Ranges and temperaments

In Music, a temperament is a system of agreement intended for an instrument with fixed sounds, built by compromise in the intention to attenuate certain disadvantages specific to the systems of pure intonation. The variations introduced by the temperaments are directly perceptible only by one trained ear, but have a total effect on perception of the piece: the instruments granted on unequal temperaments restore the melody with a harder atmosphere or melancholic person, coloured by the tonality in which they play.

For an introduction general practitioner to the context, to see musical Range.

To be able to play harmoniously in transposed tonalities, the instruments with fixed sounds are granted according to ranges which deviate slightly from the interval S harmonics, by adjustments which name temperaments . The choice of a particular system of temperaments, when it makes it possible to improve sonority of the instrument for certain tonalities, the fact with the detriment of others. These arbitrations led to varied systems.

The principal temperaments used are:

for the romantic Music and afterwards, the equal Temperament.
for the Music of the Rebirth , the Temperament mesotonic with third pure.

Between these two periods, temperaments were used (known as irregular or unequal) that some call “temperaments of transition improperly”. Actually, their characteristics specific and are adapted to the music of their time (changes of tonal colors around a central tonality). More nowadays used in practice is the Tempérament of Valotti (or “Tartini-Valotti”), because of its little marked, and thus more universal character. Finally let us quote for the Modern music , in addition to the use of the usual equal temperament, the invention of complex systems or innovating.

Origin and need for the temperaments

Consonance and dissonance

If the frequencies of the sounds and their relative intervals are quantifiable sizes, the concepts of consonance and dissonance, as for them, subjective and are based on auditive perception and the reflection that one brings to him: some define the consonance of two sound frequencies like the “pleasant” character of their simultaneous or immediately successive emission.

But, more simply, the Consonance (=sonner with) indicates the capacity of several sounds to be linked between them, related to the concept of purity of an interval. On this subject, one should not confuse purity (objective concept) and accuracy (more subjective concept) of an interval. Confusion between these two terms makes sometimes difficult a good comprehension of certain texts old (for example at Rameau).

Philosophers, acoustics experts, physicists or mathematicians of tried to seek a rational explanation to the pleasant or unpleasant character of an agreement or of another, with various successes. For some, this research appears also vain to try to explain why some like the yellow best that blue. They think that the culture plays in an important way in this appreciation (certain ears comtemporaines, alive in the environment of the moderate Gamme, can be surprised when they hear for the first time a third pure). However, the former Greeks admitted like consonants only the intervals of octave and fifth, then the Middle Ages and the Rebirth admitted the thirds gradually there before the increasing complexity of the music carries us to regard as consonants intervals which would have made squeak teeth Palestrina and its contemporaries. The intervals consonants thus arrived the ones after the others in the harmonic order (row 2: octave, row 3: fifth and quad, row 5: thirds, sixths).

The musicians and the music lovers generally agree to find consonants:

a its fundamental and one of its first Harmonic S;
two sounds which are in report/ratio of rational frequency simple (for example: 3/2, 4/3, 5/3, etc)

It is on the basis of this consensus that were elaborate the Western heptatonic range and its various alternatives.

Transpositions and agreements

“To sound just” with the ear, the musician is naturally led to interpret a piece of music by using a natural Gamme, where the notes are in harmonic relations the ones with the others. But when one grants an instrument with sounds fixes (piano, flute,…) in this manner, it can play correctly only in its Tonalité starting. It is not possible to produce an instrument with fixed sounds which sounds just in all the transpositions, and to easily be able to transpose, désaccorder should be accepted a little the instrument: these variations are the temperaments brought to the range.

This question of musical esthetics is translated rather directly into mathematical terms, owing to the fact that the sounds harmonically consonnants are in a report/ratio of frequency being expressed by a simple fraction (see Acoustics musical). There are several aspects with the question, but the two numerical main issues (which are reflected in practice musical) are those:

On the basis of a given note, twelve Quinte S differ somewhat from seven octaves. The variation (about a eighth of tone) is the Comma pythagorician or ditonic.
On the basis of a given note, four Quinte S successive does not give a third at all pure. (ex: the succession DO-SOL-RÉ-LA-MI gives SEMI which is very different from that obtained by a pure third ). the variation is the syntonic Comma.

The temperaments became necessary progressively of the development of the music, which played more and more on transposition S, modulation S, and chromatic scale, because no theoretical range was usable in practice:

they included all at least a interval unpleasant, because sounding false; the temperament does not make just it practicable, but.
they made difficult, unpleasant or impossible the Transposition; only the equal temperament allows any transposition without any coloring (useful for the accompaniment of the Chanteur S) but this the price of a “installation” of the accuracy which did not always allure the musicians.
they made difficult, unpleasant or impossible the modulation. According to the temperaments, it becomes more or less practicable, according to the temperament chosen, but also of arrival and the starting tonalities. It is advisable to specify that many old temperaments allow all the modulations which one wants, but with a strong coloring of the accuracy on the tonalities moved away, coloring that certain modern ears do not consider acceptable any more, because of the practice of the equal temperament which does not make any difference between all the tonalities.

In an additional way, the purpose of the temperaments often were complementary to make coincide the sharps and the flat, in order to improve the “jouability” of the instruments. It should however be noted that this concern did not always prevail, and that keyboard instruments which had distinct keys for two deteriorations were built before the 18th century.

The imperfections, suitable for each one of these ranges appear by the existence of “Comma S” the purpose of whose each temperament is to mitigate the sound effect and the constraints that they generate as regards composition and/or interpretation.

The temperament

The practice shows and the theory shows that it is not possible to grant an instrument to fixed sounds on several octaves by having at the same time all the intervals of octaves, fifths and thirds pure. If those were it, the intervals of second, of quad, sixth and seventh who are deduced from it would be it too.

This observation forced to find compromises to be able to practice the music on such instruments. One calls Temperament such compromises which can tend:

to be eliminated as much as possible the significant effect of the comas, while placing or by distributing those in uncommon intervals;
to simplify the musical scales by confusing the enharmonic notes ;
to allow or facilitate the transpositions and modulations.

The number of temperaments which were invented for the Rebirth and the period baroque is considerable; they can be distributed between the following categories, according to the principles implemented (but of other criteria of distribution are possible):

the temperament unequal;
temperaments mesotonic;
the equal temperament.

Let us announce for memory the temperaments by multiple division, which explored the division of the octave by a number of intervals different from twelve to try to improve the purity of certain intervals (with an interest definitely more theoretical than esthetic).

Traditional temperaments

Range pythagorician

The range “pythagorician” or of Pythagore is the oldest musical theory of the Western musical ranges. It goes up with the Greek mathematicians of Antiquity: it draws its name from Pythagore, the philosopher known in Géométrie for its famous theorem.

This range is built only on perfect fifths (report/ratio of frequency of 3/2) and octaves. To include/understand the principle of this range, it is enough to be placed in front of a piano and to leave the C on the left and to advance fifth in fifth (it is enough to move of 7 keys by counting the black keys). One obtains successively a ground , a D , a the , a semi , a if , a fa♯ , a do♯ , a sol♯ , a ré♯ , a la♯ , a F and… a C ! At the end of 12 fifths, one falls down on C located 7 octaves further. What makes say that 12 fifths are worth 7 octaves. The range pythagorician is the succession of the notes obtained by this process and which are to divide the octave into intervals coarsely equivalent. The not diésées notes are seven: the diatonic range is a “heptatonic” range. On a piano, they would be produced by the white keys. As for the chromatic range, made up of all the notes obtained except those which make almost doubled bloom (MI# and SI#) it has twelve elementary notes, and twelve intervals. The five complementary notes, on a piano, would be produced by the black keys.

But the piano cheating.

By advancing fifth in fifth, one cannot fall on 7 octaves unless shortening the last fifth known as “Quinte of the wolf”. Indeed, 12 fifths are worth 3¹²/2¹²=129.74… and 7 octaves 2⁷=128. After having gone up of twelve fifths (multiplication of the frequency by 3/2) and having lowered the result of seven octaves (division by two), the initial frequency was multiplied by (3/2) ^12 = 129.74… and divided by 2^7=128, that is to say overall by 1.0136: the result deviates from 1.36% of the initial frequency, that is to say practically a eighth of tone (23.46 hundreds, which is the definition of the coma pythagorician). The difference between SI# and C, very tiny but audible, is called the Comma pythagorician and its existence is commonly translated into what “the cycle of the fifths” (see figure) is not closed again.

One is obliged to introduce an interval of slightly false fifth (the “Quinte of the wolf”) to maintain octaves pure, which is often regarded by the musicians as necessary. In practice one would arrange oneself to defer the fifth of the wolf in a not very used interval, often MI♭ - SOL♯.

A particular range can be defined by its variations (in more or less) compared to the equal temperament. Thus, range of Pythagore presumedly established compared to a tonic " Do") will be defined by:

It is seen easily that the difference between the range of Pythagore and the moderate range is weak: the largest variation is of 12 hundreds (a half-coma), which is practically inaudible for a not informed ear.

The range pythagorician, founded on intervals of pure fifth, thus presents several defects:

the “cycle of the fifths” is not closed again, i.e. twelve fifths do not correspond exactly to seven octaves (what the existence of the coma pythagorician and the “fifth of the wolf translates”);
the major thirds that it generates are not perfectly pure what the existence of the “syntonic Comma translates”, interval height between the pure major third (5/4) and the third pythagorician (81/64) which is appreciably higher. The syntonic Comma is equal to 81/80 or 3⁴/(5 X 2⁴).

Natural ranges (Zarlino, etc)

Gioseffo Zarlino (1517 - 1590) works out one of the multiple possible natural ranges by recognizing an important place with the third interval of “pure”, and more generally with the pure intervals, i.e. correspondent with a ratio of frequency being expressed by a simple fraction.

The third is based on harmonics which multiply or divide the frequency by five, instead of a factor three as in the range of Pythagore. Between the quad and the your major , the introduction of factors " cinq" allows to work on reports/ratios 5/4 (=1,25) and 6/5 (= 1,2). These two reports/ratios are particularly simple, acoustically they sound well with the fundamental one. Lastly, since (5/4) X (6/5) = 6/4 = 3/2, one sees that their addition gives a fifth. These intervals, respectively named “third major” and “minor third” will play a leading role, with the octave and the fifth, in the construction of the natural ranges, which have many alternatives.

To build the range of Zarlino, we will express the intervals sought in function “pythagorician” of the major third then will apply the formula obtained to the major third “pure” (5/4). We already have the following intervals, notes and reports/ratios

Fondamentale = C = 1
Your major = D = 9/8 (two pure fifths transposed of an octave: 3/2 × 3/2 ÷ 2)
Minor third = MI♭ = 6/5
Third major = SEMI = 5/4 (contrary to 81/64, that is to say 4 fifths, according to Pythagore)
Quarte = F = 4/3 (as for Pythagore)
Quinte = GROUND = 3/2 (as for Pythagore)
Sixte (major) = It = 5/3 (addition of a third major and a quad : 5/4 X 4/3 = 5/3 - contrary to 27/16 according to Pythagore)
Septième (major) = IF = 15/8 (addition of a third major and a fifth : 5/4 X 3/2 = 15/8, contrary to 243/128 according to Pythagore)
Octave = C = 2.

The other intervals are calculated in a similar way, while determining, according to the range of Pythagore, a formula containing additions or third subtractions of (T), fifths (Q) and octaves (O) giving the correct result. Like the table shows it below, the fractions of the pure intervals are relatively simple, except the second minor one (16/15) and its symmetrical seventh (15/8), and especially the triton (45/32).

Compared to the moderated range, the variations of the range of Zarlino are (by supposing the range granted on a tonic of C): One sees that the variations compared to the moderate range are rather important on the thirds and sixths (about 14 hundred), like on the seventh (17 hundreds). One can in fact lira these variations in the other direction: compared to a formed range of interval pure, it is the perceptible degree of falseness on the moderate range. These intervals start to be audible for an exerted ear.

After having determined these intervals, one can check the values of the various thirds and fifths of the range of Zarlino:

the thirds all are right (report/ratio of frequencies = 5/4) except the third SOL♭-SI♭ whose report/ratio is 81/64, slightly superior;
the fifths are right (report/ratio of frequencies = 3/2) except three of them (report/ratio 40/27) which are not it (lower value): RE IT, FA♯-DO♯, SI♭-FA.

The range of Zarlino is thus particularly pure (for the melody and the agreements), but it is not easily transposable. It should be remembered that in this range, there are a major tone and a minor tone of different values. One calls Comma zarlinien the interval between these two tons: it is worth 81/80 is 1,0125; it is the syntonic coma. In the range of Zarlino, the succession of the 7 intervals constituting an octave is the following one:

your major
your minor
1/2 your diatonic
your major
your minor
your major
1/2 your diatonic

What precedes watch which the range of Zarlino cannot be used in practice when one must transpose or modulate.

Let us take the very simple example of the transposition of major C to G major. The interval GILDS in the first tonality has as a correspondent the interval GROUND it in the second, but GILDS is a major tone, and GROUND it a minor tone.

Another example: in a part exploited the range of Zarlino de Do, but transposed into Semi, the variation of the major third of +27.38 (the variation of Lab minus that the Semi one) compared to the moderated range, instead of the -13.69 will be awaited, and thus on the whole of 41.07 hundreds compared to the pure third: nearly a quarter tone!

The range of Zarlino is not the only “natural” range possible: for example, Zarlino did not include, in its range, of harmonic reports/ratios comprising figure 7 (first prime number after 2,3 and 5), because at its time, one only started to be interested physically in the accuracy of the thirds. For example, a FA♯ based on report/ratio 7/5 (that is to say 1,4) is a harmonic report/ratio much simpler than the reports/ratios approaching, deduced from Pythagore (729/512), and Zarlino (45/32). Thereafter, other theorists proposed their own system, without none being able to really have decisive advantages.

Regular temperaments

One speaks about regular temperaments when the corrections made to the intervals also apply to all, no particular interval not being musicalement right: they are thus the temperaments mesotonic and the equal temperament (which is mesotonic a private individual). The unequal temperaments are known as “irregular”.

Unequal temperaments

The idea of the unequal temperaments comes owing to the fact that, in practice musical, and especially at the time baroque before the use of the moderated range does not spread, all the intervals of fifth and major third are not also used.

One thus will try to reduce the undesirable effects of the syntonic coma, even of the coma pythagorician, by dividing them in such a way that one improves quality of certain intervals of fifths (thus of thirds), the least practiced intervals being able to be satisfied less good consonances.

The possibilities are extremely numerous and this study mobilized a great number of theorists with, each one proposing his own supposed solution to represent the best compromise: Werckmeister, Chaumont, Kirnberger, Branch, Vallotti etc

Within the framework of an unequal temperament, all the fifths (and consequently all thirds) do not have the same value in terms of reports/ratios of frequencies: each tonality thus had a “sound color” particular. Joy, sadness, serenity, melancholy, etc are expressed in the choice of supposed tonalities of better representing them: this criterion is put into practice by the large type-setters such as Bach and Couperin which attaches much importance to it. The choice of the temperament used can, contrary, being determined by the selected tonality and the modulations under consideration during the same part, some being better suitable than others.

These concerns completely disappeared since the moderate Gamme was adopted in a universal way by the type-setters. But the unequal temperaments are particularly adapted to the execution of the repertory baroque, and the specialized units usually practice them.

Temperaments mesotonic

The idea of the mesotonic Tempérament S will be to decrease all the fifths of a certain fraction of the syntonic coma, in order to make purer the resulting major thirds without to distort the fifths too much (the residual variation coming from the coma pythagorician remains always concentrated on the fifth of the wolf).

During the construction of the range pythagorician, one obtains the first major third (DO-MI) by four successive rises of fifths: DO-SOL, then SOL-RE, then RE it, finally LA-MI. This Semi pythagorician differs from Semi from the pure third from a syntonic Comma (or coma zarlinien). If thus one wants a temperament mesotonic with pure major third, it is enough to divide the syntonic coma into 4, i.e. to correct fraction 3/2 (the fifth) of the coefficient (3⁴/(5 X 2⁴))^1/4 or 3/(2 X 5^1/4), and to add the fraction corresponding to each fifth in the progression of the cycle of the fifths: the fifths will be a little false, but the thirds will be right.

Since the correction applies uniformly to all the fifths, the major thirds generated remain always equal to two tons major (the proportions are preserved) what is not the case with the unequal temperaments. It is this property of the “average tone” which is at the “mesotonic” origin of the term - one uses also the expression “temperament regular”·

If we apply this correction to the complete cycle of the fifths (that is to say 12 fifths) we will have reduced the octave of three syntonic comas on the whole (twelve quarters of comas): as the fifth of the wolf is too small of a coma pythagorician, we will add these three syntonic comas to him to preserve the pure octave, i.e. it will become now too large of (three syntonic comas minus a coma pythagorician) - value rather close to two syntonic comas because one remembers that the two types of comas have close values. Thus the fifth of the wolf remains false in the temperament mesotonic at third pure, but this time by excess.

The temperament mesotonic with quarter of syntonic coma is used the most. If it makes the thirds purer, it slightly distorts the fifths (thus besides that quads), and this is not indifferent because the ear is more sensitive to the purity of the fifths than to that of the thirds.

Other temperaments mesotonic present a better compromise by distributing “falseness” in a way more balanced between thirds and fifths: it is the case of the temperament with 1/6 or 1/8 of coma. To the extreme, the temperament with twelfth of coma makes disappear the fifth from the wolf, and is thus practically equivalent to the moderate range.

The temperaments mesotonic are practiced enough in the Baroque music, they allow acceptable modulations in tons close to the tonic.

Moderate range (equal temperament)

The moderate range is nowadays used in an almost universal way in the Western music. Only works former (approximately) to 1750 are possibly carried out nowadays in other systems, according to the practices in progress at the time of their composition.

The moderate range consists in, so to speak, “slicing the Gordian knot” of the disadvantages of all the other systems which tried compromises between accuracy of certain intervals, too not marked falseness others, possibilities of transposition and/or modulation. It quite simply consists in dividing the octave into twelve chromatic intervals all equal.

This simple idea allows all the conceivable transpositions and all modulations, since all the notes are equivalent when one regards them as tonics. It presents only one disadvantage, but which is of size and which explains the reserve of the musicians to adopt it before the period known as “traditional”: except for the octaves, all the intervals are slightly false. However, the variations are sufficiently weak to be acceptable. And the practice helping, since nowadays almost all the musics which we hear use it, this weak dissonance does not shock anybody, and in fact on the contrary the old temperaments surprise our ear when we try out them for the first time.

If one remembers that to add with the intervals amounts carrying out multiplications of reports/ratios of frequency, one sees that the octave equalizes the high chromatic semitone with the power twelve or that the chromatic semitone is worth $\ sqrt {2}$ .

The moderate range has the advantage of being completely " neutre" compared to the problems of transpositions, which precisely justify the presence of temperaments. The variations of the various ranges compared to the moderate range thus make it possible to appreciate how a particular temperament will present irregularities in its various transpositions.

Comparative tables

Comparison of the frequencies of notes of the chromatic range in various systems

First table: even

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♭.

Second table: even C

In this table:

the note C commune to 264 Hz gives 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 SOL♯ and MI♭.

Third table

In this table, the intervals are calculated starting from the preceding table:

In the “right intonation”, the fifth and the third is right, the fifth of the wolf is awfully false
In the range of Pythagore the third and the fifth of the wolf is slightly false
In the moderated range, it does not have there a fifth of the wolf; the fifths are good, and the thirds a little too large