Musical notes and ranges

One does not use in music of unspecified sounds heights but notes of well defined frequencies. When one emits the succession of these notes by increasing heights, the same melody reproduces with each interval of octave. At the interior of an interval of octave, the succession of the notes is called the range.

The notes received a proper name which is repeated with each interval of octave:

....la si ut ré mi fa sol la si ut...  

.....A   B  C   D  E   F   G  A  B  C....

The names of notes, employed in France, were proposed by GUI, monk of the Abbey of Pomposa, he born in Arezzo in Tuscany, towards the end of Xe century; they are the first syllables of half-towards psalm with Saint-Jean Baptiste whom one precisely sings on the corresponding notes of the range.

Ut queant laxis resonare fibris

Mira gestorum famuli tuorum

Solue polluti labili réatum

Sancti Johannes

Domine

The word Si(initial of Jean Saint) is allotted to the French type-setter the Mayor. One replaces Ut by do easier to sing.

In Germany and England, the notes are indicated by letters: A for la, B por si, C for do, D for re, E for mi, F for fa and G for sol.

The sol (G) engraves was initially called g then G the deformation of the letter G gave rise to the treble clef. The word range comes g (gamma).

There are several kinds of ranges. The diatonic range is major mode; there are ranges of minor mode, dorien... the greek, arabic used very particular ranges formerly.

Each range is affected of a sequence number; this number increases by a unit to each ut met while going up; in France the zero is not used: -2, -1, 1, 2, 3...

One will say by exmple ut(-1), la(4), mi(1)...

To define the frequency of all the notes it is now enough to fix the frequency of the one of them. In 1859, one adopted in France: la(3) = 435 Hz. It was necessary, indeed, to fix the frequency of la(3) which did not cease increasing; from 405 Hz (Louis XIV), it passed to 423 Hz ( the Empire); one fixed it at 435  Hz to give more glares to coppers. English one used 457 Hz, German 440 Hz for the orchestra. Currently, the frequency of la(3) is fixed at 440 Hz.

There is a scale " of the physicists " defined by ut(-2) = 16; it leads to la(3) = 430,54 but all the frequencies of C are powers of 2.

Natural or harmonic range or of the physicists or Aristoxène

This range was recommended at the beginning of XVIe century by Zarlino, priest and Italian musician. But one allots his invention to Aristoxène, Greek theorist of antiquity. It starts from the idea that by dividing a cord into 2, 3, 4 equal parts one obtains a series of sounds which, brought back in the same octave, provide the notes of the range considered. In other words, the 7 usual notes are such as their harmonics are in general of different notes of this range, from where its name. But it is also the range which a good singer (or a good violonist) naturally uses like Helmholtz and Rameau one shown since if two notes successively are emitted, the second note had already been heard in the harmonic of the preceding one. As it is not the case of the moderated range, this one is false from the musical point of view, and was regarded as a monstrosity in XVIIIe century: but it is the only usable one on the instruments with fixed sounds, if not it would be necessary to multiply the keys of the keyboard included in the octave or the keys of the wind instruments making them unplayable.

Let's study nevertheless the arithmetic reasoning which led to the natural range. Let us seek for example the harmonics of ut1 of frequency f1:

f1 = ut(1)

f2 = 2f1 = ut(2)

f3 = 3f1 = 3/2 f2 = sol(2)

f4 = 4f1 = 2f2 = ut(3)

f5 = 5f1 = 5/4 f4 = mi(3)

f6 = 6f1 = 3/2 f4 = sol(2)

All these harmonics form a triad.

The intervals of the notes of the range natural definite compared to the first, known as tonic, are written:

C     D     E     F     G     A     B     C   
1 9/8 5/4 4/3 3/2 5/3 15/8 2

One can easily calculate the successive intervals of the successive notes, one finds:

C    D      E        F      G     A     B      C

  9/8  10/9 16/15   9/8  10/9  9/8 16/15

It is seen that there are three kinds of intervals: the major tone 9/8 the minor tone 10/9 and semitone 16/15.

The semitone is worth 28 s(savarts) ; the minor tone 46 s ; the major tone 51 s . The interval between the major tone and the minor tone is 9/8 / 10/9 = 81/80 = 1 comma. It is worth in savarts 5,3 is 5.

To transpose a range means to change tonic, i.e. to repeat the melody of the range starting from a new note. A major tone corresponds with two different intervals: the musical writing would become practically impossible. It would be seen easily that C # does not coincide with D b!

It is for all these reasons, because especially of the instruments to its fixed, which one adopted the range of Bach.

Moderate range

The moderate range was imposed by famous musician J.S. Bach (1685-1750) after many discussions to which took share Rameau and of Alembert. All the pianists know " the Well-Tempered harpsichord " (1722-1744).

The notes of this range share the interval of octave in 12 equal intervals called moderate semitones.

The intervale of octave being f1/f2 = 2. It is worth in savarts:

1000 log 2 = 301,03 s

or with a sufficient approximation 300 s . Each semitone is worth thus 300/12 = 25 s. A tone is worth always 50 s.

The transposition is then easier. By taking the note C for tonic, the diatonic range represents a certain melody whose successive intervals are:

C            D            E              F              G           A             B              C

        1              1            1/2             1             1            1            1/2

If new the tonic is F, it will be enough to raise of a quad, that is to say of 5 semitones; all notes of the range, which will be always possible since one has on the keyboard of all the notes semitone in semitone. One will play then:

F            G             A           B b           C             D            E             F

        1              1            1/2             1             1            1            1/2

The moderate chromatic range is a spreading out of notes of semitone in semitone. The frequency of a note is calculated by multiplying that of the preceding note by 2^(1/12) = 1,05946.

Knowing the height of the tonic, one can thus calculate the frequency of any degree in the octave.

Range of Pythagore - minor ranges - extraeuropean ranges

For the range of Pythagore the principle of its generation is simple. A basic note is chosen; one multiplies his frequency by ratio 3/2 and so on. The frequencies are thus obtained by multiplying that of the basic note by (3/2)^n. This range is founded on planets: to each planet corresponds a note. The moon is D, Mercure C, Venus B, the Sun A , Mars  is G, Jupiter F and Saturn the E . According to Pythagore, the Earth produces a sound, since it is moving. The other planets, by their rotation around the Earth, produce they also a sound. The more the planet is moved away, the more the movement is fast, the more the sound is acute. And conversely... This theory lived nearly two thousand years. But she died abruptly by the discovery of Neptune into 1619. It was not possible to add an eighth note to this system .

By deteriorating in various ways certain degrees of the range, one can obtain minor ranges. A range is known as minor when its third and its sixth are minor. We find practically as many ranges as societies, and their degree of refinement is sometimes astonishing. Indeed, our traditional harmonic music imposes practically the range on equal semitones: it thus offers combinative rich possibilities . But in melody music, if one admits ranges with unequal degrees of natural the range type, one leads to infinite combinative of the modal music. Indeed, by beginning a scale with the second, the third... degree, one carries out as many different ranges, of which the auditive effect is each time original.

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