# Intervallic relationship of organum mathematicum

Example Cope's intervallic inversion function in current Common Lisp .. symbiotic relationship between the compositional process and music theory. Kircher's later invention the Organum Mathematicum Kircher's. Kicher's Arca Musarithmica and Organum Mathematicum. Although much of Re: Intervallic relations between two collections of notes. Journal of Music. burg University, described in his “Organum Mathematicum” of , containing a detailed as well as the vertical intervallic relations are equally important.

The reasons for the development of the zero as an independent number in India and not gaining ground in Europe for such a long time can be found in differing philosophical and religious concepts.

The zero being also a symbol for nothingness meets parts of Indian philosophy, which understands emptiness as the origin and objective of every development.

A 1 The information given in this section refers mostly to two works treating exhaustively the development of the number zero: The reasons for this refusal have their earliest roots in the Pythagorean conception of numbers around — BC. According to the Pythagorean system of thought, there is equivalence between numbers and forms; therefore a cube with a side length of zero loses its shape, and a relation including numbers that contain a zero does not make sense anymore.

For the cosmos of the Pythagorean School which only expresses itself in number proportions, the zero therefore poses a threat — an invasion of nothingness, of chaos into a perfectly designed system.

The atomists postulate an empty space between the smallest components of the world. The philosophy of Aristotle — BCwho adopted the Pythagorean views into his system, however, remained formative for the occidental thinking for nearly years. Inhe was consecrated Pope Sylvester II, but due to this attempt laid himself open to attack from the church. Yet half a century earlier, there had been efforts in Italy to introduce the zero.

This numerical series also contains an increasingly exact representation of the golden ratio through the relation of two consecutive numbers. Geometrically, the golden ratio refers to the division of a quantity such that the ratio of the larger part to the whole quantity is the same as the ratio of the smaller part to the larger part.

The golden ratio can be found in nature in several proportions and is already for the Pythagoreans an expression of perfect harmony. So, it is no coincidence that they chose a symbol for their cult in which the lines are divided in the relationship of the golden ratio, the pentagram. Writing and number systems are the basis for the abstract use of any kind of object. The basis for logical reasoning is thinking, which is assumed to be in principle consistent.

A statement conclusion may be made due to facts premises and their mutual concepts. Of course, the underlying facts must be consistent in the sense of an axiom, being a premise which is accepted as absolutely right and therefore does not require any further proof. According to Aristotle, an axiom is a sentence that is 16 2 Historical Development of Algorithmic Procedures taken for granted as valid, but it may be the basis for a proof. The conclusion or the conservation of a true statement out of the given propositions is subject to the laws of formal logic deduction that subordinates the special to the general.

A syllogism, understood as a three-part conclusion, looks according to Aristotelian logic as follows: A predicate P assigned to this quantity can in the conclusion also be assigned to the subject over the terminus medius. A well-known example for a three-part syllogism is the following conclusion: The terms used have the same meaning. Law of contradiction also law of non-contradiction: There is no statement that is both true and false at the same time. Law of excluded middle: Every statement is either true or false.

In order to receive general statements, Aristotle applied two types of induction: Imperfect induction uses a number of particular statements to get to a general statement. Induction by enumeration that can also be attributed to deduction, starts with proving a characteristic for a certain number of elements of a group in order to then prove this characteristic to be true for all other elements of this group.

Socrates used induction to infer knowledge as a general term by observing particular cases. Aristotelian logic determined the development of occidental thinking; however, from around BC, there was also a tradition of Buddhist logic in India.

Logic experienced further developments by the philosophy of the stoics from around BC. The stoa distinguished between object, theoretical image and linguistic sign. The statement became the smallest relevant part of a logical operation; 2.

The works of Aristotle fell into oblivion and only became present again in the High Middle Ages in the course of a return to antique writers. The socio-political changes in this era, such as higher agricultural production, improvement of commerce and specialization of trades led to wealth and a higher life expectancy. Reading and writing no longer were privileges reserved to the clergy.

The era of Scholasticism began with Anselm of Canterbury —putting rationality as a means of achieving knowledge alongside faith. In the 13th century, Aristotelian philosophy became, above all with Albertus Magnus around — and his student Thomas Aquinas around —an inherent part of scholastic thinking. Scholasticism tried to legitimize faith by means of rationality — logical thinking was cultivated, but mostly was an instrument for supporting Christian principles of faith.

These abstraction processes are necessary preconditions for the development of systems in whose context different algorithms for the generation of musical structure may also be applied. It is about nothing less than the mathematization of knowledge and accessing it by means of a machine for the production of logical statements.

The revolutionary concept of this approach exists in the idea that true statements can be obtained by algorithmic combinations of accepted terms. Each combination of the AM has an underlying alphabet of nine letters. The symbols from B to K are semantic carriers of expressions in different categories, such as divine attributes, categorical determinants, question words, subjects, virtues and vices, as represented in table 2.

Three diagrams and an arrangement of movable concentric circles form the working aids of the AM. The lines in the centre describe possible relations. If, for the AM, the attribute also represents something universal, the meaning changes through the combination. Because God is all-embracing in the theological context, his attributes are also universal, therefore forming the basis for everything that exists. Using the AM is a process of interpretation that happens in the context of three letters of the alphabet.

What is the term? What are essential components or manifestation of the term? Through a letter that may carry several meanings, the intellect more generally comprehends manifold meanings and also knows them [. The main problem is in the almost exclusive use of circular reasoning, meaning that what you are supposed to be proving is assumed to be true and the conclusion of the argument is implicitly or explicitly assumed in one of the premises respectively also: In addition, the ambiguous contexts make clear statements nearly impossible.

So in this system, truth can only be deduced in the context of the Christian dogmatism of that time, if at all. The underlying theological principles, however, are axioms according to Lullus and therefore an irrevocable basis of an objective establishment of the truth.

Guido of Arezzo around — contributed considerably to the development of notation, developed solmization and was an important music theorist of the medieval era. In chapter 15 and 17, he outlines a system for the automatic generation of melodies out of text material [20, p.

Letters, syllables and components of a verse are mapped on tone pitches and melodic phrases neumeswhereas groups of neumes are separated by caesurae. On the level of groups of neumes, the caesurae correspond to breathing pauses and can also be found in smaller groups in the form of pauses or held notes.

The motet became the dominant form of polyphonic vocal music in occidental music history. With kind permission of the Badische Landesbibliothek. Machaut around —Guillaume Dufay around — and Josquin Desprez around — and reached its last high point with Giovanni Pierluigi Palestrina — and Orlando di Lassus — The principle of isorhythm, invented by Philippe de Vitry and reaching its peak with Guillaume de Machaut consists of multiple repeating melodic color and rhythmic talea models that also interfere with each other and can occur in different proportions.

In order to meet the requirements of an increasingly complex polyphony, also a new concept of notation developed that, in contrast to chant and modal notation used until the early 13th century, also allowed the differentiation of rhythmic structures.

Although in chant notation, developing from neumes, notes are arranged on single text syllables, the concrete rhythmic form cannot be notated with this concept. The modal notation, however, distinguishes between some triple-timed rhythms in different modes, although complex temporal structuring is still impossible with this notation form. The decisive innovation of mensural notation lies in the ability of the system to indicate the temporal duration of a note by its shape.

The values and the modes resulting from the divisions as well as accepted combination possibilities enable a complex rhythmic repertoire.

The form of notation as known today came into being around the 16th century and stems from further developments and roundings of the mensural note values. Both mensural notation and the complex musical procedures of the motet illustrate the essential abstraction achievements of this era, ones that are also of great importance to the development of algorithmic composition. Mensural notation enables the representation of several musical parameters with a symbol and constitutes an event space for possible rhythmic constellations with constraints of the permit- 24 2 Historical Development of Algorithmic Procedures ted combination possibilities.

The taleas and colors of the motet, however, show the structure generating application of musical parameter series — a procedure which came back into use in serialism of the 20th century.

The analogies to hardware and software, data memory, program, etc. Because of the given combination possibilities, some degree of chance is involved that, however, due to the interpretation rules, provides coherent statements in the given context whose exact interpretation is left to the user.

The attempt to generate musically incontrovertible structure by applying proven or unproven sentences of any designed system on musical parameters is made time and again. What is left is the concretely produced structure, the musical information. The musical quality of a structure produced in this way as well as in all other ways has to be left aside, because even value judgments exerted by musicological discourse are inevitably subject to personal preferences or trends.

Here, too, a parallel may be drawn to a system of algorithmic composition that in most cases allows the generation of a whole class of compositions by producing a meta-structure.

This system consists of three categories of labeled wooden sticks syntagmas on which both numbers and rhythmic values are engraved. In an advanced form, style-typical material of particular musical genres can be produced.

The number columns represent levels of different modes and are arranged in groups of 2 to 12 units. These units serve to correctly transfer text passages and represent one syllable each.

Moreover, this principle shows an application of pitch classes, which have been a common representation system in the production and analysis of musical material since the 20th century. In a letter from asking for support for his plans, Leibniz wrote to his employer Duke Johann Friedrich of Braunschweig: His program for the formalization of the sciences was to be achieved through a number of preconditions: One postulate was that by combining such symbols, a logical relationship occurred. As a set of symbols, he chose the group of natural numbers.

Another logical connection could be of the following form: The characteristic numbers that cannot be divided any further must naturally be represented by prime numbers in order to enable explicit partition relations. Although it is a groundbreaking approach, the concept of characteristic numbers also exhibits a few problems regarding the application of some term attributions as well as the rep- 28 2 Historical Development of Algorithmic Procedures resentation of combinations of Aristotelian logic.

Leibniz himself has carried out some advancements of his characteristic numbers. Leibniz, however, elaborated a concrete outline of a system, regardless of the impossibility of its realization. In Leibniz, numbers become an instrument for organizing the world — even music results from a metaphysical act of calculating: Bach, who combines in The Art of Fugue or in his Goldberg variations complex structuring procedures with highest musical expression.

Indeed, its development had a long tradition, going back to the ancestor of all calculating machines, the abacus. The principle is quite simple: Numbers are represented according to their place value by balls or stones in different columns. By moving the units, addition, subtraction and also combined calculating operations can be carried out. Early forms of the abacus can already be found in Babylon.

The device described in a draft has a number of cog wheels that are capable of moving other cog wheels by one position after ten turns. With this principle, it is possible to carry out additions in the decimal system; however, it is not proven if this apparatus should represent a calculating machine or a system for gear transmission. The Pythagorean abacus is a tool consisting of a table for reading off the times multiplication table. For multi-digit multiplications, the partial products are added.

In division, the quotient is obtained stepwise by indicating all products of the respective divisor. In this case, the following division results: The last knotted string on the right side of the Quipu indicates the total amount of pages. Provided that numbers are realized as an abstract value, further representation is made within a number system enabling a clearly arranged representation of larger 2.

Symbols for number units are generated whose main counting unit represents the basis of the number system. Today, the decimal system has widely established itself.

Similar examples can also be found in English, Gaelic and other languages. Number symbols for different values are either cumulatively merged into units or symbols are used whose value is determined by their position. These systems are referred to as additive or positional number systems.

## Algorithmic Composition: Paradigms of Automated Music Generation

The basic number symbols consist of tens and ones that are merged in symbols up to the value The position of the symbol in the symbol string gives information about the respective 12 2 Historical Development of Algorithmic Procedures potency.

Between the units of sixty, a little space is often left. Similar functions are also performed by different symbols of the twenty-base vigesimal number system of the Classic era of the Maya from AD to The Egyptian number system dates back to the time of the old empire around — BC. The basic principle stays unchanged through all succeeding eras, although the hieratic from around BC on and the demotic writing system from around BC on deriving from it enable an easier representation of numbers by merging multiply used symbols to single symbols.

The hieroglyphic writing system is a decimal additive number system. The representation indeed starts depending on the notation from right to left, from left to right or from top to bottom with the smallest values; however, the potency of the base is only determined by the form of the hieroglyph and not by its position.

The four basic c Reprinted by permission of Fig. The Greek notation developed in two differing systems, making use of either special symbols from around BC on or alphabetic characters from around BC on to represent numbers. The Roman number notation 2. In the measure of length, a superior circle undoubtedly stands as a symbol for the zero. Most likely, the zero was used much earlier in Indian mathematics. With the campaigns of Alexander the Great, amongst others the Babylonian number system found its way to India that at this time used a number system similar to the Greek system.

The Babylonian system was transformed into a decimal system and since at the latest it was put down by a Syrian bishop that the Indians counted with new numbers.

It is unknown whether the zero was already used at this point; however, it is most likely that the zero was used already before the inscription of Gwalior. The achievement of the Indian mathematicians was giving the number zero in addition to its function as a placeholder an independent position amongst the numbers — the zero became a value that could be used for calculations.

The reasons for the development of the zero as an independent number in India and not gaining ground in Europe for such a long time can be found in differing philosophical and religious concepts. The zero being also a symbol for nothingness meets parts of Indian philosophy, which understands emptiness as the origin and objective of every development. A 1 The information given in this section refers mostly to two works treating exhaustively the development of the number zero: The reasons for this refusal have their earliest roots in the Pythagorean conception of numbers around — BC.

According to the Pythagorean system of thought, there is equivalence between numbers and forms; therefore a cube with a side length of zero loses its shape, and a relation including numbers that contain a zero does not make sense anymore.

For the cosmos of the Pythagorean School which only expresses itself in number proportions, the zero therefore poses a threat — an invasion of nothingness, of chaos into a perfectly designed system. The atomists postulate an empty space between the smallest components of the world. The philosophy of Aristotle — BCwho adopted the Pythagorean views into his system, however, remained formative for the occidental thinking for nearly years.

Inhe was consecrated Pope Sylvester II, but due to this attempt laid himself open to attack from the church. Yet half a century earlier, there had been efforts in Italy to introduce the zero.

This numerical series also contains an increasingly exact representation of the golden ratio through the relation of two consecutive numbers. Geometrically, the golden ratio refers to the division of a quantity such that the ratio of the larger part to the whole quantity is the same as the ratio of the smaller part to the larger part.

The golden ratio can be found in nature in several proportions and is already for the Pythagoreans an expression of perfect harmony. So, it is no coincidence that they chose a symbol for their cult in which the lines are divided in the relationship of the golden ratio, the pentagram. Writing and number systems are the basis for the abstract use of any kind of object.

The basis for logical reasoning is thinking, which is assumed to be in principle consistent. A statement conclusion may be made due to facts premises and their mutual concepts. Of course, the underlying facts must be consistent in the sense of an axiom, being a premise which is accepted as absolutely right and therefore does not require any further proof.

According to Aristotle, an axiom is a sentence that is 16 2 Historical Development of Algorithmic Procedures taken for granted as valid, but it may be the basis for a proof.

The conclusion or the conservation of a true statement out of the given propositions is subject to the laws of formal logic deduction that subordinates the special to the general. A syllogism, understood as a three-part conclusion, looks according to Aristotelian logic as follows: A predicate P assigned to this quantity can in the conclusion also be assigned to the subject over the terminus medius. A well-known example for a three-part syllogism is the following conclusion: The terms used have the same meaning.

Law of contradiction also law of non-contradiction: There is no statement that is both true and false at the same time. Law of excluded middle: Every statement is either true or false. In order to receive general statements, Aristotle applied two types of induction: Imperfect induction uses a number of particular statements to get to a general statement.

Induction by enumeration that can also be attributed to deduction, starts with proving a characteristic for a certain number of elements of a group in order to then prove this characteristic to be true for all other elements of this group. Socrates used induction to infer knowledge as a general term by observing particular cases. Aristotelian logic determined the development of occidental thinking; however, from around BC, there was also a tradition of Buddhist logic in India.

Logic experienced further developments by the philosophy of the stoics from around BC. The stoa distinguished between object, theoretical image and linguistic sign. The statement became the smallest relevant part of a logical operation; 2. The works of Aristotle fell into oblivion and only became present again in the High Middle Ages in the course of a return to antique writers. The socio-political changes in this era, such as higher agricultural production, improvement of commerce and specialization of trades led to wealth and a higher life expectancy.

Reading and writing no longer were privileges reserved to the clergy. The era of Scholasticism began with Anselm of Canterbury —putting rationality as a means of achieving knowledge alongside faith.

In the 13th century, Aristotelian philosophy became, above all with Albertus Magnus around — and his student Thomas Aquinas around —an inherent part of scholastic thinking. Scholasticism tried to legitimize faith by means of rationality — logical thinking was cultivated, but mostly was an instrument for supporting Christian principles of faith. These abstraction processes are necessary preconditions for the development of systems in whose context different algorithms for the generation of musical structure may also be applied.

It is about nothing less than the mathematization of knowledge and accessing it by means of a machine for the production of logical statements.

### Organum | Revolvy

The revolutionary concept of this approach exists in the idea that true statements can be obtained by algorithmic combinations of accepted terms. Each combination of the AM has an underlying alphabet of nine letters. The symbols from B to K are semantic carriers of expressions in different categories, such as divine attributes, categorical determinants, question words, subjects, virtues and vices, as represented in table 2.

Three diagrams and an arrangement of movable concentric circles form the working aids of the AM. The lines in the centre describe possible relations. If, for the AM, the attribute also represents something universal, the meaning changes through the combination.

Because God is all-embracing in the theological context, his attributes are also universal, therefore forming the basis for everything that exists. Using the AM is a process of interpretation that happens in the context of three letters of the alphabet. What is the term? What are essential components or manifestation of the term? Through a letter that may carry several meanings, the intellect more generally comprehends manifold meanings and also knows them [.

The main problem is in the almost exclusive use of circular reasoning, meaning that what you are supposed to be proving is assumed to be true and the conclusion of the argument is implicitly or explicitly assumed in one of the premises respectively also: In addition, the ambiguous contexts make clear statements nearly impossible. So in this system, truth can only be deduced in the context of the Christian dogmatism of that time, if at all.

The underlying theological principles, however, are axioms according to Lullus and therefore an irrevocable basis of an objective establishment of the truth. Guido of Arezzo around — contributed considerably to the development of notation, developed solmization and was an important music theorist of the medieval era.

In chapter 15 and 17, he outlines a system for the automatic generation of melodies out of text material [20, p. Letters, syllables and components of a verse are mapped on tone pitches and melodic phrases neumeswhereas groups of neumes are separated by caesurae.

On the level of groups of neumes, the caesurae correspond to breathing pauses and can also be found in smaller groups in the form of pauses or held notes. The motet became the dominant form of polyphonic vocal music in occidental music history. With kind permission of the Badische Landesbibliothek. Machaut around —Guillaume Dufay around — and Josquin Desprez around — and reached its last high point with Giovanni Pierluigi Palestrina — and Orlando di Lassus — The principle of isorhythm, invented by Philippe de Vitry and reaching its peak with Guillaume de Machaut consists of multiple repeating melodic color and rhythmic talea models that also interfere with each other and can occur in different proportions.

In order to meet the requirements of an increasingly complex polyphony, also a new concept of notation developed that, in contrast to chant and modal notation used until the early 13th century, also allowed the differentiation of rhythmic structures. Although in chant notation, developing from neumes, notes are arranged on single text syllables, the concrete rhythmic form cannot be notated with this concept.

The modal notation, however, distinguishes between some triple-timed rhythms in different modes, although complex temporal structuring is still impossible with this notation form. The decisive innovation of mensural notation lies in the ability of the system to indicate the temporal duration of a note by its shape. The values and the modes resulting from the divisions as well as accepted combination possibilities enable a complex rhythmic repertoire.

The form of notation as known today came into being around the 16th century and stems from further developments and roundings of the mensural note values. Both mensural notation and the complex musical procedures of the motet illustrate the essential abstraction achievements of this era, ones that are also of great importance to the development of algorithmic composition.

Mensural notation enables the representation of several musical parameters with a symbol and constitutes an event space for possible rhythmic constellations with constraints of the permit- 24 2 Historical Development of Algorithmic Procedures ted combination possibilities.

The taleas and colors of the motet, however, show the structure generating application of musical parameter series — a procedure which came back into use in serialism of the 20th century. The analogies to hardware and software, data memory, program, etc. Because of the given combination possibilities, some degree of chance is involved that, however, due to the interpretation rules, provides coherent statements in the given context whose exact interpretation is left to the user.

The attempt to generate musically incontrovertible structure by applying proven or unproven sentences of any designed system on musical parameters is made time and again. What is left is the concretely produced structure, the musical information. The musical quality of a structure produced in this way as well as in all other ways has to be left aside, because even value judgments exerted by musicological discourse are inevitably subject to personal preferences or trends.

Here, too, a parallel may be drawn to a system of algorithmic composition that in most cases allows the generation of a whole class of compositions by producing a meta-structure.