PRESENTATION 2

THE DEVIL’S INTERVAL

In music theory, the tritone is defined as a musical interval composed of three adjacent whole
tones. For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can
be decomposed into the three adjacent whole tones F–G, G–A, and A–B. According to this
definition, within a diatonic scale there is only one tritone for each octave. For instance, the
above-mentioned interval F–B is the only tritone formed from the notes of the C major scale.
A tritone is also commonly defined as an interval spanning six semitones. According to this
definition, a diatonic scale contains two tritones for each octave. For instance, the above-
mentioned C major scale contains the tritones F–B (from F to the B above it, also called
augmented fourth) and B–F (from B to the F above it, also called diminished fifth,
semidiapente, or semitritonus). In twelve-equal temperament, the tritone divides the octave
exactly in half.

The Science behind it:

In classical music, the tritone is a harmonic and melodic dissonance and is important in the
study of musical harmony. The tritone can be used to avoid traditional tonality: "Any tendency
for a tonality to emerge may be avoided by introducing a note three whole tones distant from
the key note of that tonality." Contrarily, the tritone found in the dominant seventh chord helps
establish the tonality of a composition. These contrasting uses exhibit the flexibility, ubiquity,
and distinctness of the tritone in music.

The condition of having tritones is called tritonia; that of having no tritones is atritonia. A
musical scale or chord containing tritones is called tritonic; one without tritones is atritonic

Since a chromatic scale is formed by 12 pitches (each a semitone apart from its neighbors), it
contains 12 distinct tritones, each starting from a different pitch and spanning six semitones.
According to a complex but widely used naming convention, six of them are classified as
augmented fourths, and the other six as diminished fifths.
Under that convention, a fourth is an interval encompassing four staff positions, while a fifth
encompasses five staff positions (see interval number for more details). The augmented fourth
(A4) and diminished fifth (d5) are defined as the intervals produced by widening the perfect
fourth and narrowing the perfect fifth by one chromatic semitone. They both span six
semitones, and they are the inverse of each other, meaning that their sum is exactly equal to
one perfect octave (A4 + d5 = P8). In twelve-tone equal temperament, the most commonly
used tuning system, the A4 is equivalent to a d5, as both have the size of exactly half an octave.
In most other tuning systems, they are not equivalent, and neither is exactly equal to half an
octave.

Any augmented fourth can be decomposed into three whole tones. For instance, the interval F–
B is an augmented fourth and can be decomposed into the three adjacent whole tones F–G, G–
A, and A–B.

It is not possible to decompose a diminished fifth into three adjacent whole tones. The reason
is that a whole tone is a major second, and according to a rule explained elsewhere, the
composition of three seconds is always a fourth (for instance, an A4). To obtain a fifth (for
instance, a d5), it is necessary to add another second. For instance, using the notes of the C
major scale, the diminished fifth B–F can be decomposed into the four adjacent intervals

B–C (minor second), C–D (major second), D–E (major second), and E–F (minor
second).

Using the notes of a chromatic scale, B–F may be also decomposed into the four adjacent
intervals

B–C♯ (major second), C♯–D♯ (major second), D♯–E♯ (major second), and E♯–F♮
(diminished second).

Notice that the latter diminished second is formed by two enharmonically equivalent notes (E♯
and F♮). On a piano keyboard, these notes are produced by the same key. However, in the
above-mentioned naming convention, they are considered different notes, as they are written
on different staff positions and have different diatonic functions within music theory.

Why is it the Devil’s note?

The tritone is a restless interval, classed as a dissonance in Western music from the early
Middle Ages through to the end of the common practice period. This interval was frequently
avoided in medieval ecclesiastical singing because of its dissonant quality. The first explicit
prohibition of it seems to occur with the development of Guido of Arezzo's hexachordal system,
who suggested that rather than make B♭ a diatonic note, the hexachord be moved and based on
C to avoid the F–B tritone altogether. Later theorists such as Ugolino d'Orvieto and Tinctoris
advocated for the inclusion of B♭. From then until the end of the Renaissance the tritone was
regarded as an unstable interval and rejected as a consonance by most theorists.
The name diabolus in musica ("the Devil in music") has been applied to the interval from at
least the early 18th century, though its use is not restricted to the tritone. Andreas Werckmeister
cites this term in 1702 as being used by "the old authorities" for both the tritone and for the
clash between chromatically related tones such as F♮ and F♯, and five years later likewise calls
"diabolus in musica" the opposition of "square" and "round" B (B♮ and B♭, respectively)
because these notes represent the juxtaposition of "mi contra fa". Johann Joseph Fux cites the
phrase in his seminal 1725 work Gradus ad Parnassum, Georg Philipp Telemann in 1733
describes, "mi against fa", which the ancients called "Satan in music"—and Johann Mattheson,
in 1739, writes that the "older singers with solmization called this pleasant interval 'mi contra
fa' or 'the devil in music'." Although the latter two of these authors cite the association with the
devil as from the past, there are no known citations of this term from the Middle Ages, as is
commonly asserted. However Denis Arnold, in the New Oxford Companion to Music, suggests
that the nickname was already applied early in the medieval music itself:

It seems first to have been designated as a "dangerous" interval when Guido of Arezzo
developed his system of hexachords and with the introduction of B flat as a diatonic note, at
much the same time acquiring its nickname of "Diabolus in Musica" ("the devil in music").

That original symbolic association with the devil and its avoidance led to Western cultural
convention seeing the tritone as suggesting "evil" in music. However, stories that singers were
excommunicated or otherwise punished by the Church for invoking this interval are likely
fanciful. At any rate, avoidance of the interval for musical reasons has a long history, stretching
back to the parallel organum of the Musica Enchiriadis. In all these expressions, including the
commonly cited "mi contra fa est diabolus in musica", the "mi" and "fa" refer to notes from
two adjacent hexachords. For instance, in the tritone B–F, B would be "mi", that is the third
scale degree in the "hard" hexachord beginning on G, while F would be "fa", that is the fourth
scale degree in the "natural" hexachord beginning on C.

Later, with the rise of the Baroque and Classical music era, composers accepted the tritone, but
used it in a specific, controlled way—notably through the principle of the tension-release
mechanism of the tonal system. In that system (which is the fundamental musical grammar of
Baroque and Classical music), the tritone is one of the defining intervals of the dominant-
seventh chord and two tritones separated by a minor third give the fully diminished seventh
chord its characteristic sound. In minor, the diminished triad (comprising two minor thirds,
which together add up to a tritone) appears on the second scale degree—and thus features
prominently in the progression iio–V–i. Often, the inversion iio6 is used to move the tritone to
the inner voices as this allows for stepwise motion in the bass to the dominant root. In three-
part counterpoint, free use of the diminished triad in first inversion is permitted, as this
eliminates the tritone relation to the bass.
 PRESENTATION 1
KEPLER’S LAWS OF PLANETARY MOTION
In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion
of planets around the Sun.

Figure 1: Illustration of Kepler's three laws with two planetary orbits.

1. The orbits are ellipses, with focal points F1 and F2 for the first planet and F1 and F3 for the
second planet. The Sun is placed in focal point F1.
2. The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover
segment A1 is equal to the time to cover segment A2.

3. The total orbit times for planet 1 and planet 2 have a ratio .

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal
intervals of time.
3. The square of the orbital period of a planet is directly proportional to the cube of the
semi-major axis of its orbit.

Most planetary orbits are nearly circular, and careful observation and calculation are required
in order to establish that they are not perfectly circular. Calculations of the orbit of
Marsindicated an elliptical orbit. From this, Johannes Kepler inferred that other bodies in the
Solar System, including those farther away from the Sun, also have elliptical orbits.

Kepler's work (published between 1609 and 1619) improved the heliocentric theory of
Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits
rather than circular orbits with epicycles.

Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System
to a good approximation, as a consequence of his own laws of motion and law of universal
gravitation.
Kepler's laws improved the model of Copernicus. If the eccentricities of the planetary orbits
are taken as zero, then Kepler basically agreed with Copernicus:

1. The planetary orbit is a circle

2. The Sun is at the center of the orbit
3. The speed of the planet in the orbit is constant

The eccentricities of the orbits of those planets known to Copernicus and Kepler are small, so
the foregoing rules give fair approximations of planetary motion, but Kepler's laws fit the
observations better than does the model proposed by Copernicus.

Kepler's corrections are not at all obvious:

1. The planetary orbit is not a circle, but an ellipse.

2. The Sun is not at the center but at a focal point of the elliptical orbit.
3. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but
the area speed is constant.

The eccentricity of the orbit of the Earth makes the time from the March equinox to the
September equinox, around 186 days, unequal to the time from the September equinox to the
March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane
through the Sun parallel to the equator of the Earth cuts the orbit into two parts with areas in a
186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately

which is close to the correct value (0.016710219) (see Earth's orbit).

The calculation is correct when perihelion, the date the Earth is closest to the Sun, falls on a
solstice. The current perihelion, near January 3, is fairly close to the solstice of December 21
or 22.

KEPLER’S LAWS:

 All planets move in elliptical orbits, with the sun at one focus.

 A line that connects a planet to the sun sweeps out equal areas in equal times.

 The square of the period of any planet is proportional to the cube of the semimajor axis
of its orbit.