Sergey Pshenichnikov, Tatiana Sotnikova
Trio Sapiens
Musical text can be represented using matrix units, like the description of verbal texts and other symbolic sequences. In the future, mathematical recognition, and creation of musical sense with substantive justification for intermediate calculations (as opposed to AI) may become possible.
Sound has four properties: pitch, duration, volume, and timbre. Timbre is not considered yet. The dictionary of the algebra of musical texts is built on the basis of musical notation for the piano.
The duration here, for the sake of brevity of the first presentation, is considered as «absolute». «Relative» is not considered, although intervals are very well studied, and their features will be needed to categorize composers.
The complexity of the musical text for the application of mathematics is explained by the desire to simplify the reading of musical notes by musicians and to minimize the use of lower and upper additional lines.
To apply text algebra to musical symbolic sequences there is no need to use a five-line staff. What is useful and familiar to musicians is «unbearably harmful» for the use of algebra. It seems advisable to use a one-line staff. In this case, the musical text becomes like the verbal text.
To solve the problem, you need to find a transformation of the canonical musical text into a «thread». And as always, for a new application of algebra, correct coordination of the subject area is necessary. In this case, each used musical notation and symbol of modern musical notation must be assigned its own serial number (natural number).
Instead of a sign, you can use the names of each note symbol - then it will be a verbal notation of musical texts written in one line «thread»).
Since the musical scale is completely represented by piano keys, the first section in height of the dictionary of musical texts consists of 88 numbered white and black keys (of which 52 are white). This eliminates the need for an octave division of the scale, octave transfer signs, keys, five alteration signs (key and random), diatonic and chromatic semitones.
All notes of the scale became fundamental in algebraic musical notation. There is an order of magnitude more of them of them than the main stages of Guido Aretinsky, but the alteration signs and names of octaves disappeared, the use of which made musical texts algebraically incompatible with verbal texts. Numbers from 1 to 88 in algebraic notation constitute a fragment of the pitch dictionary for the «thread» one-line staff.
Numbering (coordination) of notes is needed to become in the future indices of mathematical objects (matrix units), which will replace the signs of notes or their names. These matrix units are binary generalizations of integers (hyperbinary numbers). The operation of division with remainder is defined for them, as for integers. The operation will allow you to divide musical texts and their fragments into each other in order to find their similarities (multiples) and differences (remnants).
It becomes possible to calculate mathematically accurately what sounds similar or different to the ear. And automatically classify (categorize) musical texts according to their calculated characteristics (properties), without specifying or naming them a priori. For example, proof of music plagiarism may be mathematically rigorous rather than subjective expert opinion. If we define the very concept of “musical plagiarism” strictly mathematically, and not agree with a subjective opinion.
Using algebraic methods, it is also possible to recognize, for example, in the variations of Denis Matsuev’s New Year’s joke the original melody «A Christmas tree was born in the forest». With the help of sampling, Kotelnikov's theorem and Fourier analysis used in the Shazam application, for example, it is impossible to determine a piece of music in advanced and non-trivial variations of performance.
The word of the musical text is vertical consonance. In extreme cases it may consist of one note. It can also consist of notes on several staves united in canonical notation by an accolade bracket.
The names of the signs of duration and loudness in the presented algebraic notation always clearly refer to each note of the word (vertical consonance) and their effect is not transferred to neighboring consonances. In canonical notation this is not the case. There are horizontally acting signs along the staff. For example, volume signs. They can act on one note or immediately on a set of notes located along the staff. Although they are indicated once before such a group. The musician understands this and separates these two possibilities when performing.
In algebraic notation, the possibility of ambiguity (the “or” mentioned above) is excluded. For each note the volume sign is repeated. To apply mathematics, you need to place duration and volume signs in front of each note. It is known that a mathematician (or programmer) goes to bed with two glasses of water. With empty and full. This wonderful professional uses an empty glass if he doesn’t want to drink.
When formalizing, everything must be indicated explicitly. In each vertical consonance, each note is described by all the signs of musical notation that relate to it, regardless of the proximity or distance of their location to this note. For example, in a group of notes, the tails can merge into one common line or a group of straight lines to make it easier for a musician to read the musical text. There are no such lines in algebraic notation. Each note from such a group is assigned the same tail.
If it is necessary to consider the transfer of action, then in each vertical consonance the names (numbers) of the signs of duration and volume are repeated in the algebraic form of the musical text presented here.
Each note of vertical consonance is described with words from fragments of the dictionary. Multiple words from one section may be used. This depends on the detail of dividing the dictionary into sections. For example, in the duration section, could highlight a subsection of pauses.
The usual matrix unit of a verbal word has two indices. In fact, these are not two indices, but two groups of indices or two groups of coordinates. The technique of splitting one index into two, three, and so on is a common technique. This possibility is since twos, threes and other sets of indices can be numbered with natural numbers. One number instead of a set of indices.
It is extremely important that there are two groups of coordinates. In this case, the possibility of representing words by two-index matrix units remains. For clarity, groups are separated by a semicolon, and within a group, split indices separated by a comma.
In the first group on the right the first index is the number of the vertical consonance in the musical text or its fragment. The second coordinate of the first group is the note number in consonance from bottom to top. In the first group, from the left to the semicolon, the indices indicate numbers from sections of the dictionary of musical text.
The coordination of musical texts differs from the coordination of verbal texts in the order of indices and their groups. This is done to formally distinguish between musical and verbal texts. In verbal texts, first on the left are indices numbering the words in the text, then after a comma or semicolon - numbers (indices) of words in the text dictionary. Algebraically, the order of the index groups does not matter. The only difference is in the right or left division with remainder of matrix texts and their fragments.
A musical word is a vertical consonance. Each note of vertical consonance is a letter of a musical word. The order of letters in such a musical word is unimportant. Only their sum makes sense. They sound at the same time. A special case of vertical consonance is a chord. For lexicographic ordering of notes, the order of letters in a musical word is taken from bottom to top.
An example of algebraic musical text for GNOSSIENNE NO.1 by Erik Satie is given in the Appendix to the book: Sergey Pshenichnikov. Algebra of text for judgments. Self-tutorial understanding. Ridero. 2024. 159 p. ISBN 978-5-0062-3246-4.
https://ridero.ru/books/algebra_of_text_for_judgments_self-tutorial_understanding/
