] / [15] / [15/36] / [41.66p%] // [line VI]
[7] / [49] / [21] / [21/49] / [42.85...%] // [line VII]
[8] / [64] / [28] / [28/64] / [43.75%] // [line VIII]
[9] / [81] / [36] / [36/81] / [44.44p%] // [line IX]
[10] / [100] / [45] / [45/100] / [45.00%] // [line X]
[...] / [...] / [...] / [...] / [...] // [...]

This proportionally alternating data reduction, which alternates against the absolute limit <1/2 or <50%, can be used for the chain-like determination of combinations for even more complex numbers of elements of a basic inventory (e.g. phonemes or syllables) of a selected language (and for others purposes).
For example, with a total of 20 (solitary) elements of a basic language inventory (see the Sumerian language and its linguistic inventory as far as we know today), the data savings with a simple (linear) application of the method described above is already a data saving of 47.50%. This factor of possible data savings increases accordingly if the method described above is adapted into chain operations. The total capacity of the data expansion possible (for simple-linear combination chains) is already 20 placeholders (elements, e.g. phonemes or syllables). The total capacity of the data expansion possible (for simple-linear combination chains) is already 47.50/(50/100) = 95% (more on this later).
How a chain-like formation of possible (complete) combinations of a specific basic inventory of elements (e.g. a specific language) can be linked together in a chain-combinatory manner - and thus used more effectively - is explained below:

Make data yield maximally efficient through chain combinations of Cartesian products
If the results of Cartesian products in chain combination series of selected specified elements are exploited in intermediate steps and collected into new Cartesian products, enormous - maximum efficient data savings can be achieved in this way (e.g. with regard to the determination of possible combinations for phonemes and syllables for the purpose of Creation of complete phoneme and syllable databases).
For this purpose, the results (as a specific "data yield" or combination yield) of a respective specific (square) Cartesian product - always using the "backward reading option" (and the resulting mathematical procedure described above) are created into a new base stem of elements for the collection of another (square) Cartesian product and so on. In specific chain operations, this process is repeated until the desired result of data or combination yield (here: "target") is achieved: For the generation of all possible combinations of, for example, 4 solitary basic elements (e.g. phonemes or syllables), this means that this process has to be completed twice in total. This means: in order to determine all (complete) combination numbers of, for example, 4 solitaires using the backward reading option, it is necessary to carry out a Cartesian product twice in a row with the respective basic stem of elements. The basic base of elements to be raised to the self-sufficient (square) Cartesian product results from the first (square) Cartesian product. For a larger base of elements as a starting point for the first (square) Cartesian product, this process must be carried out specifically and frequently. The effort involved in data collection (e.g. for manually writing down elements and combinations (as a starting point for collecting (square) Cartesian products) increases proportionally. However, at the beginning (i.e. when collecting the first (square Cartesian product)) the effort required to generate data is specific proportionally still correspondingly low: thus, at the beginning of such chain operations, the foundations for the greatest possible (most efficient) data savings can be laid.
(later more).
Further data (e.g. when determining combinations of phonemes or syllables) can essentially be achieved by the chosen notation:
For example, an appropriately trained analyst can assign numbers to the selected specific basic inventory of phonemes or syllables and adapt this spelling to the evaluation and reading of texts. However, this step must be thought through carefully beforehand, because once trained, there is a possibility that an analyst will be strongly conditioned to the reading option acquired in this way and therefore it will only work well for a maximum of one specific language. For example, the (to our current knowledge) a total of 20 solitaires of the Sumerian language can be transcribed into numbers. The reader would simply have to learn to equate the number sequences chosen for transcribing with readable text (which in many cases requires appropriate training and.
practice may be possible). The advantage of such an approach, at least for writing (the readability of such "texts" is - as already mentioned - a different issue altogether) is data efficiency (more on this later).
The mathematical procedure used in the chain operations described above to generate possible (complete) combination databases using the principle of the (square) Cartesian product can also be explained very well using numbers for the designation of elements (theoretical is a designation Of course, this is also possible from the outset, e.g. with binary numbers; however, due to the scope required, this connection will not be explained in more detail here).

Example with a basic stem of 4 solitary elements
In order to generate all possible (complete) combinations with a basic stem of 4 solitary elements using the principle of the (square) Cartesian product, it is necessary to collect a (square) Cartesian product of specific elements twice in a row: In the first (square) In a Cartesian product, two elements are first connected to each other in a specific multiple manner. The result of such a first Cartesian product is a data set that is specifically (and maximally efficiently) reduced by using the “backward reading option”; a specific number of relevant combinations. This data set is combined into a new basic base of elements (and sorted specifically for this purpose). The selected sorting is subject to the selected and previously defined preferences (and can therefore be influenced for specific reasons).
After the generated 2nd basic stem of elements has been raised to the independent (square) Cartesian product, the possible (maximum efficient combinatorial total yield is generated, because in this second operation the K
In each operation, 2 previously combined elements (as independent, specific solitaires) are dual-combined with each other.
The sequence of the dual combination of specific solitary elements follows the following scheme:
[Step I:] a to b = a,b
[Step II]: a,b to a,b(1) = [a,b], [a,b(1)]
(see appendix 9, follows asap)

For multiple operations that result from a corresponding number of elements selected as a specific base stem, this operation method (with continuous dual combination of the resulting elements, each of which is combined to form new, independent base stems) is repeated frequently.
The main effort (here when carrying out these operational steps by hand, which can also entail corresponding health risks; please be sure to note the disclaimer attached to this article!) lies in the summary and sorting of the resulting combinations of elements - if modern Office software with table document functions are used: the step of manually writing down the data output of combinations and thus using them as new, optimally usable solitary elements for creating basic databases cannot yet be avoided (to my current, but possibly expandable, knowledge): the manual one However, recording such elements has the advantage that a large part of the data effort can be reduced in advance. This, for example, if, with appropriate expertise, the arguments of the syntax of a selected specific language can be applied to the application of the principles described above: assuming an expert can state (with absolute certainty) about a selected specific language that certain specific combinations of, for example, phonemes or syllables are superfluous because they are superfluous in the sense of the (known) syntax of a selected specific language or would never be used, then generated basic stems of combinations (of e.g. phonemes or syllables) can be thinned out in advance: the syntax of a The specific language chosen (if reliably known) is one of the main arguments for saving data when applied to the described method of creating databases for e.g. Phonemes and syllable. (However, with appropriate experience, the copy and paste functions of a specific spreadsheet software can significantly reduce the overall burden on this procedure.)

DISCLAIMER (Haftungsausschluss):
The reason I'm posting this topic in this section is because a connection to archaeomathematics automatically arises. Due to the technical possibilities, it is not easy to clearly present the necessary connections in a simple manner. I'm trying to make the best of it.
Please note that any use of the content I have written on this topic, despite careful checking, is entirely at your own risk and free of any liability on the part of me as the author. Especially for the aspects of language research and mathematics, it should be emphasized that my contributions to this topic are not peer-reviewed and I do these aspects of research purely as a hobby: I am neither a qualified mathematician nor a language researcher and have come up with all the connections myself based on logical conclusions, systematic experimentation and a little work on the subject of reading. I would particularly like to point out that my own way of presenting connections (e.g. in formulas and logical argument chains) may differ from customary international agreements. One of the most important differences in this context is the fact that I use the spelling for factorials here in the forum with this exclamation mark in front of a factorial number - and not the other way around as is usual and internationally agreed).
Please note in particular that the connections I describe in this topic are not necessarily and automatically correct (or have to be) and, despite the most careful checking, can contain errors (e.g. also formatting errors and auto-correction errors as well as grammatical errors. The ones I describe here in the topic Overall, contexts are in no way suitable as a replacement for qualified teaching or as a certified or certifiable learning aid. The texts I have written on this topic have not been proofread by others.
However, the basic principle can theoretically be transferred to e.g. phoneme inventories with any number of elements and its limits are exclusively in technical or manual feasibility. Because I think it makes sense to understand and convey the basics, I do not focus on this topic as a whole deal with the possibilities of modern IT (e.g. special software and its possible programming): in this topic I am simply showing how relatively far-reaching results can be generated using the simplest means (manually and supported by office software in the form of spreadsheets).
I must expressly warn anyone who would like to deal with this topic in more detail at this point: creating complex spreadsheet documents - e.g. in handwritten form or in the form of computer documents - can lead to extreme and even health-damaging physical stress: some of these Stress can be, for example, physical and psychological fatigue, poor ergonomic posture and strain on the tendon sheath (see e.g.: "Mouse Arm"). Please therefore pay attention to the usual warnings and note that you, as the author of this topic, are responsible for any possible consequences of a recipient's engagement with the I assume no liability whatsoever for the issues described in this topic.
Any errors that may still exist in the author's contributions can be corrected by the author at any time and without prior notice and without special highlighting/marking. The author's contributions to this topic are continuously checked for errors and, if possible, corrected promptly by the author.
The author's contributions and discussions in this topic may (unintentionally) contain serious errors in logic, conclusions, calculations, formulations and, for example, grammatical and spelling errors, among other things. In particular, the use of the content and context published by the author in this topic with regard to copyright issues and - despite the most careful check - any existing issues in this regard is entirely at the author's own risk and risk and free from any liability on the part of the author.
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In particular, the author assumes no liability for lost sales and profits with the content and context published in this topic.

Chain-operating data reduction in (square) Cartesian products
If data efficiency-optimizing numbers of combinations are generated in chain operations using (square) Cartesian products in square matrices, the following scenario arises with regard to the maximum efficient data yield:
From the tabular overview it is clear that the linear, mutually independent and non-chained data savings when applying the principle of the Cartesian product with increasing basic value (here x) in the corresponding proportional development always corresponds to a triangular number and reflects the series of triangular numbers.

[x] / [x²] / [datasavings (x² - x)/2] / [datasavings in %] / [Line]
[1] / [1] / [0] / [0.000 %] / [Line 1]
[2] / [4] / [1] / [25.000 %] / [Line 2]
[3] / [9] / [3] / [33.333p %] / [Line 3]
[4] / [16] / [6] / [37.500 %] / [Line 4]
[5] / [25] / [10] / [40.000 %] / [Line 5]
[6] / [36] / [15] / [41.666p %] / [Line 6]
[7] / [49] / [21] / [42.857... %] / [Line 7]
[8] / [64] / [28] / [43.750 %] / [Line 8]
[9] / [81] / [36] / [44.444p %] / [Line 9]
[10] / [100] / [45] / [45.000 %] / [Line 10]
[11] / [121] / [55] / [45.455... %] / [Line 11]
[12] / [144] / [66] / [45.833p %] / [Line 12]
[13] / [169] / [78] / [46.153... %] / [Line 13]
[14] / [196] / [91] / [46.428... %] / [Line 14]
[15] / [225] / [105] / [46.666p %] / [Line 15]
[16] / [256] / [120] / [46.875 %] / [Line 16]

How much data can be saved in connected and successive chain operations when applying the principle of the Cartesian product depends on how many placeholders (e.g. 20 phonemes) are used. To explain the principle, an example with 8 placeholders (e.g. 8 phonemes) is explained here: fictitiously assuming there is a language that has only 8 phonemes in its basic inventory, then 3 consecutive chain combination operations can be carried out according to the principle of the Cartesian product generate all possible phoneme combinations of such a fictitious language with the following restriction: none of the phonemes may appear in the output line more than (...follows).

Nonlinear chained operations using the Cartesian product principle (8 placeholders):

Requirements:
Basic phoneme inventory to combine (8 phonemes total, absolutely fictional):
Phonem 1 = BA
Phonem 2 = KO
Phonem 3 = JO
Phonem 4 = PAE
Phonem 5 = TOK
Phonem 6 = PAR
Phonem 7 = POI
Phonem 8 = PON

Processes to be carried out / STEPS:
For reasons of space, only the results can be listed here in the specific order chosen. The tabular overviews used to generate the Cartesian products are not shown here for reasons of space and due to the scope required. The data is first output as numerical codes and then transferred to phonetic databases, in which each digit is assigned the corresponding phoneme.

In the first combining step, each phoneme is combined with every other phoneme to form the resulting possible dual combinations (pairs):

STEP I:
Combining all placeholders into pairs in all possible combinations. The display of mirrored content is no longer necessary (use of the “Read Backwards” option; A slash separates the combinations; a double slash with a minus sign in between marks a specific sequence.)

results (36 combinations) as numbers:
1,1/2,1/3,1/4,1/5,1/6,1/7,1/8,1/-/
2,2/2,3/2,4/2,5/2,6/2,7/2,8/-/
3,3/4,3/5,3/6,3/7,3/8,3/-/
4,4/5,4/6,4/7,4/8,4/-/
5,5/6,5/7,5/8,5/-/
6,6/7,6/8,6/-/
7,7/8,7/-/
8,8

results (36 combinations) as phonems; transliterated:
BA,BA/KO,BA/JO,BA/PAE,BA/TOK,BA/PAR,BA/POI,BA/PON,BA/-/
KO,KO/JO,KO/PAE,KO/TOK,KO/PAR,KO/POI,KO/PON,KO/-/
JO,JO/PAE,JO/TOK,JO/PAR,JO/POI,JO/PON,JO/-/
PAE,PAE/TOK,PAE/PAR,PAE/POI,PAE/PON,PAE/-/
TOK,TOK/PAR,TOK/POI,TOK/PON,TOK/-/
PAR,PAR/POI,PAR/PON,PAR/-/
POI,POI/PON,POI/-/
PON,PON

(more content will follow soon)

STEP II:
From step 2 onwards, the advantages of modern electronic data processing and the use of spreadsheet calculation software begin to take effect: via copy & paste and using the option of spreadsheet calculation software, the work steps for determining all the resulting combinations, which would be extremely time-consuming to write by hand, can be carried out with the appropriate concept very short time (see appendix 10). In less than 1 hour, perhaps even in about half an hour, it is possible to determine all relevant combinations of 36X36 and create them as a spreadsheet.

Determination of the number of combinations by calculating the specific sum number (as a triangular number):
A simple but mathematically complex method, depending on the method used, to determine the number of possible combinations when using the "Read backwards" option when generating a Cartesian product (see data savings) is the following: for this, the specific sum of directly consecutive ones is simply added up Elements determined according to the principle of the series of triangular numbers and their development. This results in the corresponding number of combinations for 36X36 combinations = SUM(36) = {1+2+3+4+5+ ... + ... 32+33+34+35+36} = 666. With modern spreadsheet software, determining such totals is very easy and can be done in just a few seconds.


STEP III:
Step III is not presented here due to the scope and effort required. Elevating 666 elements to the 666X666 Cartesian product would yield a total of 443556 combinations, which when reduced by applying the Read Backward option would yield 222111 combinations.
Creating such a database using manual methods using conventional spreadsheet software can be accomplished in a relatively short period of time, perhaps even in a single day.
However, I would like to avoid this work at this point, as the main focus of this topic article is on the Sumerian language with a phoneme inventory of 20 phonemes, which requires a different approach when creating corresponding databases.

Summary:
This work may seem very laborious. However, if the individual work steps are optimized and, if necessary, transferred to many different people, it is realistically possible to create basic databases with, for example, numbers as placeholders that are suitable for transfer to databases with, for example, phonemes or syllables. Once this basic work is done, the basic databases do not need to be created again for any further action. Added to this are the modern possibilities of programmed, automated creation of such databases using software that is to be developed (or possibly already developed) specifically for this purpose.
Another advantage of using the method described above is to be able to better research the creation algorithms of the combination sequences that result in combined Cartesian products in order to determine simplifications for the development of combination series.
I also save myself here from having to verbalize the combination rows created in Step II with placeholders filled with numbers using the phonetic fantasy phonemes from the example mentioned above for illustration purposes. I think the basic principle of the possible procedure has now been sufficiently discussed and explained.

Possible procedure for creating a phoneme database with 20 placeholders as a basic inventory:
The overall procedure for creating a phoneme database with 20 placeholders as a basic inventory requires some preliminary considerations regarding the total effort involved: here the application of specific mathematics helps us in advance to avoid the worst errors and thus saves unnecessary work and time.

If we plan to create a complete, gapless phoneme database with 20 placeholders as a basic inventory, we must be aware in advance that the desired database is extremely large and requires a corresponding amount of effort to create: every logical error at the beginning will eventually take its toll over the course of the project and can take hours, destroy entire days, weeks and months of work. The basic mistakes that tend to creep in at the beginning of such an undertaking are best avoided by thinking through such a project in detail beforehand.
The main mathematical aspects that should definitely be considered before starting such a mammoth project are the following:

Determining the number of placeholders
The amount of data required later in the project depends on the preliminary determination of the number of placeholders. The data effort of such a project is specifically proportional to the number of predefined placeholders.
When determining in advance the number of placeholders and the question of what meaning should be assigned to them (e.g. phonemes, syllables), it is important to consider whether we can really be sure about how many phonemes or syllables a specific language that has been researched to date actually has. It is also essential that we are aware of the different effects that result from the application of the appropriate method with which we create such a database. Here, a distinction must be made between two basic methods in the context of my previous statements on this topic, namely, as already mentioned, between the method using the Cartesian product principle with the combined "reading backwards" option and/or the method I have described. "Telephone book method". I will discuss the “phone book method” separately in a follow-up post on this topic.
You must be aware of the following project scope when 20 placeholders are used to create databases by concatenating Cartesian products with respective intermediate result transfers when using the "read backwards" option. A crucial question is also how many placeholders of readable text lined up one after the other in a specific language (here in the example of Sumerian) will ultimately want to create. The preliminary clarification of the questions mentioned ultimately has a direct influence on which steps we decide on in the planned project.:
20 placeholders create basic inventory 20X20 = 400 combinations (without gaps) when combining 2 placeholders (basic combinations or basic combinations); With the "Read backwards" option, SUM(20) = 210 combinations are generated. The data savings when using the "Backwards read" option creates a data saving of 190 combinations in the 1st process step. The following tabular overview provides information about how this data effort series develops, if we actually aim to combine 20 placeholders with each other without any gaps to ultimately form 20 elements (placeholders) of readable text arranged in a row. The resulting superset (master set) of readable text that is output is directly mathematically dependent on the respective process step and the selected generation method:

Chain algorithmic generation of combinations with 20 placeholders basic inventory:
(Selected method: Data-efficient concatenation of Cartesian products without the "read backwards" option.)

x = number of placeholders (basic inventory)
data saving in steps; SUM(x) = [(x² - x) : 2]
AXB = generated Cartesian product
carry(x) = Carry for x in the next step; carry(x) = SUM(x)

[x] / [AXB]/ [x²] / [SUM(x)] / [carry(x)] / [line]
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
[20] / [20X20]/ [400] / [SUM(20)] / [carry(210)] / [line1]
[210] / [210X210]/ [44100] / [SUM(210)] / [carry(22155)] / [line2]
[22155] / [22155X22155]/ [490844025] / [SUM(22155)] / [carry(245410935)] / [line3]
...

The small list quickly makes it clear what an enormous amount of data we would be confronted with in such a project. Therefore, the main focus of such a project should be on every possible measure that can reduce further data expenditure. One of the most essential ways to achieve this is fundamental knowledge of the syntax of a specific language. Researching the syntax of a specific language being researched in detail is, among other things, Indispensable for the reasons mentioned here of possible data savings in decryption attempts. More about the strategies of how knowledge about the syntax of a specific language can help us to draw conclusions about possible decryption attempts or complete mapping attempts using the methods already presented here and those still to be discussed in the following.

DISCLAIMER:
Any use of the content posted by the author in this topic is entirely at your own risk and free of any liability on the part of the author.

SOURCES / BIBLIOGRAPHY:
[books]:
du Sautoy, M.: Die Musik der Primzahlen - Auf den Spuren des größten Rätsels der Mathematik. 7. Aufl. ungek. Ausg., Verlag dtv Wissen, München, 2013

Reiss / Schmieder: Basiswissen Zahlentheorie - eine Einführung (Reihe: Mathematik für das Lehramt) - Eine Einführung in Zahlen und Zahlbereiche. 2. Aufl. Verlag Springer, Berlin, Heidelberg, New York, 2007.

Schmidt / Trenkler: Moderne Matrix-Algebra. Mit Anwendungen in der Statistik. Verlag Springer, Berlin, Heidelberg, New York, 1998.

[gWiki; german-language Wikipedia]:
[gWiki1]:
Bibliografische Angaben für „Sumerische Sprache“
Seitentitel: Sumerische Sprache
Herausgeber: Wikipedia – Die freie Enzyklopädie.
Autor(en): Wikipedia-Autoren, siehe Versionsgeschichte
Datum der letzten Bearbeitung: 9. September 2023, 21:00 UTC
Versions-ID der Seite: 237177728
Permanentlink: https://de.wikipedia.org/w/index.php?ti ... =237177728
Datum des Abrufs: 1. Januar 2024, 11:03 UTC

[gWiki2]:
Bibliografische Angaben für „Georg Cantor“
Seitentitel: Georg Cantor
Herausgeber: Wikipedia – Die freie Enzyklopädie.
Autor(en): Wikipedia-Autoren, siehe Versionsgeschichte
Datum der letzten Bearbeitung: 14. Dezember 2023, 12:47 UTC
Versions-ID der Seite: 240179661
Permanentlink: https://de.wikipedia.org/w/index.php?ti ... =240179661
Datum des Abrufs: 7. Januar 2024, 11:43 UTC

[gWiki3]:
Bibliografische Angaben für „Carl Friedrich Gauß“
Seitentitel: Carl Friedrich Gauß
Herausgeber: Wikipedia – Die freie Enzyklopädie.
Autor(en): Wikipedia-Autoren, siehe Versionsgeschichte
Datum der letzten Bearbeitung: 23. Dezember 2023, 07:46 UTC
Versions-ID der Seite: 240453755
Permanentlink: https://de.wikipedia.org/w/index.php?ti ... =240453755
Datum des Abrufs: 7. Januar 2024, 15:10 UTC

(the article is currently being edited and may still contain errors. The disclaimer applies (it can be found in the first post of this topic and is also attached to the end of this topic post

The Syntax: the "second soul" of a language
Imaginatively assume that we run a small cafe or, for example, a tea house in which a small group would like to meet for a small celebration, for which we have to set up a suitable table beforehand. So that we can set the table correctly and appropriately for the customers, we have obtained the appropriate information beforehand, because we have a small problem: we only have a certain number of tea cups and coffee mugs.
Now we have to master this logistical challenge by asking ourselves in advance how we should optimally set the table, because: in the small company that we expect to soon be able to welcome, there are die-hard tea drinkers and also die-hard coffee drinkers: some of the tea drinkers As we were able to find out in advance, we categorically refuse to drink tea from coffee mugs and some coffee drinkers categorically refuse to drink coffee from tea mugs.
We also encounter another problem: some guests are left-handed, others are right-handed, but every single guest without exception makes it a point to be provided with a spoon to stir their drink and demands that it be on the right side for them have to lie.
And another strange thing is important when planning for this small company: everyone wants to sit on the opposite side of the longest table in the room, so that they can enjoy the sometimes storm-tossed Atlantic Ocean, near which our little tea house is located.
In order to master this logistical challenge, we prepare a small handwritten overview in advance for the sake of simplicity in order to avoid a faux pax in our company. We write abbreviations for the names of each of the company's customers on a piece of paper. Next to each name abbreviation, we write in brackets on the correct side of the name abbreviation the side on which each guest's spoon should be placed. On the blank page of each name abbreviation we write in brackets whether each guest needs a coffee mug or a tea cup in order to be able to determine the total number of drinking vessels required. For this we use the abbreviations "sp." for "spoon"; "te." for “tea” and “co.” for “coffee”) We get the following result:

Karl = (sp.)KAR(co.)
Berty = (te.)BER(sp.)
Sven = (te.)SV(sp.)
Jamie = (sp.)JAM(te.)
Oliver = (sp.)OLI(co.)
Hansi = (te.)HA(sp.)
Kathrina = (co.)KAT(sp.)

Now we create our seating plan for the small company in the hope that everything goes smoothly. We are satisfied with our planning results and write them down again on a separate piece of paper that we place on the counter where our assistant can see the piece of paper with the plan. We have now written the following on the piece of paper (to avoid problems, we simply took the liberty of putting all the coffee drinkers in one group and the tea drinkers right next to them in the other group):

(sp.)OLI(co.)(sp.)KAR(co.)(co.)KAT(sp.)(sp.)JAM(te.)(te.)BER(sp.)(te.)SV(sp.)(te.)HA(sp.)

Afterwards we go on a lunch break. When we return from our lunch break, our assistant is standing behind the counter and, waving our previously made piece of paper in his hand, asks teasingly: "Hey!? Have you invented a new language? I don't understand a word and can't make sense of it."

Let us now take a closer look at what our fictitious seating plan can mean in combinatorial terms with regard to the syntax of a language and what benefits can be derived from this knowledge:
Assuming our seating plan was a series of phonemes from a purely fictitious language and assuming the information in the seating plan in brackets were syntax information from our fictitious language: for this I substitute the abbreviations "sp." for "spoon", "te." for “tea” and “co.” for "cofee" with (optional in this example) digits:

1 = "sp." for "spoon"
2 = "te." for "tea"
3 = "co." for "coffee"

it results:
(1)OLI(3)(1)KAR(3)(3)KAT(1)(1)JAM(2)(2)BER(1)(2)SV(1)(2)HA(1)

I can then derive a (here purely fictitious) syntax for our fantasy language from the construct of phonemes and instructions in brackets:
For the sake of simplicity, the basic rule that should always be applied is simply: "Like likes to associate with like."

By applying this (fictitious) basic rule, a syntax (as a fantasy product) is created that can be derived from the construct (our original seating plan): we can now derive a small list from the syntax, which will be of great help to us in future combinatorial projects with our fictitious language, because the present syntax helps us to save large amounts of data to be processed if we want to determine all the phoneme combinations that can be formed with the present fantasy phonemes.
Due to our fictitious basic rule, the result of the modified construct is that something can be seriously wrong here: if the basic rule is "like to like", certain phonemes must not be arranged in the way we find here (one could In a figurative sense, it also means that the original seating plan does not fit and has to be changed for some reason because our guests also want to sit right next to each other based on preferences that are incomprehensible to us.
Now, because of the problem, we have to think about whether and how we can actually solve the newly created problem:

Unfortunately, we have no choice but to first check what options we actually have and whether we have any satisfactory options at all. To make this possible, we use combinatorics: we first create a small list with all the preferences as a sorted overview:

From our original seating plan:

(1)OLI(3)(1)KAR(3)(3)KAT(1)(1)JAM(2)(2)BER(1)(2)SV(1)(2)HA(1)

becomes like this:
(sorted alphabetically and numerically ascending; is based first on the left bracket of a phoneme)
(1)JAM(2)
(1)KAR(3)
(1)OLI(3)

(2)BER(1)
(2)HA(1)
(2)SV(1)

(3)KAT(1)

I call this characteristic of the lineup property description. In a software-based programming syntax in this - or similar - way, countless different properties can theoretically be assigned to elements (as I suspect; I don't have any basic knowledge of programming and creating software).

Now let's imagine that the waiter had had the same idea - for whatever reason; Maybe to get his cafe business running better, maybe because he wants to save time in the long run or he's just simply interested in combinatorics. And the story continues: The teahouse owner decides to buy a little book about combinatorics and, after reading the first few chapters, tries around a bit. He quickly comes up with the idea of trying out how many possible combinations there are in order to arrange the letter abbreviations for his guests. After reading a little more in his little book to find out whether someone had already had this idea before him, he quickly found what he was looking for, because it is, so to speak, a basic combinatorial repertoire.
The cafe owner, let's just call him Jim for the sake of simplicity, is horrified when he learns about the calculation rule for determining the combinations and carries out the corresponding calculation. It turns out that Jim can combine the abbreviations of his guests, which can theoretically be used to form a fictional language, an astonishing 7^7 times and each time comes out with a very unique combination, including the reversed ones.
Because Jim picked up a little something in math class that wasn't a given for his time at school and wasn't always up to him, Jim can now calculate this task relatively easily using his cash register, which is installed on a laptop - with the calibration seal from them Tax authorities and all the trimmings.
(Sometimes in his childhood math class, Jim was more concerned with counting the teacher's hits on the students' hands than making sure he remembered the formulas the teacher scribbled on the board and shouted about for the future - which is why the cane was within reach on the teacher's desk.)
So Jim now calculates the number of possible combinations with 7 placeholders - i.e. 7 elements that can be combined with each other which I call placeholders). Jim is amazed and immediately frustrated when he reads the number displayed by the calculator application on his laptop: 7*7*7*7*7*7*7 = 823543 can be read on the application's display.
Jim wonders how he is going to try out such a huge number of combinations in seven limited time and what would happen if he had a really large party with idiosyncratic seating plans. But Jim doesn't dismiss the topic because Math actually interests him - despite the terrible lessons from his childhood. So Jim continues to think about whether and in what way the problem can perhaps be solved. But Jim doesn't want to read the booklet he got straight away, but instead wants to try out what possible solutions he can think of for himself.
After thinking about it for a bit and wasting time (which he can't really afford to do as a cafe owner), Jim lets his gaze wander thoughtfully through the cafe, which is empty after closing time. Finally, his eyes settle on the phone book lying on the counter next to the cash register.
As if by inspiration, Jim reaches for the telephone book, opens it, leafs through it a little and, as if with a small flash of inspiration, realizes that telephone books cannot contain all possible letter combinations with the letters of our alphabet as names: in telephone books, like that Jim thinks to himself, there must be an incredibly large amount of waste, because whoever is hot with a last name, for example, Bbbbbbbbb or Kkkkkkkkk or something like that.
In order to check whether his idea is perhaps nonsense, he checks the idea with a small calculation - and in fact: Jim comes up with an impressive number of possible combinations with his calculations if he - like his own last name Bristlowe - assumes 9 placeholders ( But let's be honest - Jim needed a little help from time to time, leafed through the booklet and did a little research on the topic on the Internet). So Jim finally figured out how to calculate the whole thing - which can be quite a headache.
In order to approach the topic gradually and carefully, Jim first tries to calculate how many different combinations in total (including all mirror-inverted combinations) are possible if you start from 26 placeholders (letters) for a standard alphabet and include umlauts and other punctuation marks, etc . leaves out. However, Jim quickly realizes that the matter is not that simple: he notices that he not only has to take into account the number of letters but also the question of how many placeholders - as letters - should be strung together in order to try out all possible combinations.
So Jim - in order not to overdo it - first tries around with 2 placeholders with 26 different letters and quickly comes to the conclusion that there must be 26*26 combinations that are possible with 26 letters with 2 placeholders. Jim immediately starts wrote down the combinations, because it is still affordable to do so by hand and is a very good exercise for him as a cafe owner and not just a way to pass the time. A lax thought that came about carelessly and thought too quickly, as will soon become clear. Maybe Jim was just too tired from the previous lunch when he thought about it. Ultimately, however, Jim quickly experiences a nasty surprise: there seem to be many more possible combinations with the conditions he has set, because Jim's writing down seems to never end. While writing, Jim quickly realizes that just from the first letter of the alphabet, A, in pairs with all the other letters of the alphabet, he already gets 26 possible combinations, namely:
(a,a), (a,b), (a,c), (a,d), (a,e), (a,f), (a,g), (a,h) (a,i), (a,j) (a,k), (a,l), (a,m), (a,n), (a, o), (a,p), (a,q), (a,r), (a,s), (a,t), (a,u), (a,v), (a,w), (a,x), (a,y), (a,z)

And then it hits Jim like a lightning bolt: if he gets 26 different combinations for a single letter of the alphabet in combination with all the other letters of the alphabet - out of 26 different letters in total - then the chosen task must be a very a much larger number of possible combinations than Jim can even imagine. This puzzle ultimately captivates Jim so much that he can't stop working on the task when he gets home after work. Until late at night - which is certainly not a good idea for him as a cafe owner - Jim sits at the kitchen table at home on his laptop and tries things out. Eventually, however, he finds out in time enough to at least get a little sleep for the next day. "I really shouldn't recommend this action to anyone", Jim murmurs as his eyes almost close to himself. When Jim finally decides to go to bed to save the next day, he has a brief moment of frustration: while brushing his teeth, he leafs through the little book he got and is shocked to find that his calculation, which he spent half the night working on is printed in razor-sharp writing at the back of the inside cover of the booklet, meaning he could have saved himself all the work. "I'll probably never forget this calculation method again," Jim mutters in frustration and gargles a little more with his toothpaste before he completes the half-night he's spent and puts it aside so that he can start the new day somewhat fresh.
The next morning, having already arrived back at his little cafe and standing behind the counter, he once again looks over what is printed in the back cover of his little book. He grabs a pen and a piece of paper lying nearby and writes down the calculation instructions. Jim is almost finished copying the little formula when his assistant, a Norwegian exchange student named Sigurd, comes in. Jim briefly lifts his eyes from his notes and, with a slight nod of his head, greets Sigurd with a brittle “Good morning.”
"Well, boss? Got up on the wrong foot first today?", Sigurd teases and teases. Jim doesn't let himself be deterred and finishes copying the formula. On the piece of paper in front of Jim it is now written:

26*26 combinations with 2 placeholders

For this, Jim translated the formula in his own way, which he can easily remember:

In each step, multiply the number of placeholders multiplied by the number of placeholders.

"So with 26 letters and 2 placeholders there are 26*26 combinations
With 26 letters and 3 placeholders, there are 26*26*26 combinations
With 26 letters and 4 placeholders, there are 26*26 *26*26 combinations
etc. etc.

or can be written like this (as powers):
26 elements, 2 placeholders = 26^2
26 elements, 3 placeholders = 26^3
26 elements, 4 placeholders = 26^4
etc.etc."

Below that, Jim has copied down a small exemplary tabular overview that explains how the principle of forming combinations is actually called the "telephone book principle" in the little book (although - as Jim thinks - that can't be entirely true ). "I'll probably have to read about that again in peace - but whatever," Jim murmurs, "let's just write it down first.":

e.g. 3 elements, 3^3 combinations {(1,1,1), (1,1,2), (1,1,3), (1,2,1), (1,2,2), (1,2,3), (1,3,1), (1,3,2), (1,3,3), (2,1,1), (2,1,2), (2,1,3), (2,2,1), (2,2,2), (2,2,3), (2,3,1), (2,3,2), (2,3,3), (3,1,1), (3,1,2), (3,1,3), (3,2,1), (3,2,2), (3,2,3), (3,3,1), (3,3,2), (3,3,3).

"And, boss? Are you playing the lottery unsuccessfully again?" Sigurd asks as he passes by, taking a look at Jim's note and tying his apron around himself.
"No", Jim grumbles. "I'm practicing the high science of combinatorics here, wise guy."

"What I take from this is that you can definitely tell me how many ways I can tie my apron with a simple knot, if you're that clever."

"Of course I know that", Jim says, and starts to calculate. "I wasted half the night yesterday, you know? So you better not mess with the wrong guy. The same principle applies to your question as to the question of how many possibilities there are in letter combinations, for example. You just have to transfer the principle to the apron wrapping technique. You can tell your buddy, who is an archaeologist, that he can also use the same principle if he happens to find - let's say three - fragments of something and doesn't know in which order he should arrange them together to create something reconstruct. I have no idea, for example - let's say - 3 pretty old pieces of clay with writing on them or something."
"It's called CERAMIC Shards Improved Sven Jim."
"Thank you, Professor, I had actually forgotten that with all the hustle and bustle and the fact that I have to do all the work alone while you are still puzzling over how you can tie your apron in 27 different ways - theoretically, but not practically."

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The next night, too, Jim spends a lot of time (he just can't help it and will probably pay the price the next morning). But the topic simply fascinates him too much. However, he is an experienced businessman enough to follow the brothers' spirit and example Wilbur and Orville Wright, the co-pioneers of human flight, and not to neglect his company too much despite all his fascination with science. Jim roughly calculates that when he goes to bed at 4 in the morning he will still get enough sleep is to master the next day. Jim lives in a quiet residential area, where such thoughts are not possible without calculating imponderables. But the worst thing that can happen is that the cats in the street perform courtship capers at night and lead to this with their meowing The night will be even shorter. But that's what Jim makes it count because nothing saves the next day better than a good cup of tea in the morning - that was Jim's father's motto and Jim has adopted it and found that his father was right in his foresight had kept.
When Jim opens his recently acquired little book about combinatorics and cryptology over a late-night cup of tea to continue reading, he is initially taken aback: instead of information about the question of how the letters of an alphabet with 26 letters (with the general omission of the umlauts) differ in terms of efficiency Data saving can be used combinatorially and cryptologically, the author of the booklet first continues with a chapter on music. Jim can't make any sense of it yet, but starts reading the chapter in the quiet confidence that he didn't waste the money on the book in vain. After all, the little book was already of some use to Jim: he was able to show off a little to his assistant with the knowledge he had acquired in a single day through experimentation and in a single night through reading.
And this is what Jim reads in the little book:
"Anyone who has ever wanted to know what music and combinatorics have in common and why the causality between the cause of the creation of music and the effect of music is completely irrelevant with regard to today's copyright law and artistic freedom as well as the protection of so-called intellectual property If you need a new kind of discussion, it's best to continue reading here:
There is a lot of discussion these days about copyright protection law, about strikes by artists and authors in the USA and about the justification of fears about so-called artificial intelligence - the so-called AI. AI is currently on the rise and seems to be implanting itself in our lives like a global colony of hard-working ants, although the fear, which at first seems justified, is that these are mutated super ants with superpowers that will one day exterminate us - or something like that ; the proverbial tables would then be turned, dear humanity) - cannot (or should not) be swept away from the table."
Strictly speaking, however, music is nothing other than combinatorics, if we want to look at music from a mathematical point of view and can live with our sensitivities, that music is an incredibly important part of people's self-expression, creativity and self-perception - and therefore an intrinsically important cultural part of our everyday life. But let's be honest: Today even the most unmusical people can generate music with just a few clicks - and not just since yesterday: cyber intelligence has been on the rise for a long time, but the invention of notation itself is even more ancient: without combinatorial Basic ideas musical notation would never have been created and therefore no one would read it today.
When discussing music, we ideally have to distinguish between the "emotional experience" of music (and making music) and the technical aspect of music: a trumpet, for example, would not produce appropriate notes (assuming the trumpet player knows what he is doing), if the master brass instrument maker had not known how to make the trumpet based on highly mathematical and physical processes.
A lot of people can generate music combinatorially using the simplest means: no matter how unmusical they may be, they don't have to be able to play any instrument at all - not even the hackneyed comb that is typical of silent film clichés - as we will see later. Actually, people who deal with the creation of music using combinatorial methods don't actually have to know much about music. And all of this without any artificial intelligence. The sometimes exaggerated fear of AI is therefore unimportant. Karl Lagerfeld is said to have said that it is pointless to defend yourself against a trend that cannot be stopped anyway (something like this or something like this):
A look at the always ready Wikipedia (aside from the annoying calls for donations, which have not yet algorithmically calculated in AI fashion when you don't want to be disturbed while reading Wikipedia articles), is enough to generate music combinatorially using the simplest means. For this we only need the names for - let's say - three basic chords (whether you know what a chord actually is doesn't matter at all, because we can later look for a good friend or acquaintance who ideally knows what a chord is Chords are and how to play them.
If we were fans of the Beatles (and at least a little musical) and were wondering which three chords the Beatles played most often, we would have an interesting question to answer: unfortunately, I personally cannot answer this question (who could?).
And just because I don't really know much about music myself, I choose a very freely individually interpreted type of "very modern free jazz" and simply take three common chords from jazz and play with them in combination. Let's see whether something really audible comes out of it in the end (I'm also deliberately ignoring whether musicians could actually play one or two of the things that come out of it in terms of chord progressions and, for example, fingering technique, because I actually have no real idea about musical instruments.

In order to generate (probably more or less playable and audible) music from these three chords using combinatorial methods, I proceed as follows:

Step 0: Select three chords:
9, ♯11, 13 = A, Cis, E

(According to the German language Wikipedia, corresponds to the sound material of the Mixo#11-Skala; see [gWiki4, Akkordsubstitutionen])

Combinatorial process I:
(the whole thing only takes a few minutes if you have the right experience and use modern spreadsheet software)

Step 1: Select the method for combining 3 placeholders:
Here I choose the telephone book method already mentioned (which creates complete combination sequences); i.e. the most data-intensive method

Step 2:
I create a table form (which can also be handwritten if necessary, but doesn't necessarily have to be the case these days) with 7 columns and 27 rows. I name the first 3 columns (Columns I, II and III) as Column I, Column II and Column III.
I name columns IV, V, and VI with a', b' and c'. Finally, I name the 7th column “Carry Forward”.

Step 3:
I enter the numbering in front of the first column at row height for 27 rows in a row (I can also insert an extra column for this if I want.

Step 4:
In columns I, II, III I enter the already mentioned series of combinations with 3^3 = 27 different consecutive combinations.

Step 5:
I now assign a chord to each number in columns I, II, III (as an actual placeholder) (which number stands for which chord doesn't matter in the example)

Step 6:
In columns IV, V and VI I exchange the numbers with the same assignment as in columns I, II and III (e.g. in order) with the selected chords.

Step 7:
Finally, I create a (here so-called) carry by entering the results from columns IV, V and VI as a "block" from each individual line in columns IV, V and VI into the respective line of the carry behind it .

- take a look at the image attachment with figure number 1 -

Combinatorial process II:
Now I use the carryover from the first combinatorial process and use each individual carryover in a new combinational process like its own placeholders.
In the following I use the special method of calculating two quantities to form the Cartesian product. This means that I "cross" set A with set B and get a corresponding number of elements as a new independent set. Since I am dealing with an output of 27 elements as a result of the combinatorial process number I, I cross 27 elements with themselves in the combinatorial step II. The result is 27*27 or 27^27 = 729 combinations with each Length of 6 combined elements preserved as independent chord progressions. This means that with the two steps described I get somewhat more extensive chord progressions.
For this I proceed as follows:

(...)

The next morning, Jim wakes up with his head on the kitchen table with a flattened face.
"Oh man," he clears his throat as he pushes himself up from the table, "how could you be so stupid as to spend a night at my age like a rock star in his prime. But not with friends and so on, but with a book. I never thought I'd be able to experience that again."
Jim stretches his stiff arms up and moves his upper body back and forth so that the tired old bones crack.

He then looks at the final result of his work from the last night, which was a complete nightmare, before he fell asleep on the beach and didn't hear the wake-up call from his cell phone.
At this point he looks at his cell phone (sorry, today we call it a smart phone), which is next to a notepad full of scribbles. The smartphone shows 06:06 and Jim wonders how he managed to wake up just in time and without an alarm clock.
"It's all a question of sportsmanship," Jim murmurs proudly, "I don't seem to be so old now that I'm forgetting the virtues of my youth. Back then there were stupid smartphones that didn't even beep when you went to the toilet - and if your alarm clock was broken, old boy. You had to wake up like that if you didn't want to get in trouble with the boss, that idiot, right?
Despite the stupid time change, the fact that every four-legged creature is used to wanting to comfortably cross the street at 5:00 in the morning as usual, Jim was 40 years and a few crushed after his 18th birthday, quite well in the race, 8 minutes late. He could live with that.
"Wait a minute," Jim murmured, "that's just that many percent. Man, I was always pretty good at mental arithmetic: something over 1 percent and under 100 percent, based on my thumb. Oh, what the heck."
Jim reached for the piece of paper he had scribbled all over it last night and thought about the resulting mental outpourings again:
For Sigurd, it said in scribbled writing, "Hey, low-paid, part-time, temporary Hi-Wi, are YOU kind enough to rub this in your buddy the archaeologist's face before you get off work tonight? I'd be interested to know what he thought about it holds:

Jum-Dei-Del
(archaeologist song with 3 phonemes)
(1, 1, 1) Yumm, Yumm, Yumm
(1, 1, 2) Yum, Yum, Dei-Do
(1, 1, 3) Yum, Yum, Dei-Do
(1, 2, 1) Yum, Dei-Do, Dum

"Oh man. I am a real HERO. How would Paul like that? I wouldn't even ask Ringo. Real rock stars couldn't have done this better", Jim said to himself.

Below that there were a few scribbled notes on the piece of paper:

Woynych manuscript
Carrier pigeon message
Nixdorf Museum
Alan Turing (Oh man... - Dude. What did they do to you back then despite everything you did for us... - you played pretty well, Benedict. Almost as good as Stephen Hawking back then.
Enigma.
Damned. If I had known that back then when I was running around like a blind chicken in TOmb Raider I in the Plaast des Midas.
Molly told me that I didn't have time to water her flowers so she could fly to Mallorca again (I should have told her that 10 years ago)...
Linear (A)
what on earth is cuneiform???

Jim got up, shuffled to the kitchenette and made an extra strong pot of coffee to save the day.
When the first drops of the Kichkstarter trickled into the pot underneath, which was already completely calcified at the edges, in hopeless situations, Jim reached for his little book to quickly skim through another chapter; and this is what the author wrote somewhere in the first third of the booklet, which Jim had now worked through:

"How we can compose music in C major without great knowledge of music
Imagine you want to compose a song, let's say in C major (whatever that may be for you). How do we do this when we have combinatorics as a method and nothing else? Because, for example, we are extremely unmusical and have no idea how to play an instrument or anything like that: well, that's actually quite simple. What we need for this, however, is a little bit of mathematics and appropriate perseverance or, for example, spreadsheet software (I strongly recommend the software, because writing too much by hand can be very unhealthy and lead to long-term consequential damage, so please overdo it under no circumstances and take care of yourself, dear reader).
You should start with the following - and this step is crucial for the overall amount of subsequent work. However, I am explaining the example here exclusively using a pure series of tones and ignoring what makes music so tasty; These are: note lengths, pauses, intonation and so on (there is not enough space here to discuss these connections in the necessary detail and the publisher nbun doesn't pay that well for my time).
If you decide to choose the key of C major with its 8 different tones scale, you will need a table form with 8 different placeholders. And now the first calculation task begins: How many work steps are necessary to combinatorially arrange 8 different elements in all possible constellations? Well - if you have worked through the previous chapter carefully instead of constantly going to make coffee, for example, you should now know how it works:
First we ask ourselves which methods and in which sequences we want to use in order to be able to work as efficiently and data-efficiently as possible: if we remember the chapter about Alan Turing, then we know that this is particularly important when we deal with cryptological combinatorics.
In the example discussed here, we now begin by combining all placeholders with each other into all dual combinations (i.e. in pairs) by elevating the placeholders to the Cartesian product AXB.
But be careful: this method does not create a complete database. In order to create a complete database, we would have to choose the much more complex telephone book method. The method shown here is just a small example of how you can create quite a few combinations according to a certain scheme with a certain small number of placeholders. The disadvantage of the method with the Cartesian product, which is determined in complex chain algorithms, is that this method only produces a very specific section of the total possibilities. In this case, a lot can be done with this excerpt. I will explore the difference between the two methods in more detail in another chapter.
In the example discussed here, we now begin by combining all placeholders with each other into all dual combinations (i.e. in pairs) by elevating the placeholders to the Cartesian product AXB. This results in 8 x 8 = 8^2 = 64 different combinations of tone pairs (dual or binary combination).
Now we already know the simple trick of how we can save a large amount of data to be generated with the appropriate number of placeholders if we use the "backward reading" option discussed in the last nChapter when collecting the Cartesian Product (because we read every output from mlinks to the right or from right to left).
When we have obtained the 8^2 dual combinations and create the remainder by freeing them from the excess of combinations when using the "Read backwards" option, thus reducing them to the bare minimum, we finally get the "tea grounds", i.e the resulting combinations, the number of which can be calculated according to:

A(2)XB(2) = x² - [(x² - x) : 2] = 8² - [(8² - 8) : 2] = 64 - [(64 - 8) : 2] = 64 - 28 = 36

Of course, if you are still familiar with this procedure (otherwise, look it up), you can also use the "Gaussian method" that I have mentioned and determine the master sum of the number 8^2, or in a modern table calculation software With little effort, we can draw up a series of numbers from 1 to 36 and form the sum of them. Et voila: the second step of the chain combination is finished, if we now use the resulting 36 elements as placeholders to form the Cartesian product. And then we have all of them possible tone combinations in length of 4 tones of the C major tone series.
If we paid attention, remembered appropriately and were "smart" (without meaning that in a judgmental way), we already knew in the last step described that we would consider the combinations determined in the first step of the combination series (as pairs) as one for the sake of simplicity Consider placeholders (and write them down accordingly in the output) so that the collection of the Cartesian product in the second step is easier and more data-efficient.
The big advantage in this case (if we use the decimal system to write it down) is - because we only combine 8 different placeholders with each other - that we can do without the comma between the digits when writing down the pairs of numbers (if we want to).
For later evaluation of the results, we can simply remember that each digit - even if the digits are written one after the other - represents an independent tone from the key of C major. In this context we read, for example, the numbers 1 and 3 written one after the other, which can be read like the number 13, but in this case they do not mean the number 13, but 1,3.
With appropriate experience, this takes less than an hour with modern spreadsheet software; maybe just about half an hour.
If this procedure is too confusing for you (there are no commas between notes), then simply write the numbers with a comma in between.
But be careful: this method does not create a complete database. In order to create a complete database, we would have to choose the much more complex telephone book method. The method shown here is just a small example of how you can create quite a few combinations according to a certain scheme with a certain small number of placeholders. The disadvantage of the method with the Cartesian product, which is determined in complex chain algorithms, is that this method only produces a very specific section of the total possibilities. In this case, a lot can be done with this excerpt. I will explore the difference between the two methods in more detail in another chapter.
Now, after just two combinatorial steps, we already have all the tone combinations possible with the selected key of C major with 4 consecutive tones each (with the limitations mentioned above, because the method does not produce a complete output; quite different from the much more complex telephone book method).
In the last step it gets a little more complicated, but after that we've already done it:
The final step would be a little more complicated and labor-intensive, but after that we would have it done. I will save the detailed explanation of the last, third step here due to the scope requirements and the lack of space here. But this much can be said: The reduced result from the 2nd work step is raised to a Cartesian product with itself. We get exactly... combinations from the... reduced combinations from the 2nd step after completing the 3rd step, because:

1st step = 8X8 = 8^2 combinations altogether; 8^2 - {[(8^2) - 8] : 2} reduced = 36 combinations
2nd step = 36X36 = 36^2 combinations altogether; 36^2 - {[(36^2) - 36] : 2} reduced = 666 combinations
3rd step = 666X666 = 666^2 combinations altogether; 666^2 - {[(666^2) - 666] : 2} reduced = 222111 combinations

Here is a small excerpt of the results from the completed second step in the transmission in tone sequences.
Example:Row 23 of the second form in horizontal direction and column 2 in vertical direction gives the following combination result: 4512
In the translation here (in the simplified reading discussed with numbers of the decimal system) this means the tone sequence = 4512 = F,G,C,D from the key of C major.

at: C = 1; D = 2; E = 3; F = 4; G = 5; A = 6; H = 7; C´ = 8

If we were to continue with this algorithmic procedure in a correspondingly data-intensive manner, we would ultimately create a considerable specific selection (with correspondingly definable gaps) of all possible tone sequences with the tone type C major: the result depends exclusively on programming and computing capacity and computing effort. Although there are still a few combination tricks that can be used, people are already limited by this method with only a few placeholders. An appropriately programmed machine is not interested in the possible tendonitis of a prolific writer who writes too much (and therefore health-threatening) by hand (please don't copy it!).

To show you, dear reader, how important this topic will be in the future for our societies and for example - let's say - archeology, which, for example, also deals with historical written records, some of which are still not deciphered today I will explain to you in the next section of the chapter. There - in order to be able to build on the previous example - I take another excursion into the world of music from a combinatorial perspective. With this I am anticipating a combinatorial trick that works quite well and is very easy to use these days, as you will see. I hope you will like the trick, because it is really very useful - almost anyone who can read and write and ideally has modern spreadsheet software (which is strongly recommended for health reasons) can do this trick apply.

How we can make our lives as combinatorics easier with modern spreadsheet software:
Anyone who is interested in combinatorial theory and deals with combinatorics has a great modern advantage that many combinatorists did not even dare to dream of just a few decades ago: the modern (and certainly no longer so young) spreadsheet calculation software. Typically, modern spreadsheet software is alphanumeric and optimized for the use of the decimal system: in a traditional spreadsheet software (with alphabetically sorted columns and rows) we draw in the work grid of the application, which is now almost ubiquitous worldwide, a simple series of numbers across several rows or columns, the numbers (as digits) are automatically output in a numerically stringent order and sorted according to the decimal system. The fact that the software uses the decimal system is combinatorially and cryptologically crucial and is extremely gratifying when we know how we can make combinatorial use of this connection to the subprogramming of the software: this is quite simple, as I will show below.
To show you, dear reader, how important this topic will be in the future for our societies and for example - let's say - archeology, which, for example, also deals with historical written records, some of which are still not deciphered today I will explain to you in the next section of the chapter. There - in order to be able to build on the previous example - I take another excursion into the world of music from a combinatorial perspective. With this I am anticipating a combinatorial trick that works quite well and is very easy to use these days, as you will see. I hope you will like the trick because it is really very useful - almost anyone who can read and write and ideally has modern spreadsheet software (which is strongly recommended for health reasons) can do this trick apply.
Just this quickly in this context: for combinatorics it would - in my opinion at least - be downright fantastic if one day there were freely accessible spreadsheet software in which the place value system for calculations could be set up in an uncomplicated and diverse way. Personally, I haven't come across such software yet, but perhaps I'm already too old-fashioned and simply haven't researched the topic thoroughly enough. But even if you specialize, you can't know everything and take care of everything, because there are only 24 hours in a day. Maybe something like this has been around for a long time and I just haven't noticed because I prefer to spend so much time on the basics and the rest of the time making tea."

Jim suddenly stopped reading. Leaning against the worktop of the kitchenette, where the coffee machine was panting away at arm's length in the usual manner and was already on its last legs, Jim glanced over at the kitchen table, where his laptop was still open from the almost completely "drunken" night. "Geez Meier," Jim said to himself in astonishment, "I have a spreadsheet like that on my laptop too. I practically live from being able to calculate how much money I as an entrepreneur won't have left over at the end of the year and how happy the tax office will be again in the next quarter. "Such crap that I have to go, but Jim, cheer up: That's what the Wright brothers did and it was ultimately successful!"
Jim quickly opened the cupboard above the counter, picked out his favorite coffee mug and poured himself some coffee. As he left his apartment, he hastily tried to sip a little more of the coffee, which was still too hot, but then had to leave the cup. trying to slip into his summer jacket at the same time, willy-nilly left on the mailbox on the wall. "Unfortunately, if the postman doesn't dare to throw me today's bills, I can't help him either. Maybe that's just my luck," Jim grumbled grumpily and hurried towards the garden gate to the street.

"Boss? The toilet paper is out again. Is someone stockpiling for the next state of emergency or have you simply forgotten to buy some? Let me guess: Of course you forgot again but are now going to teach me that this is actually my job , right?"
Sigurd stood in front of the counter and tied his apron around himself, prepared for the new day. In his casual manner, his dreadlocks gave him an even more daring impression of rebelliousness in a conventional working world, but at the same time something new-fashionably balanced that brought a breath of fresh air into something as banal as going to work.
“Are you actually so cheeky with your professors at university or are you just doing things with me - or is that some Scandinavian directness thing?” Jim snapped back and closed the cash register after the morning count.
"As a certified assistant, don't you trust me? It doesn't look like someone got in here at night and plundered your basically empty cash register anyway.", ask Sigurd.
"No, my dear, don't worry, it's just a tax office thing. Never heard of inventory?"
“Sure, I’ve heard of inventory before”, Maybe your summer jacket in January just irritated me. All right then. I'm already going shopping for toilet paper. Do you give me money or should I advance it to you again?"
"No, here you have it, you greedy bastard," replied Jim irritably, "but it's my last money note this morning. You'll have to deal with that for the day."
"All right. I already know."
"While I'm looking at you: You're studying computer science, right?", asked Jim, "is there any spreadsheet software available anywhere these days where you can set up the value system yourself?"
"Who wants to know something like that?", asked Sigurd as he let the cafe door snap shut behind him, already on his way to the supermarket around the corner.
“Next time, will you take off your apron first,” Jim shouted afterwards, “or do you want to embarrass us?”
Jim let his gaze wander around the small room of the cafe with its sparse furniture. Everything was ready for this morning. The small cafe on the corner of the busy street was still empty at this time. Soon the first construction workers would rush in and out again for a hasty breakfast break. So Jim took advantage of this little opportunity and grabbed a little book to read a little before a strenuous day.

"How we can make our lives as combinatorics easier with modern spreadsheet software (Part II):
So. If you are interested in combinatorics and would like to know how to use very simple methods to generate an incredible amount of combinatorial data with modern, conventional spreadsheet calculation software, here's how to do it. In order to illustrate the topic better and make it a little more interesting, I again choose the combination of different placeholders, which are assigned pitches from the C major scale.
The special thing about the method explained below is the application of the decimal system as a place value system to combinatorics: imagine that you are analyzing a basic inventory of 10 different elements (for example letters, phonemes, syllables, or pitches, just to name a few). to give a few examples) then you can do this very easily using conventional spreadsheet software. To do this, simply carry out the following combinatorial steps:

Step 1:
Define an assignment for each placeholder: In the example, I choose the eight directly consecutive basic tones from the key of C major, which are known to be the tones (or pitches) C, D, E, F, G, A , H, C. and because we are talking about 10 space owners at this point, I do a special trick, it will have an effect in the subsequent combinatoir analysis: In the basic inventory I remain 2 free, with elements Placeholders. To assign the placeholders, I choose two different pause characters: I assign half a pause to one remaining placeholder and a whole pause to the other remaining placeholder. To describe the pause characters as elements, I choose the following notation (variable and because it fits well into the resulting extensive continuous text image of the combination lines for better differentiation):

X = whole break (whole)
Y = half break (half)

In addition to notable pitches, rest symbols play an important role in music notation. Now, in step 1, I have made all the necessary preparations to seamlessly create all possible combinations of elements with relatively little effort: because I use the "telephone book method" in the following step, I create the complete series of combinations with all the selected elements as seamlessly combinatorial possibilities that can be created using placeholders. This means: in a relatively short space of time, I create virtually all of the music that can be notated in the chosen key of C major according to the specified specifications and can therefore also be set to music in this sense (or rather, "musical composition"): the predetermined prerequisites are, however, of crucial importance: it Only combinations with a length of 10 elements (placeholders) are output.
In a further step you could (if you want and have the resources and knowledge for this) program software with which the results from the following step 2 can be combined with each other in a variety of ways, including passages of more than 10 elements each arise. But now to the second step:

Step 2:
To carry out step 2 explained here, it makes sense to first make a small calculation: for this we need to know (which I have already mentioned elsewhere) that numbers of combinations result from the following scheme when using the telephone book method:
1 placeholders = 1^1 = 1*1 = 1 combinations
2 placeholders = 2^2 = 2*2 = 4 combinations
3 placeholders = 3^3 = 3*3*3 = 27 combinations
4 placeholders = 4^4 = 4*4*4*4 = 256 combinations
5 placeholders = 5^5 = 5*5*5*5*5 = 3.125 combinations
6 placeholders = 6^6 = 6*6*6*6*6*6 = 46.656 combinations
7 placeholders = 7^7 = 7*7*7*7*7*7*7 = 823.543 combinations
8 placeholders = 8^8 = 8*8*8*8*8*8*8*8 = 16.777.216 combinations
9 placeholders = 9^9 = 9*9*9*9*9*9*9*9*9 = 387.420.489 combinations
10 placeholders = 10^10 = 10*10*10*10*10*10*10*10*10*10 = 10.000.000.000 combinations

- This section is currently being revised - it still contains logic and context errors. Please be patient -

Based on the calculation above, it is understandable that I am only showing the example discussed here of the combinatorial analysis of 10 elements to be combined with each other as a placeholder as an example and in extracts. Although it is actually possible to create the entire form for such a combination series using modern tabular calculation software (depending on the computing capacity of a computer and software), this would require a certain amount of time. It is therefore more interesting to show the general method.
To carry out the second step, I now create a completely conventional series of numbers at the beginning of a table document in vertical directon: here in the example, for better classification, I also placed a positional number bar above the series of numbers, which I have a grayish background (take a look at Figure 2). which is currently being created.

To carry out the second step, I now create a very conventional row of numbers at the beginning of a table document: here in the example, I also placed a positional number bar above the row of numbers for better classification, which I put on a gray background.
Then comes a quite labor-intensive step, depending on how far the procedure needs to be continued. To create the table form, I only select the first few rows: I now drag the number row starting from the first number in the first cell in column 1 down to a number row of any length (this is where software capacity, endurance and trick 17 decide). How long the row of numbers will be and when the software will go on strike - trying is more important than studying: I always do this by drawing the first numbers in the row in a vertical direction and then fixing the left mouse button with a piece of adhesive tape, otherwise I would die In the long run, the tendon sheaths of the fingers and arm will be ruined. So please take responsibility for your health if you try what is described above).
When the table is hereby completed to the desired specified size, the elements contained in the table as placeholders are simply mentally replaced by the elemental assignment of the placeholders. In more complex table programming steps, which I will not explain here due to the scope required, the selected elements for assigning the placeholders can also be entered directly into the table. With regard to reading the table, however, it is easier from the outset to retrain yourself in a different way of reading in which the defined meaning of the elements is replaced by numbers as placeholders. For example, if I see the combination series 12345678910 in front of me, I read in the transfer: C, D, E, F; G, A, H, C, Y, X
I read the numbers output in the table in the vertical direction as if there were commas between the individual digits as separators or something similar. For example, the number output 123456789 in the table at the corresponding line position in the vertical direction is read from in a special way and way the sequence: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 which in our example can be translated equivalently as: CDEFGAHCYX or also C, D, E, F,. G, A, H, A, Y. X.
With this simple trick, we now see before us all the possible combinations for the placeholders we previously filled with elements - in this case, pitches from the key of C major and the selected rest symbols, and could theoretically and practically set the exemplary example selected here to music accordingly as long as we consider the possibilities for this.
According to the scheme shown here, we could, for example, select any number size between 1 and 10^10 as conditions under the aforementioned specifications and make “music” out of it."

The next morning, Jim stood behind the counter of his small cafe, randomly tapping the keys on the cash register. With each key press, the cash register made a beep, beep, beep-op or bi-diet. Jim really got going with what he was doing.
Sigurd stalked in with his long herring bones, which he called legs. In reality, they were more like the long, muscular, well-trained legs of a passionate touring cyclist. The helmet on Sigurd's Rasta mop seemed somehow too small and a little helpless in the face of the youthful hair that spilled out from underneath him.
Jim eyed Sigurd's tanned and toned calves.
"Sun-tanned calves in the winter? How do you manage that? And don't start telling me about climate change."


(For everything described here, take a look at Figure 2, which follows).


(more on this topic later, post is currently being processed)

DISCLAIMER:
Any use of the content posted by the author in this topic is solely at your own risk and free of any liability on the part of the author.

BIBLIOGRAPHY:
Tobin, James: Die Eroberung des Himmels: Die Gebrüder Wright und die Anfänge der Fliegerei. (Reihe: Vom Pol zum Äquator - Die abenteuerlichen Reisen der großen Entdecker und Eroberer), Verlag Das Beste, Suttgart, Zürich, Wien, 2004 (Readers Digest, Deutschland, Schweiz, Österreich).

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