Temperament by multiple division

In the Theory of the Western music, a temperament by multiple division consists of a division of the octave in addition to twelve interval S elementary.

Temperaments by multiple division

Several theorists conceived temperaments based on a division of the octave in addition to twelve elementary intervals. Noting that division in twelve equal intervals does not lead to the purity of the intervals of fifth and third, they sought if a division of the octave in a number different of intervals did not make it possible to approach this ideal purity. In fact, several diagrams of division were thus determined, which sometimes make it possible to also improve quality of the other notes. It is besides an obviousness which more the elementary interval is small, better is the approach of the purity: just as a scale in mm a better measurement gives than a scale in cm.

The methods are varied but several of them implement the following technique:

1. to determine two “traditional” intervals $I\_1$ and $I\_2$ connected to the octave (noted $O$) by a relation of the type: $O\; =\; n\_1\; \backslash \; cdot\; I\_1\; +\; n\_2\; \backslash \; cdot\; I\_2$,
   $n\_1$ and $n\_2$ being integers;
2. to determine two integers $N\_1$ and $N\_2$ such as $\backslash \; frac\; \{N\_1\}\; \{N\_2\}$ is nearest possible to $\backslash \; frac\; \{I\_1\}\; \{I\_2\}$;
3. if the two reports/ratios above were rigorously equal, there would exist an interval $i$ such as $I\_1\; =\; N\_1\; \backslash \; cdot\; i$ and $I\_2\; =\; N\_2\; \backslash \; cdot\; i$. There would be thus $O=n\_1\; \backslash \; cdot\; N\_1\; \backslash \; cdot\; I\; +\; n\_2\; \backslash \; cdot\; N\_2\; \backslash \; cdot\; i$ or $O=\; (n\_1\; \backslash \; cdot\; N\_1\; +\; n\_2\; \backslash \; cdot\; N\_2)\; \backslash \; cdot\; i$ or finally $i=\; \backslash \; frac\; \{O\}\; \{(n\_1\; \backslash \; cdot\; N\_1\; +\; n\_2\; \backslash \; cdot\; N\_2)\}$
4. one defines $i$ then, calculated like above, as bases temperament.

This method introduced, inter alia, the temperaments with 19,31 and 53 elementary intervals per octave, but much of other ideas were developed to lead to the ideal temperament - which does not exist.

The temperaments by multiple division do not have legitimacy that if they bring a true advantage in terms of musical quality. It is in general more of theoretical curiosities that systems really implemented and having been used as support with philosopher's stones. It can be practiced simply only with the voice or certain instruments with variable intonation (family of the violins, certain coppers) or by instruments with fixed sounds conceived for them and comprising “exotic” devices such as keyboards slipping, double keyboards, divided keys etc Their little of practical success is also due to the difficulty of the play which requires a specific training of the artist: the play is worth hardly the candle of it!

Temperaments based on 5 your S and 2 Semitone S

These temperaments consider that the octave is divided into 5 tons and 2 semitones, the latter being close to half of a tone - say that a tone will lie between 1,5 and 2,5 times the semitone.

If the proportion is of two, one is brought back to the octave divided into 12 equal intervals, because 12 = 5 X 2 + 2 X 1: it is the equal Tempérament usual with twelve equal intervals per octave.

While varying this proportion, one determines, inter alia, the temperaments with 19,31,43 and 53 intervals per octave:

• 19 = 5 X 3 + 2 X 2: the proportion is of 3/2 and the tone is worth 1,5 semitones;
• 31 = 5 X 5 + 2 X 3: the proportion is of 5/3 and the tone is worth 1,666… semitones;
• 43 = 5 X 7 + 2 X 4: the proportion is of 7/4 and the tone is worth 1,75 semitones;
• 53 = 5 X 9 + 2 X 4: the proportion is of 9/4 and the tone is worth 2,25 semitones.

One can as note as numbers 19,31 and 43 are form 12 X N + 7.

Obviously, more the elementary interval dividing the octave is small, more one is likely to obtain a temperament approaching the right intonation as well as possible. However, one runs up then against the question, very concrete, of the “jouability” about instruments with hand drive operation (keyboard, wind instruments with key). It is this essential aspect which prevented the real use of these temperaments, remained for the majority in a theoretical state of construction - an electronic and/or data-processing instrumentation can on the contrary carry them out.

Temperament with 19 equal intervals

The equal division of the octave in 19 intervals has the effect generally more closely of approaching the intervals diatonic of the right intonation - which are the reference in the field of hearing and the music - that the temperament equal to 12 semitones (usual moderate range).

The approximation is even almost perfect with regard to the minor third and the major sixth. But the other chromatic intervals are rather less good.

In the table below, one deferred only the degrees which offer a favorable comparison.

Temperament with 31 equal intervals (or of Huyghens)

Recommended - but not invented - by Huyghens, this temperament can be introduced simply by considering the proportion of 5/3 between the tone and the semitone.

It is that this proportion is practically that which exists between the tone and the semitone diatonic of the Tempérament mesotonic with ¼ of syntonic coma.

In this last temperament, indeed:

• the moderate fifth is worth 5^1/4…
• the tone T, either 2 decreased moderate fifths of an octave is worth 5^1/2/2, approximately 1,118034
• the semitone diatonic D, or 3 decreased octaves of 5 moderate fifths is worth 2³/5^5/4, approximately 1,069984
• one can check that T³ and D⁵ have a very close value, approximately 1,4.

So now one considers the temperament with 31 equal intervals, with a tone which is worth 5 of them and one semitone diatonic which is worth 3 of them, its fifth is worth 18 of them and it thus equalizes 2^18/31.

One can then calculate the interval enharmonic si$\backslash \; sharp$-do.

• in the range pythagorician, it acts of the Comma ditonic.
• in the temperament of Huyghens, it is worth 7 octaves minus twelve fifths:
  

27/(2^18/31) ¹², or, after simplifications

2^1/31 i.e. the elementary interval of Huyghens, which thus plays at the same time the elementary part of interval and coma! In complement of this temperament, the Dutch scientist recommended a system of slipping keyboard allowing to concretize it but which remained in state of a theoretical and mechanical curiosity.

Temperament with 43 equal intervals

This temperament present of interesting characteristics and the characteristic to be able to be introduced by a very different reasoning: to see below Temperament of Saver

Temperament with 53 equal intervals (or of Holder)

This temperament is directly resulting from the Gamme pythagorician.

Calculations - detailed in this last page - show that an octave is divided there into a sum of 5 apotomes and 7 filed. If one takes account of what, approximativement, the apotome is worth 5 Comma S pythagoricians and filed it in is worth 4, the octave is worth approximately 53 comas, that is to say 5 X 5 + 7 X 4.

This naturally brings to introduce the elementary interval I , very near to the coma pythagorician, such as i⁵³ = 2: the temperament obtained is regular and as well as possible approaches the principal intervals of the range pythagorician, with a possible adjustment on the level of the thirds, which can be made almost pure - because the syntonic coma is itself close to the pythagorician.

The theoretical interest of this division of the octave was highlighted as of Antiquity. The Chinese theorist Ching Fang (78-37 before JC) indeed discovered that 53 pure fifths are very close to 31 octaves ( N.B. number 31 gives also a possibility of division of the octave - to see above). He calculated the difference between these intervals which can be approximate, with 6 decimals, by fraction 177147/176776. Well later, this observation was again made by the mathematician and theorist of the music Nicolaus Mercator (about 1620-1687) which gave the formula 3⁵³/2⁸⁴ of it: it is the Comma of Mercator. The coma of Mercator is a low value (approximately 3,615 hundreds) which, distributed on the 53 fifths, becomes negligible in practice. Holder in addition noticed that this temperament makes it possible to approach very close the pure third (with a variation of 1,4 hundreds) - all these characteristics thus make an excellent temperament of it, from the theoretical point of view.

Temperament of Saver

Joseph Saver , erudite mathematician and founder of the Acoustic although deaf person (!) a particular temperament starting from a property of the natural Gamme imagined diatonic and while profiting from an algebraic fortunate coincidence.

Let us consider the reports/ratios of frequencies of the eight “right” notes of an octave:

• C 1
• RE 9/8
MID
• 5/4
• F 4/3
• GROUND 3/2
• 5/3
• IF 15/8
• C 2

it is easy to calculate the corresponding intervals diatonic:

• GILDS 9/8
• REMI 10/9
• MI-FA 16/15
• FA-SOL 9/8
• GROUND IT 10/9
• LA-SI 9/8
• SI-DO 16/15

Thus an octave (O) equalizes 3 tons major (T = 9/8) + 2 tons minor (T = 10/9) + 2 semitones diatonic (D = 16/15).

Saver defines the “tone average” T_m such as O = 5 X T_m + 2 X D
then from there, the minor semitone D = T_m - D
and finally the coma of Saver C = D - D.

The numerical values (possibly approximate) are the following ones:

• your major 1,125
• your minor 1,1111…
• your means 1,1194…
• 1/2 your major 1,0667…
• 1/2 your minor 1,0495…
• coma 1,0164…

One can then calculate how much there are comas of Saver in an octave, and calculation is done using the logarithms and shows that an octave is worth 42,6211… comas of Saver, value very close to 43. One finds, but by another process, the temperament with 43 equal intervals by octaves mentioned above.

There is another mathematical characteristic: the logarithm at base 10 of 2 is worth 0,30103 gold 301 (that is to say 1000 times this logarithm) equalizes 43 X 7. Thus, Sauveur can introduce a unit of musical interval, the heptaméride such as:

• the coma of Saver is worth 7 heptamérides,
• an octave is worth 301 heptamérides,

and, by approximation,

• an average tone = 7 comas = 49 heptamérides
• one 1/2 your major = 4 comas = 28 heptamérides
• one 1/2 your minor = 3 comas = 21 heptamérides

(one can check, while translating into heptamérides, that the octave is worth 5 tons average + 2 major semitones).

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