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The Complexity of Songs
Nikolaos Petros Triantafyllidis, AEM:1843
Theory of Computation
The Complexity of Songs
Concept proposed by Donald Knuth
Published in 1977 on ACM SIGACT news.
And in 1984 on ACM Communications
In-Joke
An attempt to determine the spatial complexity in
the texts of songs
•  A fair attempt in being funny…
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Donald Knuth
•  The father of Analysis of
Algorithms
•  The Art of Computer
Programming
•  TeX
•  Turing Award 1974
•  Pretty badass…
The Complexity of Songs
•  Computer Science concepts are applicable to
areas that have nothing to do with computers
•  Popular songs in terms of modern Complexity
Theory
•  Could try an information theory approach too but…
•  Big O…!!!
•  Missed it?
The Complexity of Songs
•  A song of a length n usually requires a text of length
~n
•  Too many songs à Too Much text, No memory L
•  So our ancient ancestors invented refrain
•  Refrain provides a complexity of cn, c<1
•  (wow)
Lemma 1
•  Let S be a song containing m verses of length V and
a refrain of length R where the refrain is to be sung
first, last, and between adjacent verses. Then, the
space complexity of S is (V/(V + R)) n + O(1) for fixed
V and R as mà+oo.
PROOF
•  Some math follows…
•  The length of S when sung is
n =R+(V+R)m
(1)
while its space complexity is
c = R + Vm.
(2)
By the Distributive Law and the Commutative Law ,
we have
c= n- (V+R)m + mV
=n-Vm-Rm +Vm (3)
=n-Rm.
The lemma follows.
The Complexity of Songs
•  Not bad but we can do better…
•  Jewish song “Ehad Mi Yode’a”
•  A cumulative song structure that produces a
complexity of:
O( n)
•  But we can go further down thanks to Mr. Old
McDonald...
Lemma 2
•  Given positive integers a and X, there exists a song
whose complexity is
(20 + ! + " ) + n / (30 + 2 ! ) + !(1)
PROOF
PROOF (2)
The Complexity of Songs
•  That was really pleasant but let’s leave some of the
pleasure to the audience…
•  Other songs fall into that category too…
That f*cker is (20 + ! + " ) + n / (30 + 2 ! ) + !(1)
The Complexity of Songs
•  An improvement was proposed for a song with a
complexity of
O( 3 n)
•  Wrong definition, the actual complexity was
O( n / log n)
•  Minor improvement but showed that the square
root barrier could be broken
The Complexity of Songs
•  Square root complexity is rather fine but can we go
further?
•  Alcohol is always the solution…
•  Beer time!
Theorem 1
•  There exist songs of complexity O(log n).
PROOF
•  Consider the schema
Vk = TkBW’,’
TkB’;’
If one of those bottles should
happen to fall,
Tk-1BW’.’
where
B = ‘bottles of beer’
W = ‘on the wall’
and where Tk is a translation of the integer k into
English.
PROOF (2)
•  It requires only O(m) space to define Tk for all k < 10m
since we can define
Tq*10m+r = Tq ’times 10 to the’ Tm ‘plus’ Tr ,
for 1 ≤ q ≤ 9 and 0 ≤ r ≤ 10m-1.
(14)
Therefore the songs Sk defined by
So = ε, Sk = VkSk-1 for k ≥ 1
(15)
have length n ≈ k log k, but the schema which defines
them has length O(log k); the result follows.
The Complexity of Songs
•  Logarithmic complexity is a charm but we can go
even further
•  In Knuth’s own words:
•  “However, the advent of modern drugs has led to
demands for still less memory, and the ultimate improvement of Theorem 1 has consequently just
been announced”
•  (don’t do drugs)
Theorem 2
•  There exist arbitrarily long songs of complexity O(1).
PROOF
•  (due to Casey and the Sunshine Band). Consider
the songs Sk defined by (15), but with
Vk = 'That's the way,' U 'I like it, ' U
U = 'uh huh,' 'uh huh'
The Complexity of Songs
•  Last remark on the paper:
•  It remains an open problem to study the complexity
of nondeterministic songs.
Applications
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Education (Introduction to Complexity Theory)
Fun (for a rather special audience…)
Songwriting schools
Quantum computation
Exploration of space
Cosmology
Human Cloning
I am lying.
Further Study
•  Prof Kurt Eisemann of San Diego State University has
made some remarks about the impact of the
hidden c constant.
•  Proposes an O(0) approach
•  "When the Mayflower voyagers first descended on
these shores, the native Americans proud of their
achievement in the theory of information storage
and retrieval, at first welcomed the strangers with
the complete silence. This was meant to convey
their peak achievement in the complexity of songs,
namely the demonstration that a limit as low as c =
0 is indeed obtainable."
Further Further Study
•  I would propose another approach to achieve O(0)
…
•  Numbers say that the most popular song on the
planet for the current year comes from far in the
east
•  Thus:
Oppan Gangnam Style
•  You can’t speak
Korean
•  Your audience can’t
understand Korean
•  O(0)…!
Furthest Study
•  What is the complexity of MUSIC?
•  Bearing in mind that it is largely a matter of
definition
•  We might take a long shot in trying to get some
insight on the complexities of musical information
•  An Information theory approach could be
employed
•  Let’s define the musical pattern as the basic unit of
information.
The Complexity of Songs
•  Since the dawn of time musicians have devised
tricks to help them organize their musical expression
and reduce the amount of actual musical text that
had to be memorized
•  Scales, Chords, Rhythms, Forms and a broad set of
Empirical Rules/Traditions
•  These can be found in both folk/popular and art
music of the western world.
•  So what makes a musical opus more or complex?
•  Personal estimate : The ability of the human ear
(trained or not) to recognize patterns and connect
them to a set of rules
Popular Songs
•  Popular songs throughout the history of mankind vary
dramatically but common patterns can be detected in
large within the songs of the same period
•  Different songs of the same period can easily be
reduced into the same class of songs that share the
same ‘recipe’ that produces them.
•  Rather simple and easy listening forms and chord
progressions
•  Since the era of J.S.Bach the western human ear has
been tuned on the most simple cadences I-IV-I, I-V-I and
the 4/4 time
•  Popular songs of the last half of the 20th century songs
tend to converge into uniformity
•  Axis of Awesome – Four Chord Songs
Blues/Folk Music
Relied upon the experience of the musicians
Largely improvised
Traditions and forms are the main focus
A lot to remember but the actual musical text to
memorize is minimal
•  The same patterns occur in different songs and are
rather easy to spot
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Minimal/Experimental/
Electronic Music
•  Patterns are extremely straight forward and
recognizable
•  Repeated by machines in large extent
•  Thank you Tiesto you solved our problem!
•  Minimal classical music can have a pattern
repeated for several minutes (or hours!) on a single
instrument
•  But how complex are those patterns after all?
Jazz
•  Totally Improvised
•  The set of rules is broader than that of classical
music
•  The set of traditional songs performed over the
decades is huge
•  The harmonies are complicated and difficult for the
ear to grasp
•  The concept of entropy seems to take effect
•  Complexity hits a peak
•  Still produced by a recipe
Classical Music
Tons of dense musical information
Technically challenging
Years to master
Kinda impossible to memorize the extent of the
classical works
•  Rather easily explainable…
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Iannis Xenakis (math music)
•  Sounds like pure noise
•  Largely hated (sentenced to death by the greek
government…….)
•  The amount of entropy is large
•  But it also has a set of rules that produces and
explains it
•  Very formal. Very mathematical.
Free Jazz/Free Impro
No rules
No Forms
No Rehearsals between musicians
No repetitive patterns
The amount of entropy can make the listener eat
their own liver
•  Do we have a winner?
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Conclusions
•  People will always find ways to compress
information
•  People will always find ways to make information
non compressible
•  Diversity is complexity
•  Repetition is simplicity
•  Beauty can be found on either end
•  We can apply our knowledge on nearly anything
•  Computer Science rules!
THANKS A TON!
J