Natural Range

A range is known as natural when the sounds which compose it (in an interval of octave) result from the harmonic of the tonic note . Because of this definition, one also speaks about range of the physicists . This definition is sufficiently vague to allow alternatives, but they have jointly to make play a big role with the third interval of major.

The natural ranges, just like that of Pythagore, have important disadvantages as regards transposition and of modulation: of this fact they are not used in practice.

This article gives in extenso the theory of the ranges known as “natural” . 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 Western music.

Introduction

The existence of the harmonics was shown by the scientist acoustics expert Joseph Sauveur and was used for Jean-Philippe Rameau to work out his theory of the traditional harmony. For example, it justifies the preeminent place of the major triad (DO-MI-SOL) in what these three sounds are organically dependant between them as first harmonics of a fundamental note, C with the lower octave, and by applying the principle of the equivalence of the octaves. Thus, for this author, the harmony preexists to the melody and constitutes quintessence even music.

There is not a doubt that the phenomena of Consonance were identified by the first musicians before the mathematicians do not work out a theory of it. The first natural ranges, created in an empirical way, thus preceded certainly by very a long time the Gamme pythagorician, building algebraic rather complex.

The range pythagorician is built starting from a particular harmonic, the fifth, then by successive rises of fifths the number of times necessary to traverse a complete octave. The sounds obtained by this method are increasingly complex harmonics of the fundamental sound. It was also seen that this method does not make it possible to directly find the quad which is however a very simple harmonic (4/3) of this one (and obligatory complement of the fifth). The range pythagorician, moreover, result of remarkable theoretical speculations, is not without defects:

the problem of the Coma, solved for want of anything better by the “fifth of the Wolf”, prohibited certain combinations of notes and certain modulations;
certain very intuitive intervals, and particularly the major third (DO-MI) are not generated in a perfect way, and sound, actually, enough forgery.

From where attempts of the theorists to implement other methods, based on other considerations.

Harmonic or partial sounds

A continuous invariable musical sound results from the superposition (or combination) from a simple sound and its harmonic sounds whose frequencies are multiple entireties of its own frequency. One calls it second harmonic the double sound of frequency, third the triple sound of frequency etc For a its inharmonic variable , one calls his components partial sounds. They are very close (generally acuter) to the harmonic frequencies, but are not multiple entireties of the fundamental sound.

If one leaves C 0 by taking his frequency like unit:

partial n°1: frequency 1 (=DO 0)
partial n°2: frequency 2 (=DO 1)
partial n°3: frequency 3 (=SOL 1)
partial n°4: frequency 4 (=DO 2)
partial n°5: frequency 5 (=MI 2, pure third)
partial n°6: frequency 6 (=SOL 2)
partial n°7: frequency 7 (resembles Sib 2)
partial n°8: frequency 8 (=DO 3)
partial n°9: frequency 9 (=RE 3)
partial n°10: frequency 10 (=MI 3, pure third)
partial n°11: frequency 11 (intermediary between F and Fa#)
partial n°12: frequency 12 (=SOL 3)
partial n°13: frequency 13 (resembles It)
partial n°14: frequency 14 (resembles Sib 3)
partial n°15: frequency 15 (resembles If 3)
partial n°16: frequency 16 (=DO 4)

etc

The names of the notes above correspond to the heights defined in the range of Pythagore except for the SEMI ones. As it is seen, the note GROUND is a harmonic of the note C, but not of that which precedes it in its octave: C 0 for GROUND 1, C 1 for GROUND 2 etc Donc the interval of fifth (report/ratio 3/2) connects two notes - C 1 and GROUND 1 for example - of which acutest is not a harmonic of most serious; however both are harmonics of the same third notes more serious. It is thus by an abuse language, that the principle of the equivalence of the octaves authorizes, that one can state that GROUND is a harmonic of C. It is as by convenience as, in the same way, one will consider in what follows as in harmonic report/ratio of the sounds whose relative frequencies are in rational report/ratio one compared to the other: there exists a sufficiently serious note then (but perhaps inaudible!) they are both of partial truths.

The purely harmonic intervals are those for which the report/ratio of frequency is a Integer. The elementary relations are those which get along easily in the first harmonics produced by a string instrument or copper. These intervals are defined gradually starting from the fundamental note by increasingly complex reports/ratios:

the octave (multiplication or division by two);
the fifth and its interval reverses the quad (multiplication or division by three);
the third and its reverse the sixth (multiplication or division by five).

The following interval, seventh or second (multiplication or division by seven) appear a little false with the ear, and the prime number following (multiplication or division by eleven) falls exactly halfway between the quad and the increased quad. ; Complexity of a natural interval The harmonic intervals are all the more obvious with the ear that the division or multiplication factor is a factor first small. To fix the ideas, one can define the complexity of a musical, definite interval like a simple report/ratio of frequencies, in the following way:

the complexity of unison is null;
When one multiplies or divides a report/ratio by two, the complexity of an interval of octave is increased by a unit;
When one multiplies or divides a report/ratio by three, complexity is increased by two
And more generally, when one multiplies or divided by N , N being first, complexity is increased by n-1 .

With this definition, a report/ratio of fifth (3/2) has a complexity of three (2+1), and a report/ratio of third (5/4) has a complexity of 6 (4+1+1).

In practice, the ear tends to assimilate the intervals at reports/ratios of low complexity. So the totality of the usual sounds can be restored by factors which utilize only 2,3 and 5, factor 7 not being differentiated in practice from Sib, and factor 11 of a complexity too much large to be really perceptible in harmony.

Harmonic intervals

By repeating these elementary intervals which correspond to factors 2,3 and 5 (and possibly 7), and by privileging the reports/ratios of low complexity systematically, one obtains the following frequencies, where are superimposed the intervals purely pythagorician (fifths and quads) and the other natural intervals - primarily built on the third and the sixth (in practice, the seven multiples and eleven are not necessary).

One sees on the table according to whether several harmonic intervals can be proposed for certain notes (third, sixth,…), and that the least complex interval is then that which utilizes a multiple of five (what justifies the definition of the " complexité"). It is also seen that whatever the interval retained for Fa# (the Triton), his complexity is high in any case, which joined its reputation of particularly dissonant interval. The column of the " Écarts" give the variation (in " cent" , i.e. in hundredth of tone) of the pure frequency compared to the frequency with equal temperament (a variation of about ten hundreds is in practice inaudible, but changes the coloring of the note). It is seen that an equal temperament gives correct fifths and quads (with a margin of 2 hundreds), but of the too high thirds (thus too " dures" or " brillantes") compared to the harmonic, and contrary to the too low sixths (too much " sourdes" or " ternes").

The theory

The simplest harmonic, the fifth (interval C - GROUND, report/ratio of frequencies = 3/2) is the base of the Gamme of Pythagore which is very old and probably at the origin of our division of the octave in seven notes: C - RE - MID- F - GROUND - - IF. The fifth has as a complementary interval the quad (interval GROUND - C or C - F, report/ratio of frequencies = 4/3).

The octave, the fifth and the quad have reports/ratios of frequency which all are of the form (n+1) /n - N being an integer. The interval which separates the quad and the fifth is the tone or second major (interval F - GROUND or C - RE, report/ratio of frequencies = 9/8): it is same form (n+1) /n.

But the method of Pythagore produces an interval of third which sounds relatively false (report/ratio of frequencies 81/64), from where the idea to substitute to him an interval of the form (n+1) /n which of either very near, or 5/4 - the major third “pure”. The difference between these two intervals is the syntonic Comma .

On the basis of this new major third, one from of deduced directly:

with the quad: the interval of major sixth (interval C -, report/ratio of frequencies = 5/3 is 5/4 X 4/3)
at the fifth: the interval of seventh (interval C - IF, report/ratio of frequencies = 15/8 is 5/4 X 3/2).

We thus determined the seven notes of the diatonic range.

The construction of the ranges known as “natural” will consist in dividing the octave, in the most regular possible manner, by using for intervals the rational reports/ratios (with the mathematical direction of this term) simplest possible.

There exists an infinity of reports/ratios of the type n/m (N and m being integers representatives of the frequencies) giving values ranging between 1 (it tonic or fundamental) and 2 (the octave). It thus will be necessary to choose, in this infinite whole of possibilities, the most suitable reports/ratios, which can be done by the pure intuition or by setting rules a priori.

The numbers seven (diatonic notes) and twelve (chromatic notes) released by the range of Pythagore - chronologically the first consistent theory of division of the octave - were in a way more or less conscious of the numbers to be found in the establishment of natural ranges: the chance actually provided for it…

By introducing “of force” the interval of quad into the range pythagorician, this one includes/understands in particular intervals 2/1 (octave), 3/2 (fifth), 4/3 (quad) and 9/8 (your major), all the form (n+1) /n.

After the quad and before the your major , this form algebraic would give us 5/4 (=1,25), 6/5 (= 1,2), 7/6 (1,166666?) and 8/7 (=1,142857?).

The two first are particularly simple, acoustically they sound well with the fundamental one and they are very close to certain intervals to the range pythagorician, those which we indicated by I₄ (= 1,265625) and I₉ (=1,201354). 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.

One calls syntonic coma the existing interval between the major third “pure” (5/4 = 1,25) and the third pythagorician (3⁴/2⁶ = 1,265625): its value is 81/80, that is to say 1,0125, slightly lower than the coma pythagorician.

Gioseffo Zarlino (1517 - 1590) works out one of the multiple possible natural ranges by recognizing an important place with the third interval of “pure”.

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).

According to Pythagore, the sixth (27/16) is the addition of a third major (81/64) and of a quad (4/3) since 27/16 = (81/64) X (4/3).

Let us apply this formula for Zarlino: the sixth is worth (5/4) X (4/3) = 5/3.

According to Pythagore, the seventh (243/128) is the addition of a third major (81/64) and of a fifth (3/2) since 243/128 = (81/64) X (3/2).

Let us apply this formula for Zarlino: the seventh is worth (5/4) X (3/2) = 15/8.

According to Pythagore, the your major is half of the third major because (9/8) ² = 81/64. We cannot make so that (9/8) ² = 5/4 and must thus replace a your major by a your minor such as your minor X (9/8) = 5/4. The your minor (REMI interval) is worth thus 10/9.

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, then by replacing in this formula the value of the third pythagorician by that of the third pure . We will treat, as illustration, notes DO♯ and RE♯, with a simplified symbolism which one will interpret easily:

DO♯ (P) = 2087/2048 = 2T-Q from where DO♯ (Z) = 25/24

RE♯ (P) = 19683/16384 = 2T + Q - O from where RÉ♯ (Z) = 75/64

etc

Certain notes, which can appear useless (examples: MI♯, DO♭ etc) find their use in the development of the musical modes and for the modulation.

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.

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.

In addition, 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.

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.

The other natural ranges have similar disadvantages.

As for the range of Pythagore, these musical problems encouraged the theorists to consider new principles of division of the octave.

Table of synthesis

See too

Articles in relation

Ranges and temperaments

Temperament by multiple division
Its (physics)

External bonds

Construction of the range of Zarlino

Bibliography and sources

Deviates Dominique: the temperament musical, philosophy, history, theory and practical , International Musical Bookstore, Marseilles (second edition 2004).
Moreno Andreatta : " Algebraic methods in music and musicology of the XXe century: theoretical, analytical aspects and compositionnels" , thesis, EHESS/IRCAM, 2003 (available on line to the address: http://www.ircam.fr/equipes/repmus/moreno/).
Edith Weber: Resonance in the musical scales, revision of Edmond Costère, Re-examined musicology, T.51, N°2 (1965), pp. 241-243 - DOI: 10.2307/927346
Edmond Costère: Laws and styles of the musical harmonies, Paris, PUF, 1954.
Edmond Costère: Died or transfiguration of the harmony, Paris, PUF, 1962.
Franck Jedrzejewski: Mathematics of the acoustic systems. Contemporary temperaments and models, Harmattan, 2002.
Guerino Mazzola: " The Topos Geometry off Musical Logic" (in Gerard Assayag and Al (ED.) Mathematics and Music, Springer, 2002, pp. 199-213).
Guerino Mazzola : The Topos off Music, Birkhäuser Verlag, Basel, 2003.
François Nicolas: " When the mathematical algebra helps to think (and not only to calculate) combinative the musicale" , Seminar, Ircam, February 2003 (available on line to the address: http://www.entretemps.asso.fr/Nicolas/TextesNic/mamux.html).
E. Lu, G. Mazzola and T. Noll (ED.), Prospects off Mathematical and Computer-Aided Music Theory, EpOs, University of Osnabrück, 2004.

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