25 Music
25.1 EvenTempered
Scale
The piano has 88 keys that include 7 octaves and 4 extra keys. Each
octave on the musical scale contains 7 white keys and 5 black keys. The notes
increase in whole and in other cases half step multiples. The notes in
brackets are black piano keys and are a half step above the previous note.
A [A+=B] B C
[C+=D] D [D+=E] E F [F+=G] G
[G+=A]
Sometimes there are half steps between white piano keys as in B&C and
E&F and between all white to black keys. The +’s denote sharps and
the –‘s flats. The note at each octave has double the frequency of
the previous octave and is thus a harmonic. We add suffix numbers to the notes
to specify a particular octave. For example, A4 is harmonic to A3 in the
following table. A wave is harmonic to a lower frequency wave if the lower
frequency waveform’s zero amplitude points are also of amplitude zero in
the higher frequency wave.
Program 251: Three Harmonics
Here we plot the sum the 1^{st},
2^{nd}, and 3^{rd} harmonics:
ezplot('sin(x)+sin(2*x)+sin(3*x)')
To play the harmonics in Matlab, one can use
the sound function, which expects normalized input between –1
≤ y
≤
1.
x = .1:.1:10000;
y = 1/3 .*
(sin(x)+sin(2*x)+sin(3*x));
sound(y)
For a higher sounding
harmonic
y =
sin(x)+sin(3*x)+sin(5*x);
Now make the third term the 2^{nd}
harmonic of the second term and one will hear the double harmonic which has a
greater gap sound within.
y =
sin(x)+sin(3*x)+sin(6*x);
Contains higher
overtones.
y =
sin(x)+sin(6*x)+sin(8*x);
x = .1:.1:10000;
y = 1/3 .*
(sin(3*x)+sin(6*x)+sin(12*x));
sound(y)
To obtain a dial tone like sound, simply add
more and more sin waves:
y =
1/5.*sin(3.2332*x)+sin(7.23423*x)+sin(13.32523*x)+sin(8.2324*x)+sin(2.424*x));
sound(y)
One can superimpose sounds in Matlab, by
running multiple ‘sound(y)’s back to back and then listen to them
terminate onebyone until a single sound is heard.
Now we
can plot the repeating pattern for the harmonics:
Figure 251: Three Harmonics
Harmonics have a repetitive pattern too short to distinguish any
differences while listening. By mixing nonharmonics, we obtain chords that may
repeat over much longer periods so that one may hear cycles. By adding many
additional nonharmonics we can get dial tone sounds. Though there is no period
, one can see evenly spaced peaks which produce a repetitive like sound, a dial
tone.
Program 252: Creating Stereo Sound
y1 = 1/3 .* (sin(2*x)+sin(4*x)+sin(8*x));
y2 = 1/3 .* (sin(3*x)+sin(6*x)+sin(12*x));
y = [y1' y2'];
sound(y)
Table 251: Frequencies of the Third Octave
Note

Frequency (Hertz)

A3

220

B3

246.9

C3

261.6

D3

293.7

E3

329.6

F3

349.2

G3

392

A4

440

How can we calculate these steps? We denote a waveform with
frequency f0 and its harmonic with frequency f1. By definition:
f1/f0=2.
We also note that all of the frequencies between will have the same
multiple (denoted by h) from the previous frequency. Say f0 is A3 than A3+ =
A3*h. Now to the computation of h:
Equation 251: Step Frequency Multiplier for the EvenTempered
Scale
f1 = h * h * h * h * h * h * h * h * h * h * h * h *
f0
f1/f0 = h^12
2 = h^12
h = 12^{th} root of 2 =
1.059463094
The half step always has a fixed ratio in the EvenTempered Scale
that is h. For example C3/B3 = 261.6/246.9 =
1.059463094.^{[2616]} Now we can
fill in the frequencies of the sharps and flats as well:
Table 252: Sharp and Flat Frequencies of the Third
Octave
Note

Frequency (Hertz)

A3

220

A3+B3

233.1

B3

246.9

C3

261.6

C3+D3

277.2

D3

293.7

D3+E3

311.1

E3

329.6

F3

349.2

F3+G3

370.0

G3

392.0

G3+A4

415.3

A4

440

25.2 Pythagorean
Diatonic Scale
The Diatonic scale differs from the Equal tempered scale in that the
multiples between frequencies are fractions. Pythagoras developed the scale
based on fractions while studying the tonal relationships between string lengths
on a monochord. He did not consider a multiple of length based on an irrational
numbers. A frequency of the wave of a string will change inversely with its
length. Hence given two strings, the second string half the length of the
first; the frequency of the second string will be twice that of the first.
The frequencies of the First, Fourth, Fifth, and Octave are known
through observation. The derivation of the rest follows based on the Fifth.
D2 = 3/2 f * 3/2 = 9/4 f
D1 = 9/4 f * ½ = 9/8 f
C2 = 4/3
f * 3/2 f = 2f
A3 = 9/4f * 3/2 = 27/8 f
A2 = 27/16 f
Table 253: Diatonic Frequencies of the Octave
Note

Position

String Length Ratio

Frequency

C1

First

l

F

D1

Second

8/9 l

9/8 f

E1

Third

64/81 l

81/64 f

F1

Fourth

¾ l

4/3 f

G1

Fifth

2/3 l

3/2 f

A2

Sixth

16/27 l

27/16 f

B2

Seventh

128/243 l

243/128 f

C2

Octave

½ l

2 f

Since there are six whole tones in the space of an octave, we’d
expect 9/8 ^ 6 => 2 but it = 2.027, so the tones are not evenly distant.
Equal temperament does away with this problem.
Pythagoras found that the
octave, fifth, and forth had the most harmonious sounds. A song’s key is
the octave forward from the starting note. On the Diatonic Scale, the multiples
between the same notes may vary with Octave. Hence, the same piece of music
played in a different key tends to sound quite different unlike when playing a
piece with the EvenTempered Scale.
This problem with uneven multiples
manifests itself in other areas. For example, 12 fifths doesn’t equal 7
octaves, though they are the same key on the piano. This is known as the
Pythagorean comma.^{[2617]} This
imbalance in the frequency distance between keys means that it is impossible to
construct Pythagorean tuned piano.
The frequency of C1 = f
The frequency of C8 = 2^7 f = 128f from Octave multiplication.
The
frequency of C8 from multiples of fifths is 3/2 ^ 12 f = 129.74f
The
problem doesn’t manifest significantly when playing music in a single
octave, but when playing music across octaves one cannot for example tune the
8^{th} octave to the 7^{th} fifth since they are not compatible.
One cannot build a Pythagorean piano.
25.3 Fractals
In the kabbalistic Tree of Life, the microcosm reflects the macrocosm, the
sefirot in man reflect the sefirot of Gd. Man is created in the image of Gd.
This recursive principle is the essence of fractals, which are repeated images
within images.
Figure 252: Microcosm reflecting the Macrocosm
Tobias Cohen in Maaseh Tuviyah shows how the microcosm of man reflects the
macrocosm of a
house.^{[2618]}
“The natural world is composed of fractals that are the trails,
or visual representation, left by the dynamic (continuously moving) systems that
created them. A tree is fractal in nature – a single branch resembles the
entire tree, a smaller twig resembles the branch, even the structures of the
veins in the leaves exhibit the same branching process as the tree itself.
Granted, the patterns at different scales are not exact replicas of one another,
but the idea is the same. The patterns created by the rocks of a rugged
coastline also repeat themselves at different scales – a snail crawling
along the nooks and crannies of the rocks on the shore traverses a meandering
path similar to the one you would take on a hike along the cliffs, which in turn
is similar to the general outline of the coast viewed from an airplane. Natural
fractals reach a point where the pattern (at the biological level, anyway)
stops; however, in a true scaling fractal (for instance, the Koch snowflake),
this detail continues ad
infinitum.”^{[2619]}
25.4 Fibonacci
Ratio
The Fibonacci ratio has an aesthetic appeal to the human mind. There are
five black keys for every 8 white keys in a piano Octave. The partitioning of
great paintings is often done according to the ratio. Also the lengths of
different movements in a classical piece of music will often follow this ratio.
Equation 252: Fibonacci Ratios
1/1
1/2
2/3
3/5
5/8
8/13
133/377 = 0.618037
25.5 Television
A kabbalist should not have a TV, yet there are spiritually uplifting
programs such as ‘Little House on the
Prairie’.
^{[2616]} Math and Music
Harmonious Connections, Trudi Hammel Garland and Charity Vaughan Kahn, Dale
Seymour Publications, page
40.
^{[2617]} Math and Music,
page 62.
^{[2618]} http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2175063/
^{[2619]} Math and Music
Harmonious Connections, Trudi Hammel Garland and Charity Vaughan Kahn, Dale
Seymour Publications, page 132.