Download Modern Western Tuning System - Digital Commons @ Kent State

Mathematical
Investigation
Scott Marino II
September 1
2013
An Investigation
of Music & Math
 Introduction
 In my course of education the fields of Music and Math have fascinated me by their
separation in description, math being considered a science and music a subject of art.
However I now see them both as an art and a science. I am a cellist, pianist, and
trumpeter and in my pursuit to master these instruments I began to blur the lines
between math and music seeing music in a more mathematical light, recognizing the
distances between notes on a cello’s finger board and opening the piano to see the harp
and the length and thickness of stings; I started to think that math was in essence the
basis of all music. For my mathematical investigation I wanted to see how exactly
mathematical concepts existed in music and then to see if I could derive these concepts
from my playing of cello.
 Mathematics is the fundamental building block of sound and in its musical form an even
larger array of number phenomena exist. Although the mathematical properties of
sound had been studied by the ancient Chinese, Mesopotamian, and Egyptian societies,
the scholars of the Pythagorean School is ancient Greece were the first to investigate
and formulate musical scales and tuning systems in terms of numerical ratios and
functions. They had believed that all nature consists of harmony arising out of the small
integers of 1, 2, 3, and 4; regarding them as the source of all perfection as did Confucius
from China and Indian theorists. These numbers are crucial in the equations of musical
scales and tuning systems.
 Tuning Systems
 Pythagorean Just Intonation
 The Pythagorean Scale is based on the ratio of 3:2 this ratio is found in the
harmonic series and when multiplied by any frequency will produce a note
known as the fifth in relation to it. Pythagoras had based his scale system upon
this ratio because it is one of the most consonant or pleasing to the human ear
since it uses what they referred to as pure harmonic integers or ratios using the
numbers 1, 2, 3, and 4.
 Harmonic Series (Music): A harmonic is a wave that is an integer multiple of a
fundamental frequency. It produces a pattern that can be formulated into an
(
) , where the frequency of a harmonic
arithmetic sequence of
is the fundamental frequency plus (n-1) multiples of the fundamental. The
ratio of the fifth is obtained using the ratio
(ascending 5th) or
(descending 5th) where n correlates to the column of harmonics (figure 2).
Harmonics
1
2
4
5
3
6
7
2
4
8
8
Note
16
32
64
128
Octave (1)
17
34
68
136
Minor second (2-)
9
18
19
36
38
72
76
144
152
Major second (2)
Minor third (3-)
10
20
40
80
160
Major third (3)
11
21
22
42
44
84
88
168
176
Perfect fourth (4)
Augmented fourth (4+)
Diminished fifth (5-)
23
46
92
184
12
24
48
96
192
Perfect fifth (5)
13
25
26
50
52
100
104
200
208
Augmented fifth (5+)
Minor sixth (6-)
27
54
108
216
Major sixth (6)
14
28
29
56
58
112
116
224
232
Augmented sixth (6+)
Minor seventh (7-)
15
30
60
120
240
31
62
124
248
Major seventh (7)
Augmented seventh (7+)
32
64
128
256
Octave (8)
16

The wavelength of a harmonic can be expressed by integrating the equation for
frequency of a harmonic into the equation for wavelength and creating an
explicit formula.

 where f is the frequency ratios of harmonics
hence
(
(
)
)
 If the speed of sound was not known and the wavelength of the first
harmonic was known, a arithmetic sequence could also be used to represent
the wavelength of a harmonic

Mathematics of Pythagorean Just Intonation
 The Pythagorean Tuning system uses the fifth ratio and powers of it along with
powers of ratio 2:1, the ratio of the octave.
 An issue arose with this tuning system. When applied to extended fifths and
creating a whole chromatic scale, no group of 3:2 notes (fifths) will fit evenly into
a 1:2 ratio octave.
Note




Gb
Interval from C
Diminished fifth
Db
Formula
Frequency Ratio
1.404663923
( )
( )
Minor second
( )
( )
1.05349742
Ab
Minor sixth
( )
( )
1.580246714
Eb
Minor third
( )
( )
1.185185185…
Bb
Minor seventh
( )
( )
1.777777…
F
Perfect fourth
( )
( )
1.33333…
C
Unison
1
G
Perfect fifth
1.5
D
Major second
( )
1.125
A
Major sixth
( )
1.6875
E
Major third
( )
( )
1.265625
B
Major seventh
( )
( )
1.8984375
F#
Augmented fourth
( )
( )
1.423828125
C
Octave
2
In the formulas, the ratios 3:2 or 2:3 represent an ascending or descending
perfect fifth, while 2:1 or 1:2 represent an ascending or descending octave.
In order to create a 12-tone chromatic scale Pythagoras had to omit the Gb
(which is supposed to be the same note as F#, however this tuning produces two
different frequency rations for these notes) only using the 12 remaining notes to
comprise a complete chromatic scale.
The untrue fifth which does not justly adhere to the 3:2 ratios from F# to Db is
left out of just intonation and is out of tune, being slightly smaller by what is
known as a Pythagorean Comma (a quarter tone). This interval was given the
name the wolf interval due to its displeasing nature.
This mathematical phenomenon made it impossible for these two notes to be
played simultaneously in the music. Based on which note a Pythagorean scale
was formulated as its tonic, the wolf interval will shift. Due to the shifting
ambiguity of the wolf interval, it made it impossible to play all keys in tune.
F#-Gb

A Solution to the Problem
 Pythagoras had proposed that instead of using both ascending and descending
fifths to formulate a chromatic scale that only ascending fifths be used, which
would, as he thought eliminate the existence of the wolf tone as well as the off
pitch augmented fourth-diminished fifth twin tones (same tone). By doing so
Pythagoras thought he would resolved the problem of not being able to
transpose to other keys.
Note
C
Interval from C
Unison
Formula
or ( )
G
Perfect fifth
( )
( )
1.5
D
Major second
( )
( )
1.125
A
Major sixth
( )
( )
1.6875
E
Major third
( )
( )
1.265625
B
Major seventh
( )
( )
1.8984375
F#
Augmented fourth
( )
( )
1.423828125
Db
Minor second
( )
( )
1.067871094
Ab
Minor sixth
( )
( )
1.601806641
Eb
Minor third
( )
( )
1.201354980
Bb
Minor seventh
( )
( )
1.802032471
F
Perfect fourth
( )
( )
1.351524353
C
Octave
( )
( )
2.027286530
( )
Frequency Ratio
1


Pythagoras had resolved the problem of the wolf tone and the frequency ratio
discrepancy concerning the augmented fourth-diminished fifth. However he was not
able to resolve the issue of playing in all keys in tune due to the fact that his new
mathematics left the octave interval a Pythagorean comma (quarter tone) out of
tune. Where the true just ratio of an octave is 2:1 while Pythagoras’ fifth (3:2) ratio
based scale produces an octave that is 2.027286530:1, hence leaving the problem of
transposition unresolved yet. This affirmed the inability to fit a series of 3:2 ratio
fifths evenly among a 2:1 octave interval.
The inability to achieve equal ratios prompted composers to consider what specific
key they wanted to write in as depending on which note (frequency) they wanted to
use as the tonal center, unique dissonances would ensure based on the shifting
properties of the wolf tone. Composers often referred to a keys mood or colour as
some would contain more dissonant or consonant intervals.
 Equal Temperament(12TET):
 Equal temperament is a tuning system which uses logarithms in order to have all
pairs of adjacent notes be equidistant from another (same frequency ratio). The
most common and relevant of the Equal Temperament tuning systems is the 12TET
which has 12 tones within the distance of an octave. Other Equal Temperaments
exist such as 5, 7, 19, 22, and 31TET however for the purpose of understanding the
math behind this system 12TET is the best representation as it solved the issues of
Just Intonation and is the most popular tuning system in modern time.
 History of 12TET
 The founding of 12TET is often attributed to Chinese scholar Zhu Zaiyu (1584)
and Dutch mathematician Simon Stevin (1585); both of their studies were
independent from one another and occurred concurrently
 Zhu Zaiyu
 Chinese culture had for centuries practiced a tuning system which was
based on five equidistant notes where the first and last where at a ratio
of 1:2 ; it was known as the Pentatonic scale.
 Zaiyu was a Music theorist and was interested in developing another
equal tone scale. He accurately calculated the value of the 12th root of
two using the properties of generalized continued fractions which
states:
Where; An is the numerator and Bn is the denominator,

Zaiyu then applied the properties of generalized continued fractions
to that of roots to the nth degree where can be restated as
+y then the proceeding is true.

√
√(
)
Hence the 12th root of 2 is approximated by;
√(
)
(
( )
=

√
(
(
(
)
)
( (
)
( )
(
)
)
( (
(
)
)
)
)
( ( )
( )
)
1.059463028
*The actual value of the twelfth root of two has been more accurately calculated
using logarithms to the value 1.059463094359295264561825.
The Modern 12-TET Tuning System
Interval Name
Exact Value
Decimal Value
Unison
1.000000
( √ )
Minor second
1.059463
( √ )
Major second
1.122462
( √ )
Minor third
1.189207
( √ )
Major third
1.259921
( √ )
Perfect fourth
1.334840
( √ )
Augmented fourth
1.414214
( √ )
Perfect fifth
1.498307
( √ )
Minor sixth
1.587401
( √ )
Major sixth
1.681793
( √ )
Minor seventh
1.787797
( √ )
Major seventh
1.887749
( √ )
Perfect octave
2.000000
( √ )
 Application in Real World
 I have been instructed in the western style of music, my teachers had never told me
explicitly that I was using the 12-TET tuning system. When practicing scales I would be told

to play any of the major scales and although they had different names they all sounded the
same. It hasn’t been until now that I understand the mathematics behind it.
I wanted to see how accurately me playing a C-Major scale is to the frequency values
calculated using 12-TET. I used sound analysis software that recorded data from wave
patterns and displayed a spectrogram (A spectrogram, or sonogram, is a visual
representation of the spectrum of frequencies in a sound). The frequency I will use will be
an average of the two obtained from the spectrogram and wave pattern.
Note C:
(
)
⁄
263.72 *All frequency calculations follow suite
Note: D
(
)
Note E:
(
)
Note F:
(
)
Note G:
(
)
Note A:
(
)
Note B:
(
)
Note C:
(
)
Note
Wave Data
(Hz)
263.72
290.94
328.18
344.84
397.73
439.59
495.93
528.08
C
D
E
F
G
A
B
C

Spectrogram
(Hz)
262.02
293.16
330.47
350.87
392.57
439.24
495.14
525.20
Average (Hz)
262.87
292.05
329.33
347.86
395.15
439.42
495.54
526.64
Calculated Values of Notes in a C-Major Scale
 To find the frequency of a note using the 12-TET tuning system the following
equation can be used
( √ )
 In this formula refers to the value of the desired pitch, refers to the
value of a predetermined reference pitch (typically A440), and whole
number values given to the desired and assigned pitch. These numbers refer
to consecutive integer numbers assigned to the numbered key they are on a
standard piano.
 Where

( )
and where if

( √ )
( √ )
 Calculations follow same process with respective
Note
C
D
E
F
G
A
B
C
values.
Calculate Frequency 12-TET (Hz)
261.63
293.66
329.63
349.23
392.00
440.00
493.88
523.25
 Comparison of Frequencies
Slope of Experimental Data: 38.59
Slope of Calculated Data: 38.17
(
)
 Conclusion
 In conclusion through my investigation I found a greater understanding of the tuning/scale
systems that I use on a daily basis. I also determined that when practicing my scales in real life I
do not play them perfectly in 12-TET but rather I have a percent error of 1.10%. Through more
practice I hope to become mathematically accurate with my scales now that I know the history,
mathematical concepts, and principles of determining the frequency ratios between notes in 12TET .