Chord Analysis in the Key of Math and Physics
By
Jenny Fromme
Table of Contents
1. Introduction
2. Background
3. Results
4. Conclusions
Appendices
References
Introduction
Which is more musically pleasing, a Mozart piano sonata or a toddler pounding
away at the keys of a piano? Which is more enjoyable to listen to, a symphony
orchestra or an elementary school strings concert? Why is one more favorable to listen
to than the other?
When I started college, I began as a music education major. In the music theory
classes, one question that I constantly revisited was why do certain notes sound good
together? Through all of my music study, I searched for some sort of concrete answer
to that question. Unfortunately, most answers gave either a strictly musical reason or a
strictly physical reason, and as a result, most answers failed to adequately answer the
question for me.
When I became a math major, this question of why certain musical notes sound
good together stayed with me. As I gained more knowledge and background in
mathematics, my intuition told me that the answer lied somewhere in between music
and physics and that likely, the answer could be explained mathematically. Thus, my
research focuses on trying to find a mathematical explanation for musical consonance
and dissonance.
The rest of this paper is divided into 3 sections. The first section gives the
background information necessary for understanding the problem from a musical,
mathematical and physical standpoint and also offers some history on the study of this
idea.
The second section describes the procedures used to analyze this phenomenon
and gives the results gained from the research. The last section gives possible
interpretations of the data collected and offers some ideas for further study.
2
Background
In order to study musical chords, several fundamental musical ideas must be
understood. The first musical element is pitch. In musical terms, pitch refers to the
highness or lowness of a sound. A sound wave is an oscillatory disturbance of some
medium that travels as a wave through that medium, such as a pressure wave
propagation through the air (Hecht, 2000). In the simplest case, such a wave can be
modeled with a sinusoidal function, such as y=sin(x).
With the sinusoidal wave, the pitch is represented by the period, or the amount of it
takes for one cycle of the wave to occur, measured in seconds per cycle. Inversely, one
could measure the number of oscillations which occur in one second. This is called
frequency, measured in Hertz, which is cycles per second. Thus, frequency is the
inverse of period. Lower pitches have a lower frequency and higher pitches have a
higher frequency (Benade, 1960).
Now that we have pitches, we can begin to build chords. Simply stated, a
musical chord is just several pitches sounded simultaneously. More precisely, we
define a musical chord as consisting of three or more unique pitches, none of which are
the same note in a different octave (Kamien, 1996). Mathematically, this means no two
pitches in the chord have frequencies that are related by a factor of two or a power of
two. Measuring frequencies in Hertz (cycles per second) we require that
3
For all pairs of pitches f i and f j in a chord,
there exists NO integer n such that
fi  2 n f j
The intervals, or frequency differences, between the pitches of a chord
determine the chord’s quality, with the most common chord qualities in order of
diminishing consonance being major, minor, diminished and augmented (Rossing,
Moore & Wheeler, 2002). A major chord is a chord with a root note, a note a major
third (or 4 half-steps) above it and a note a perfect fifth (or 7 half-steps) above the root
note. A half-step is the smallest interval between notes in western music, such as D to
D# or E to F. A minor chord is built using a root note, a note a minor third (or 3 halfsteps) above it and a note a perfect fifth (or 7 half-steps) above the root note. The
diminished chord is made up of a root note and a minor third (3 half-steps) above it and
a note a diminished fifth (or 6 half-steps) above the root note. The augmented chord is
made up of a root note, a note a major third (or 4 half-steps) above it and a note an
augmented fifth (or 8 half-steps) above the root note. The spellings of a C chord under
each of these qualities is listed in the table below.
Major
G
Spellings of a C Chord
Minor
Diminished Augmented
G#
G
Gb
E
C
E
Eb
Eb
C
C
C
Analyzing chords constructed from three unique pitches is more complicated
than it might first appear. The complications arise as pairs of pitches in the chord
interact with each other and with the listener’s perception of those pitches. When two
pitches are sounded simultaneously, the pitches created by the sum and differences of
the sinusoidal waves are also perceived (Benade, 1990). The sum and difference
components can be seen in the equation of the sum of two sinusoidal functions:
4
cos2f 2 t   cos2f1t   2sin 2  f 2  f1  t2 sin 2  f 2  f1  2t 
The presence of these tones were first discovered around 1714 by violinist Giuseppe
Tartini, who used the difference tone to create pitches lower than the range of his violin,
thus the difference tone is often called the Tartini tone (Grout & Palisca, 1996). As we
shall see below, these perceived sum and difference frequencies will play an important
role in the relative consonance or dissonance of a given chord.
Now that we have an understanding of chords, we can talk about chord
consonance and dissonance. Before we discuss our ideas on consonance and
dissonance, we should discuss how people have previously attempted to these terms.
Consonance is the pleasing quality of musical tone or chord. Hermann Ludwig
Ferdinand von Helmholtz in 1877 described consonance by referring to Ohm’s
acoustical law, which stated that the ear performs a spectral analysis of sound,
separating a complex sound into its various partials (Rossing, Moore & Wheeler, 2002).
A complex sound is a tone that includes a vibration at the frequency sounded-the
fundamental, and at multiples of that frequency-the partials, corresponding to higher
order modes of vibration of the system. The frequency’s partials are the integer
multiples of the fundamental frequency formed by taking the original wavelength of a
pitch, the fundamental, and dividing it into parts equal to the ratios given by the
harmonic series: ½, 1/3, ¼, ….1/n.
The analysis of these complex tones is also known as Fourier Analysis, after the
mathematician Joseph Fourier (1768-1830). Fourier formulated the mathematical
theorem that any periodic vibration, however complicated, can be built up from a series
of simple, sinusoidal components, by choosing the proper amplitudes and phases of
these components (Brown & Churchill, 2001). The Fourier Transform takes any
periodic function, f(t), and transforms it into the distribution of frequencies, F(f), that
make up the function
F f  



f t  e
 i 2 f t
dt
Conversely, dissonance is the unpleasing quality of musical tone. Helmholtz
also concluded that dissonance occurs when partials of the two tones produce 30-40
beats per second. A beat is a pulsating sound that is produced when two pitches are
5
close together. The closer the two pitches are, the slower the beats are. These beats are
simply the perceived difference frequencies, or Tartini tones described above.
There is yet one more complication, namely, how to divide the octave into
discrete steps, i.e. the mapping of note names onto specific frequencies. While this may
seem trivial, the question of how to best do this has been debated for over 2000 years
and is still disputed today. As we shall see, it also plays a role in mathematically
assessing the consonance and dissonance of musical tones.
The study of how music and mathematics relate has been traced by to the
Ancient Greeks. Pythagorus was one of the first people to study the connection
between mathematics and music. Boethius, the author of the first music theory text,
written in the early years of the sixth century AD, wrote that Pythagorus observed the
relation of the weights of hammers used by blacksmiths and the tones that the strikes of
the hammers against their anvils produced. When a hammer was half the weight of
another hammer, it produced a pitch one octave higher (or double the frequency) than
the pitch produced by the hammer twice its weight. Pythagorus then went on to find the
ratios between the weights of other hammers and the intervals produced by those
pitches, focusing on creating true fourths and fifths, intervals common to Greek music
(Appendix 1). Boethius’s music theory text, De institutione musica (The Fundamentals
of Music), consisted of three parts. The first part consists of three books, which was
based off of the writings of the Pythagoreans, the second part contains the writings of
Euclid and Aristoxenus, and the last part is based on the writings of Ptolemy. Most
readers came away with the message that music was a science of numbers and that
numerical ratios determined the melodic intervals, consonances, composition of scales
and the tuning of instruments and voices.
The set of ratios used for tunings as described by Boethius was the most
commonly used tuning until around 1025-28 AD when Guido of Arezzo simplified the
ratios determined by Pythagorus by dividing a string into sections. In additional to
being numerically simpler, these simplified tuning ratios also made intervals of thirds
and sixths more pleasingly in tune at the expense of the consonance of fourths and
fifths. Since the Church though intervals of thirds and sixths represented the Holy
6
Trinity and were, therefore, more divine, and since the Church was the major influence
of musical development at the time, the new simplified tuning quickly gained
prominence. These tuning were rigorously developed in 1482 by Spanish
mathematician and music theorist Bartolome Ramos de Pareja. This system was then
further developed by Franchino Gaffurio (1451-1522) into the system of tuning that is
now known as the just intonation system. Though the just intonation system simplifies
the ratios and improves the sound of the thirds and sixths, the sound of the fifths and
fourths were compromised.
The just intonation system was predominant until 1722 when Johann Sebastian
Bach wrote a set of 24 preludes and fugues in all 24 keys, Das wohltemperirte Clavier
(The Well-Tempered Piano), thus needing equal intervals between all pitches on the
piano. In addition, most of the notes which created semitones between the musica ficta
(white keys on the piano plus Bb), were dropped thus giving us our current 12 note
chromatic scale. This new system created 12 equal intervals per octave with a constant
frequency ration of 2^(n/12) between sequential notes. This system is referred to as the
equal temperament system and is the system universally accepted today. There is still
much controversy today in the music world as to which scale system is more pleasing
and useful, the just temperament system or the equal temperament system (Grout &
Palisca, 1996).
Results
Now that we have defined some of the basic ideas, we can begin to look at our
research question, is there a mathematical trend in musical chords that correspond with
consonance and dissonance. The first part of the research focused on finding the sum
and difference frequencies for major, minor, diminished, and augmented chords using
the equal temperament scale. The sum and difference frequency for each combination
of two notes from the three that made up each chord was taken, thus giving three sum
and three difference frequencies. These frequencies were then compared to the
frequencies of actual pitches and a percentage of error was determined between the sum
and difference frequencies and the frequencies of the actual pitch. If the error
7
percentage was less than one percent, the sum or difference frequency was considered
significant. This was repeated twelve times, each time building the chord off each of
the twelve musical notes. Note that there is no difference between major chords built
off different pitches in the equal temperament system since the difference in frequency
between the pitches are equal.
Equal Temperament
Since the frequency ratios of all sequential notes are the same in the equal
temperament system, we can take one example, e.g. the C chord, and use it as a
generalization for the rest. The significant pitches produced from the C major chord
were from the sum of C and E, which resulted in a D (22/12), the sum of C and G, which
resulted in an E (24/12), the difference between C and G, which resulted in a C (2) and
the difference between E and G, which resulted in a B (211/12). The C and G that were
perceived are pitches which are part of the original C major chord and as a result,
reinforced those notes, and the D and B notes are other notes in the scale, but are not
present in the chord.
SUMS
DIFFERENCES
G
F#
E
F
B
E
D#
D
C
D
C#
C
For each of the chord qualities under the equal temperament system, different
notes were reinforced and perceived. As stated above, for the major chord, the base
8
note and the 24/12 were reinforced and the 22/12 and 211/12 were perceived. For the minor
chord, the base note and the 27/12 were reinforced and the 24/12 and the 25/12 were also
perceived. For the diminished chord, no notes were reinforced, but the 27/12 and the
210/12 notes were perceived and for the augmented chord, the 28/12 note was reinforced
and the 26/12 note was perceived. Unfortunately, though, since the intervals between
notes in the equal temperament system are irrational frequencies, none of the sum and
difference frequencies matched up exactly to actual frequencies that represented pitches.
Regardless, there seems to be a correlation between consonance and occurrences of sum
and difference frequencies.
Just Intonation
Then the entire process was repeated using notes from a scale created using the
just intonation system. Since there are more than twelve musical notes in a diatonic
scale created using the just intonation system, where there were two ratios that could
represent a single note in the equal temperament system, the simpler of the two ratios
were chosen. This was done because overtime, a common trend of new tuning systems
was toward simpler frequency ratios.
Unlike in the equal temperament system, frequency ratios between sequential
notes in the just intonation system do not stay constant. Therefore, one example of a
chord could not be used to represent all chords. Thus, sum and differences were
generated for chords built from all possible base notes. The results of these sums and
differences (along with the sums and differences from the equal temperament chords
and the complex tones to follow) are listed in Appendix 2.
Among all the major chords formed using each of the twelve tones as a root
note, three principle patterns of sum and difference frequencies were observed. In one
group of major chords the sum and differences reinforced the root note and the 5/4 note.
In another group, the sum and differences reinforced the root note and the 5/4 note. In
addition, the 9/8 note was also perceived. In the last group of major chords the root
note and the 5/4 note were reinforced with the 15/8 note being perceived.
9
The minor chords had only one pattern run through some of the chords, which
was the root note and the 6/5 note being reinforced with the 4/3 and the 8/5 notes being
perceived. The rest of the minor chords which didn’t fit this single pattern also didn’t
seem to form any other clear pattern amongst themselves. Similarly, the augmented and
diminished chords both didn’t seem to form any patterns of reinforced or perceived
notes.
Overall, with chords built using a just intonation tuning system, as consonance
decreased, the number of notes reinforced and perceived decreased. Also, as
consonance decreased, the number of patterns between different chords of the same
quality lessened and those patterns that were observed became less apparent.
Complex Tones
So far, the research has focused on looking at pitches made up of a single
frequency. This was necessary to see what effect sum and difference tones had on the
chords, but real musical tones are more complicated than this. Musical tones are made
up of the frequency at the pitch being played, and the pitches that result from multiples
of the given frequency. These notes are called overtones. This presented several
problems. One was that there is essentially no upper limit to the overtone frequenciesthe overtones go on forever, so the decision was made to cut the overtones off after the
seventh overtone. Also, the amplitudes of the overtones differ between instruments.
Thus, the Fourier transform of the sound of several instruments was taken. Several
instruments including a cello, trombone, clarinet, trumpet, voice, flute were recorded
playing an “A” at 220 Hz as indicated below in their resulting Fourier transforms.
Flute at 220 Hz
10
Voice at 220 Hz
Cello at 220 Hz
Trumpet at 220 Hz
Clarinet at 220 Hz
Trombone at 220
Hz
There was not only a difference in the amplitude of the overtones between the
different instruments, but there was also a difference in the range of frequencies
sounded at each overtone (seen as the sharpness of the peaks). Thus, in addition to, the
sum and difference frequencies between pairs of two pitches from the original three
chord pitches, but also between the notes in the different overtones and between the
range of the frequencies sounded at each overtone frequency.
When the sum and difference frequencies of the overtones were looked at,
results occurred which were similar to the results of the chords without overtones in that
as consonance decreased, so did the number of pitches that reinforced the notes of the
chord. Over two-thirds of the significant pitches formed by the sums and differences of
the overtone frequencies reinforced the notes of the chord. Also, the lower sum and
difference frequencies tended to reinforce the root pitch and the note a third above it
11
while the higher sum and difference frequencies tended to reinforce the note of the
chord that was a fifth above the root in all four common chord qualities.
Conclusion
This research shows that there is a correlation between the perceived
consonance of a musical chord and the types of sum and difference frequencies that are
produced by those tones. In each of the cases, as consonance decreased, the number of
overall sum and difference frequencies created decreased and the number of sum and
difference frequencies that reinforced the pitches in the original chord also decreased.
Since the differences between chords formed using the equal temperament system and
chords formed using the just intonation system were looked at, we can also conclude
from our results that if as the number of frequencies of sum and difference pitches that
match up to frequencies of actual pitches increases, consonance increases, then the just
intonation system of tuning provides more consonant tunings of chords than the equal
temperament system of tuning does.
One inference that can be derived from our data is that since the frequency that
relates to the pitch a major seventh (11 half-steps) above the root note often appeared in
the sum and difference frequencies, this could explain why seventh chords were
developed and occur frequently in music. Another inference that could be made is that
the switch from the just intonation system to the equal temperament system facilitated
the development of atonal music, or music that does not conform to our standard
notions of a scale or key. This inference is supported by the fact that the widespread
switch in tuning systems occurred just prior to the development of atonal music and that
none of the sum and difference frequencies of a chord created using the equal
temperament system match up to actual pitches, thus desensitizing listeners of the time
to dissonance.
There are many extensions to the research presented here. One major extension
could be to look at the amplitudes of the overtone frequencies produced by different
instruments and establish a weight to the sum and difference frequencies produced by
the different combinations of overtones. This could help to show which overtones are
12
better perceived by the listener and perhaps give a more reliable way to set an upper
boundary for the study of overtones which are significant to the perception of
consonance and dissonance. If this study of amplitudes is done, one could study a
chord formed by producing different notes of a chord on different instruments and by
looking at the sum and difference frequencies between the instruments. This could be
extremely useful in the study of orchestration as a means of determining the best
voicing for an arrangement of a piece of music.
There are several other extensions. One extension would be to try to use the
sum and difference frequencies to better simulate musical sounds created electronically.
Another extension would be to try to find a set of ratios that would create a musical
scale which would have the optimal amount of sum and difference frequencies that
matched up to actual pitches. Also, pieces of music could be analyzed by not only
looking at the pitches written, but also at the sum and difference pitches perceived and
possibly derive a deeper understanding of certain works.
13
References:
Benade, Arthur H. (1990). Fundamentals of Musical Acoustics. 2nd revised ed. New
York: Dover Publications, Inc.
Benade, Arthur H. (1960). Horns, Strings, and Harmony. Garden City, NY:
Doubleday and Company, Inc.
Brown, James Ward & Ruel V. Churchill. (2001). Fourier Series and Boundary Value
Problems. 6th ed. New York: McGraw-Hill Higher Education.
Grout, Donald J. & Claude V. Palisca. (1996). A History of Western Music. 5th ed.
New York: W. W. Norton and Company.
Hecht, Eugene. (2000). Physics: Calculus. 2nd ed. Pacific Grove, CA: Brooks/Cole.
Kamien, Roger. (1996). Music: An Appreciation. 6th ed. New York: McGraw-Hill.
Rigden, John S. (1977). Physics and the Sound of Music. New York: John Wiley &
Sons.
Rossing, Thomas D., F. Richard Moore & Paul A. Wheeler. (2002). The Science of
Sound. 3rd ed. San Francisco: Addison Wesley.
14
APPENDIX 1
This table shows the relationships between the note names and their corresponding
ratio in each tuning system. In the equal temperament system, some of the notes
are combined and thus share the same ratio (Rigden, 1977).
Note Name
C
C#
Db
D
D#
Eb
E
E#
Fb
F
F#
Gb
G
G#
Ab
A
A#
Bb
B
B#
Cb
C
Pythagorean ratio Just intonation ratio
1
1
2187/2048
25/24
256/243
16/15
9/8
9/8
19683/16384
75/64
32/27
6/5
81/64
5/4
177147/131072
125/96
8192/6561
32/25
4/3
4/3
729/512
45/32
1024/729
36/25
3/2
3/2
6561/4096
25/16
128/81
8/5
27/16
5/3
59049/32768
225/128
16/9
9/5
243/128
15/8
531441/262144
125/64
4096/2187
48/25
2
2
15
Equal temperament ratio
1
2^(1/12)
2^(2/12)
2^(3/12)
2^(4/12)
2^(5/12)
2^(6/12)
2^(7/12)
2^(8/12)
2^(9/12)
2^(10/12)
2^(11/12)
2
APPENDIX 2
This appendix shows the sum and difference frequencies for each set of two notes within a
major
chord and its overtones formed using the equal temperament system of tuning
as an exmaple
of the work done to find the sum and difference frequencies. Due to the length
and number of
spreadsheets necessary to find all the data we collected, the other
spreadsheets are not printed
here. For more information on the other spreadsheets, please
contact the author.
Sums
Frequency Actual Pitch Percent Error
1100
C#/Db-6
0.79
1320
E-6
0.11
1540
1.82
1760
A-6
0.00
497.18263
B-4
0.66
774.36526
1.24
1051.54789
C-6
0.48
1328.73052
E-6
0.77
1605.91315
2.36
1883.09578
A#/Bb-6
0.98
2160.27841
2.65
549.62756
C#/Db-5
0.86
879.25512
A-5
0.08
1208.88268
2.83
1538.51024
1.92
1868.1378
A#/Bb-6
0.19
2197.76536
C#/Db-7
0.90
2527.39292
1.52
1320
E-6
0.11
1540
1.82
1760
A-6
0.00
1980
B-6
0.23
717.18263
2.61
994.36526
B-5
0.66
1271.54789
2.13
1548.73052
1.24
1825.91315
2.12
2103.09578
C-7
0.48
2380.27841
1.30
769.62756
1.87
16
Differences
Frequency Actual Pitch Percent Error
660
E-5
0.11
880
A-5
0.00
1100
C#/Db-6
0.79
1320
E-6
0.11
57.18263
1.90
334.36526
1.42
611.54789
1.75
888.73052
A-5
0.98
1165.91315
D-6
0.75
1443.09578
2.56
1720.27841
2.31
109.62756
A-2
0.34
439.25512
A-4
0.17
768.88268
1.96
1098.51024
C#/Db-6
0.93
1428.1378
2.19
1757.76536
A-6
0.13
2087.39292
C-7
0.27
440
A-4
0.00
660
E-5
0.11
880
A-5
0.00
1100
C#/Db-6
0.79
162.81737
1.23
114.36526
1.90
391.54789
G-4
0.11
668.73052
1.42
945.91315
1.44
1223.09578
1.75
1500.27841
1.35
110.37244
A-2
0.34
1099.25512
1428.88268
1758.51024
2088.1378
2417.76536
2747.39292
1540
1760
1980
2200
937.18263
1214.36526
1491.54789
1768.73052
2045.91315
2323.09578
2600.27841
989.62756
1319.25512
1648.88268
1978.51024
2308.1378
2637.76536
2967.39292
1980
2200
2420
1157.18263
1434.36526
1711.54789
1988.73052
2265.91315
2543.09578
2820.27841
1209.62756
1539.25512
1868.88268
2198.51024
2528.1378
2857.76536
3187.39292
2420
2640
1377.18263
1654.36526
C#/Db-6
A-6
C-7
A-6
B-6
C#/Db-7
A#/Bb-5
F#/Gb-6
A-6
B-5
E-6
G#/Ab-6
B-6
E-7
F#/Gb-7
B-6
C#/Db-7
B-6
F-7
A#/Bb-6
C#/Db-7
E-7
G#/Ab-6
0.86
2.24
0.08
0.23
2.83
1.69
1.82
0.00
0.23
0.79
0.52
2.48
0.78
0.49
2.30
1.13
1.41
0.19
0.06
0.75
0.15
1.78
0.03
0.25
0.23
0.79
2.85
1.51
2.61
2.83
0.66
2.14
2.13
0.94
2.88
1.87
0.23
0.86
1.55
2.24
1.61
2.85
0.11
1.43
0.41
219.25512
548.88268
878.51024
1208.1378
1537.76536
1867.39292
220
440
660
880
382.81737
105.63474
171.54789
448.73052
725.91315
1003.09578
1280.27841
330.37244
0.74488
328.88268
658.51024
988.1378
1317.76536
1647.39292
220
440
660
602.81737
325.63474
48.45211
228.73052
505.91315
783.09578
1060.27841
550.37244
220.74488
108.88268
438.51024
768.1378
1097.76536
1427.39292
220
440
822.81737
545.63474
17
A-3
C#/Db-5
A-5
A#/Bb-6
A-3
A-4
E-5
A-5
E-4
#N/A
E-4
E-5
B-5
E-6
G#/Ab-6
A-3
A-4
E-5
G-5
C#/Db-5
A-3
A-4
C#/Db-6
A-3
A-4
G#/Ab-5
0.34
1.00
0.17
2.77
1.96
0.15
0.00
0.00
0.11
0.00
2.40
1.71
1.79
1.95
1.94
1.53
2.79
0.23
#N/A
0.23
0.11
0.04
0.06
0.84
0.00
0.00
0.11
2.57
1.23
1.13
1.90
2.38
0.11
1.30
0.73
0.34
1.03
0.34
2.06
1.00
2.14
0.00
0.00
0.95
1.60
1931.54789
2208.73052
2485.91315
2763.09578
3040.27841
1429.62756
1759.25512
2088.88268
2418.51024
2748.1378
3077.76536
3407.39292
2860
1597.18263
1874.36526
2151.54789
2428.73052
2705.91315
2983.09578
3260.27841
1649.62756
1979.25512
2308.88268
2638.51024
2968.1378
3297.76536
3627.39292
1817.18263
2094.36526
2371.54789
2648.73052
2925.91315
3203.09578
3480.27841
1869.62756
2199.25512
2528.88268
2858.51024
3188.1378
3517.76536
3847.39292
831.54789
1108.73052
1385.91315
1663.09578
C#/Db-7
D#/Eb-7
A-6
C-7
A#/Bb-6
F#/Gb-7
G#/Ab-6
B-6
E-7
F#/Gb-7
G#/Ab-7
C-7
D-7
E-7
A#/Bb-6
C#/Db-7
A-7
G#/Ab-5
C#/Db-6
F-6
G#/Ab-6
2.28
0.40
0.12
1.11
2.64
2.29
0.04
0.20
2.86
1.66
1.89
2.49
2.31
1.83
0.52
2.72
2.48
2.55
0.78
1.91
0.70
0.19
1.75
0.06
0.28
0.75
2.81
2.61
0.06
0.94
0.44
1.16
2.10
1.14
0.27
0.83
1.58
2.26
1.64
0.06
2.69
0.11
0.00
0.79
0.11
268.45211
8.73052
285.91315
563.09578
840.27841
770.37244
440.74488
111.11732
218.51024
548.1378
877.76536
1207.39292
220
1042.81737
765.63474
488.45211
211.26948
65.91315
343.09578
620.27841
990.37244
660.74488
331.11732
1.48976
328.1378
657.76536
987.39292
1262.81737
985.63474
708.45211
431.26948
154.08685
123.09578
400.27841
1210.37244
880.74488
551.11732
221.48976
108.1378
437.76536
767.39292
277.18263
554.36526
831.54789
1108.73052
18
#N/A
A-4
A-2
A-3
A-5
A-3
C-6
C-2
D#/Eb-5
B-5
E-5
E-4
#N/A
E-4
E-5
B-5
B-5
D#/Eb-3
B-2
A-5
C#/Db-5
A-3
A-4
C#/Db-4
C#/Db-5
G#/Ab-5
C#/Db-6
2.54
#N/A
2.71
1.55
1.15
1.77
0.17
1.01
0.68
1.14
0.25
2.71
0.00
0.35
2.40
1.11
1.71
0.77
1.79
0.32
0.26
0.23
0.45
#N/A
0.45
0.23
0.04
1.45
0.22
1.41
2.02
0.96
0.30
2.07
2.82
0.08
0.59
0.67
1.72
0.51
2.16
0.00
0.00
0.11
0.00
1940.27841
2217.46104
606.81019
936.43775
1266.06531
1595.69287
1925.32043
2254.94799
2584.57555
1385.91315
1663.09578
1940.27841
2217.46104
2494.64367
883.99282
1213.62038
1543.24794
1872.8755
2202.50306
2532.13062
2861.75818
1940.27841
2217.46104
2494.64367
2771.8263
1161.17545
1490.80301
1820.43057
2150.05813
2479.68569
2809.31325
3138.94081
2494.64367
2771.8263
3049.00893
1438.35808
1767.98564
2097.6132
2427.24076
2756.86832
3086.49588
3416.12344
3049.00893
3326.19156
1715.54071
C#/Db-7
A#/Bb-5
F-6
G#/Ab-6
C#/Db-7
D#/Eb-7
A-5
A#/Bb-6
C#/Db-7
C#/Db-7
D#/Eb-7
F-7
F#/Gb-6
D#/Eb-7
F-7
G-7
D#/Eb-7
F-7
A-6
C-7
G#/Ab-7
1.82
0.00
2.55
0.44
1.70
1.74
2.61
1.66
2.03
0.79
0.11
1.82
0.00
0.23
0.45
2.55
1.60
0.44
0.68
1.70
2.37
1.82
0.00
0.23
0.79
1.16
0.73
2.43
2.65
0.38
0.55
0.09
0.23
0.79
2.85
2.88
0.45
0.22
2.55
1.34
1.60
2.74
2.85
0.11
2.59
1385.91315
1663.09578
52.44493
382.07249
711.70005
1041.32761
1370.95517
1700.58273
2030.21029
277.18263
554.36526
831.54789
1108.73052
1385.91315
224.7377
104.88986
434.51742
764.14498
1093.77254
1423.4001
1753.02766
277.18263
554.36526
831.54789
1108.73052
501.92033
172.29277
157.33479
486.96235
816.58991
1146.21747
1475.84503
277.18263
554.36526
831.54789
779.10296
449.4754
119.84784
209.77972
539.40728
869.03484
1198.6624
277.18263
554.36526
1056.28559
19
F-6
G#/Ab-6
G#/Ab-1
C-6
C#/Db-4
C#/Db-5
G#/Ab-5
C#/Db-6
F-6
G#/Ab-2
A-6
C#/Db-4
C#/Db-5
G#/Ab-5
C#/Db-6
F#/Gb-6
C#/Db-4
C#/Db-5
G#/Ab-5
G-5
G#/Ab-3
C#/Db-4
C#/Db-5
C-6
0.79
0.11
1.01
2.60
1.86
0.50
1.89
2.31
2.69
0.00
0.00
0.11
0.00
0.79
2.11
1.01
1.26
2.60
1.37
1.86
0.40
0.00
0.00
0.11
0.00
1.60
1.35
1.13
1.42
1.72
2.48
0.28
0.00
0.00
0.11
0.63
2.11
2.76
1.01
2.77
1.26
2.00
0.00
0.00
0.93
2045.16827
2374.79583
2704.42339
3034.05095
3363.67851
3693.30607
3603.37419
1992.72334
2322.3509
2651.97846
2981.60602
3311.23358
3640.86114
3970.4887
2269.90597
2599.53353
2929.16109
3258.78865
3588.41621
3918.04377
4247.67133
988.88268
1318.51024
1648.1378
1977.76536
2307.39292
2637.02048
1648.1378
1977.76536
2307.39292
2637.02048
2966.64804
2307.39292
2637.02048
2966.64804
3296.2756
2966.64804
3296.2756
3625.90316
3625.90316
3955.53072
4285.15828
497.18263
994.36526
1491.54789
A#/Bb-7
B-6
E-7
F#/Gb-7
G#/Ab-7
B-7
B-7
B-5
E-6
G#/Ab-6
B-6
E-7
G#/Ab-6
B-6
E-7
F#/Gb-7
E-7
F#/Gb-7
G#/Ab-7
F#/Gb-7
G#/Ab-7
B-7
B-4
B-5
F#/Gb-6
2.34
1.07
2.49
2.44
1.23
0.97
2.31
0.86
1.16
0.56
0.73
0.34
2.43
0.49
2.31
1.44
1.05
1.95
1.91
0.84
1.45
0.11
0.00
0.79
0.11
1.82
0.00
0.79
0.11
1.82
0.00
0.23
1.82
0.00
0.23
0.79
0.23
0.79
2.85
2.85
0.11
2.31
0.66
0.66
0.78
726.65803
397.03047
67.40291
262.22465
591.85221
921.47977
277.18263
1333.46822
1003.84066
674.2131
344.58554
14.95798
314.66958
644.29714
1610.65085
1281.02329
951.39573
621.76817
292.14061
37.48695
367.11451
329.62756
659.25512
988.88268
1318.51024
1648.1378
1977.76536
329.62756
659.25512
988.88268
1318.51024
1648.1378
329.62756
659.25512
988.88268
1318.51024
329.62756
659.25512
988.88268
329.62756
659.25512
329.62756
57.18263
114.36526
171.54789
20
C-4
D-5
C#/Db-4
#N/A
D#/Eb-5
D-4
F#/Gb-4
E-4
E-5
B-5
E-6
G#/Ab-6
B-6
E-4
E-5
B-5
E-6
G#/Ab-6
E-4
E-5
B-5
E-6
E-4
E-5
B-5
E-4
E-5
E-4
1.83
1.27
2.81
0.23
0.76
1.18
0.00
1.12
1.60
2.22
1.35
#N/A
1.13
2.32
2.65
2.85
2.00
0.08
0.52
2.08
0.78
0.00
0.00
0.11
0.00
0.79
0.11
0.00
0.00
0.11
0.00
0.79
0.00
0.00
0.11
0.00
0.00
0.00
0.11
0.00
0.00
0.00
1.90
1.90
1.79
1988.73052
2485.91315
2983.09578
3480.27841
549.62756
1099.25512
1648.88268
2198.51024
2748.1378
3297.76536
3847.39292
606.81019
1213.62038
1820.43057
2427.24076
3034.05095
3640.86114
4247.67133
B-6
D#/Eb-7
F#/Gb-7
C#/Db-5
C#/Db-6
G#/Ab-6
C#/Db-7
G#/Ab-7
0.66
0.12
0.78
1.14
0.86
0.86
0.75
0.86
1.66
0.75
2.69
2.55
2.55
2.43
2.55
2.44
2.43
6.98
228.73052
285.91315
343.09578
400.27841
109.62756
219.25512
328.88268
438.51024
548.1378
657.76536
767.39292
52.44493
104.88986
157.33479
209.77972
262.22465
314.66958
367.11451
21
A-2
A-3
E-4
A-4
E-5
G#/Ab-1
G#/Ab-2
G#/Ab-3
C-4
F#/Gb-4
1.90
2.71
1.79
2.07
0.34
0.34
0.23
0.34
1.14
0.23
2.16
1.01
1.01
1.13
1.01
0.23
1.13
0.78