Matematičeski osnovi na obšta teorija na muzikata:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | Bulgarian |
| Veröffentlicht: |
Plovdiv
Akad. Izdat. "Nova Magnaura"
2001
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Abstract |
| Beschreibung: | PST: Mathematical fundamentals of general music theory. - In kyrill. Schr., bulg. - Zsfassung in engl. Sprache |
| Beschreibung: | 180 S. Ill., graph. Darst., Notenbeisp. |
| ISBN: | 9549081311 |
Internformat
MARC
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| 008 | 050223s2001 adgl |||| 00||| bul d | ||
| 020 | |a 9549081311 |9 954-90813-1-1 | ||
| 035 | |a (OCoLC)645274776 | ||
| 035 | |a (DE-599)BVBBV019708892 | ||
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| 100 | 1 | |a Kožucharov, Ilija |d 1949- |e Verfasser |0 (DE-588)140574158 |4 aut | |
| 245 | 1 | 0 | |a Matematičeski osnovi na obšta teorija na muzikata |c Ilija Kožucharov |
| 264 | 1 | |a Plovdiv |b Akad. Izdat. "Nova Magnaura" |c 2001 | |
| 300 | |a 180 S. |b Ill., graph. Darst., Notenbeisp. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a PST: Mathematical fundamentals of general music theory. - In kyrill. Schr., bulg. - Zsfassung in engl. Sprache | ||
| 650 | 0 | 7 | |a Mathematik |0 (DE-588)4037944-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Musiktheorie |0 (DE-588)4040876-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Musiktheorie |0 (DE-588)4040876-0 |D s |
| 689 | 0 | 1 | |a Mathematik |0 (DE-588)4037944-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m Digitalisierung BSBMuenchen |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013036306&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung BSB Muenchen |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013036306&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Abstract |
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Datensatz im Suchindex
| _version_ | 1804133161953656832 |
|---|---|
| adam_text | СЪДЪРЖАНИЕ
Предисловие към изданието
.................................................... 11
1.
Въведение
.................................................................. 13
2.
Ситуиране на проблема
....................................................... 15
2.1.
Цел
....................................................................... 15
2.1.1.
Ситуиране на целта
........................................................... 15
2.1.1.1.
Методика
-
методология
....................................................... 15
2.1.1.1.1. Ars -
compendium
pragmaticum
-
scíencia artis
...................................... 16
2.1.1.1.2.
Принцип на „хипотетичния дивак ................................................
17
2.1.1.1.3.
„Откъде знам това, което считам, че знам?
........................................ 18
2.1.1.1.3.1.
Скептицизъм
-
„Парадокс на Нелсън
............................................. 19
2.1.1.1.4.
Емпирична и математическа
индукция
............................................ 20
2.1.1.1.4.1.
„Вярвам, за да знам или „Знам, за да вярвам ?
.................................... 20
2.1.1.1.5.
Статистика
-
вероятност
....................................................... 21
2.1.1.1.6.
Редукционизъм
.............................................................. 22
2.1.1.1.7.
Редукция
1 :
безкрайно
-
крайно
................................................. 23
2.1.1.1.8.
Редукция
2:
непрекъснато
-
прекъснато
........................................... 24
2.1.1.1.9.
Редукция
3
(традиционна
&
лоша): количество
-
качество
............................. 24
2.1.1.1.10.
Комбинаторика
.............................................................. 25
2.1.1.1.11.
Причина
-
следствие
.......................................................... 26
2.1.1.1.12.
Антиномия:
точно
-
неточно
.................................................... 27
2.1.1.1.12.1.
Теория на размитите множества
................................................ 27
2.1.1.1.12.2.
Парадигма
„Холопов
-
парадигма
„Колмогоров
.................................... 29
2.1.1.1.12.3.
Точна вселена
-
неточна снимка
................................................ 30
2.1.1.2.
Обобщения за
2.1.1.1. - 2.1.1.1.12.3.............................................. 30
2.1.2.
Постижимост на целта
........................................................ 31
2.1.2.1.
физика
.................................................................... 31
2.1.2.1.1.
Ентропия
.................................................................. 31
2.1.2.2.
Биология
.................................................................. 31
2.1.2.2.1.
Обратна връзка
............................................................. 32
2.1.2.2.2.
„Мултидименсия и „Черни кутии
............................................... 32
2.1.2.3.
Медицина
.................................................................. 33
2.1.2.3.1.
Експертни системи
........................................................... 33
2.1.2.3.2.
Медицинска статистика
....................................................... 34
2.1.2.3.3.
Приблизителна
импликация
.................................................... 34
2.1.2.4.
Психология
................................................................. 35
2.1.2.4.1.
Модел и моделиране
.......................................................... 35
2.1.2.4.2.
Извод по аналогия
........................................................... 36
2.1.2.4.3.
(Много)
факторен
анализ
....................................................... 36
2.1.2.4.4.
„Мисля и чувствам, ерго не знам: човек ли съм или машина?
......................... 36
2.1.2.5.
Лингвистика
................................................................ 37
2.1.2.5.1.
Честотни речници
............................................................ 37
2.1.2.5.2.
Валентни речници и графи
..................................................... 38
2.1.2.5.3.
Автоматизиран машинен превод
................................................ 39
2.1.2.5.4.
Вериги на Марков
............................................................ 40
2.1.2.6.
...идруги
.................................................................. 40
2.1.2.6.1.
Икономика
................................................................. 40
2.1.2.6.2.
Социология
................................................................. 41
2.1.2.6.3.
История
................................................................... 42
2.1.2.6.3.1.
Контент
анализ
.............................................................. 42
2.1.2.7.
Изкуствознание
.............................................................. 43
2.1.2.7.1.
Поетика, проза
.............................................................. 43
2.1.2.7.1.1.
Пространство на вълшебните приказни елементарни събития
.......................... 43
2.1.2.7.1.2.
Дигитална поезия и дигитална проза
............................................. 44
2.1.2.7.2.
Графика,
живопис
............................................................ 45
2.1.2.7.2.1.
формули на
естетическата оценка
............................................... 45
2.1.2.7.2.2.
Семантичен диференциал
...................................................... 47
2.1.2.7.2.3.
фрактали ..................................................................
48
2.1.2.7.3.
Музика
.................................................................... 49
2.1.2.7.3.1.
От Питагор до Марен Мерсен
................................................... 49
2.1.2.7.3.1.1.
Принудително връщане към скептицизма
.......................................... 50
2.1.2.7.3.1.2.
Отново върху историческата линия: Питагор
-
Марен
Мерсен
........................... 51
2.1.2.7.3.2.
XX
век
..................................................................... 52
2.1.2.7.3.3.
Българско участие
............................................................ 60
2.1.2.8.
Обобщения за
2.1.2.1.-2.1.2.7.3.3............................................... 62
2.2.
Задачи
..................................................................... 62
3.
Опит за решаване на проблема
................................................. 63
3.1.
Аксиоматика
................................................................ 63
3.1.1.
Хипотетико-дедуктивен метод и операционални дефиниции
............................ 63
3.1.2.
Избор на аксиоми
............................................................ 64
3.1.2.1
Изисквания към аксиомите,
метатеория
........................................... 64
3.1.3.
Аксиоми на музикалната теория
................................................. 64
3.1.3.1.
Коментари
.................................................................. 64
3.1.3.1.1.
Първа аксиома
.............................................................. 65
3.1.3.1.2.
Втора аксиома
.............................................................. 66
3.1.3.1.3.
Трета аксиома
.............................................................. 66
3.2.
Експеримент в едномерност
.................................................... 66
3.2.1.
Закон за скока
.............................................................. 67
3.2.1.1.
Хипотеза
................................................................... 67
3.2.1.2.
Проверка
.................................................................. 67
3.2.1.2.1.
Насочена случайност
......................................................... 68
3.2.1.2.2.
Опит
с
една
тенорова
мелодия
.................................................. 68
3.2.1.2.3.
Сума на поелементните разлики
................................................ 70
3.2.1.2.4.
„Добри , „лоши и
„?
скокове
.................................................. 71
3.2.1.3.
Изводи
.................................................................... 72
3.3.
Експеримент в тримерност
..................................................... 72
3.3.1.
Представяне на многомерни данни
............................................... 72
3.3.1.1.
Работен пример на тримерност
.................................................. 73
3.3.1.1.1.
Метрическа позиция
.......................................................... 73
3.3.1.1.2.
Ритмическа стойност
.......................................................... 73
3.3.1.1.3.
Мелодически
интервали и преходи
............................................... 74
..3.1,1.4.
Метро-ритмо-интервалика
..................................................... 75
3.3.2.
От силует към отворен край
.................................................... 76
3.3.2.1.
Ствол на силуета
............................................................. 76
3.3.2.1.1.
Предварително лимитиране на източниците
........................................ 76
3.3.2.1.1.1.
Палестрина и кризата на католицизма
............................................ 76
3.3.2.1.1.2.
Палестрина и Тридентския събор*
................................................ 76
3.3.2.1.2.
Набиране на данните
.......................................................... 77
3.3.2.1.2.1.
Крива
с и
без насищане
....................................................... 78
..3.2.1,3.
Първи резултати
...........................................................,. 78
3.3.2.1.3.1.
Метрическа субординация
..................................................... 78
3.3.2.1.3.2.
Ритмическа субординация
..................................................... 80
3.3.2.1.3.3.
Интервална субординация
..................................................... 80
3.3.2.1.3.3.1.
Низходящ Палестринов синдром
................................................ 80
3.3.2.1.3.4.
Quasi
-четвърто измерение в тримерността
......................................... 81
3.3.2.2.
Развитие от
III
аксиома
........................................................ 82
3.3.2.2.1.
Честотен речник на предфункционално ниво
(preF)
................................... 83
3.3.2.2.2.
Честотен речник на нулево функционално ниво
(F3/0)
................................ 83
3.3.2.2.3.
Положително време
-
отрицателно време
......................................... 85
3.3.2.2.4.
Алгоритъм за изчисляване на интрахомотетията
.................................... 85
3.3.2.3.
Компютърна Палестринова мелодия
.............................................. 86
3.3.2.4.
Честотен речник на първо функционално ниво
(F2/1)
................................. 87
3.4.
Опит в предверието на експерименталната музикална социална психология
............... 88
3.4.1.
Постановка на експеримента
.................................................... 88
3.4.1.1.
Странично наблюдение
............................ 88
3.4.1.2.
Признание с проекция в
III
аксиома
.............................................. 88
3.4.2.
Пространство на
експеримента
................................................. 90
3.4.3.
Първоначални
констатации
..................................................... 90
3.4.4.
Анализ на
прогнозите
......................................................... 92
3.4.4.1.
Прогноза
№1 ................................................................ 92
3.4.4.2.
Прогноза
№2 ................................................................ 92
3.4.4.3.
Прогноза
№3 ................................................................ 92
3.4.4.4.
Прогноза
№4 ................................................................ 92
3.4.5.
Ентропийност на
функционалността: (отново)
за и против
............................. 93
3.4.5.1.
Междинен
извод за ентропийните поведенчески тенденции на социума
.................. 94
3.4.5.2.
За
опасностите отхиперанонсиране
на
ентропийния показател
........................ 94
3.4.5.3.
За
съотношението: „ентропия / максимална ентропия
............................... 95
3.4.5.3.1.
За парадокса: „по-голямо пространство
-
по-голям конкорданс
........................ 96
3.4.6.
Проба с
поелементните разлики .................................................
97
3.4.7.
За
изчислителните
неточности,
породени
от неизчислителни неточности
................. 98
3.5.
Обобщения за
3. - 3.4.7........................................................ 98
4.
Заключение
.................................................................100
5.
Приложения
................................................................101
5.1.
Приложение
№1 :
Из
„Въведение в композицията
от Богуслав Шефер
...............-------101
5.2.
Приложение
№2:
Из „Канон
№3
за
цигулка
и пиано от Илия Кожухаров
.................108
5.3.
Приложения
№№
За,
36
и Зв:
Съотношения
между
„добри , „лоши и
„?
скокове .........
110
5.4.
Приложение
№4:
Базисен
масив
отдании за Палестриновия мелос
....................113
5.5.
Приложение
№5:
Ранжиране по
тегло
на
присъствие
на едномерните позиции в тримерното
Палестриново пространство на
preF
-ниво
.......................................
142
5.6.1.
Приложение №6а: Ранжиране по
тегло
на
присъствие
на
точките
в тримерното Палестриново
пространство на
Fo
-ниво
....................................................
143
5.6.2.
Приложение
№66:
Ранжиране по
тегло
на
присъствие
на
точките
в тримерното Палестриново
пространство на
Fo
-ниво
в
съответствие
с метрическата им позиция
..................146
5.7.1.
Приложение №7а:
Компютърно симулирана
мелодия в Палестринов
стил, базирана върху
данните
от
Fo
-ниво
.........................................................150
5.7.2.
Приложение
№76:
Компютърно симулирана мелодия в Палестринов стил, базирана върху
данните от
Fo
-ниво,
с
нанесени корекции в пространствата около паузите
.............151
5.8.
Приложение
№8:
Ранжиране по тегло на присъствие на част от функциите
в двумерно
Палестриново пространство на
Fi
-ниво ........................................
152
5.9.
Приложение
№9:
Динамика на един експериментален социален музикално-психологически
процес
..................................................................154
6.
Библиография
...............................................................160
7.
Предметен указател
...........................................................163
8.
Именен указател
.............................................................174
9.
Content
&
Summary
(Съдържание и резюме на английски език)
.........................175
9.
CONTENTS AND SUMMARIES
The contents of the present work is divided into small sections. This is done to ease the readers (most of them probably musicians),
and to help them quickly orientate in the text, including cases when they decide to go back through already read episodes. The English
translation of the contents would help those readers, who do not know Bulgarian, to preview the set of problems in this work and also to
order a translated version, if they wish. This work will be eventually published in english according to the aggregate interest. With the same
purpose, there are some summaries and short notes added to the different parts of the contents.
Foreword to the Edition
..................................................................... 11
1.
Introduction
.............................................................................. 13
For the musicians and people of art as a whole, it is a tradition to stay distant to the exact science. Indirectly, this is because
of their fear of abolishing the aureole of art in cases of eventual positive results when it is studied and observed more
accurately.
2.
Situating the Problem
...................................................................... 15
2.1.
Purpose
................................................................................. 15
To prove the necessity of mathematization of theoretical musicology and, on that base, to define some main points of a
general music theory.
2.1.1.
Situating the Purpose
...................................................................... 15
2.1.1.1.
Methods
-
Methodology
.................................................................... 15
Mathematisation
of theoretical musicology may be interpreted in two aspects:
First aspect: mathematisation consists of searching for concrete mathematical ways and theories, related to the solving of
concrete music theory tasks.
; Second aspect: mathematisation fundamentally changes the entire basis of music theory, using a deep, structure-defining
mathematical help.
2.1.1.1.1. Ars -
Compenrium pragmaticum
-
Scientia artis
.................................................. 16
2.1.1.1.2.
Principle of the „Hypothetical Savage
......................................................... 17
2.1.1.1.3.
„From Where I Know, What I Think I Know?
..................................................... 18
2.1.1.1.3.1.
Scepticism
-
„Nelson s Paradox
.............................................................. 19
2.1.1.1.4.
Empirical and Mathematical Induction
......................................................... 20
A large number of scientists incline to extrapolate results from empirical induction to the degree and size of results, achieved
by mathematical induction as a theoretical procedure. Objects of mathematical induction have an ideal character, and being
so they can fit a certain range of exactly pre-defined regulations. That is the reason why, it is possible those objects to be
non-contradictedly characterized with absolutizing phrases such as: „everything , „all , „nothing . On the basis of empirical
induction, it is only possible to reach such results, that are of a statistical probable character.
2.1.1.1.4.1.
„I Believe to Know or „I Knowto Believe ?
..................................................... 20
2.1.1.1.5.
Statistics
-
Probability
...................................................................... 21
2.1.1.1.6.
Reductionism
............................................................................ 22
2.1.1.1.7.
Reduction
1:
Infinite
■
Finite
................................................................. 23
2.1.1.1.8.
Reduction
2:
Continuous
-
Discrete
........................................................... 24
2.1.1.1.9.
Reduction
3:
(Traditional
&
False): Quality
*
Quantity
............................................. 24
2.1.1.1.10.
Combinatorics
............................................................................ 25
2.1.1.1.11.
Cause
-
Effect
............................................................................ 26
People tend to name some phenomena „reasons for some other „consequences only because the first (with any degree of
stability) appear or happen before the second. In all similar cases, those people ought to have in mind the following:
1.
A thing can be pointed as a reason for something else, only in cases, when the person has a preliminary knowledge of all
things in the world of things.
2.
In any other cases, a phenomenon can only conditional;/ be interpreted as a reason for the occurrence of another
phenomenon. This is true when we speak about a system of finite number of events, which may not contain the real reason.
3.
Within a system with a finite number of dimensions, which also contains
afinite
number of events, a single event can only
be taken
(!)
for more probable reason, according to its frequency of appearance previous to the following event.
2.1.1.1.12.
Antinomy: Accurately
-
Inaccurately
........................................................... 27
2.1.1.1.12.1.
Theory of Fuzzy Sets
....................................................................... 27
2.1.1.1.12.2.
Paradigm „Holopov
-
Paradigm „Kolmogorov
.................................................. 29
Holopov: „By its nature the main object of musical analysis is inexact (its quantity cannot be measured). Except for that the
exact methods appear to be in a deep relation to the apparatus of the inexact .
Kolmogorov: „I belong to those utmost desperate cyberneticians, who consider that there is no problem, impossible to solve
with the means of cybernetics .
2.1.1.1.12.3.
Accurate Universe
-
Inaccurate Picture
......................................................... 30
2.1.1.2.
Summary for
2.1.1.1. - 2.1.1.1.12.3............................................................ 30
1.
Methodological problems take more and more serious place in the field of science problems and this tendency indicates
all symptoms of durability and stability.
2.
For music science it would be convenient to have its own theory which to enable the description and exploration of any
musical phenomenon without preliminary stylistic and other limits similar to those; on its turn, this would provide a possibil¬
ity for each result to be compared and commensurable with any other of that kind.
175
3.
Building up music theory, using models of mathematical axioms appears to be rational; it would
iii
into the context of the
wide-spreading tendency of scientific reductionism.
4.
It would be expedient for mathematical borrowings to be orientated (at least at a preliminary stage) towards the math¬
ematics of the finite.
5.
For the musical theorist it would be wise to stick to the following regulations:
a) concepts, such as „cause , „effect , etc. should be interpreted only as links between the components of a concrete
described system of musical elements;
b) only these musical-and-theoretical terms which cover measurable objects can be accepted as defined correctly.
2.1.2.
Achievability of Purpose
.................................................................... 31
The thesis of achievability of the stated purpose can be supported with examples from other sciences, that have an object of
study, similar to the object of musical science.
2.1.2.1.
Physics
................................................................................. 31
During years, physicians have been facing huge problems while examining complex objects, such as macro-systems and
macro-bodies. Those objects contain a great number of identical microunits. Thus physicians lay down the foundations of
statistical physics. On its base they manage to determine a variety of properties of the explored objects, using the character¬
istics of their units and the forms of interaction of those units.
2.1.2.1.1.
Entropy
................................................................................. 31
2.1.2.2.
Biology
................................................................................. 31
2.1.2.2.1.
Feed-back
............................................................................... 32
2.1.2.2.2.
„Multidimension and „Black Boxes
.......................................................... 32
From the second half of our century biologists more and more frequently come to the conclusion, which can be approxi¬
mately formulated this way:
„The principle of classical physics, according to which, only one variable quantity is allowed to be changing, while others
stay constant, appears to be inapplicable to huge and extremely complex biological systems. Those systems are con¬
structed of multiparametrical, hierarchically subordinated dynamic subsystems, which are so tighly correlated with one
another, that when changing one characteristic feature, many others change in response (reticulated causality). Such sys¬
tems are to be explored with the means of certain .¡multidimensional analysis .
2.1.2.3.
Medicine
................................................................................ 33
2.1.2.3.1.
Expert Systems
........................................................................... 33
2.1.2.3.2.
Medical Statistics
......................................................................... 34
From a long time ago medical man from all over the world had overcome the fallacy that mathematics cannot help them in
studying and curing illness. The main thesis of the enemies of mathematical invasion in medicine is: „The objects of medical
studying are extremely complex, constanly changing, with a lot of specific features. That is why those objects cannot be
placed whitin margins of a certain mathematical outline. The modern condition of medicine shows that those statements
have been made by people, who absolutely do not know the possibilities of a certain mathematical branch of science.
Approximate Implication
.................................................................... 34
Psychology
.............................................................................. 35
Model and Modeling
....................................................................... 35
Conclusion by Analogy
..................................................................... 36
(Multijfactoral Analysis
.................................................................... 36
(Multijfactoral analysis is a procedure, based on statistical methods. These are methods of measuring a large number of
characteristic features of a given representative excerpt, finding out to what degree those features are related in between and
to what extend they (as a whole) may appear distinctive towards analogous characteristic features of other multitude of
objects.
2.1.2.4.4.
„I Think and Feel, So I Do Not Know: Am I a Man or a Machine?
..................................... 36
Frequently, when modeling mental activity, it is considered, that this activity could be interpreted as a manifestation of
multiaspect functions, which had accepted different meanings in a certain multidimensional area. The characteristic features
of mental activity situated on their levels of display form the so called area of atomic psychical units. Such an approach
contains an excellent and successfully working scientific idea: the complex phenomenon is disintegrated into a multitude of
elementary phenomena, and thus, when necessary, reduction is performed (infinite-finite, continuous-discrete).
2.1.2.5.
Linguistics
.............................................................................. 37
2.1.2.5.1.
Frequency Dictionaries
..................................................................... 37
2.1.2.5.2.
Valency Dictionaries and Graphs
.............................................................. 38
2.1.2.5.3.
Automated Machine Translation
............................................................
¡.
. 39
2.1.2.5.4.
Markoff s Chains
.......................................................................... 40
Markoff s Chains help the interpretation of language (as well as many other processes) to be performed as a row of events.
In their multitude, each event is conditionaly interpreted as a reason of the following.
2.1.2.6. ...
and Others
............................................................................ 40
2.1.2.6.1.
Economics
.............................................................................. 40
2.1.2.6.2.
Sociology
............................................................................... 41
2.1.2.6.3.
History
................................................................................. 42
2.1.2.6.3.1.
Content Analysis
.......................................................................... 42
2.1.2.7.
Art Criticism
............................................................................. 43
2.1.2.7.1.
Poetics, Prose
............................................................................ 43
176
2.1.
.2.3.3.
2.1
.2.4.
2.1.2.4.1.
2.1
?4?
2.1
?43
2.1.2.7.1.1. Space
of the Magic Tale Elementary Events
2.1.2.7.1.2.
Digital Poetry and Digital Prose
.................................................. 77
2.1.2.7.2.
GraphicArt, Art of Painting
..................................................................
T*
2.1.2.7.2.1.
Aesthetic Evaluation Formulas
............................................................... 45
Let us, for example, take a work of art and name everything we know in it „order
(0)
and all that we do not know chaos (C)
It seems possiple to measure the „aestetic evaluation (AE) with the following formulae:
AE =
1/(10
-CJ
+1).
Accordig to the formulae, the maximum „aestetic evaluation will be achieved in cases when there
¡s a
balance between the
„order and the „chaos
(50% 0 - 50%
C). Anyway, this can be taken only as a hypothesis. On other hand, according to the
starting definitions mentioned:
0 + 0 = 1,
and
G
= 1 - 0,
and
/(|()|) /(|201|+1).
If we add a coefficient „m in order to balance the proportion between the natural human simultaneous polar aspirations for
conservatism
(0)
and adventure (C), last version of the formulae will look this way:
AE = 1/(|2.m.0-1|+1).
2.1.2.7.2.2.
Semantic Differential
......................... 47
2.1.2.7.2.3.
Fractals
............................................................................... 48
2.1.2.7.3.
music
.....................................................
і.!!!! !!!!!!!!!!!!!!!!!!!!!
49
2.1.2.7.3.1.
From Pitagor to
Maren Mersenne............................................................. 49
2.1.2.7.3.1.1.
Necessary Return to the Scepticism
........................................................... 50
2.1.2.7.3.1.2.
Upon the Historical Line Again: Pitagor
- Maren
Mersenne
......................................... 51
2.1.2.7.3.2.
XX Century
.............................................................................. 52
2.1.2.7.3.3.
Bulgarian Participation
.................................................................... 60
2.1.2.8.
Summary for
2.1.2.1.-2.1.2.7.3.3............................................................. 62
1.
Mathematization of science is a general phenomenon. It concerns natural sciences, as well as scientific spheres of the
humanitarian area.
2.
The complexity of the explored objects is not an obstacle for examining them with mathematical methods.
3.
The first step in the complex phenomena research is the attempt to disintegrate them into simple atomical units. Whatever
successful or not, this atempt helps exploring and describing complex phenomena in a multlfactoral aspect.
4.
The statistical-probability studies take a particular part in mathematical intervention when studying complex objects.
5.
The control of the derived regulations is performed with models including (mostly used) algorithmical computer models.
6.
In many cases, founding some own research methods, appears to be a way, quicker than searching for already known
methods in tha concrete sphere. This, leads .to .the creation and at the same.time .the .existence of many, different .scientific
ways of approach, terminological systems etc.
2.2.
Tasks
................................................................................... 62
1.
To construct a system from basic concepts and procedures. This system should have enough universality, in order to be
able to play the role of a general music theory and at the same time, to be in general compared to the tendencies of scientific
development from the second half of the twentieth century.
2.
After solving the first problem, we must use only the precisely chosen ideas, theories and methods from the spheres of
exact science.
3.
The proposed theoretical system, should be experimentally checked.
3.
An Attemptto Solve the Problem
.............................................................. 63
3.1.
Axiomatics
..............................................................................
63
3.1.1.
Hypothetical-Deductive Method and Operational Deffinitions
........................................ 63
3.1.2.
Choice of Axioms
.........................................................................
64
3.1.2.1.
Requirements for the Axioms, Metatheory
........................·.............................
64
3.1.3.
Music Theory Axioms:
......................................................................
64
It is stated that:
1.
When we speak of a musical event (or musical act) we should understand each acoustic act, which can be registered and
evalueted.
2.
We accept that there is a multidimensional, of finite number of dimensions, discrete area of musical events (Sm), and
those events cannot happen outside its outline.
3.
A function is each real or thinkable relation among discrete positions in Sm.
64
The chosen axioms may provoke disagreements. In some of the axioms, somebody may see only elements of definitions.
Such critics come from the presumption, that the axiom should, by priority, serve for establishing the act of the existence of
something, and after that to describe its form and name.
If the third axiom was defined the following way: „It is possible to establish a real or thinkable relation, between each-two
discrete positions in Sm, and this relation is called a function , it would be easy to notice, that in an aspect, rich
m
content
thfsdefiiiton
S noľprovide
new infomation, despite the fact, that in this form, the axiom indicates an act of existence of
concrete events, which are objects of axiomatisation. 65
3.1.3.1.1.
First Axiom
.............................................................................. 66
3.1.3.1.2.
Second Axiom
...........................................................................
177
3.1.3.1.3.
Third Axiom
...................................................·------- .....
¿Ł
3.2.
An Experiment in Mono-Dimensionality
........................................................
^
3.2.1.
Skip s Rule
..............................................................................
öï
3.2.1.1.
Hipothesis
...............................................................................
67
If we accept that there is doubt in the objective ruling properties of a certain regulation (for example, the „Rule of the Skip and
its Balancing ), it would be possible to allow a hypothetical parallel between the melodical movement and the traectory of a
Brown s whit. Let s imagine that the Brown s whit moves in a certain cyllinder, the distance between „bottom and „top of
which may be randomly changeable. If this distance grows smaller, cases of natural ricochet around „bottom and „top
should increase their number as a percentage, which obviously will make the
impresión
that the movement of the Brown s
whit is submitted to the „Skip s rule .
3.2.1.2.
Verification
.............................................................................. 67
3.2.1.2.1.
Controlled Probability
...................................................................... 68
Each process can be simulated with the help of a computer if:
1.
The process can be disintegrated into separate microprocesses, which are following one another.
2.
The probability of those microprocesses (including their conditional probability) is known.
3.2.1.2.2.
An Experiment with a Tenor Melody
........................................................... 68
3.2.1.2.3.
Sum of the Differences between Corresponding Elements
.......................................... 70
3.2.1.2.4.
„Bad , „Good and
„?
Skips
............................................................... 71
3.2.1.3.
Conclusions
............................................................................. 72
Approximately
75%
of the „good skips (in other words, those which are followed by a neutralizing movement in the opposite
direction) are the skips which Palestrina would have written, without having knowledge of the „Skip s rule .
3.3.
An Experiment in Three-Dimensionality
........................................................ 72
The analysis of the Palestrina s
melos
is reviewed in three aspects:
1.
Metric characteristic,
2.
Rhythmic measurment,
3.
Melodically-interval appearance and transition.
3.3.1.
Presentation of Multi-Dimensional Data
........................................................ 72
3.3.1.1.
Experimental Example in Three-Dimensionality
.................................................. 73
3.3.1.1.1.
Metrical Position
......................................................................... 73
3.3.1.1.2.
Rhitmical Value
.......................................................................... 73
3.3.1.1.3.
Melodical Intervals and Transitions
........................................................... 74
3.3.1.1.4.
Metro-Rhythmo-lntervalness
................................................................ 75
Formation of a guasi-theorem: Musical style is a frequency distribution of musical multidimensional points.
3.3.2.
From a Silhouette to an Open End
............................................................ 76
3.3.2.1.
Stem of the Silhouette
..................................................................... 76
3.3.2.1.1.
Preliminary Limitation of the Sources
.......................................................... 76
3.3.2.1.1.1.
Palestrina and the Crisis of the Catholicism
..................................................... 76
3.3.2.1.1.2.
Palestrina and the Trident Council
............................................................ 76
3.3.2.1.2.
Data Accumulation
........................................................................ 77
The process of melodical data set up (the three-dimensional melodical points) is stopped when reaching the quantity of
115
117
points.
3.3.2.1.2.1.
A Curve with and without Saturation
........................................................... 78
3.3.2.1.3.
First Results
............................................................................. 78
The accumulated data shows that Palestrina uses
15
metrical positions,
23
different rhythmical values and
16
different
melodical intervals, respectively transitions (from tone to pause, from pause to tone and from tone to the silence after the
end of work). This automatically leads to the conclusion, that the capacity of Palestrina s three-dimensional area, inside
which his melody is disposed, includes at least
5 230
metro-rhitmo-interval points. In fact Palestrina uses only
336
of them
(or
6.09%).
Such a strong restriction may be an additional reason for naming Palestrina s style „strict .
3.3.2.1.3.1.
Metrical Subordination
................................................................... .. 78
3.3.2.1.3.2.
Rhythmical Subordination
................................................................... 80
3.3.2.1.3.3.
Interval Subordination
..................................................................... 80
3.3.2.1.3.3.1.
Descending Palestrina s Syndrome
........................................................... 80
Many explorers of Palestrina s work have pointed out the perfect ballance between ascending and descending melodical
movements. Unfortunately the computer analysis did not confirm this. The entire database was totaly analysed by a single
criteria: direction of movement (all descending movements were compared to all ascending movements). The interval called
melodical second was asumed as a standard measure. The results showed that the descending movement dominates with
6 483
seconds (about
926
octaves, if we accept that an octave contains
7
seconds) over the ascending.
3.3.2.1.3.4.
Quasi-Forth Dimension in Threedimensionality
................................... .............. 81
3.3.2.2.
Development from Third Axiom
............................................................... 82
There are some corollaries, which can be derived from the third axiom:
1.
No onedimensional point upon the co-ordinates of Sm can be accepted as a function: no connections can be made inside
It, because each connection requires minimum two components.
2.
Each multidimensional point in Sm is a function in the accepted sense. The multidimensional musical point has the sense
of an atomic musical unit. So, the onedimensional point included in the composition of the multidimensional one may be
178
interpreted as a sub-atomic musical unit.
3.
Each link between a multidimensional point and some other
multi- or
one-dimensional point is also a function in the
accepted sense.
4.
For the purpose of systematize of the functional manifestations, it is convenient to use some functional levels-
•
pre-functional level (preF)
-
on this level only the manifestations of an onedimensional point are registered without its
relation with other one- or multi-dimensional points in Sm;
•
basic (zero) functional level (Fo)
-
on this level all functional manifestations inside a multidimensional point are reqis-
tered; M
•
first functional level
(Fi)
-
on this level all functional manifestations between two functions from Fo in Sm are registered
(this is the relation of a multidimensional point from Fowith another multidimensional point from Fo; in cases when Sm is
onedimensional
-
this is the relation of an onedimensional points from Fo with another
ofthat
kind).
•
second functional level (Fz)
-
on this level all functional manifestations of a function from
Fi
with a function from Fo in
Sm are registered (this is the relation of a pair multidimensional points from
ñ?
with a multidimensional point from Fo);
•
the next levels are constructed according to the listed above scheme.
5.
The musical act of creation appears to be a selection of different functions from a multitude of existing functions. This
selection is followed by mutual configuration of those functions and it runs simultaneously at all levels.
6.
What we call style is a manifestation of certain distribution of musical functions (multidimentional points or complexes of
such points). Style has a functionality on all possible levels, nevertheless we realise it or not.
7.
Many musical-and-theoretical concepts may be considered as restricted private forms of a style manifestation. For
example, the concepts „mode , „modality , „tonality (without being alternative to one another), cover phenomena, which
appear to be limited realizations of a style in the accepted sense. In most cases the restriction refers to the choice of the
onedimentional area of tonal height, which practically is one single sector of the defined multidimensional musical area.
8.
From technological point of view and aiming to make things clearer, it is reasonable to add double index, next to the
denotation F. The first number of the index displays the measurement of the area, and the second one
-
the functional level.
For example: H/i means
-
fivedimensional area, first functional level.
3.3.2.2.1.
Frequency Dictionary on Pre-Functional Level (preF)
.............................................. 83
3.3.2.2.2.
Frequency Dictionary on
Basic-(Zero-)
Functional Level (F3/0)
...................................... 83
The frequency dictionary of the threedimensional (metro-rhythmo-interval) points of the Palestrina melodies, descendingly
arranged according to their frequency, is a descending curve with a particulary expressed pick and plain part (Appendix
№6a). This dictionary puts out a lot of questions. For instance: If we assume that Palestrina s mind and hand have been lead
by a complex of restrictive norms, then we should ask: „Where does the border line between regular and irregular phenom¬
ena lie? .
3.3.2.2.3.
Positive Time
-
Negative Time
............................................................... 85
3.3.2.2.4.
An Algorhythm for Calculating the Intrahomothety
................................................ 85
3.3.2.3.
Computer Simulation of Palestrina s Melody
.................................................... 86
The computer generates a Palestrina s style melody using the basic data in Appendix №6b. Some samples are shown in
Appendix №7a. The following notions have to be considered:
1.
The computer does not have any „knowledge about the descending Palestrina s syndrome while generating a melody.
After being instructed (inside each pause the machine had to ascend with a melody second) the results from Appendix №7b
were received.
2.
The computer does not have any „knowledge about the scopes of the separate vocal parts.
3.
The computer worked with generalized data, extracted not only from episodes with polyphonical character, but from
episodes with chordal character, as well.
3.3.2.4.
Frequency Dictionary on First-Functional Level (¥w)
.............................................
87
3.4.
An Experiment in the Entrance of Experimental Social Music Psychology
............................... 88
3.4.1.
Prepapation for the Experiment
...............................................................
88
A group of students was told that on the blackboard they would see a note with a certain heigh in
С
major and with a certain
rhythmical value in a tempo
„moderato
and in size
4/4.
Their task was to decide what a continuation (a tone with a certain
height and rhythmical value) they would place as the most natural appendix to the first tone, according to certain limits
concerning: the scope limits of the soprano, the alteration signs ban and the limits for the quantity of different note values.
After the students made their first prognoses and wrote them down, they were written another tone next to the first, and again
the students had to decide which was the most natural continuation of the new two-toned melody and to make another
prognose.
Following this scheme, the experiment spread up to
8
bars. (The students did not know for how long the experi¬
ment would last.) 8g
3.4.1.1.
Outside Observation
.......................................................................
88
3.4.1.2.
Projection in the Third Axiom
.................................................................
gQ
3.4.2.
Experimental Space
..................................;......; ; · : · ;.
V.
....................
The discrete network of the
bidimensional
area of the experiment is shown on illustration
14.
^
l are dispersed upon the different positions among the
112
points of the area ,n such a way, that
entropy
W
■
O Ld
maximum concordance
I
/У
(Стах
= 100%).
The values of the entropy and the concordance are calculated at each step of the
experiment,
ih-, rv-r
:
dance value is calculated using the formula:
Ci
= (1 -
(Ei/Emax(76)))
. 100 (%).
(The values of the concordance at Emax(ii2) are shown on each graphic in Appendix
№9.)
The experiment enables the check of the students readiness of finding out places for inner melodical repeatance of whole
motives, phrases, and musical semisentences (including variations), and their readiness to „reconcile with or „rebel against
a monotonous rhythmic and the resticted tonal capacity of the slowly opening melody, or to what extend the growth of the
length of the offered text leads to a solidarity in the students oppinion.
3.4.4.
Analysis of the Prognoses
............................■··......................................
s
3.4.4.1.
Prognosis
№1................................................·...........................
92
3.4.4.2.
Prognosis
№2 ............................................................................
92
3.4.4.3.
Prognosis
№3............................................................................
92
3.4.4.4.
Prognosis
№4...........................................................·................
92
3.4.5.
Entropy of Functional Manifestations: (Once More) „For and Against
................................. 93
3.4.5.1.
Interstitial Conclusion About Entropical Behaviour Tendencies of Society
.............................. 94
3.4.5.2.
About the Danger of Overestimation of the Entropy Index
........................................... 94
3.4.5.3.
Aboutthe Ratio: Entropy
/
Maximum Entropy
.................................................... 95
3.4.5.3.1.
About the Paradox: Bigger Space
-
Bigger Concordance
........................................... 96
3.4.6.
A test with the Sum of the Differences between Correspondings Elements
.............................. 97
3.4.7.
About the Computing Inexactness, Caused by Non-computing Inexactness
............................. 98
3.5.
Summary for
3. - 3.4.7...................................................................... 98
1.
The proposed here foundations^ a General Music Theory are built upon the concept of the multidimensional space and
the functionality, as a form of relation among elements, described with the means of such a space. This theory is indifferent
towards the analysed data, and this provides a high universality of the method.
2.
This foundations are consonant with the tendencies of the scientific development nowadays.
3.
The choice of statistical-probability methods, multidimentional discrete space, the mathematics of the finite etc., gives a
chance for the proposed ideas to lean on reliable not musical methods and theories.
4.
The results of the carried out experiments show that the proposed theoretical principles are reliably at vyork. It seems
absolutely possible some new and specifically necessary secondary concepts and procedures to be defined from the main
concepts and procedures.
5.
Each result is transperant. We can allways check: the way of receiving the result, the intput and the output, the experimen¬
tal precision.
6.
With the help of the proposed analitical and synthetical methods, we can achieve results, some of which support and
some reject the famous traditional principles.
7.
The
indépendance
of the proposed theoretical system towards the musical material, being its object, opens a possibility
for a high degree of precision when applied to comparative analytical procedures.
8.
When using a formalised music theory, a possibility for accumulation, enrichment, saving and using of big databases is
opened. On their turn, those database would allow the creation of musical expert systems. They can be efficiently applied in
automated education on musical-and-theoretical disciplines.
9.
In order to sufficiently develop, the proposed theoretical principles need a rich complex of experimentally confirmed data
from the sphere of the musical psycology.
4.
Conclusion
...............................................................................
-JOO
5.
Appendix
................................................................................101
5.1.
Appendix
№1:
From „Introduction to Composition by Boguslav Scheffer
............................101
5.2.
Appendix
№2:
From „Canon
№3
for Violin and Piano by Ilia Kozhouharov
..........................108
5.3.
Appendix №3a, 3b and 3c: Correlation Among „Bad , „Good and
„?
Skips
.........................110
5.4.
Appendix
№4:
Main Database of Palestrina s
Melos
............................................113
5.5.
Appendix
№5:
Distribution of The Onedimentional Positions in the Threedimentional Palestrina s Space
on preF level According to Their Weight
......................................................142
5.6.1.
Appendix №6a: Distribution of Points in the Threedimentional Palestrina s Space on Level Fo
According to Their Weight
........................................... 143
5.6.2.
Appendix №6b Distribution of Points in the Threedimentional Palestrina s Space on level Fo
...............
According to Their Weight and Metrical Position
......................................... 146
5.7.1.
Appendix №7a: Computer Simulation of a Palestrina s Style Melody, Based Upon the Data
f
roni
Level Fo
.....150
5.7.2.
Appendix №7b: Computer Simulation of a Palestrina s Style Melody, Based Upon the Data from Level Fo
with Correction in the Areas Around the Pauses
..................... 151
5.8.
Appendix
№8:
Distribution of a Part of the Functions in the Bidimentional Palestrina s Space on Level
Fi
.....
According to Their Weight
...............................
152
5.9.
Appendix
№9:
Dynamics of a Certain Experimental Social Musical-Psychological Process
.................154
6.
Bibliography
................ ................_-
7.
Subject Index
............... .......................................................
J
JJ
8.
Name Index
........................ . . . . . . . . . . . ....................................... 174
9.
Content
&
Summary
........ ......................................................
]Lz
.............................................................175
180
( _ Bayerische ]
|
| any_adam_object | 1 |
| author | Kožucharov, Ilija 1949- |
| author_GND | (DE-588)140574158 |
| author_facet | Kožucharov, Ilija 1949- |
| author_role | aut |
| author_sort | Kožucharov, Ilija 1949- |
| author_variant | i k ik |
| building | Verbundindex |
| bvnumber | BV019708892 |
| ctrlnum | (OCoLC)645274776 (DE-599)BVBBV019708892 |
| format | Book |
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| id | DE-604.BV019708892 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T20:04:19Z |
| institution | BVB |
| isbn | 9549081311 |
| language | Bulgarian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013036306 |
| oclc_num | 645274776 |
| open_access_boolean | |
| owner | DE-12 |
| owner_facet | DE-12 |
| physical | 180 S. Ill., graph. Darst., Notenbeisp. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Akad. Izdat. "Nova Magnaura" |
| record_format | marc |
| spelling | Kožucharov, Ilija 1949- Verfasser (DE-588)140574158 aut Matematičeski osnovi na obšta teorija na muzikata Ilija Kožucharov Plovdiv Akad. Izdat. "Nova Magnaura" 2001 180 S. Ill., graph. Darst., Notenbeisp. txt rdacontent n rdamedia nc rdacarrier PST: Mathematical fundamentals of general music theory. - In kyrill. Schr., bulg. - Zsfassung in engl. Sprache Mathematik (DE-588)4037944-9 gnd rswk-swf Musiktheorie (DE-588)4040876-0 gnd rswk-swf Musiktheorie (DE-588)4040876-0 s Mathematik (DE-588)4037944-9 s DE-604 Digitalisierung BSBMuenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013036306&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung BSB Muenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013036306&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Abstract |
| spellingShingle | Kožucharov, Ilija 1949- Matematičeski osnovi na obšta teorija na muzikata Mathematik (DE-588)4037944-9 gnd Musiktheorie (DE-588)4040876-0 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4040876-0 |
| title | Matematičeski osnovi na obšta teorija na muzikata |
| title_auth | Matematičeski osnovi na obšta teorija na muzikata |
| title_exact_search | Matematičeski osnovi na obšta teorija na muzikata |
| title_full | Matematičeski osnovi na obšta teorija na muzikata Ilija Kožucharov |
| title_fullStr | Matematičeski osnovi na obšta teorija na muzikata Ilija Kožucharov |
| title_full_unstemmed | Matematičeski osnovi na obšta teorija na muzikata Ilija Kožucharov |
| title_short | Matematičeski osnovi na obšta teorija na muzikata |
| title_sort | matematiceski osnovi na obsta teorija na muzikata |
| topic | Mathematik (DE-588)4037944-9 gnd Musiktheorie (DE-588)4040876-0 gnd |
| topic_facet | Mathematik Musiktheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013036306&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013036306&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kozucharovilija matematiceskiosnovinaobstateorijanamuzikata |