Scala naturale en
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The content of this page is very delicate. It is directed at musicians, however, instead of speaking their language and calling the intervals by name (minor sixth (m6), augmented fourth (A4), etc.), they will be translated into mathematical formulae, which will be incomprehensible at first sight. The mathematical structure that generates musical scales as a succession of intervals is elementary, even though it looks "repellent". It involves only basic operations (actually only multiplication and division) and can be constructed step by step, which is what we will do.
How is the natural scale constructed?
The natural scale was initially created due to a greater need for consonance in certain intervals (thirds) with respect to the Pythagorean scale. The "naturalness" of the scale, along with its aesthetic foundation, has a physical foundation that is well illustrated by the theory of harmonics. This foundation is based on the ability of a string with a fixed length to generate several frequencies simultaneously, all of which are multiples of the frequency of the fundamental note.
 The sounds of the natural scale come from the series of natural harmonics of its reference note.
This series can be generated according to the following rules:
 a reference note is chosen and its frequency is multiplied by 2, 3, 4, etc.;
 to bring the generated notes back to the initial octave, their frequencies are divided by , where "n" is the number of octaves from the initial note to the final note;
 then, any double notes obtained are eliminated. The remaining problem is to decide how many distinct notes to include in the scale. Tradition suggests seven for the diatonic natural scale and twelve for the chromatic natural scale.
For example, if we start with the note C, we obtain:
Harmonic no.  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  ... 

Ratio  1:1  2:1  3:2  2:1  5:4  3:2  7:4  2:1  9:8  5:4  11:8  3:2  13:8  7:4  15:8  2:1  17:8  9:8  19:18  5:4  ... 
Note (approx.)  C  C  G  C  E  G  Bb  C  D  E  Gb  G  Ab  Bb  B  C  C#  D  D#  E  ... 
Two observations are immediately evident:


The diatonic natural scale
In theory, the diatonic natural scale (i.e. without altered notes) should be made up of a series of the natural harmonics of seven notes without alterations. However, as shown in the previous harmonic series, the note A is actually not present. It would have corresponded to the 27th harmonic obtained four octaves above the initial C; i.e. to a ratio of 27:16. However, this ratio is well approximated by the "simple" ratio 5:3, which differs from 27:16 by only 21.6 cents (about a quarter of a temperate semitone).
Comparison with the Pythagorean scale




The chromatic natural scale
As with every diatonic scale, the limited number of notes offers a limited range of melodic possibilities. This problem can be overcome by increasing the number of notes making up the scale. Obviously, the new added notes, which can greatly enrich the melodic possibilities, must not compromise the advantages of the diatonic scale. In particular, they must:
 preserve the consonance of the most important intervals (octave, fifth and third) as much as possible;
 make the consecutive steps of the scale as uniform as possible;
 not be of an excessive number, so as not to have frequencies too close together.
The choice of frequency ratios for the construction of the chromatic scale is not unique. Nor is the number of semitones contained in an octave (there are temperaments with up to 88 semitones, which obviously violate the third abovementioned criterion). Once it has been decided to keep only some of the harmonic ratios listed above, an element of arbitrariness is introduced. The element can be:
 to fix the number of harmonics to consider for the generation of all the tones;
 to fix the number of "small" (and prime) numbers with which to construct all the ratios. In particular, we can see the Pythagorean scale as a natural scale constructed upon only two harmonics (the octave and the fifth).
It is for this very reason that, in the effective constructing of the chromatic natural scale, various criteria are possible:
 choosing the distance between the third and fourth step of the diatonic scale (minor second) as a semitone, we obtain: (4/3):(5/4)=16/15. The semitone thus obtained is called the just diatonic semitone (equal to about 112 cents). At first sight, this choice, using a semitone already present in the diatonic scale, appears to give the scale uniformity. However, this uniformity is actually illusory due to both the diversity of the major and minor tone and the need to maintain consonance between the most harmonically important intervals (e.g. the frequency of F# and Bb were chosen to form a perfect fifth with B and Eb). This would actually cause the formation of other semitones that weaken the uniformity of the consecutive steps of the scale.
 Another criterion could be based on a generative rule that uses small prime numbers (e.g. 2,3,5) to construct intervals. In this case, we must calculate the ratio between a fifth and two consecutive thirds . To distinguish it from the just diatonic semitone, this semitone is generally called the just chromatic semitone (equal to about 70.672 cents).
(N.B. The following tables approximate the measurements in cents to the unit: this calls for a slight "shift" in the calculation of the amplitudes of the semitones. For example, between Bb and B the amplitude seems to be 70 cents. If we had used the exact value of 70.672, this would not have happened!


It is interesting to observe that, beyond the diversity of the adopted criteria:
 the semitones that come into play in the two scales are almost indistinguishable to the ear;
 the scale with the chromatic semitone is more uniform, if we adopt the number of different semitones contained in the scale as our criterion of uniformity;
 the scale with diatonic semitones is more uniform, if we adopt the variance with respect to the "average semitone" with an amplitude of 100 cents as our criterion of uniformity. Semitones with a maximum (133.238) and a minimum (70.672) amplitude are less used in this scale.
Disadvantages
The disadvantages of the chromatic natural scale are the same as those of the Pythagorean scale:


Visit the page Comparison of scales, to listen to examples comparing the various scales.
The problem of tonality change
The natural scale is the scale that allows us to apparently have "more consonant" intervals for a given tonic and is also the scale in which people spontaneously tend to tune when singing solo or in chorus. The reasons can be found in the fact that some key intervals have particularly coincident overtones (for a review of physiological theories, see the page Physiology of the auditory system).
However, both the Pythagorean scales and natural scales are not invariant by translations. This mean that starting the scale with a note other than C and keeping the pitches of the sounds as they were obtained in the scale in C, the sequence of the various types of semitone in the new scale changes completely.
In Greek music, this variation was used to generate seven distinct musical modes and each music genre was executed in only one specific mode. However, with the advent of polyphonic music and the concept of tonality, this property of the natural scale makes impossible the practice called modulation, which is the changing of the tonic of a scale within the same piece of music.
Simply:
 a fretted instrument tuned to the natural scale in C plays well only in the tonality of C. To change the tonic, the instrument must be changed or retuned.
The problem can be partly resolved with fretless instruments (e.g. violin), in which the performer can impose the necessary variations required by the change of tonality. This, however, is impossible with fretted instruments. In this case, a change of tonality requires a complete retuning of the instrument.
On the page about equal temperament, you will find the solution to this problem. You will be amazed by the contrast that there is between the simplicity of the solution found (make all the semitones equal) and the lateness with which it was introduced. This was probably due to the complete separation that existed for centuries between music "theorists" (mathematicians and philosophers who had derived the musical scale from generative mathematical principles) and musicians (for whom the search for harmony and consonance has always been the "natural" fruit of musical experimentation carried out in the field and not influenced by abstract principles).