Reflection of Christ’s monogram and Christianity in music

Presentation of the work "Fugue of Constantine" for string ensemble. Polyphonic, harmonic means, which the composer used to reflect the monogram of Christ in work. Analysis of the symmetry of the product, which is applied through the Fibonacci number.

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University of Nis

Reflection of Christ's monogram and Christianity in music

fugue string ensemble polyphonic

Danijela Stojanovic,

PhD in Pedagogy, Professor, Vice-dean for Art and Science of Faculty of Arts

Natasha Nagorni Petrov,

PhD in Pedagogy, Professor, Faculty of Arts

Nis, Republic of Serbia

Abstract

A significant place in history took place in 2013, marking the celebration of the 1700th anniversary of adopting the Edict of Milan, the victory of Christianity over the paganism. Nis, as the birthplace of Constantine the Great, was therefore honored to hold the central celebration of the year. Many historians, scientists and artists have contributed to the great event. One of them, contemporary Serbian composer Dragan TomiC, inspired by this historical event, wrote a composition «Constantine Fugue» for the string ensemble on this occasion.

The paper is based on a short analytical presentation of the composition, with a special emphasis on the polyphonic and harmonic means used by the composer to reflect Christ's monograph in this piece. The choice of vertical and horizontal elements initiated the fugue as a piece form, while imitation, variation and chromatization of the theme, as well as polyphonic techniques of stretto and inversion, are the elements used to achieve the symbol reflection in this musical piece. A special aspect of the paper regarding the reflection of Christ's monograph is the analysis of the piece symmetry, which is applied through the Fibonacci number, in the form of a golden cross-section.

Keywords: Dragan Tomic, Christ's monograph, polyphonic elements, golden cross-section.

Аннотация

Отражение монограммы Христа и христианства в музыке

Даниела Стоянович, кандидат педагогических наук, профессор кафедры музыкального искусства, заместитель декана по художественно-научным вопросам факультета искусств, Университет в Нише (г. Ниш, Республика Сербия).

Наташа Нагорни Петров, кандидат педагогических наук, профессор кафедры музыкального искусства факультета искусств, Университет в Нише (г. Ниш, Республика Сербия)

В 2013 году произошло знаменательное событие в истории - празднование 1700-летия принятия Миланского эдикта, победы христианства над язычеством. Поэтому город Ниш, как место рождения Константина Великого, имел честь провести центральное празднование юбилея. Многие историки, ученые и художники внесли свой вклад в столь великое событие. Одним из них является современный сербский композитор Драган Томич. Вдохновленный таким грандиозным историческим событием, он написал по этому поводу сочинение «Фуга Константина» для струнного ансамбля.

Данная статья основана на кратком аналитическом изложении композиции, с особым акцентом на полифонические и гармонические средства, которые композитор использовал для отражения монограммы Христа в своем произведении. Выбор вертикальных и горизонтальных элементов положил начало фуге как форме произведения, в то время как подражание, вариация и хроматизация темы, а также полифонические методы стретто и инверсии использовались для отражения символа в этом музыкальном произведении. Отдельная особенность статьи, касающаяся отражения монограммы Христа, - это анализ симметрии произведения, которая применяется через число Фибоначчи, в виде золотого сечения.

Ключевые слова: Драган Томич, монограмма Христа, полифонические элементы, золотое сечение.

Main part

Introduction. The Roman State represents a unique example of a constant long duration in history, of a total of 12 centuries, with occasional rises and falls. As the most powerful empire, politically, economically, and militarily, the Roman Empire was at the height of its power in the 1st and 2nd centuries. The enormous economic potential of the state made it possible to enormously enrich the Roman aristocracy, in favor of which the citizens enjoyed. The Romans forced the conquered countries and nations to accept the Roman system, with the cult of the gods, and the Latin language as official and public [9]. The Romans did not impose other changes on subordinate nations, except when domestic religion or ruler advocated anti-Roman policy. Until Galerius' Edict of religious tolerance from 311, great persecutions were carried out against Christians for seven years, by three emperors: Maximus, Deacon, and Diocletian [13]. On the basis of Galerius' published edict, repressive measures against Christians have been abolished, allowing them to live in their own faith, build churches provided they do not violate the state order and pray to their God for the good of the emperor, the state and their own good. In Edict, Galerius' recognition that the policy of force against Christians was a mistake and a wrong move was conceived. The Roman Empire was divided into the East, ruled by Licinius, a legitimate August, and Maximinus Daia, an arbitrary ruler, and into the West, which was divided between the two Augusts, Constantine and Maxentius, an illegitimate one and another one who was a usurper. Their conflict was unavoidable, because the victor would rule the entire western part of the Roman Empire. During the war preparations, Constantine made an alliance with Licinius, a legitimate August, making him a fiancйe to his sister Constantia, there by securing his support. [9] In the desire to be the only ruler of the Western Roman Empire, Constantine confidently moves towards Rome, subduing all the cities on that road.

1. Inspiration for composition of Constantine fugue

The city of Nis (Naissus), as the birthplace of Constantine the Great (Flavius Valerius Aurelius Constantinus), was chosen as the central place for the celebration of the 1700th anniversary of the proclamation of the Milan Edict. Contribution to this great event was given by many historians, scientific researchers, visual art and music artists, inspired by the historical events in 312 and 313 (Constantine's victory at the Milvian Bridge in October 312 and the proclamation of the Milan Edict on religious tolerance in 313). Among them is the contemporary composer from Nis, Dragan Tomic, a professor at the Faculty of Arts in Nis, who inspired by the historical figure of Constantine the Great and the symbolism of Christ's monogram, composed the one-movement program work of Constantine Fugue.

The composition was premiered in 2012 by Concertante String Orchestra in Moscow at the 5th Euroorchestra Festival, under the conductorship of Milena Injac, PhD, professor at the Faculty of Arts in Nis. The next year, in 2013, within the framework of the project «Nis composers,» the composition was recorded on a CD, in the performance of the string ensemble Fairy sounds, under the conductorship of Zoran Stanisavljevic, MA, a professor at the Faculty of Arts in Nis. Among these presentations, on the occasion of the celebration of the 1700th anniversary of the Milan Edict, the composition Constantine

Fugue has been so far performed a great many times for various manifestations, both in Serbia and abroad, and by many different performers. These data indicate that the composition has timeless character, because it possesses all the predispositions for unlimited existence, just like the very historical event that still inspires many artists today to create new works, equally in all segments of art.

2. Constantine's dream - Christ's monogram

The news that Constantine was invincible quickly rushed to Rome. The historical moment for Christianity is the Battle of the Milvian Bridge in October 312, when, after months of war, Constantine's army was exhausted and without hope of victory. Then, in a dream, after appealing to God, a sign with the inscription «With this sign you will win» (lat. In hoc signo vinces) appeared in the sky to Constantine. The sign in the form of Christ's monogram was composed of the Greek initial letters of Christ's name X and P, placed one over the other and crossed [3]. The record of Constantine's biographer Evgenius of Caesar, to whom Constantine told his experience also testified to this event: «He said that at about noon, when the day began to calm down, he saw with his own eyes a light sign in the form of a cross in the sky, in the form of the sun, with the inscription «For this victory.» He was amazed by this scene, and also was his army, which testified this miracle. And as he contemplated the significance of that vision, the night fell suddenly, then Jesus Christ appeared to him in the dream with the same sign that he had already seen in the sky, asking him to make a similar sign and to use it as a protection in all the encounters with the enemy» [3].

The conviction of Constantine's dream and the uncritical nature with which it is accepted makes it still alive and current. The dream confirms and extends the message of the meaning of Christ's monogram with the mystical element of Christ's immediate experience and his recommendation for marking the shields of Constantine's soldiers. With the positive significance of the symbol in the vision before the battle and the announcement of the outcome of the event, Constantine's dream belongs to the prophetic dreams and represents the publication of the divine secret that encourages interpretation [3]. Inspired by the divine sign, in the battle of the Milvian Bridge, Constantine convincingly defeated Maxentius' army.

The dream Constantine had dreamed of and that had envisioned his victory, converted him into a Christian. Perhaps that dream did not determine Constantine's conversion, but it proves that he himself just decided to convert or if he had already been converted to publicly state it [1]. Critics in historiography take a different stand on the question of Constantine's acceptance of Christianity. In discussing possible motives for his conversion, critics have cited several: voluntary acceptance of Christianity, political benefit, religious conviction, progress and rebellion against the old system (Edward Gibbon). Critic Palank thinks that Constantine accepted the pagan monotheism of his father, and that, only for the benefit, he wanted to articulate Christianity in this religious system. Although he was a turning point between pagan and Christian civilization and the empire, most saw him as a true believer who initiated a new way of thinking and created a new mentality among people in the whole empire [14].

k4M09535 www.fctosearctrcofn

Figure 1. Christ's Monogram

As the winner and the undisputed ruler of the Western Roman Empire, the next year, in 313 he meets with Licinius, a legitimate August in Milan to keep his promise, as he promised his sister as a pledge of alliance. Then an agreement was reached between them, known as the Milan Edict, which represents the edict of religious tolerance, which allows Christians to confess freely and to no longer fear the persecutions that were marked by the three previous centuries. In the Church history, the text of Evgenius about the Milan Edict was recorded: «Since the two of us, I, Constantine August and myself, Licinius, a legitimate August, happily met in Milan, and since we considered everything related to the public good, among all the things that seemed to us to be useful in many things for all, we decided to give priority, and to put first what is related to the respect of gods and godliness, and to allow both Christians and everyone else the freedom to follow the religion they want, so that everything that is in heaven as benevolent to us and to all those who are under our authority. Today, therefore, we have made a salutary and just decision that absolutely no right be denied the right to choose and follow the divine service of the Christian religion, and that everyone should be free to turn his mind to the religion he considers to be in accordance with his attitude, so that the deity is benevolent to us, takes care of us and gives us the care and protection. Following this principle, we have agreed to bring this recipe so that what is in our previous decisions concerning Christians and which are devoted to your devotion, and which seems completely opposite and strange to our mildness, he abolished and that at the same time everybody of those who have the above-mentioned commitment to protect Christian religion, he can freely and easily hold it without difficulty» [9].

3. Analytical presentation of the composition

The composition Constantine Fugue is of a program character, written for string ensemble. It can be divided into three main parts, each of them containing fewer parts:

1) Introduction (Fanfare, Coral part and Word part),

2) Central part (Fugue: exposition, development part and ending part) and

3) Code (Coral Part andWord part).

To create a composition, the author chooses a fugue as a form, the most adequate form whose musical means can «tell» the historical event - Constantine's winning battle at the Milvian Bridge, as well as with the selected harmonic and polyphonic elements provided by the fugue, displays the Christ's monogram and demonstrates the influence of this symbol on Constantine, his victory and conversion into Christianity.

The introductory part of the composition (1-11. t) was entrusted to solo cello, on the tonal basis of d-moll, melodic line of ambitus of which is less than two octaves (quartdecime). Punctuated rhythms and trio movements in melody are the means by which the sound of a fanfare is conveyed to mark the beginning of the battle. At the same tonal basis (d-moll) in the Coral section (12-23. t), with the completion of the entire string ensemble, the chord-harmonic structure is formed, in which the cello and the double bass perform the role of a pedal per ton D, while above them in the sections of violin and viola unfolds the downward chromaticity in the moll quintachord, from the Gis to Dis tones. Through the facture of the Coral Part of the composition, Constantine's dream and his vision of the symbol of Christ's monogram are conveyed bydynamic, rhythmic and harmonic means. The third part of the composition introduction (23-35. t) entrusted to the cello (Vc 1), with the melody volume in the interval of reduced quart (Gis-c), as well as the rhythmic move in the eighths and the longer note values at the end of the phrase, give to this piece the role of Word part of modulating character (from d-moll in a-moll). Symbolically, the Word part represents the text appearing to Constantine in the sky «With this sign you will win.» Believing that this symbol symbolizes immortality and the victory over death, in this way is painted his decision to turn to a Christian, his own conviction that God will provide him with a victory that will be the victory of Christ, the true commander of his army [1]. Above the Word part in the cello section, the modulation is complemented by colored chromatic elements in the violin and viola sections, announcing the central part of the composition Fugue.

Example 1

A schematic

presentation of composition Constantine Fugue

INTRODUCTION

CORAL

WORD

FUGUE

CORAL

WORD

PART

PART

PART

PART

Fanfare

exposition

development

ending

part

part

1-11.

12-22.

23-35.

36-74.

75-125.

126-167. 168-176.

177-184.

To describe Constantine's battle, the author uses a polyphonic form of fugue, in which the elements necessary for the reflection of a historical event in musical arts are applied: imitation, stretto, inversion, variation of the theme, pedal. The Fugue (36-167. t) begins with the a-moll tonality and contains all three standard parts: exposition, development part, and ending part.

The exposition (36-75. t) is presented in as four - voice, in the form of a dux and comes, with a shorter interval movement between the third and fourth theme expositions. The theme for the fugue is of a heterophone-doubling structure [12], because its core is divided into two melodies, one on the lying tone A in the form of a pedal, and the second one containing the second movement.

Theme (dux in a-moll) pocoapococresc.

Table 1. Schematic presentation of the Fugue exposition

tact

36-42.

43-44.

45-50.

51.

53.

STRETO

60-64.

Interval

movement

65-74.

Vln 1

THEME (in Dis)

varied II part of theme

Vln 2

THEME (in Cis)

Л

CP

Vla

THEME (in Cis

varied II part of theme

Л

doubled THEME in 8

Vc 1

THEME

(in H II part)

KS

THEME (in Cis)

Л

CP

Vc 2

THEME

(in A)

KS

KS

CP

THEME (in Cis)

Л

doubled THEME in 8

Cb 1

THEME (in H)

KS

THEME (in Cis)

Л

CP

Cb 2

THEME

(in A)

KS

KS

CP

THEME (in Cis)

Л

doubled THEME in 8

With the chosen harmonic and polyphonic elements, the beginning of the battle, the manifold of Constantine's army, is conceived. The appearances of the dux and comes represent the attack and the attack of dense first ranks of easier infantry, as well as other ranks of infantry with heavy weapons. The creation of a tension of fighting, depicting the collision of conflicting parties, a swarm of swords and shards of shields, in addition to the second imitation of the theme, was achieved also by complexing the facture by completing the entire performer's apparatus. In the third appearance of the theme in the form of a dux in violin 2 and viola, the streto technique was applied with the cello 1 and 2 sections and double bass 1 and 2, with a two-tact interval. Using this polyphonic technique, a picture of the very fight of the first rows of infantry and the lateral rows of the cavalry, which was mounted on the wings of the army, is shown [4]. The fourth appearance of the theme as comes in dismoll, in violin section 1, is reinforced by doubling of the theme in violins, violoncelloes 2 and contrabasses 2, while the other part of the theme varies, which can symbolically convey the redeployment of the ranks of the army after the loss imposed to the opposite side.

The repetition of the theme exposition is regular, from the deepest to the highest voice, and the imitation interval is the second:

1) Dux - cello 2 and double bass 2 (a-moll)

2) Comes - double bass 1 and cello 1 (h-moll)

3) Dux - Violin 2 and Viola (cis-moll)

4) Comes - violin 1 (dis-moll).

The development part (75-125 t), as the central and most extensive part of the fugue, represents the culmination of the battle using different polyphonic means. By homorythmic facture at the beginning of the development part, with motifs from the theme of the second-downward movement in doubles, confident Konstantin, the unique organization of the army and readiness for victorwere painted. The application of streto technique on the one hand, in the cello sections of violoncello 1 and 2 and contrabass 1 and 2, and, on the other hand, in violin sections 1 and 2 and viola, was organized through three links, in the form of a modulatory second-downward sequence, in eis-moll, e-moll, and dis-moll, leading to the culmination of the piece, the reflection of the symbol of Christ's monograma.

Table 2. Stretto in the development part

tact

75-84.

85-87.

88.

STRETO

89-91. 92. 93. 94.

95-96.

Vln 1

Я

(for 2Т)

KS

THEME (in Eis)

THEME (II part in E)

THEME

(II part in Dis)

Vln 2

Я

Я

KS

THEME (in Eis)

THEME (II part in E)

THEME

(II part in Dis)

Vla

Я

Я

KS

THEME (in Eis)

THEME (II part in E)

THEME

(II part in Dis)

Vc 1

Я

Я

THEME (in Eis)

THEME (II part in E)

THEME (II part in Dis)

KS

Vc 2

Я

Я

THEME (in Eis)

THEME (II part in E)

THEME (II part in Dis)

KS

Cb 1

Я

Я

THEME (in Eis)

THEME (II part in E)

THEME (II part in Dis)

KS

Cb 2

Я

Я

THEME (in Eis)

THEME (II part in E)

THEME (II part in Dis)

KS

The final part of the fugue (125-167. t) represents the reminiscence of the introductory part of the composition, because it begins with a motif that reflects the sound of fanfare, marking the victory. The theme is presented in total four times, as well as in the exposition, with a smaller difference in tonality and fidelity of exposition. All four times violoncellos 1 and 2 and contrabass 1 and 2 bring the theme, for the first time in a-moll, the second time in h-moll, with a chromatized theme, the third time in cis-moll, and the fourth time again in a-moll. At the fourth exposition of the theme, theme reversal in the violin 1 and 2 violin sections was simultaneously applied. At all times, the pedal is in the viola section, which is still in duration from development part.

Table 3. The ending part of the fugue

tact

126-134.

135-139.

140-147.

Interval

movement

148-155.

155-164.

Interval

movement

165-167.

CORAL

PART

168-176.

WORD

PART

177-184.

Vln 1

fanfare

CP

Л

THEME

in inversion

Л

Vln 2

fanfare

CP

Л

THEME

in inversion

Л

Vla

pedal

pedal

pedal

pedal

pedal

Л

Vc 1

THEME

(in A)

THEME

(varied I part of theme)

THEME

(in Cis)

Л

THEME

(in A)

Л

Vc 2

THEME

(in A)

THEME

(varied I part of theme)

THEME

(in Cis)

Л

THEME

(in A)

Л

Cb 1

THEME

(in A)

THEME

(varied I part of theme)

THEME

(in Cis)

Л

THEME

(in A)

Л

Cb 2

THEME

(in A)

THEME

(varied I part of theme)

THEME

(in Cis)

Л

THEME

(in A)

Л

The end of the composition (168-184. t) is presented in the form of the Coral Part, the homophonie facture in d-moll, with which the composition began, and the Word part, completely identical melody lines as in the introductory part of the composition. The reprise in the final part, as a reminiscence of the introductory part, symbolically represents the confirmation of Constantine's faith in the victory before the beginning of the battle and its positive outcome in the end.

The symbol of Christ's monogram is depicted in the development part of the fugue in the form of a fragment, which appears five times in total, in varied forms, in the section of the first violin (100-125. t).

Table 4. A schematic presentation of the culmination part and reflection of Christ's monogram

tact

100-108.

109.

110.

111.

112-113.

114-117.

118-119.

120-125.

Vln 1

fragment

fragment

fragment

Ж as fanfare

fragment

fragment

Л

THEME (I part)

Л

THEME

(II part)

THEME (in D)

Vln 2

Л

THEME (I part)

Л

THEME

(II part)

Ж as fanfare

THEME (in D)

Vla

Л

pedal

pedal

pedal

pedal

pedal

Vc 1

Л

THEME (II part)

THEME in inversion

Vc 2

Л

THEME (II part)

THEME in inversion

Cb 1

Л

THEME (II part)

THEME in inversion

Cb 2

Л

THEME (II part)

THEME in inversion

The culmination part of the fugue, and the entire composition, is depicted through the reflection of Christ's monogram, as the moment of the most fierce struggle, in which the Constantine's army is, believing that the symbol engraved on their shields will bring victory to them.

For the reflection of the symbol of Christ's monogram, which contains five arms, the author repeats the musical fragment five times in total, in accordance with the monogram. Every time the fragment is shown in a different way, with a variety of accompanying sections, which makes all the total facture different each time. The melody line of the fragment is basically built from the motif of the theme and motif from the introductory part of the composition.

Forms of reflecting the symbol of Christ's monogram:

1) First symbol arm - first exposition of the fragment (100-108. t): fragment (Vln 1) + theme motifs in other sections = homophone facture.

First exposition of the fragment

2) Second symbol arm - second exposition of the of the theme (Vc1, Vc 2, Cb 1, Cb 2) in the third part fragment (109-110. t): of the fragment = polyphonic facture.

Shortened fragment (Vln 1) + first part of the theme (Vln 1, Vln 2) + pedal (Vla) + the second part

Second exposition of the fragment

3) Third symbol arm - third exposition of the fragment (111-117. t): Shortened fragment (Vln 1) + streto with the other part of the theme (Vln 1, Vln 2) + motif fanfare (golden cross-section 113. t) + pedal (Vla) + theme motifs (Vc1, Vc 2, Cb 1, Cb 2) = polyphonic facture.

Third exposition of the fragment

4) Fourth symbol arm - Fourth exposition of the fragment (118-119. t):

Shortened fragment (Vln 1) + theme (Vln 1, Vln 2) + pedal (Vla) + theme in inversion (Vc1, Vc2, Cb 1, Cb 2) = polyphonic facture.

5) Fifth symbol arm - fifth exposition of the fragment (120-125. t):

6) Example 6

Fourth exposition of the fragment

Varied fragment from previous exposition (Vln 1) + pedal (Vla) + theme in inversion (Vc1, Vc2, Cb 1, Cb 2) = polyphonic facture + extension = homophonous facture leading to the final part of the fugue.

Fragments in the role of the reflection of Christ's monogram

Starting from the thesis that art is a question of the balance and relation of parts to the whole, the problem of proportionality is shown as one of its key factors. Hemenway [2] points out that paintings, sculptures, architectural and musical achievements in their appearance are organized according to a hidden meaning for proportions. Therefore, the task of interpreting the influence and effects of this proportion within a certain work is set.

In the domain of music, the golden section and the Fibonacci series (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…) are considered from the aspect of the structure of instruments, acoustics and compositional structure. Another aspect of the observation of the golden cross-section in music refers to the perception of their effects in a particular composition. Although this process can be applied to almost all of the music components, consideration of the correlation between this proportion and the formal pattern is a particularly important analytical field, since to the greatest extent it indicates the balance, the harmonious proportionality and the uniqueness of the concept of work [2].

The golden cross-section can be expressed numerically as a constant whose size is 1.6180339…, but its meaning in nature is not so much in its numerical value as in the proportion it determines. The only natural arithmetic scale or proportion that can be obtained with only two elements is expressed by the formula a:b = (a+b):a, where the smaller size, according to the greater, is in relation as larger according to the sum of both sizes. This proportion is called the Golden proportion.

And the composition Constantine Fugue has a golden cross-section. It can be noticed in the second part of the 114th tact, or more precisely 113.75 tact of the composition, between the third and fourth exposition of the fragment. If we take into account that the golden cross-section constant is 1.6180339…, and that the composition has a tot easy to see the golden cross-secti equation by determining that a = 113.75, b = 70.25. The obtained equation of the golden cross-section will be: (113.75 + 70.25) = 1.617582…. = 1.618 113.75

The golden cross-section of the composition

The composition Constantine Fugue is an example of a reflection of a historical event and symbols in musical art. The victory of Constantine's army at the Milvian Bridge, the conversion of Constantine the Great into a Christian, the symbolism of Christ's monogram, was presented by DraganTomic with adequate harmonic and polyphonic means. The composition is written in the form of orchestral fugue. Its formal three-dimensionality was basically achieved by a combination of homophones and polyphonic factures. The themeof the fugue is exposed in various ways: shortening, variation, chromatization, inversion, in streto. The pedal, in the way of the Baroque fugue, was applied in the viola sections. The symbol of Christ's monogram is shown, especially with the harmonic - polyphonic means, through the fragmentary structure in different ways, in which the golden cross-section is applied. All of these elements are one of the ways of contemporary musical interpretation of the historical event of the 17th century, thus achieving a very timeless duration.

Literature

1. Ven Pol. Kada je nas svet postao hriscanski. - Novi Sad: Mediterran Publishing, 2013. - 187 p.

2. Vesic D. Zlatni presek i Fibonacijev niz u muzici - osobenosti Bartokovog stvaralackog opusa // Zbornik radova Akademije umetnosti. Muzika u nauci i praksi. - Novi Sad: Akademija umetnosti. - 2014. - №2. - P 198-209.

3. Jовановиh Б. Легенда о Константиновом сну // Зборник радова Ниш и Византща. - Ниш: Универзитет у Нишу, 2013. - №XII. - P. 507-516.

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6. МиличиЪ Д., Дрндарски М., Холод А. & ЛазиЪ З. Фибоначи и златни пресек у интердисциплинарно] настави // Иновацще у настави. - 2014. - XXVII, №4. - P 86-91.

7. Milojkovic J. Zlatni presek. Master rad. - Beograd: Matematicki fakultet Univerziteta u Beogradu, 2009. - 83 p.

8. Novakovic J. Pitagorejska geometrija. Master rad. - Beograd: Matematicki fakultet Univerziteta u Beogradu, 2016.

9. РадиЪ Р Константин Велики - надмоЬ хришЪанства. - Београд: Еволута. - 2010. - 209 p.

10. Radic R., Stojiljkovic D. & Zakarija F. Trijumf hriscanstva. - Beograd: Dan Graf, Zavod za udzbenike, Sluzbeni glasnik, 2013. - 95 p.

11. Sabo A. Ispoljavanje simetrije u muzickom obliku - pitanja metodologije analize. Doktorska disertacija. - Beograd: Interdisciplinarne studije, Univerzitet umetnosti u Beogradu, 2012. - 263 p.

12. Скребкова Ф.М. Фактура у музици. - Ниш: Виша музичка школа, 1996. - 72 p.

13. Фер] анчиЪ С. Константин. - Београд: Службени гласник, 2013. - P. 306-337.

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15. ЧаировиЪ И. Константин Велики као хришЪанин-однос неких историографа према аему и аегово] вери // Зборник радова Ниш и Византща. - Ниш: Универзитет у Нишу, 2009. - №VIII. - P. 471-478.

References

1. Ven Pol. Kogda nash mir stal khristianskim [When our world became Christian]. Novi Sad, Mediterran Publ., 2013. 187 p. (In Serb.).

2. Vesic D. Zolotoe sechenie i posledovatel'nost' Fibonachchi v muzyke - kharakteristiki tvorchestva Bartoka [The golden ratio and the Fibonacci sequence in music are characteristics of Bartok's work]. Trudy Akademii khudozhestv. Muzyka v nauke i praktike [Proceedings of the Academy of Arts. Music in science and practice]. Novi Sad, Academy of Arts Publ., 2014, no. 2, pp. 198-209. (In Serb.).

3. Jovanovic B. Legenda o mechte Konstantina [The legend of the dream of Constantine]. Trudy Nisha i Vizantii [Proceedings of Nis and Byzantium]. Nis, University of Nis Publ., 2013, no. XII, pp. 507-516. (In Serb.).

4. Keravica N. Nepobedimyy Konstantin [Invincible Constantine]. Beograd, Evoluta Publ., 2011. 132 p. (In Serb.).

5. Kostic G. Hi-Ro: monogramma Khrista ili astrokod Konstantina [Hi-Ro: Christ's monogram or Constantine's astro codex]. Trudy razvitiya astronomii u serbov VII [Proceedings Development of Astronomy in Serbs VII]. Бeograd, Ру^ер БошковиЪ Publ., 2014, no. 13, pp. 963-974. (In Serb.).

6. Milicic D., Drndarski M., Holod A. & Lazic Z. Fibonachchi i Zolotoy krest v mezhdistsiplinarnom obuchenii [Fibonacci and the Gold Cross in Interdisciplinary Teaching]. Prepodavanie innovatsiy [Teaching Innovation], 2014, XXVII, no. 4, pp. 86-91. (In Serb.).

7. Milojkovic J. Zolotoe sechenie. Magisterskaya dissertatsiya [Golden ratio. Master's dissertation]. Beograd, Faculty of Mathematics of the University Publ., 2009. 83 p. (In Serb.).

8. Novakovic J. Pifagoreyskaya geometriya. Magisterskaya dissertatsiya [Pythagorean geometry. Master's dissertation]. Beograd, Faculty of Mathematics of the University Publ., 2016. (In Serb.).

9. Radic R. Konstantin Velikiy - prevoskhodstvo khristianstva [Constantine the Great as the dominance of Christianity]. Beograd, Evoluta Publ., 2010. 209 p. (In Serb.).

10. Radic R., Stojiljkovic D. & Zakarija F. Triumf khristianstva [The triumph of Christianity]. Beograd, Dan Graf; Zavod za udzbenike, Sluzbeni glasnik Publ., 2013. 95 p. (In Serb.).

11. Sabo A. Ispol'zuyemoye metodologicheskoye issledovaniye i analiz metodologii analiza: doktorskaya dissertatsiya [Used methodological research and analysis of the analysis methodology. Diss. of Dr. Sci]. Beograd, Interdisciplinary Research, University of the Arts Publ., 2012. 263 p. (In Serb.).

12. Skrebkova F.M. Faktura v muzyke [The texture of the Music]. Nish, Higher School of Music Publ., 1996. 72 p. (In Serb.).

13. Ferjanchih S. Konstantin [Konstantin]. Beograd, Sluzhbeni glasnik Publ., 2013, pp. 306-337. (In Serb.).

14. Chairovic I. Preobrazhenie i kreshchenie tsarya Konstantina [Transfiguration and the baptism of Emperor Constantine]. Trudy Nisha i Vizantii [Proceedings of Nis and Byzantium]. Nish, University of Nish Publ., 2007, no. VI, pp. 369-378. (In Serb.).

15. Chairovic I. Velikiy kak khristianin - otnosheniye nekotorykh istoriografov k nemu i ego vere [Great as a Christian - the approach of some historiographers to him and his faith]. Trudy Nisha i Vizantii [Proceedings of Nis and Byzantium]. Nish, University of Nish Publ., 2009, no. VIII, pp. 471-478. (In Serb.).

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