Pythagorean proportions

If every known thing has a relationship with number, it means that number is a sharp gnosiological tool, for reality has a mathematical structure
According to tradition, Pythagoras was the first who understood the link among music, mathematics, and geometry. The scales, the tones, thre rules of harmony as we use them today were all conceived by him. Every sound is a number and every piece of music is a mathematical equation. Notes reflect the planets and music reflect their cosmic dance, which Pythagoras could hear. The entire musical construct served as a philosophical teaching meant to guide man in his development and reincarnations.
The Pythagorean philosopher searches for those relationships that connect the plane of sensible reality to the numerical structure that sustains it. The idea of a secret relationality, which tightens all things, finds a peculiar resonance in the expression of logos - the speaking entity who speaks the world into existence.
There is harmony between number and things, because there is a link between the knowledge of things and the possibility of describing them through numerical relationships, which indicate how the part is strung to the whole.
Despite the importance of Pythagoras, he is seldom quoted in the books about music. Nowadays, players are instructed on how to play their instruments blindly, but have no clue of what they are doing and why. Composers receive a technical aseptic and standardized training, which omits the why's likewise. Therefore, our society lost contact with the roots of music completely and the old tradition survives only by a few fortunate ones. We will try to remedy to this, presenting the reader with the core of the secret teachings.
At the beginning of music exploration, it was universally perceived that octaves made a scale. The issue was how to divide it. Many attempts followed, until we came to the 7 notes and 12 semitones (7-12 abbreviated). Pythagoras contributed substantially in defining the "proportions" of the scale. He had a chord with a fixed tension. Then, he applied weighs to it and let it sound. He saw that, when weighs were changed, the frequency of sound was affected. Multiples of weigh allowed the notes in the scale to form.
In some countries - notably in the east - they still divide the scale into a much higher number of notes, up to 36. Somebody says that they have a much more trained ear than we have, as they are able to distinguish a much smaller interval. However, all tend to refer to the 7-12 subdivision. Smaller intervals are mostly used in vocalizations that always revolve around the 7-12 system. In a certain way, smaller subdivisions also happen in string instruments. It is said that an A flat is not the same note as a G sharp, because one is a little waning and the other is a little rising. Strings can use this trick to confer their melodical line a more suffered and lived character.
After the establishment of the 7-12 system, it became a matter of choosing the proper order of tones and semitones. Many modes were experimented - notably major, minor, Doric, Phrygian, Lydian, Mixolydian, Aeolian. Practically, with reference to the C major scale, every other mode started with a different key. This affected the sequence of tones and semitones.
Eventually, only the major and minor modes won the test of time. The major mode has a character of determination, clarity, pomp, shine. The minor mode induces sadness. The other modes give rise to mixed feelings, usually ending up in boredom. It seems that those modes cannot sustain themselves.
Far from being only a matter of taste, there is a technical explanation of this phenomenon, which can be traced back to Pythagoras. When we play a note, not only its linked frequency is played, but also all the upper harmonics. Each harmonic has half the volume of the previous harmonic.
We can do the following experiment with a piano. We can gently press as many harmonics as we can, starting from the fundamental note (the low C), so that only the keys go down but no sound is produced. Now, we lift the finger over the fundamental and hit strongly for a moment, rising it at once again. We will hear the fundamental note for a moment, as it will be damped by the muffler felt. However, we will realize that the harmonics are still playing, despite we did not hit them. They started playing by themself. As we pressed their keys, we removed the muffler felt from them, so that they were free to do their work.
Piano Harmonics
If we repeat this experiment with non harmonics, we will see that they do not start playing by themselves. They need to be harmonics.
The timbre of an instrument is determined by the arrangement harmonics take. Some harmonics may be missing, some will be stronger and some weaker. This unique combination determines the sound of the instrument. Brass instruments are rich in higher harmonics, which confer to them that shiny, metallic, and rough character. The flute is the exception, as it has no harmonics. It plays only almost perfect sine waves. Strings have a wide range of harmonics.
Now, if we take the entire sequence of harmonics, we see that, predominantly, the major scale is built. Another way to build it is in intervals of a perfect fifth, taking into account its symmetry: F → C → G → D → A → E → B.
This is a clear mathematical explanation why the major mode is so natural to us, as it has a perfect architecture. The issue comes when we try to explain the minor mode, as the sequence of its harmonics does not exist in nature. It will be noted that, if we reverse the sequence of harmonics in a major scale, we obtain the minor scale.
Therefore, we have two powerful modes. One - the major - mimics nature, is bestowing, shiny, merciful, victorious. The other - the minor - goes against natural flow. Thus, it shows judgment, peril, fight, anguish, sadness.
Now, we need to explain the implications of the 7-12 system. Pythagoras related the 7 notes to the 7 planets. Today, we know that there are more than 7 planets. However, we can think that they are the meaningful planets. If a planet cannot be seen with a naked eye, then it will likely have no appreciable influx over the earth, as it is too distant. Therefore, only 7 planets are considered. Each planet can become a scale in the major and minor mode. Thus, it is double-faceted. We could say that it has good and evil.
12 are the subdivisions of a sphere, such as it is used in astrology. They are the 12 zodiacal signs - i.e. the subdivision of the heavenly vault. Each semitone corresponds to 30 degrees of the vault.
Therefore, music is a representation of the divine comedy of the Macrocosm - i.e. the Work of Creation. We can imagine the semitones as the heavenly vault where the planets - i.e. the tones - wander. Harmony is the discipline that studies the the sequence and interaction of chords. Chords are built upon a fundamental tone. It is possible to build a chord over every note in the scale. However, ony 3 chords are the main ones. All the others are sub- or derived chords.
By the way of modulation - i.e. introducing alterations - we change from one scale to another. Therefore, when we increase a note with a semitone and the vault spins, the planets follow and another scale is established.
As we talk this heavenly language, the stars express in their houses, generating different meanings. We could examine those meanings the same way we draw a horoscope in astrology. As this is a universal language, even if the listener (or the performer) is not aware of the meaning, this is still perceived at a subconscious level.
The performer, especially, must be aware that, when he plays, he established a connection between heaven and earth (the public) and he is moving the holy spheres. Music must be treated with respect and wisely.

  1. Nicomachus of Gerasa: Manual of harmonics
  2. Nicomachus of Gerasa: Introduction to arithmetics
  3. Iamblichus: Life of Pythagoras
  4. The Pythagorean Sourcebook and Library
  5. Jason Scott Nicholson: An introduction to Pythagorean arithmetic
  6. Melanie Richards: Pythagoras and music
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