**Music and Mathematics**

Ancient Greece

In ancient Greece, music and mathematics were strongly connected. In the curriculum of the Pythagorean School, music was placed on the same level as arithmetic, geometry, and astronomy.

The Greeks perceived music as a quantitative science of sound and harmony

Concepts developed by the ancient Greeks include:

a tuning system developed by Pythagoras

a system of consonance and dissonance

octave model scale systems

musical terminology

"The important thing to realize is that numbers and math are not cold and lifeless, and that music, which is a tangible incarnation of numbers, reflects in its beauty and emotion some of the beauty and emotion in the world of mathematics" -Harvey Reid

The main mathematical difference between ancient and modern scales on a chord is that the ancient scales concentrated on three harmonies provided by simple ratios, whereas the even-tempered scale of today is based on a logarithmic division of the chord (the n-th semitone is obtained by the n-th power of the 12-th root of 2).

The key to harmonic ratios is hidden in the famous Pythagorean tetractys, a pyramid made up of the first four numbers in which their proportions reveal the intervals of the octave, the diapente, and the diatessaron.

Pythagoras applied his newly found law of harmonic intervals to all the phenomena of nature, even going so far as to demonstrate the harmonic relationship of the planets, constellations, and elements to each other.

Even-tempering makes all the notes of the scale slightly out of tune, and divides the error equally among the scale notes to allow complex chords and key changes.

Even-tempered intervals cannot be expressed as a ratio, for the 12th root of 2 is an irrational number, and so musicians learn to tune an instrument by training their ear

The human ear prefers the 'pure' Pythagorean intervals, however, a tempered scale is necessary to produce complex, chordal music.

The integer 2 is represented with the octave. The integer 3 is represented as the musical 5th. the next note in the harmonic series, corresponding to the number 4 is a second octave (2 times 2). The number 5 produces a new note, called the musical 3rd. The number 6 produces a note an octave higher than the 5th and 7 produces the first dissonant note in the harmonic series. if you build a musical system out of these integer notes, it is called the Pythagorean scale.

Pythagorean intervals and their derivations:

Interval Ratio Derivation

Unison 1:1 Unison 1:1

Minor Second 256:243 Octave - M7

Major Second 9:8 (3:2)^2

Minor Third 32:27 Octave - M6

Major Third 81:64 (3:2)^4

Fourth 4:3 Octave - 5

Augmented Fourth 729:512 (3:2)^6

Fifth 3:2 (3:2)^1

Minor Sixth 128:81 Octave - M3

Major Sixth 27:16 (3:2)^3

Minor Seventh 16:9 Octave - M2

Major Seventh 243:128 (3:2)^5

Octave 2:1 Octave 2:1

The following table shows how the standard intervals of Pythagorean tuning are derived primarily from superimposed fifths, thus having ratios which are powers of 3:2

Pythagorean tuning defines all notes and intervals of a scale from a series of pure fifths with a ratio of 3:2.

The one flaw of this system is that the fourth or fifth between the Eb-G# will be out of tune. This is because 12 perfect fifths do not round off to an even octave, but surpass it by a small ratio known as a Pythagorean comma.

**Pythagorean Tuning**

Consonance and Dissonance

The ancient Greeks discovered that to a note with a given frequency only those other notes whose frequencies were integer multiples of the first could be properly combined

Example: if a note with a frequency of 220 Hz were played, notes of the frequencies of 440, 660, 880, and so on would sound best when played together with the first

Octaves

The Greeks saw in the octave a 'cyclic identity.' The following ratios build the musical fifth (2:3), fourth (3:4), major third (4:5), and minor third (5:6), which are all significant in the creation of chords.

The system of dividing an octave into twelve equal semi-tones was introduced by Johann Sebastian Bach.

**The Pythagorean Scale**

Fibonacci and the Golden Ratio

The sequence of Fibonacci ratios converges to a constant limit, called the golden ratio (0.61803398...).

This golden ratio can be applied to various areas of the arts, especially painting and photography, where the pictures length or width are divided following the golden proportion.

In music, Fibonacci ratios are either used to generate rhythmic changes or to develop a melody line

In most of Mozart's piano sonatas, the relation between the exposition and the development and recapitulation (a section of musical sonata form where the exposition is repeated in an altered form and the development is concluded) conforms to the golden proportion.

The sensations in solving a mathematical problem seem to be similar to those appearing when performing a musical work.

"...children playing the piano often show improved reasoning skills like those applied in solving jigsaw puzzles, playing chess, or conducting mathematical deductions" (Motluk, 1997:17).

Relating to the Renaissance, Baroque, Classical, and Romantic periods, there are many similarities between the evolution of mathematics and the evolution of music.

Connecting Math and Music

http://videos.howstuffworks.com/science-channel/29645-what-the-ancients-knew-pythagoras-and-chords-video.htm

http://www.maa.org/audio%20clips/stewartlecture2010.mp3

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