Mathematics and Music: Seeking Harmony

* Monograph presented by Larissa Suarez Peres
at the University of Greater ABC

1. Introduction: Musical Scales and Mathematical Relations

From the acoustic point of view, the sounds used for music production (except for the sounds of some percussion instruments) have certain physical characteristics, such as well-defined oscillations (frequencies) and the presence of harmonics. In this case, well-defined oscillations are understood to mean that a musical sound most often occurs sustainably (little or very much), so that its oscillation characteristic is maintained for some or many cycles, unlike noise and other nonmusical sounds.

With regard to the presence of harmonics, it should be remembered that most musical sounds do not occur only in their simplest mode of vibration (fundamental mode), but are always composed of this mode (fundamental) and others, called harmonic modes, that they are nothing more than the vibrating body also oscillating with whole multiple frequencies (x2, x3, x4, etc.) of the fundamental mode frequency.

Harmonics present in a sound are extremely important components in the musical process, both in the formation of musical scales and in musical harmony. Because of these natural characteristics, sounds with different pitches (frequencies) when set at the same time can create aesthetically different auditory sensations.

At first glance, we can understand that two sounds that maintain a full relationship between the values ​​of their fundamental frequencies will certainly result in a natural or pleasant auditory sensation, because their harmonics are in "sympathy" or "consonance". In the specific case where the fundamental frequency of one sound (f1) is twice the fundamental frequency of another (f2), the first is said to be one octave higher than the second (f1 = 2. F2).

If we want to generate two different, perfectly consonant musical sounds, they must maintain an octave ratio, where all the harmonics of the loudest sound will be in perfect harmony with the lowest sound. However, sounds generated simultaneously at some other non-octave intervals can produce a pleasing sensation to our ears, as they also contain a good deal of coincident harmonics, which is actually the so-called fifth range, which maintains a 3: 2 ratio. .

Of course, if only octave and fifth intervals were used to create sounds in music, the result would be quite poor due to the scarcity of notes. Thus, several civilizations sought to develop, scientifically and experimentally, frequency ranges within the octave range, with which to build their music. These ranges are called musical scales, and there are a variety of them, based on different criteria for the definition of notes.

Interval Relationship
minor tuesday
sixth minor
6:5 (1,200)
5:4 (1,250)
4:3 (1,333)
3:2 (1,500)
8:5 (1,600)
5:3 (1,667)
2:1 (2,000)

Consonant Ranges

In addition to the eighth and fifth, other sound ranges are also considered aesthetically consonant by most authors, and are presented in the table above. It is noteworthy that the intervals in question were represented by their mathematical relations with respect to the harmonic relationship. Take the case of the fifth interval as an example: its frequency is equal to the frequency of the third harmonic of the reference note (three times the frequency of the fundamental), and is divided by two so as to lower an octave to fall within it. eighth of the reference note, hence the 3: 2 ratio.

The harmonics of a musical note are precisely those partial sounds that make up its sonority, and the Harmonic Series of that same note is characterized by the sequence of such sounds ordered from bass to treble. The sound of an instrument or a human voice becomes brighter the higher its richness in superior harmonics, which makes us assign adjectives to the sound produced by certain instruments is directly associated with the distribution of the harmonics of that sound.

Concerning the production and use of harmonics, wind instrument performers can obtain the following and fundamental harmonics by blowing their instruments more intensely, just as string players produce harmonics by playing a string lightly at appropriate points. , which causes it to vibrate in certain sections associated with the harmonic to be highlighted.