International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 1

ISSN 2229-5518

TUNING OF MUSICAL NOTES

THROUGH MATHEMATICS

ANUSHREE SAH, SAURABH RAWAT, BHASKAR NAUTIYAL, ADIL AHMED, OKESH CHHABRA

Abstract— Mathematics, an enigma in numbers and calculations, often accompanied by feelings of rejection and disinterest while Music, a flow with emotions, feelings and life. Motivation for investigating the connections between these two apparent opposites’ poles is attempted in this paper. A correlation between Mathematics and Music is shown.. Music theorists sometimes use mathematics to understand music. Mathematics is "the basis of sound" and sounds itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical". In today’s technology, without mathematics it is difficult to imagine anything feasible. In this paper we have discussed the relation between music and mathematics. How piano keys are interrelated with mathematics, frequencies are correlated and discussed. With the aid of mathematical tools, regression, geometric progression, tuning frequency can be calculated and further related to key of piano used to produce that particular frequency . This paper will also be helpful for music seekers and mathematician to understand easily a relationship between Mathematics and Music from a mathematician prospective.[12]

Index Terms— Piano frequencies, regression, geometric progression,piano keyboard, tuning, sine wave.

——————————  ——————————

1 INTRODUCTION

he patterns that exist between math, language, and music have prompted numerous studies to be commissioned to establish their inter- relationship. Music is a series of notes
that are played in accordance to a pattern and math’s too works in a similar way. In math’s a result always remains fi- nite despite the various ways in which you can add, multiply, subtract, and divide numbers.
The same can be said about music. Notes can be combined in an endless variety of groupings but the number of notes and sounds that exist are finite. It is these patterns and combina- tions that make music and math very similar.[11].
Mathematics and music are interconnected topics. “Music gives beauty and another dimension to mathematics by giving life and emotion to the numbers and patterns.” Mathematical concepts and equations are connected to the designs and shapes of musical instruments, scale intervals and musical compositions, and the various properties of sound and sound production. This paper will allow exploring several aspects of mathematics related to musical concepts.
The ancient Greeks figured out that the integers correspond to musical notes. Any vibrating object makes overtones or har- monics, which are a series of notes that emerge from a single vibrating object. These notes form the harmonic series: 1/2,
1/3, 1/4, 1/5 etc. The fundamental musical concept is proba-
bly that of the octave. A musical note is a vibration of some-
thing, and if you double the number of vibrations, you get a
note an octave higher; likewise if you halve the number of
vibrations, it is an octave lower. Two notes are called an inter- val; three or more notes is a chord. The octave is an interval
common to all music in the world. Many people cannot even distinguish between notes an octave apart, and hear them as the same.
A musical keyboard is the set of adjacent depressible levers or keys on a musical instrument, particularly the piano. Key- boards typically contain keys for playing the twelve notes of the Western musical scale, with a combination of larger, long-
er keys and smaller, shorter keys that repeats at the interval of an octave. Depressing a key on the keyboard causes the in- strument to produce sounds, either by mechanically striking a string or tine (piano, electric piano, clavichord); plucking a string (harpsichord); causing air to flow through a pipe (or- gan); or strike a bell (carillon). On electric and electronic key- boards, depressing a key connects a circuit (Hammond organ, digital piano, and synthesizer). Since the most commonly en- countered keyboard instrument is the piano, the keyboard layout is often referred to as the "piano keyboard".
The twelve notes of the Western musical scale are laid out
with the lowest note on the left; The longer keys (for the seven
"natural" notes of the C major scale: C, D, E, F, G, A, B) jut
forward. Because these keys were traditionally covered in ivo-
ry they are often called the white notes or white keys. The
keys for the remaining five notes—which are not part of the C
major scale—(i.e .,C♯, D♯, F♯, G♯, A♯) are raised and shorter.
Because these keys receive less wear, they are often made of
black colored wood and called the black notes or black keys.
The pattern repeats at the interval of an octave.

1.1 Piano Keyboard [10]

For understanding of this paper, it is important to have some knowledge of the piano keyboard, which is illustrated in the following diagram. This keyboard has 88 keys of which 36 (the top of the illustration), striking each successive key pro- duces a pitch with a particular frequency that is higher than the pitch produced by striking the previous key by a fixed interval called a semitone. The frequencies increase from left to right. Some examples of the names of the keys are A0, A0#, B0, C1, C1#. For the purposes of this paper, all the black keys will be referred to as sharps (#). In this paper different fre- quencies of piano are discussed, how they are produced peri- odically with the use of Regression Analysis and Geometric Progression.

IJSER © 2012

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 2

ISSN 2229-5518

Diagram illustrates different key numbers, key names and their corresponding frequencies in piano keyboard. From key numbers 1 to 12 frequencies are given, but from 13 to 24 they form the same pattern but double the initial values and from
25 to 36 values are thrice of initial values and so on.
the left shows the first 12 keys of a piano. The table down shows the frequency of the pitch produced by each key, to the nearest thousandth of a Hertz (Hz).


Fig. 2 Different Frequencies on Piano Keyboard
Fig. 1 Piano Keyboard[10]

The frequencies of all successive pitches produced by striking the keys on a piano keyboard form a pattern. The diagram on

2 REGRESSION

In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed. Regression is of two types,
Linear Regression and Exponential Regression
A linear regression produces the slope of a line that best fits a

IJSER © 2012

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 3

ISSN 2229-5518

single set of data points. For example a linear regression could be used to help project the sales for next year based on the sales from this year.

An exponential regression produces an exponential curve that best fits a single set of data points. For example an exponen­ tial regression could be used to represent the growth of a pop­ ulation. This would be a better representation than using a linear regression.

Best fit associated with n points

(-';, y1),(x2,Y2 ),........... .(xn,Yn)

has exponential formula,

y =arx

Taking log both sides logy= loga + xlogr Equating withY= mx + b

Slope m = logr

Intercept b = loga

Best fit line using logy as a function of x.

r =10m

a= lOb

Equating above table with Regression analysis, taking key number as x and frequencies as y =log (frequency).

Equations for exponential regression

Slope=

nL (.ry)-(L x)(Ly)

m = 2 2

nLx - (Lx)

12x 126.61342-(78) x (18.92718)

m=-------------------- -------

12x650-(78i

m =.0250821

Intercept=

b = ( y)- m( x)

n

b =1.41423135

y=mx+b

y = .025082lx+ 1.41423135

3 GEOMETRIC PROGRESSION

The frequencies of pitches produced by striking the piano keys can also be modeled by a geometric sequence. The model can be determined by using a pair of keys with the same letter and consecutive numbers; for example, AO and A1, or B1 and B2, or G2# and G3#. Each pair of consecutive keys with the same letter has frequencies with a ratio of 2:1. In other words, the frequency of A1 (55.000 Hz) is double the frequency of AO (27.500 Hz), the frequency of A2 (110.000 Hz) is double the frequency of A1 (55.000 Hz), and so on. In mathematics, age­ ometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54

... is a geometric progression with common ratio 3. Similarly

10, 5, 2.5, 1.25, is a geometric sequence with common ratio 1/2. The sum of the terms of a geometric progression, or of an ini­ tial segment of a geometric progression, is known as a geo­ metric series. Thus, the general form of a geometric sequence is

a, ar, ar2 , ar3 , ar4 , . . .

ao,a1,a2 ...............an-1

Or

a,apa . ... ...........an_

nth term will be

an =arn-1

Where a is the first term and r is the common ratio.

a

r =-1

a

Referring table no. 2.1

So if we take frequencies of key numbers in geometric pro­

gression then first term will be 27.500 and second term will be

29.135, so r = 29.135/27.500 = 1.05945

an =arn-1

an = 27.500(1.05945)n-1

4 COMPARISON OF FREQUENCIES THROUGH REGRESSION ANALYSIS AND

GEOMETRIC PROGRESSION.

IJSER !b) 2012 http://www.ijser. org

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 4

ISSN 2229-5518

TABLE 1

COMPARISON OF FREQUENCIES

KN = Key Name, GP = Geometric Progression

IJSER © 2012

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 5

ISSN 2229-5518

As we can observe from the table, moving from key numbers 1 to 12, Geometric progression is more effective in determining values of frequencies near to actual frequencies. But as we proceed further towards 13, onwards to higher numbers, Re- gression Analysis is the method to count upon in determining frequencies quite close to actual frequencies.

If a single key produces a frequency of 783.5Hz, than which is this key.
From Regression analysis y = .0250821x + 1.41423135
a = amplitude of the sound wave in pascals
f = frequency of the sound wave in Hertz (Hz)
t = time in seconds
A violin is being tuned with an A440 tuning fork (f = 440 Hz). The sound wave that is produced by the tuning fork has an
amplitude of 0.10 Pa. This sound wave can be represented by the function

P  0.10 sin(880t)

Plotting P(Pa) on y axis and taking t on x axis, following graph
is obtained.

10 y  f

= 783.5
After solving x= 58.99.
From geometric progression analysis

a  27.500(1.05945)n1

n

.5  27.500(1.05945) 1

783 n

After solving n= 59.00
So 59 = 12x5 -1 = equal to 11 = G1 Key
Or 783.5 = 48.99( G1) x16 =783.9 = a multiple of frequency of
key G1.
So any frequencies produced by the piano can be related to
given key.

5 TUNING

Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones. Tuning is usually based on a fixed refer- ence, such as A = 440 Hz. Out of tune refers to a pitch/tone that is either too high (sharp) or too low (flat) in relation to a given reference pitch. While an instrument might be in tune relative to its own range of notes, it may not be considered 'in tune' if it does not match A = 440 Hz (or whatever reference pitch one might be using). Some instruments become 'out of tune' with damage or time and have to be readjusted or re- paired. Some instruments produce a sound which contains irregular overtones in the harmonic series, and are known as inharmonic. Tuning may be done aurally by sounding two pitches and adjusting one of them to match or relate to the other. A tuning fork or electronic tuning device may be used as a reference pitch, though in ensemble rehearsals often a piano is used (as its pitch cannot be adjusted for each rehears- al).
When tuning a musical instrument such as a piano, a violin, or a guitar, a tuning fork or an electronic tuning device is used to produce a pure pitch at a specific frequency.
The tuning fork used most often is for the key A4 (f = 440 Hz) and is referred to as an A440 tuning fork. A tuning fork pro- duces a sound wave whose pressure variations
can be modelled by a sinusoidal function of the form.

P  a sin(2ft )

Swhere P = pressure variation of the sound wave in pascals
(Pa)
Another tuning fork with a different frequency is being used to tune a particular pitch (key) on a piano. The tuning fork is struck twice, with a different force each time. A graph repre- senting each of the sound waves produced is shown on the coordinate plane below.

From the above graph ,when t= 1/1024 then P= 0.15Pa and when t = 3/1024 then P= -0.15Pa

IJSER © 2012

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 6

ISSN 2229-5518

So as

P  a sin(2ft )

[4] Beer, Michael (1998): How do Mathematics and Music relate to each other? (Retrieved in june 2004 from: http://perso.unifr.ch/-

0.15Pa  a sin(2f 

1 )

1024

------------(i)

michael.beer/mathandmusic.pdf) .

[5] Lesser, Larry (2002): Favorite quotations on Math & Music, (retrieved in June

 0.15Pa  a sin(2f 

Solving (i) and (ii)

3 )

1024

------------(ii)

2004 from:

http://www.math.armstrong.edu/faculty/lesser/Mathemusician.html)).

[6] Rueffer, Claire (2003): Basic Music Theory and Math. The Circle of Fifth. (Re- trieved in july 2003 from:http://www.geocities.com/-ctempesta04/fifth).).

2f 

3

1024

   2f 

1

1024

[7] BIBBY, N (2003) Tuning and temperament; closing the spiral. In J . FAUVEL, R . FLOOD, and R , WILSON (eds), Music and Mathematics ; From Pythago-

ras to Fractals, chap 1, pp 13-27, Oxford University Press, Oxford.

f  256Hz

So tuning fork is of frequency 256Hz, and the key of piano which has been tuned is,
From Regression analysis
y = .0250821x + 1.41423135

10 y  f  256

After solving x= 39.63
From Geometric progression

[8] GARLAND, T .H & KAHN, C. V (1995) Math & Music: Harmonious Con- nections. Dale Seymour Publications, Polo Alto.

[9] ROTHWELL, J, A (1977) , The Phi Factor, Mathematical Properties in Musical forms. University of Missouri, Kansas City.

[10] Pure Mathematics 30, Student Project : Mathematics and Music, http://education.alberta.ca/media/614443/feb2007_pmath30_project.pdf

[11] WWW.Musicedmagic.com

[12] .En.wikipedia.org/wiki/musical tunning

[13] Mathemusic- number and notes,A mathematical approach to musical fre- quencies.

a  27.500(1.05945)n1

256  27.5(1.05945)n1

Solving n = 39.63
So 39.63= 12x3 +4 = equal to 4 = C4 Key
Or 256 = 32.7009(C1) x8 =256 = a multiple of frequency of key
C1
So any frequencies produced by the piano can be related to
given key.

6 CONCLUSION AND FURTHER WORK

In this paper we have elaborated the fact that music and mathematics are interrelated. The different frequencies used in music are based on mathematical calculations. Paper discusses two cases, first where a frequency was known and another with unknown frequency. After calculating frequency through graph, the required key in piano is found with the help of re- gression and geometrical progression.. Both the methods are effective and have produced desired results.. Further work in determining frequencies and their pattern can be done through Fourier Transform. This paper will help both music seekers as well as mathematical intellectuals a belief that both mathematics and music are interconnected.Tunning frequency can be calculated, with further finding of piano key which is producing the desired result.

REFERENCES

[1] Chugunov, Yuri (1985): Harmony of jazz, Moscow.

[2] Fuchs, Laszlo (1973): Infinite Abelian Groups, in: Academic Press, vol.. [3] Huntley, H. E: (1970): The Divine Proportion, Dover.

IJSER © 2012

http://www.ijser.org