Ancient Greek mathematician, philosopher created the Pythagorean comma


Pythagoras. Credit: public domain

The Greek mathematician and philosopher Pythagoras, who lived 2,500 years ago, also applied his genius to music throughout his distinguished career, creating the Pythagorean comma as part of music theory, and his genius is still recognized today. nowadays.

The Pythagorean theorem remains one of the fundamental ideas in the field of mathematics and is still taught in schools around the world.

The influence of the ancient Greek thinker, born on the island of Samos in the year 570 BC.

Grecian Delight supports Greece

The philosophy of Pythagoras influenced both Plato and Aristotle, and through them his ideas were fundamental in Western philosophy. However, a lesser-known contribution that the brilliant theorist gave to the world is the “Pythagorean comma”, the small interval (or comma) existing in the Pythagorean chord between two enharmonically equivalent notes such as C and B♯, or D â™­ and C♯. .

The Pythagorean comma is part of essential music theory

Its “comma”, known to music theorists around the world, is used to express the measures between musical notes over a scale.

In music theory, a comma is a very small interval which is really just the difference resulting from the tuning of a note in two different ways. The unqualified comma word refers to the syntonic comma, which can be defined, for example, as the difference between an F♯ tuned using the D-based Pythagorean tuning system and another F♯ tuned to l using quarter-comma based on D. mesodic tuning system.

Intervals separated by the 81:80 ratio are considered the same note because the Western 12-note chromatic scale does not distinguish between Pythagorean intervals and 5-limit intervals in its notation. Other intervals are considered commas due to the enharmonic equivalences of a tuning system. For example, in 53TET, B7 ♭ and A♯ are both approximated by the same interval although they are separated by a septimal kleisma.

The word “comma” comes from the Latin of the Greek κόμμα, from the first * κοπ-μα = “an act of cutting”.

In the same chord system, two enharmonically equivalent notes (such as G♯ and A ♭) can have a slightly different frequency, and the interval between them is a comma. For example, in extended scales produced with a five-limit tuning, an A ♭ tuned as a major third below C5 and a G♯ tuned as two major thirds above C4 are not exactly the same note, because they would be of equal temperament. The interval between these notes, the diesis, is an easily audible comma (its size is greater than 40% of a semitone).

Commas – the difference in size between two semitones

Commas are often defined as the difference in size between two semitones. Each mesodic temperament tuning system produces a 12-tone scale characterized by two different types of semitones (diatonic and chromatic), and therefore by a comma of one size. It is the same for the Pythagorean agreement.

One Pythagorean comma is equal to the frequency ratio (1.5) 12⁄27 = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, which is about a quarter of a semitone (between 75:74 and 74:73). The comma that musical temperaments often refer to is the Pythagorean comma.

Pythagorean comma over C using Ben Johnston’s notation. The note represented as lower on the staff (B♯ +++) is slightly higher in pitch (than C â™®).

Pythagorean comma (PC) defined in the Pythagorean chord as the difference between semitones (A1 – m2), or the interval between enharmonically equivalent notes (from D â™­ to C♯). The diminished second has the same width but an opposite direction (from to C♯ to D â™­).

The Pythagorean comma can also be defined as the difference between a Pythagorean apotome and a Pythagorean limma (that is, between a chromatic semitone and a diatonic semitone, as determined in the Pythagorean chord), or the difference between twelve perfect perfect fifths and seven octaves, or the difference between three Pythagorean ditones and one octave (this is why the Pythagorean comma is also called a ditonic comma).

The diminished second, in the Pythagorean chord, is defined as the difference between limma and apotome. It therefore coincides with the opposite of a Pythagorean comma and can be considered a descending Pythagorean comma (for example from C♯ to D ♭), equal to approximately −23.46 cents.

As described in the introduction, the Pythagorean comma can be derived in several ways:

The difference between two enharmonically equivalent notes in a Pythagorean scale, such as C and B♯ (About this soundPlay (help · info)), or D ♭ and C♯ (see below). Or it could be the difference between the Pythagorean apotome and the Pythagorean limma.

It can also be obtained by the difference between twelve perfect perfect fifths and seven octaves or the difference between three Pythagorean ditones (major thirds) and an octave.

A perfect straight flush has a frequency ratio of 3: 2. It is used in the Pythagorean chord, with the octave, as a standard to define, with respect to a given initial note, the frequency ratio of any other note.

“Apotome” and “limma” are the two types of semitones defined in the Pythagorean chord. The apotome (about 113.69 cents, for example from C to C♯) is the chromatic semitone, or augmented unison (A1), while the limma (about 90.23 cents, for example from C to D â™­ ) is the diatonic semitone, or minor second (m2).

A diton (or major third) is an interval made up of two major tones. In the Pythagorean chord, a major tone has a size of about 203.9 cents (9: 8 frequency ratio), so a Pythagorean saying is about 407.8 cents.

The life of Pythagoras strewn with trials, struggles despite the outburst

Around 530 BC, the Greek philosopher went to Croton (today Crotone) in southern Italy and founded a school. The initiates of the schools of Pythagoras took an oath to secrecy and led a community and ascetic life.

This lifestyle involved a number of dietary bans, including vegetarianism, although some modern researchers doubt that he argued for the diet.

The mysterious personality of Pythagoras was noticeable during his teaching. Strangely, no grades or questions were allowed in his lessons, which is why much of his work is lost. There is no additional information, even on the famous Pythagorean theorem.

It is also not known whether Pythagoras invented this theorem on his own or with the help of his students.

He died around 495 BC.

Another ancient account states that Pythagoras was killed for political reasons, when supporters of his opponents Cylon and Ninon attacked the Pythagoreans in a meeting.

After his death, the house of the Greek philosopher was transformed into the Sanctuary of Demeter, and the road leading there became a Sanctuary of the Muses – a fitting transformation, as the great man was responsible for explaining the workings behind the more fleeting of all arts, music.

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