Moderate Range

This article exposes the theory of the moderate range in detail. For a simplified presentation and of synthesis, to see the article Ranges and temperaments which gives also an overall picture of the ranges of the traditional Western music.

The moderate range - expression synonymous with “temperament equal” (which is current name) is the system of division of the octave most commonly used nowadays in the Western music and the musics which of it result. The principle is to cut out the octave in twelve equal chromatic intervals without being worried, could one say, of the Consonance between them of the sounds thus determined.

NB One can also speak about “equal temperament” when the octave is divided into more than twelve equal intervals (31, 53, etc These temperaments is approached in Tempérament by multiple division). What follows relates to only the temperament equal to twelve intervals with octave.

Genesis of the equal temperament, theory evolutionist

In fact, the number of twelve intervals per octave, bases our music, does not have anything fortuitous. In the multiplicity of the intervals which our hearing can appreciate, the theory of the Gamme pythagorician made it possible to distinguish a “pallet” from seven principal sounds and five sounds sharps which constitute the Cycle of the right fifths, however without closing again it absolutely - remains a residual variation are equivalent to the difference between seven octaves and twelve fifths: the Coma pythagorician. The range of Pythagore was the support of the music of the Middle Ages.

Until the end of the Middle Ages, one recognized like consonnants only the intervals of unison, octave, fifth and quad. The major third was excluded from it, because it is true that the third pythagorician is relatively false, being higher than the third just of a syntonic Comma, which makes a crippling difference. At that time, one started to admit the consonance of the third ; the number of seven diatonic notes per octave firmly established, one could formalize “natural” ranges - of which that of Zarlino - rather close to the range pythagorician but allotting a big role to the third and other simple harmonic reports/ratios. The intervals dividing the octave remained unequal and another disadvantage emerged: the division of the major third in a major tone and a minor tone different. The modulation was not practiced - and hardly could the being.

The temperaments mesotonic were imagined, probably, in XVIe century to obtain rather right thirds divided into two tons equal: they let remain a more or less marked fifth of the wolf like certain unequal thirds; however some of them approached the equal temperament more or less. These temperaments made it possible to extend the possibilities of modulation until in distant tonalities - but not without sound defects: these regular temperaments implied brutally intervals different (less consonants) from the others. Certain type-setters knew to use about it.

If one traverses the cycle of the fifths in the direction going down - and not going up - one meets other faded sounds (flat) which do not coincide exactly with the sounds sharps, because the semitones of the range of Pythagore which are the apotome (of value 2187/2048) and filed it (256/243) are not equal and differ besides from a coma. Always it is that they are sufficiently close so that the various temperaments imagined during XVIe, XVIIe and XVIIIe centuries aim at simplifying the range by assimilating them. Thus since the end of the Rebirth, the keyboard instruments in general have keyboards with seven natural keys (steps) and five faded keys (pretenses) per octave.

The assimilation of the diésées and bemolized notes - which consisted in fact to privilege one compared to the other - did not imply in any way the equality of all the chromatic intervals. It had even become the craze of the theorists of the XVIIIe century to discover the unequal temperament which would present the best possible compromise between the accuracy of the octaves, the fifths and the thirds, it not limitation of the use of the various tonalities and the infinite possibilities of transposition and modulation. But this research on the paper and in the discussions of living room was extremely far away from the concrete reality of the music.

A quite moderate keyboard?

With the temperaments of XVIIIe S., each tonality had a “sound color” differentiated - whose Johann Mattheson made an inventory - to which the large type-setters - among whom Bach and Couperin - paid more the great attention. These unequal temperaments mobilized an imagination and a considerable inventiveness and some of them is of a remarkable ingeniousness. Much however presented almost more disadvantages than advantages. The “quite moderate Clavier” of J.S. Bach was composed to be the proof of the feasibility of one of these unequal temperaments, in opposition to the others which were not practicable. Perhaps conversely, the two collections of Jean-Sebastien Bach entitled “Das wohltemperierte Klavier” were composed to show these possibilities of playing in all tons them major and minor by the use of the equal temperament, without modifying the agreement of the instrument - although this assumption makes debate. Indeed, the compositions in the twenty-four let us tons also could be made up to show the differences of stamp between the tonalities, or their absence in a moderate range.

Bradley Lehman, an American musicologist, has just shown that Bach left in the drawings of its manuscript the precise structure of the temperament which he recommends: containing fifths moderated with 6th of coma pythagorician, and some others with 12th. (English article only)

Historical development

One can reasonably think that a temperament relatively close to the equal temperament was used since the beginning of XVIe century for the agreement of the hoop and string instruments: lute, guitar, viol, etc the hoops laid out through the handle of these instruments imply in theory identical reports/ratios of sound portions of cords granted on different notes (for example, for the acoustic guitar, Mi, Ré Ground If, Mi). So all the intervals have to be equal between them. However, there remains possible that the hoops were not always laid out equally. A temperament close to certain old temperaments was realizable on the handles of lutes, and a fortiori on the handles of viols, whose curve allows more subtleties. For parts grouping of the viols and a harpsichord, one can think that this last was granted according to the equal temperament, only possible for the viols. This method of agreement required an auditive tolerance to accept relatively false fifths and especially thirds. It is the opposite which occurs nowadays. The ears accustomed to the equal temperament are surprised when they redécouvrent a harmonic interval just.

Another assumption, attested by certain old sources, proposes this track of auditive tolerance: the harpsichord was granted with a temperament better than the hooped instruments, the diversity of sonority of the instruments and the margin of auditive sensitivity absorbing the differences.

The progressive generalization of the equal temperament is relatively recent (two centuries and half at most) and rests on a true paradox: this system is - musicalement speaking - roughest which is (quite lower than certain temperaments mesotonic or unequal); it is however that which saw its advent during the extraordinary development of the European music of the middle of the XVIIIe century at the beginning of XXe, in esthetics as different as those from Haydn, Chopin or Stravinsky.

It is in fact that the concept of Consonance, and sets of consonances lost at that time its importance, with the profit of a musical language then under development full, and who sought the maximun openings, while basing itself on bricks which one wanted now universal: interchangeable and capsizable intervals at will, in a universe entirely domesticated by the man. Finished the natural intervals of which we could now free us, since human industry and education enabled us to recognize like new absolute values the moderate intervals always similar, and thus easily memorable and répétitibles of the equal temperament.

It remains still to stress that the peremptory necessity for an equal temperament truly strict has raison d'être only with the arrival of the Serial music (initiated by Arnold Schoenberg and the school of Vienna), which finally is the first to draw part of its completely homogeneous characteristics fully, by being assimilated it. The moderate range is indeed purely a series (a series of notes homogeneously distributed, but deprived of consonances common others that approximate), and not the result of a harmonic construction such as are the other systems.

It should well be noted that up to one recent time (according to the mediums), the majority of the tuners of piano always took care to color the temperament (i.e.… to make it slightly unequal!), in order to make more sensitive the tonal music played above by a differentiation of the tonalities. It is probable that certain still do it.

Final victory?

The disaffection of the current public for the music (erudite) of its time, and its diving masses some in the marvellous landscapes of the old Musique now opens the ears of all with new values which call in question the assets of the equal temperament. The accuracy gains there, and one again understands pure thirds in the string quartets of the academies!

The equal temperament, on its side, is triturated, peeled (systems with multiple sounds) expanded (temperament equal to perfect fifths), amended… it should well be recognized that actually, one forever played exactly in equal temperament, which remained only one reference. Only the piano, the guitar, sometimes the organ, have summers really faithful to the temperament equal… like all the synthetizers to good (or less good) market, which do not make it possible any to change the agreement of the instrument.

One has to expect in the course of XXIe century for the emergence of new systems of agreement, which would be freed for example from the pure octave… certain type-setters work already in these directions.

The theory

It is simpler: to divide the octave into twelve intervals - semitones - equal. It has legitimacy only as it produces sounds not too distant from those of the “right intonation”; the practice of the musician and his listener makes the remainder.

Since to add with the intervals amounts carrying out multiplications of reports/ratios of frequencies, the octave equalizes the semitone raised with the power twelve. Thus, the semitone is worth $\ sqrt {2}$ (approximately 1,0595). The moderate fifth equalizes 7 semitones, that is to say 2^7/12 .

One can as consider as the coma pythagorician is distributed according to twelve equal shares between the twelve fifths of the cycle. The coma pythagorician is worth 3¹²/2¹⁹: the twelfth of coma is worth thus (3¹²/2¹⁹) ^1/12 or 3/2^19/12. The moderate fifth (decreased pure fifth of a twelfth of coma) is thus worth (3/2)/(3/2^19/12) that is to say 2^19/12-1 or 2^7/12 : we find the same result.

Vincenzo Galilei proposed, for the semitone which is at the same time diatonic and chromatic, the approximate value: 18/17 (this high number with power 12 is worth 1,98555995… very near to 2.

Marin Mersenne determined the proportion $\ sqrt {\ sqrt {2 \ over3- \ sqrt2}}$ which approaches it still more precisely and which can be built with the rule and the compass.

At the XVIIIe century, Johann Philipp Kirnberger discovered that the fifth of the equal temperament is excessively close to the interval of 5 octaves decreased by 7 pure fifths and a pure third. The difference between the two is so tiny that, translates into report/ratio of frequencies, it is necessary more than six decimals to see that it is different from unison. This interval is undetectable with the ear.

See also Its (physics) > Frequency and height.

Musical qualities of the moderate range

The moderate range allows the modulations ad infinitum - it is besides the reason of its general adoption. It standardizes the semitones, diatonic or chromatic (this property does not show through in the musical notation - to see the articles relating to the Solfège). The fifth of the wolf disappears as all colorings from the tonalities which become equivalent in the same mode.

Separately the octaves, all the intervals are (slightly) false.

the fifths result relatively right, from the decreased pure fifth of a twelfth of coma pythagorician, value low;
the quads are slightly too large (even reason);
the thirds are better than the third pythagoricians, too much large, who are reduced of a third of coma pythagorician, therefore of a a little higher fraction of the syntonic coma. They are nevertheless still far away from the purity.
the same remark applies to the sixths, too small;
the seconds (or let us tons) move away from a sixth of coma of the value right): they also lose in purity too small).
the seventh are too large of a sixth of coma, consequently.

See the table below

These disadvantages, quite real, could not prevent the musicians from joining there, because the advantages in terms of composition and expressivity carried it.

The general adoption of the temperament equal to the recent last centuries is also explained by an esthetic evolution of art in general. To the brightness of the colors baroques the harpsichord corresponds, with its crystalline lens, granted in unequal temperament, with rather pure intervals. To softness melancholic person of the romantic period the piano corresponds, with the less definite sonority, softer and wrapped, which opens the door with the more approximate but regular intervals of the equal temperament.

The moderate range is, of all, most difficult to grant: contrary to the other systems or the instrumentalist must establish in a precise way of the consonances (to which hearing is very sensitive), to grant an instrument in equal temperament it must establish equal dissonances all inside an octave, which is appreciated by faculty as from the “beats” a second. This technique is out of reach practice of the amateur: the equal temperament is at the origin of the trade of tuner of Piano S. With a little practice, the amateur owner of a Clavecin can rather easily grant its instrument according to one of the temperaments bequeathed by the 18th century.

Comparison of 3 systems of division of the octave

NB In this table:

the note C commune to 264 Hz gives it to it to 440 Hz (current Diapason) in the right intonation.
the natural ranges are represented by the “right intonation” starting from C.
the range of Pythagore is assembled in such way that the fifth of the wolf is between ground # and semi ♭.

NB In this table:

the note is common to IT to 440 Hz (current Diapason)
the natural ranges are represented by the “right intonation” starting from C
the range of Pythagore is assembled in such way that the fifth of the wolf is between SOL♯ and MI♭.

Articles in relation

Ranges and temperaments

square Root of two

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