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The Fractal Properties of Byzantine Music
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F. Hindeleh1
Grand Valley State University
J. Sears2
Eastern Michigan University
L. Copenhaver3
Grand Valley State University
November 1, 2012
1 [email protected]
2 [email protected]
3 [email protected]
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Abstract
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The songs found in Byzantine music possess a great cultural significance, a
beautiful sound, and wonderful mathematics. We study one representative
song from each of the eight tones in Byzantine music and demonstrate that
each one exhibits a fractal relation between the frequencies of successive notes
for a given note interval.
1
Historical Introduction
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Byzantine music is an ancient form of writing music with roots stemming
all the way back to Pythagorus. The nature of the music gives it a unique
monophonic sound. Today, the music’s main purpose is for worship in the
Eastern Orthodox Church, though in the past it had some entertainment
purposes outside of the Church. Byzantine music began to grow in 330
A.D., the same year that the emperor Constantine enforced the practice
of toleration for Christianity in the Roman Empire. In its earliest forms,
Byzantine music was influenced by the music of the age, including Jewish
and Greek music. The Greek musical heritage of the music provided the
mechanism for dividing the scales, using the technique which stemmed all
the way back to Pythagorus himself. In the Jewish synagogue, the Psalms
were chanted using only the human voice. This is where the monophonic
sound of Byzantine music originated from [9]. Through songs, the Biblical
stories could be presented to everyone, including those who could not read
and putting the stories to a melody made the stories even easier to recall [9].
The music was passed along vocally from generation to generation without
being written down. As the church began to spread, the singer’s memories
were no longer trusted to accurately recall the chants. So, the music was
written down around the 9th century. The written form of Byzantine music
has evolved over the years, but the form that is used today developed around
the last quarter of the 12th century [1].
2
Reading the Music
There are eight different Tones to Byzantine music, each with a different
feeling or emotion attached to it and a different musical sound. The eight
different Tones arose from the eight ecclesiastical modes that were already
present in the 8th century [1]. Each Tone has certain rules governing the use
of sharps and flats.
Byzantine music behaves a bit differently than Western musical notation,
which uses the five line staff. There are different rules and symbols used in
Byzantine music. At the beginning of each piece, there is a symbol which
tells the chanter the Tone of the piece, which note to start with, and another
symbol for the tempo of the piece. Instead of telling the chanter what note
to sing next, Byzantine music uses symbols to tell the chanter how many
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steps to move up or down along the music scale from the previous note.
The Diatonic scale used in Byzantine music is very similar to the one
used in Western music [14]. This is the scale we use most of the time. It
begins with N η, which corresponds to the note C. See Figure 1 for the names
of the rest of the notes along the Diatonic scale for Byzantine music. Some
Tones will use scales that start with notes other than N η. These scales are
also very similar to the scale used in Western music, with the exception of
the Soft Chromatic scale.
Figure 1: The Byzantine music scale. The notes on the left are the notes
from Byzantine music and the notes on the right are the corresponding notes
in Western music.
In the western chromatic equal-tempered scale, the notes are √
all equally
12
spaced, with the ratio of frequencies of consecutive notes being
2. Using
symbols, Byzantine music redefines the frequencies of notes on the scale in
many musical pieces, in a way that gives it its unique sound. This
√ creates new
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scales in which the frequencies of consecutive notes is close to 2. Musically,
this difference is significant, but for the purpose of our mathematical analysis
it does not.
To get a feel for how Byzantine music works, we will demonstrate the
process using a small section of a song from Tone 1, shown below.
Figure 2: A line of a song from Tone 1.
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The symbol at the beginning tells us that we should start at the note Πα,
which corresponds to the note D. The next symbol is called an ison and
tells us that we do not move up or down. So the first note is Πα. Then we
move up the scale 3 steps to the note ∆ι which corresponds to the note G.
And now we have two ison in a row, which means that the next two notes
would also be ∆ι. We then move down one step to Γα or F. From there, we
move up one step back to ∆ι. And we move up one step again to K, or A.
However, the voice changes a little on this note. Instead of being K for the
entire duration of the note, the voice ”flickers” or moves up in pitch, then
down in pitch, and back to K very quickly. The flickering is beautiful and
gives a unique sound to Byzantine music, but difficult to replicate on an
instrument. For the purposes of this paper, we chose to ignore the voice
flickering as it affected only a few notes. Summarizing the notes gives us
Πα, ∆ι, ∆ι, ∆ι, Γα, ∆ι, K. Anyone with a knowledge of both Byzantine
music and Western music can translate between the two forms. Now that
we have a basic grasp on how the music works, we can begin analyzing the
music. For a more complete resource for the rules in Byzantine music see
[15].
3
Analysis
Music is not a random occurrence. Randomly playing notes will not
produce music. The notes in a musical piece are carefully ordered and
timed to produce the beautiful music we all enjoy. We can look at the
successions between notes from a mathematical perspective to gain some
insight into music. In [2] K. Hsü and A. Hsü analyzed a few classical
compositions from Bach and other composers and argued that compositions
were fractal if the percentage incidence frequency (F ) of the note interval
(i) between successive notes could be defined by the relation
F =
c
iD
(1)
where D is the fractal dimension and c is a constant of proportionality. We
can think about Byzantine music in a similar fashion.
First, consider our fractal relation in Equation 1. We can take the log of
both sides and simplify to make the equation easier to manipulate. Instead
of working with equation 1, we will work with the following equation:
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Table 1: Frequencies in Hertz and MIDI frequencies for notes in Byzantine
music
Note Frequency (Hz) MIDI Frequency
Nη
261.6
60
Pα
293.7
62
Boυ
329.6
64
Γα
349.2
65
∆ι
392
67
K
440
69
Zω
493.9
71
Nη
523.3
72
log(F ) = log(c) − D log(i)
(2)
For our analysis, i represents a note interval, or the difference in frequencies
two consecutive notes. For instance, if we have N η followed by Zω, i would
be (71 − 62) = 9. The variable F is the percentage incidence frequency in
the musical piece. This means that F keeps track of how many times each
note interval i occurs.
To begin our analysis of the songs, we first translated each song note by
note and used the JFugue package for JavaTM provided in [3] to play the
songs to check that our translations were correct. We chose to use
JavaTM due to the discrete nature of the piano. The keys on the piano are
already set in their frequencies, and the frequencies in use, naturally, are
the ones used for Western music. However, in Byzantine music, we often
need frequencies that can not be found on the piano. The JFugue package
in JavaTM gave us the flexibility with the frequencies that the piano could
not offer. We then converted each note to its corresponding MIDI number.
For example the MIDI number for the note N η is 60. The rules governing
the raising and lowering of notes are unique to each tone. We have provided
only the standard note frequencies here. The rest of the MIDI frequencies
as well as the frequencies in Hertz can be found in Table 1 [3].
We will use a song we chose from Tone 7, ”Inna Hirodos” in Appendix A as
an illustrative example to demonstrate the process used for our analysis.
The symbol at the beginning tells us that we are in Tone 7, and that the
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Table 2: Table of results for the representative song from Tone 7
Interval (i) Incidence % Incidence Frequency (F )
0
102
38.93129771
1
47
17.9389313
2
104
39.69465649
3
1
0.381679389
4
2
0.763358779
5
5
1.908396947
6
0
0
7
1
0.381679389
Total
262
100
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first note will be Γα. So, the first note will be Γα, as well as the two notes
after it. So the first two differences are 0. Then, we move up three steps to
Zω[, using the rules governing sharps and flats for Tone 7. The next
difference would be |70 − 65| = 5, using the values from Table 1. We have
summarized the results below in Table 2. The songs in Byzantine music
also have another part, which is called the ison. The ison is a base tone
which stays fixed throughout most of the song, but changes a little.
Including the ison would add many differences of 0, which we did not
include in our calcuations. The ison ads beauty and richness to the music,
but would change the calculations little.
From Figure 2 we can see that a large portion of our notes repeat, that is
the interval between them is 0. To avoid complications we shifted
everything along the x-axis, that is we looked at log i + 1 instead of log i.
Plotting the points using a log − log plot and using least squares to give us
the line of best fit, we obtain Figure 3.
The equation for the line is 1.873 − 2.189i. This means that
101.873
i2.189
F =
The correlation for the data is −0.838. This means that the line of best fit
is closely correlated with our data, which means that we can represent the
song from Tone 7 using the fractal relation in 1
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Figure 3: The line of best fit for Tone 7 from [7]
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So, we can express the succesions of notes for our representative song from
Tone 7 as a fractal.
We also analyzed 7 other songs, one from each tone. Some of the songs are
in Arabic, while others have been translated to English. Obtaining copies of
Byzantine music sheets presented a challenge. A language challenge, since
most of the resources for Byzantine music are in Greek and Arabic. And an
availability challenge, since most of the resources are difficult to locate. The
individuals at St. Anthony’s Monastery are experts in Byzantine music and
have worked hard to translate many of the songs into English.
We chose ”Lord I Have Cried” from Tone 1 [6]. The fractal relation in
Table 3 is given by
F =
101.690
.
i1.853
The correlation for Tone 1 is −0.779.
From Tone 2, we chose ”Athimi Ya Nifsi” [8]. The fractal relation in Table
3 is given by the equation
F =
101.625
.
i1.124
The correlation for Tone 2 is −0.76.
Our results for ”Ayuha Al Mukhalles” from Tone 3 [5] can be found in
Table 3 and are given by the fractal relation
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101.7
F = 1.38 .
i
F =
101.889
.
i2.438
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The correlation for Tone 3 is −0.806.
We analyzed ”Dogmatic Theotokion” from Tone 4 [10]. The results are
shown in Table 3 and are given by the fractal relation
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The correlation for Tone 4 is −0.878.
From Tone 5, we chose the song called ”Awed by thy Beauty” [11]. The
results are given by Table 3 and given by the fractal relation
101.827
i2.373
F =
The correlation for Tone 5 is −0.843.
We chose another version of ”Dogmatic Theotokion” for our representative
song from Tone 6 [12]. The results are given by Table 3 and the fractal
relation
101.747
i2.139
F =
The correlation for Tone 6 is −0.843.
Finally, the song we chose to represent Tone 8 is called ”It is Truly Right”
[13]. The results are given by Table 3 and the fractal relation
101.674
i1.714
F =
The correlation for Tone 8 is −0.771.
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Table 3: The results for all 8 tones.
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Conclusion
By studying 8 representative songs, one from each of the 8 Tones in
Byzantine music and converting each note to its corresponding Musical
Instrument Digital Interface (MIDI) number, we have demonstrated that
the 8 songs we studied exhibit a fractal relation and are strongly correlated.
We have only explored a very small sample of the songs found in Byzantine
music, but we conjecture that all the songs in Byzantine music will exhibit
a fractal pattern because it is composed monophonically and each one of
the eight Tones has its own pattern.
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Acknowledgements
We would like to thank the following people and organizations:
• David Koelle for developing the JFugue software used in our project
and answering our Java questions.
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• Father Ephraim and the individuals at St. Anthony’s Monastery for
answering questions and their work on the Divine Music Project
which made the Byzantine Music available to us. There are few
sources available in English for Byzantine music. Many are written in
Greek and Arabic. The individuals at St. Anthony’s Monastery have
resources in both Greek and English.
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• Nichoals Ayoub, the Protosaltese at the Antiochian Archdiosese, for
providing music and proofreading.
References
[1] D. Conomos. A Brief Survey of the History of Byzantine and
Post-Byzantine Chant. Retrieved June 30, 2009, from St. Anthony’s
Orthodox Monastery website:
http://www.stanthonysmonastery.org/music/History.htm.
[2] K. Hsü, A. Hsü. (1990). Fractal Geometry of Music. Proceedings of the
National Academy of Sciences of the United States of America. 87,
938-941.
[3] D. Koelle (2008). The Complete Guide to JFugue: Programming Music
in JavaTM . Retreived from http://www.jfugue.org/.
[4] Indiana University School of Music Center for Electronic and
Computer Music. (August 31, 2003). MIDI Note Number to Equal
Temperament Semitone to Hertz Conversion Table. Retrieved from:
http: // www. indiana. edu/ ~ emusic/ hertz. htm .
[5] M. El Murr. Holy Week, Al Ossbou Al Muqaddas (Arabic). Beirut,
Lebanon. pp.116-117.
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[6] St. Anthony’s Orthodox Monastery The Divine Music Project.
Retrieved July 24, 2009, from St. Anthony’s Orthodox Monastery’
website: http: // www. stanthonysmonastery. org/ music/
Vespers/ b2100_ Lord_ I_ have_ cried. pdf .
[7] M. El Murr. (1972) Spiritual Pipe: Part 1, Al Mizmar Al Rohi: Al
Juzo’ Al Awal (Arabic). Beirut, Lebanon. pp 125.
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[8] M. El Murr. (1972) Spiritual Pipe: Part 1, Al Mizmar Al Rohi: Al
Juzo’ Al Awal (Arabic). Beirut, Lebanon. pp. 225.
[9] Byzantine Music History. P. Papadakis; [cited September 8, 2009].
Available from:
http: // liturgica. com/ html/ litEOLitMusDev1. jsp .
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[10] St. Anthony’s Orthodox Monastery The Divine Music Project.
Retrieved July 24, 2009, from St. Anthony’s Orthodox Monastery’
website: http: // www. stanthonysmonastery. org/ music/
Mysteries/ b4004_ Beauty. pdf .
[11] St. Anthony’s Orthodox Monastery The Divine Music Project.
Retrieved July 24, 2009, from St. Anthony’s Orthodox Monastery’
website: http: // www. stanthonysmonastery. org/ music/
Vespers/ b2440_ Dogmatikon. pdf .
[12] St. Anthony’s Orthodox Monastery The Divine Music Project.
Retrieved July 24, 2009, from St. Anthony’s Orthodox Monastery’
website: http: // www. stanthonysmonastery. org/ music/
Vespers/ b2640_ Dogmatikon. pdf .
[13] St. Anthony’s Orthodox Monastery The Divine Music Project.
Retrieved July 24, 2009, from St. Anthony’s Orthodox Monastery’
website: http: // www. stanthonysmonastery. org/ music/ Chrys/
b0800_ It% 20is% 20truly% 20right% 20-% 20Mode% 208% 20-%
20Anastasios. pdf .
[14] St. Anthony’s Orthodox Monastery The Divine Music Project.
Retrieved September 17, 2009, from St. Anthony’s Orthodox
Monastery’ website:
http: // www. stanthonysmonastery. org/ music/ Chromatic. htm .
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[15] Y. Yazigi. (2001). Foundations of Byzantine Music. St. John of
Damascus Institute of Theology, El Koura, Lebanon.
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Tone 7, Inna Hirodos [7]
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Figure 4: A representative of Tone 7, ”Inna Hirodos” from [7]
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