Mathematics and Music: In Search of Harmony (part 9)

5. Math in the Emergence of Temperament

Temperament in the general sense means a scale in which all or almost all ranges are slightly inaccurate but not distorted. They end up valuing certain intervals over others, in the sense that the former correspond to the same frequency ratios as the others.

The Pythagorean temperament is the oldest temperament used in the West. In the Renaissance and early Baroque, unequal temperaments prospered in which the highest natural thirds were given priority. Still other temperaments arose in this period that were surpassed as the music extended to all shades.

At the end of the Middle Ages and the beginning of the Renaissance, music went down paths that summoned its detachment from melodic conceptions towards the conquest of a mainly harmonious character. The trajectory of western music led the music to freedom of mosulation not only for close tones, but for different tonal scenarios, that is, the compositions of that time, intimated the freedom of tone transposition.

Temperament did not occur as a sudden process, but developed in several ways. In the early sixteenth century, as attempts to fill natural gaps in a relatively symmetrical manner always encountered the fatal coma at some point (a term used to refer to the small remainder that always occurred when trying to fit pure intervals into an integer). octave), achieving harmonic perfection only at restricted intervals, some kind of partial temperament was dominated especially in the key instruments.

The Pythagoras and Zarlino ranges made it possible to construct slightly asymmetrical scales that were unable to fully respond to the cultural needs of the late Renaissance and early Baroque, which would, from the point of view of the above-mentioned difficulties, establish a liberating support for the song called equal temperament.

From the mathematical point of view, the problem was to find a factor f corresponding to the semitone interval that after multiplying 12 times a frequency f0 corresponding to a given note, reached its octave for frequency 2. Based on geometric progression - octave = 2/1; semitone = 2 1/12 -, Euler researched a tuning system that allowed composers to modulate to and from any of the 12 tonal centers (corresponding to the 12 notes of the temperate scale, # = reb, re, re # = mib , fa, fa # = solb, sun, sun # = lab, la, la # = sib, si) without distortions generated by corresponding intervals which were hitherto asymmetric at different scales. From the mathematical point of view, the problem was represented by the following equation:

f0.f.f.f.f… f = f0f12 = 2.f0

After some simple algebraic operations, it is not difficult to conclude that the value of f should assume a value of 2 1/12.

Therefore, the grades on this scale have the following frequency relationships with the starting grade:

Of re mi do Sun there si of
1 21/16 21/3 25/12 27/12 23/4 211/12 2

At this point, it would also be worth raising the question why choose 12 notes out of 300 different sounds within an octave that can be discriminated by the trained human ear. Probably, the division proceeded in this way out of respect for a certain continuity on the Greek scale, whose construction process - path of farms - was presented in such a way that the path outlined there assumed, less than octaves, maximum approximation of the note. after 12 cycles, referring to the 12 notes.

Thus we realize that if the frequency relationship is simple, then the sound of the corresponding range is beautiful, which naturally suggests doubt about the reciprocal as well as many other discussions.