In details

Mathematics and Music: In Search of Harmony (Part 2)

Any vibratory movement of air at the ear entrance corresponds to a musical tone that can always and uniquely be displayed as a sum of an infinite number of simple vibratory movements corresponding to the partial sounds of this musical tone. The first components in the Harmonic Series correspond to the frequencies associated with the first Fourier Series terms which thus determine ratios of small integers related to Pythagorean consonances, both a string and air columns in wind instruments have the characteristic of vibrating not just as a whole, but still simultaneously as two halves, three thirds, four quarters and so on.

From the mathematical point of view, it is observed that the strength of each harmonic will contribute to the construction of the form of periodic vibration that relates to the timbre of sound.

In musical instruments, harmonics are exploited and used in a variety of ways, wind instruments obtain harmonics of a particular sound by blowing them more intensely, while stringers can make a single string vibrate in corresponding sections. at certain harmonics by lightly tapping at maximum points that inhibit lower harmonics.

In almost all the peoples of antiquity there are manifestations of these two fields in separate. The conquering power of music is already expressed in Greek mythology in Orpheus, whose song accompanied by lyre supported the rivers, tamed beasts and moved stones. Mathematics has also been present since ancient times, for example in the counting of things. The interaction between these areas becomes strongly manifested from the need to equate and solve consonance problems, in the sense of seeking scientific foundations capable of justifying such concept.

Concerning the organization of musical scales, it has occurred in different ways in different peoples and times, but with some aspects in common. The Greeks developed the tetracords and then scales with seven tones.

Musical theorists such as Pythagoras, Arquitas, Aristoxenus, Erastosthenes devoted themselves to constructing scales by developing different affinity criteria. For example, by valuing the perfect fifth intervals as well as using only numbers from 1 to 4 to obtain fractions of the string to generate the scale notes, Pythagoras set up a pitch using fifth paths to obtain the scale notes.

Arquitas builds his scale based on fractions of the string resulting from harmonic and arithmetic averages of those found by Pythagoras in the monochord experiment. Erastosthenes elaborated the differentiation between arithmetically calculated intervals in the Aristoxen manner, from intervals calculated by reason.

2.1. The Monochord Experiment and Music at the Pythagorean School

The first signs of marriage between mathematics and music came in the sixth century BC when Pythagoras, through experiments with monochromatic sounds, made one of his most beautiful discoveries, which gave birth to the fourth branch of mathematics at the time: music. .

The principal musical theorists of the Pythagorean school were Pythagoras and Philolaus in the pre-classical period, as well as Architas, Aristoxen and Aristotle in the classical period.

Possibly invented by Pythagoras, the monochord is an instrument consisting of a single string extended between two easels fixed on a board or table and also has a movable easel placed under the extended string and the musical pitch of the sound emitted when played. Pythagoras sought length ratios - ratios of integers - that produced certain sound intervals. He continued his experiments by investigating the relationship between the length of a vibrating string and the musical tone it produced. This Pythagorean experiment is the first experiment in the history of science to isolate any device for artificially observing phenomena.

Pythagoras noted that pressing down on a point at ¾ the length of the rope relative to its end - which is equivalent to reducing it to ¾ its original size - and then tapping it a quarter above the pitch of the string. whole string. Pressed at 2/3 of the original size of the string, it was heard a fifth higher and ½ was the octave of the original sound.

From this experience, the intervals are called Pythagorean consonances. So if the original length of the string is 12 and if we reduce it to 9, we will hear the fourth to 8, the fifth to 6, the eighth.