Equivalence of the octaves

Concerning the melody component of the music, one calls equivalence of the octaves - or identity of the octaves or relationship of the octaves - the postulate according to which the interval of octave - more precisely, the octave right - constitutes the privileged framework of the musical heights, which reproduces in a cyclic way, whatever the scale adopted.

General information

From an Acoustic point of view, the octave right corresponds to the reports/ratios of Fréquence S following: 2/1 while going up, and 1/2 while going down - for example, if two notes has and B are at a distance from octave, one can write: “= B/2 has”, that is to say: “B = A * 2”. The octave is the interval separating the two first Harmonique S of a its: it is probably the reason for which this interval seems to be the tallies melody common to all the musical systems using of the given scales.

  • the “melody matrix” consisted the octave, contains a “finished” number of names of notes repeating “ad infinitum” according to the axis of the Fréquence S. Thus two sounds separated by an octave, though of height S different, are felt like very strongly related : this intuitive report - the justification Acoustique will be of course posterior - quite naturally resulted in assigning a even name to the two heights in question.

For example, in the range containing the following notes: C, D, semi, ground and the - called pentatonic range, i.e., range made up of “five degrees”, or “five notes” -, one will not find “a sixth” name of note after the the , but another C , starting point of a new cycle, and whose Fréquence is the double of that of the initial C located an octave below.

When men and women - or children - sing same the melody, the interval of octave which separates the voice is often felt like an interval of unison.

  • the principle of the equivalence of the octaves was theorized by Jean-Philippe Rameau, but its applications are largely former to the 18th century, and find in many musical systems belonging to civilizations very distant from/to each other.

Preliminary observations on the intervals

Subsequently, one will speak about intervals between two notes in two different ways

  • by their name (third, Quinte, octave, Quarte)
  • by a number which is the relationship between their two Fréquence S

The interval of an octave is of 2 because the frequency of the acutest note is twice larger than that of the most serious note. By playing these two notes together, one obtains a very harmonic sound, one of the notes basing itself in the other. It is said that the octave is neutral because it does not generate a difference, so much so that one takes the practice to give the same name to two separate notes of an octave (see the article on the equivalence of the octaves).

The interval of the fifth is 3/2 because the frequency of the acutest note is equal to the frequency of the lowest note multiplied by 1,5. It is this interval which particularly interests us in the range pythagorician.

The interval of the quad is of 4/3; the quad is the complement of the fifth compared to the octave - one says also his “inversion”, expressed differently: “a quad plus a fifth are worth an octave”.

The interval of the third called also diton is of 34/26 because the frequency of the acutest note is equal to the frequency of the lowest note multiplied by 34/26. One will see in the mathematical part the reason of this report/ratio.

The interval separating the highest note from the quad and the highest note of the third is called filed and is worth:

\ frac {4/3} {\ frac {3^4} {2^6}} = \ frac {2^8} {3^5} .

Consequences of the principle in the Western tonal system

The tonal Système of the Western music rests on the principle of the equivalence of the octaves. The following rules are the direct consequence.

Seven names of note

The diatonic range, on which the Western music is founded since centuries, consists of seven names of notes: C, D, semi, F, ground, the and if .

Redoublings of interval

An interval and its redoubling are regarded as harmonically equivalent.

For example, the following harmonic intervals - the perfect fifth C-ground and the twelfth just C-ground - are equivalent.

Inversions of interval

An interval and its inversion are regarded as related.

For example, if the third major ascending C-semi is regarded as the inversion of the minor sixth downward C-semi , it is because the two intervals, though concerning different height S, produce same the degree - semi in this example.

Inversions of agreement

Different the states from the same agreement - fundamental state, first inversion, second inversion, etc - is also felt as connected, because, constituted of the same notes - cf the theory of the fundamental Basse of Rameau. It is advisable to specify however, that the traditional harmony treats sometimes these various states like different agreements, when those do not have same the tonal function.

  • On the other hand, different the provisions of agreements containing the same notes - provided the low remains the same one - is equivalent from the point of view of the tonal function of the agreement in question.

For example, the agreement C-semi-ground and the agreement C-semi-ground-C are analyzed like same a Accord of three notes - major triad, with C like Fondamentale -, even if the first contains “three sounds”, and the second, “four”.

See too

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