CORELLATION OF MUSIC WITH MATHS
Music
theorists sometimes use mathematics to understand music, and although music has
no axiomatic foundation in modern mathematics, mathematics is "the basis
of sound" and sound itself "in its musical aspects... exhibits a
remarkable array of number properties", simply because nature itself "is
amazingly mathematical".Though ancient Chinese, Egyptians and
Mesopotamians are known to have studied the mathematical principles of sound,
the Pythagoreans of ancient Greece were the first researchers known to have
investigated the expression of musical scales in terms of numerical ratios,particularly
the ratios of small integers. Their central doctrine was that "all nature
consists of harmony arising out of numbers".
From the
time of Plato, harmony was considered a fundamental branch of physics, now
known as musical acoustics. Early Indian and Chinese theorists show similar
approaches: all sought to show that the mathematical laws of harmonics and
rhythms were fundamental not only to our understanding of the world but to
human wellbeing.Confucius, like Pythagoras, regarded the small numbers 1,2,3,4
as the source of all perfection.
The
attempt to structure and communicate new ways of composing and hearing music
has led to musical applications of set theory, abstract algebra and number
theory. Some composers have incorporated the golden ratio and Fibonacci numbers
into their work.
Time,
rhythm and meter
Without the boundaries of rhythmic structure – a fundamental equal and
regular arrangement of pulse repetition, accent, phrase and duration
– music would not be possible.^{[9]} In Old
English the word "rhyme", derived to "rhythm", became
associated and confused with rim – "number"^{[10]} – and modern
musical use of terms like meter and measure also reflects
the historical importance of music, along with astronomy, in the development of
counting, arithmetic and the exact measurement of time and periodicity that is
fundamental to physics.
Musical
form
Musical form is the plan by which a short piece of music is extended.
The term "plan" is also used in architecture, to which musical form
is often compared. Like the architect, the composer must take into account the
function for which the work is intended and the means available, practicing
economy and making use of repetition and order.^{[11]} The common
types of form known as binary and ternary ("twofold"
and "threefold") once again demonstrate the importance of small
integral values to the intelligibility and appeal of music.
The word "rhyme" was not derived from "rhythm"^{[}^{contradiction}^{]} (see Oxford
and Collins dictionaries) but from old English "rime". The spelling
of "rime" was later affected by the spelling of "rhythm",
although the two are totally different.
Frequency
and harmony
Chladni figures produced by sound vibrations in
fine powder on a square plate. (Ernst Chladni, Acoustics,
1802)
A musical scale is a discrete
set of pitches used in
making or describing music. The most important scale in the Western tradition
is thediatonic scale but many
others have been used and proposed in various historical eras and parts of the
world. Each pitch corresponds to a particular frequency, expressed in hertz
(Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an
interval of repetition, normally the octave. The octave of any pitch refers to a
frequency exactly twice that of the given pitch. Succeeding superoctaves are
pitches found at frequencies four, eight, sixteen times, and so on, of the
fundamental frequency. Pitches at frequencies of half, a quarter, an eighth and
so on of the fundamental are called suboctaves. There is no case in musical
harmony where, if a given pitch be considered accordant, that its octaves are
considered otherwise. Therefore any note and its octaves will generally be
found similarly named in musical systems (e.g. all will be called doh or A or Sa,
as the case may be). When expressed as a frequency bandwidth an octave A_{2}–A_{3} spans
from 110 Hz to 220 Hz (span=110 Hz). The next octave will span
from 220 Hz to 440 Hz (span=220 Hz). The third octave spans from
440 Hz to 880 Hz (span=440 Hz) and so on. Each successive octave
spans twice the frequency range of the previous octave.
Because we are often interested in the relations or ratios between the
pitches (known as intervals) rather than the
precise pitches themselves in describing a scale, it is usual to refer to all the
scale pitches in terms of their ratio from a particular pitch, which is given
the value of one (often written 1/1), generally a note which
functions as the tonic of the scale.
For interval size comparison cents are often
used.
The exponential nature of octaves when measured on
a linear frequency scale.
This diagram presents octaves as they appear in the
sense of musical intervals, equally spaced.
Common name

Example
name
Hz

Multiple
of fundamental 
Ratio
within octave 
Cents
within octave 
Fundamental

A_{2},
110

1x

1/1 = 1x

0

Octave

A_{3}
220

2x

2/1 = 2x

1200

2/2 = 1x

0


Perfect Fifth

E_{4}
330

3x

3/2 = 1.5x

702

Octave

A_{4}
440

4x

4/2 = 2x

1200

4/4 = 1x

0


Major Third

C♯_{5}
550

5x

5/4 = 1.25x

386

Perfect Fifth

E_{5}
660

6x

6/4 = 1.5x

702

G_{5}
770

7x

7/4 = 1.75x

969


Octave

A_{5}
880

8x

8/4 = 2x

1200

8/8 = 1x

0

Tuning
systems
5limit tuning, the most common
form of just intonation, is a system of
tuning using tones that are regular number harmonics of a
single fundamental frequency. This was one of
the scales Johannes Kepler presented in
his Harmonices Mundi (1619) in connection with
planetary motion. The same scale was given in transposed form by Scottish
mathematician and musical theorist, Alexander Malcolm, in 1721 in his 'Treatise
of Musick: Speculative, Practical and Historical',^{[12]} and
by theorist Jose Wuerschmidt in the 20th century. A form of
it is used in the music of northern India. American composer Terry Riley also made use
of the inverted form of it in his "Harp of New Albion". Just
intonation gives superior results when there is little or no chord progression: voices and other
instruments gravitate to just intonation whenever possible. However, as it
gives two different whole tone intervals (9:8 and 10:9) a keyboard instrument
so tuned cannot change key.^{[13]} To calculate
the frequency of a note in a scale given in terms of ratios, the frequency
ratio is multiplied by the tonic frequency. For instance, with a tonic of A4 (A natural
above middle C), the frequency is 440 Hz, and a justly tuned fifth above it (E5) is simply
440×(3:2) = 660 Hz.
The first 16 harmonics, their names and frequencies, showing the
exponential nature of the octave and the simple fractional nature of nonoctave
harmonics.
The first 16 harmonics, with frequencies and log frequencies.
Semitone

Ratio

Half Step


0

1:1

480

0


1

16:15

512

16:15


2

9:8

540

135:128


3

576

16:15


4

600

25:24


5

640

16:15


6

45:32

diatonic tritone

675

135:128

7

720

16:15


8

8:5

768

16:15


9

5:3

800

25:24


10

9:5

864

27:25


11

15:8

900

25:24


12

2:1

960

16:15

Pythagorean tuning is tuning
based only on the perfect consonances, the (perfect) octave, perfect fifth, and
perfect fourth. Thus the major third is considered not a third but a ditone,
literally "two tones", and is (9:8)^{2} = 81:64, rather
than the independent and harmonic just 5:4 = 80:64 directly below. A whole tone
is a secondary interval, being derived from two perfect fifths, (3:2)^{2} =
9:8.
The just major third, 5:4 and minor third, 6:5, are a syntonic comma, 81:80, apart from
their Pythagorean equivalents 81:64 and 32:27 respectively. According to Carl Dahlhaus(1990,
p. 187), "the dependent third conforms to the Pythagorean, the
independent third to the harmonic tuning of intervals."
Western common practice music usually
cannot be played in just intonation but requires a systematically tempered
scale. The tempering can involve either the irregularities of well temperament or be
constructed as a regular temperament, either some form
of equal temperament or some other
regular meantone, but in all cases will involve the fundamental features
of meantone temperament. For example, the root of
chord ii, if tuned to a fifth above the dominant, would be a major
whole tone (9:8) above the tonic. If tuned a just minor third (6:5) below a
just subdominant degree of 4:3, however, the interval from the tonic would
equal a minor whole tone (10:9). Meantone temperament reduces the difference
between 9:8 and 10:9. Their ratio, (9:8)/(10:9) = 81:80, is treated as a
unison. The interval 81:80, called the syntonic comma or comma of
Didymus, is the key comma of meantone temperament.
In equal temperament, the octave is
divided into twelve equal parts, each semitone (halfstep) is an interval of
the twelfth root of two so that
twelve of these equal half steps add up to exactly an octave. With fretted
instruments it is very useful to use equal temperament so that the frets align
evenly across the strings. In the European music tradition, equal temperament
was used for lute and guitar music far earlier than for other instruments, such
as musical keyboards. Because of this
historical force, twelvetone equal temperament is now the dominant intonation
system in the Western, and much of the nonWestern, world.
Equally tempered scales have been used and instruments built using
various other numbers of equal intervals. The 19 equal temperament, first proposed
and used by Guillaume Costeley in the 16th
century, uses 19 equally spaced tones, offering better major thirds and far
better minor thirds than normal 12semitone equal temperament at the cost of a
flatter fifth. The overall effect is one of greater consonance. 24 equal temperament, with 24 equally
spaced tones, is widespread in the pedagogy and notation of Arabic music. However, in
theory and practice, the intonation of Arabic music conforms to rational ratios, as opposed to
the irrational ratios of equally
tempered systems. While any analog to the equally tempered quarter tone is entirely
absent from Arabic intonation systems, analogs to a threequarter tone,
or neutral second, frequently occur.
These neutral seconds, however, vary slightly in their ratios dependent
on maqam,
as well as geography. Indeed, Arabic music historian Habib Hassan Touma has written that
"the breadth of deviation of this musical step is a crucial ingredient in
the peculiar flavor of Arabian music. To temper the scale by dividing the
octave into twentyfour quartertones of equal size would be to surrender one
of the most characteristic elements of this musical culture."^{[14]}
The following graph reveals how accurately various equaltempered scales
approximate three important harmonic identities: the major third (5th
harmonic), the perfect fifth (3rd harmonic), and the "harmonic seventh" (7th
harmonic). [Note: the numbers above the bars designate the equaltempered scale
(i.e., "12" designates the 12tone equaltempered scale, etc.)]
Note

Frequency (Hz)

Frequency
Distance from previous note 
Log frequency
log_{2} f 
Log frequency
Distance from previous note 
A_{2}

110.00

N/A

6.781

N/A

A♯_{2}

116.54

6.54

6.864

0.0833 (or 1/12)

B_{2}

123.47

6.93

6.948

0.0833

C_{3}

130.81

7.34

7.031

0.0833

C♯_{3}

138.59

7.78

7.115

0.0833

D_{3}

146.83

8.24

7.198

0.0833

D♯_{3}

155.56

8.73

7.281

0.0833

E_{3}

164.81

9.25

7.365

0.0833

F_{3}

174.61

9.80

7.448

0.0833

F♯_{3}

185.00

10.39

7.531

0.0833

G_{3}

196.00

11.00

7.615

0.0833

G♯_{3}

207.65

11.65

7.698

0.0833

A_{3}

220.00

12.35

7.781

0.0833

Below are Ogg Vorbis files demonstrating the difference between
just intonation and equal temperament. You may need to play the samples several
times before you can pick the difference.
·
Two sine
waves played consecutively – this sample has halfstep at
550 Hz (C♯ in the just
intonation scale), followed by a halfstep at 554.37 Hz (C♯ in the equal
temperament scale).
·
Same two
notes, set against an A440 pedal – this sample
consists of a "dyad". The lower
note is a constant A (440 Hz in either scale), the upper note is a C♯ in the
equaltempered scale for the first 1", and a C♯ in the just intonation scale
for the last 1". Phase differences
make it easier to pick the transition than in the previous sample.
Connections
to set theory
Musical set theory uses some of the concepts from mathematical set theory to organize
musical objects and describe their relationships. To analyze the structure of a
piece of (typically atonal) music using musical set theory, one usually starts
with a set of tones, which could form motives or chords. By applying simple
operations such as transpositionand inversion, one can discover
deep structures in the music. Operations such as transposition and inversion
are called isometries because they preserve the intervals between
tones in a set.
Connections
to abstract algebra
Expanding on the methods of musical set theory, some theorists have used
abstract algebra to analyze music. For example, the notes in an equal
temperament octave form anabelian group with 12
elements. It is possible to describe just intonation in terms of
a free abelian group.
Transformational theory is a branch
of music theory developed by David Lewin. The theory allows
for great generality because it emphasizes transformations between musical
objects, rather than the musical objects themselves.
Theorists have also proposed musical applications of more sophisticated
algebraic concepts. Mathematician Guerino Mazzola has
applied topos theory to music,^{[}^{citation needed}^{]}though the result
has been controversial.^{[}^{citation needed}^{]}
The chromatic scale has a free and transitive action of the cyclic group , with the action
being defined via transposition of notes. So
the chromatic scale can be thought of as a torsor for
the group