Range pythagorician

The range pythagorician is a musical Gamme particular, built on the Cycle of the fifths. For a presentation of synthesis, to see the article Ranges and temperaments which gives also an overall picture of the ranges of the traditional Western music.

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

The range pythagorician has for principal disadvantages:

• the fifth of the wolf , unusable interval;
• of the intervals of major third (DO-MI) which are enough far from the pure consonance;
• of let us tons diatonic which is not equal and which makes problematic the Transposition (the play of the same piece with a different tonic) and the modulation (the change, even temporary, of tonality during the same piece).

So it is practically not used nowadays.

General information and history

Historically, the first traces of a musical construction in octaves and perfect fifths go back to Chinese antiquity. Attribution in Occident of this type of construction in Pythagore seems to go back to the Middle Ages whereas it does not seem to have contributed directly to the establishment of such a scale. It does nothing but base one thought, which tries to include all pheonomenes of the universe, based on the first four numbers: 1,2,3 and 4. Indeed, these four simple numbers form the ratios of the intervals of octave (2/1), fifth (3/2) and quad (4/3).

But the theorization of the heptatonic range by the school of the pythagoricians is former two centuries to the contacts, moreover very indirect, between the worlds Mediterranean and Chinese which could follow the conquests of Alexandre the Large one. Thus, it is almost certain that it was done without reference to preceding Chinese, who gave rise to besides a pentatonic Musique very different.

The Harmonie of the spheres, theory of origin pythagorician based on the idea that the universe is governed by harmonious numerical reports/ratios, and that the distances between planets in the geocentric representation of the universe correspond to musical intervals.

Through the disturbed period of the cruel invasions, the continuity of the ancient Greek musical tradition was ensured in Europe, inter alia, by the songs of the Christian liturgy - the Gregorian chant taking its source as of the 7th century under the impulse of the Gregoire pope the Large one.

Parallel to the development of this scale in the Occidental culture, we also find it in the Arab writings, in particular in “the brothers of the purity” (Ikhwan Al-Safa) and Al-Kindī.

The range pythagorician was gradually forsaken in low Moyen-âge when one started to regard as consonnant the interval of third.

The approach of the construction of the range pythagorician can be done on considerations of acoustics or mathematics.

Musical introduction

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. The notes with flat are obtained by a cycle of successive perfect fifths, not while going up, but while going down.

But the piano cheating (see moderate Range). The mathematical part of this article proves the impossibility of this equality. Indeed, 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 by 1.0136: the result deviates from 1.36% of the initial frequency, either practically a eighth of tone (23.46 hundreds, or a coma).

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♯.

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. 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.

One can approach this construction by acoustics, while trusting within the meaning of hearing, or by mathematics.

Construction by acoustics

The Oreille allows, in an intuitive and very precise way (by the absence of Battement S), to identify an interval of octave or fifth.

On the basis of an unspecified and sufficiently low note, one determines his fifth then the fifth of the note obtained, in a reiterated way. If one repeats this process 12 times, one realizes that the final note is “so to speak” the seventh octave of the starting note. It is what one calls the “Cycle of the fifths” i.e. 12 fifths are equivalent about to 7 octaves. In fact there is a weak difference between these two intervals, which one calls the “Comma pythagorician” (see diagram).

These observations are at the base of the range “pythagorician”. One can start again the process, by lowering the successive notes obtained of an octave when the interval of fifth makes us leave the first octave: when 12 intervals of fifths are assembled, one will have had to lower 7 times of an octave to find, except for the coma pythagorician, the initial note. It, by doing this, will have been noticed that the successive notes obtained are distributed about uniformly in the interval of an octave. We will have thus built the “range pythagorician” with 12 intervals, therefore twelve notes, will have included/understood in the octave. The conservation of a pure octave is regarded by many musicians as impossible to circumvent.

The last fifth of our cycle (or another unspecified) will have to thus be preserved slightly different from the others, and will be relatively false: it is called the “Quinte of the wolf”; it will give place to different “the Tempérament S” the purpose of which is to attenuate to the maximum the disadvantages of them.

Mathematical description of the range

See also: Equivalence of the octaves

By taking again the example of the preliminary, one can build the continuation of the first 12 fifths which give us 12 notes that we will call by convenience B, C, D, E, F, G, H, I, J, K, L, m on the basis of the named basic note “has”. (NB. these letters do not have any relationship with the Anglo-Saxon notations).

These 12 notes are presented in the table below, with their variation with the starting (A) note.

The calculation of the last variation gives approximately 129,746 whereas the variation of 7 octaves gives 2⁷ = 128 exactly. The two variations are different. The notes obtained have a relative variation of

$\backslash \; frac\; \{3^\; \{12\}\}\; \{2^\; \{12\}\}\; /2^7\; =\; \backslash \; frac\; \{3^\; \{12\}\}\; \{2^\; \{19\}\}\; \backslash \; approx\; 1,0136$.

That is to say a difference of 1,36%. This variation is called the coma pythagorician . It is audible for many people, although very weak. To fall down on 7 right octaves, it is necessary to reduce a fifth (for example the last): instead of taking a variation of 3/2, one takes a variation of

$\backslash \; frac\; \{2^7\}\; \{\backslash \; frac\; \{3^\; \{11\}\}\; \{2^\; \{11\}\}\}\; =\; \backslash \; frac\; \{2^\; \{18\}\}\; \{3^\; \{11\}\}\; \backslash \; approx\; 1,480$

One of the reproaches which one can make with this continuation of note it is that it leaves large “holes” since fifth in fifth is advanced. But we as noticed as two separate notes of an octave bear the same name. It is thus enough to lower all these notes of the sufficient number of octaves so that they find all in the same octave. It is necessary for that to divide their frequency by a power of 2.

By carrying out this operation and by ordering the notes by their variation growing, we obtain the following table

The 12 notes of our range are thus obtained. One can be interested in the interval between two consecutive notes. It is enough for that to submit the report/ratio of their frequencies. We obtain the following table then

It is noted, and this fact is remarkable, that one does not obtain that two possible values for the variations

• 3⁷/2¹¹ either the apotome
• 2⁸/3⁵ or the filed

one can calculate that the apotome is higher than filed of a coma pythagorician.

The interval between the note has and C notes it being the major tone, this one equalizes a apotome + one filed: the “semitones” in the range pythagorician are thus not of the same value!

But this range offers another disadvantage, the reversed variation of the fifth (which should be the quad) does not appear in the list (variation of 4/3). The variation which approaches some more (of a coma) is 3¹¹/2¹⁷. One thus replaces this variation by variation 4/3. That returns in fact to give again with the last fifth '' its real size and to move the fifth of the wolf with before last position ''.

One obtains the following table then where the last line also changed. For more legibility, one will replace the differences between two consecutive notes by the letter has for apotome and L for filed.

It does not remain any more that to name the notes, that is to say 7 simple names for the simplest fractions (those whose numerator and denominator are smallest) and 5 names “deteriorated” for the most complex fractions (which is the last obtained in our cycle of the fifths). The first seven notes (“natural” notes) divide the octave into seven unequal intervals: it is about the heptatonic range. The faded notes make it possible to obtain finer and almost equal intervals between them: it is about the chromatic range.

This gives for the range of C

It is noticed whereas the variations of a apotome occur when the same note is preserved and that it is deteriorated: one calls this variation a chromatic semitone .

It is noticed in addition that the variations of one filed appear between two notes not bearing the same name: one calls this variation a semitone diatonic .

The process above is theoretical. In practice - if as well is as one still grants instruments to fixed sounds according to the range of Pythagore - one arranges oneself to place the “fifth of the wolf” on an uncommon interval where it is not likely to appear - in general: SOL♯ - MI♭. To note that the intervals “spanning” the fifth of the wolf are themselves false and to avoid.

the flat

It was noticed that the range which has just been built comprises only diésées notes.
C with DO♯ : 1 apotome (1/2 your chromatics)
DO♯ with D: 1 filed (1/2 your diatonic)
C with D: 1 your major

One can define, by reversing the order of the apotome and of filed in the GILDED major tone, a new intermediate note, RÉ♭ such as
C with RÉ♭ : 1 filed (1/2 your diatonic)
RÉ♭ with D: 1 apotome (1/2 your chromatics)
C with D: 1 your major

The notes with flat are obtained, on the basis of C, by downward fifths successive (contrary to the sharps, by ascending fifths). One has thus, in the order:
C - F - SI♭ - MI♭ - LA♭ - RÉ♭ - SOL♭ - DO♭ - FA♭ - SI♭ ♭ - MI♭ ♭ - LA♭ ♭ - RÉ♭ ♭

Starting from DO♭ (compare to IF) the notes obtained are assimilated to the diatonic notes with the coma near:
FA♭ = MI
SI♭ ♭ = the
MI♭ ♭ = RE
LA♭ ♭ = SOL
RÉ♭ ♭ = DO

The bemolized notes are lower of a coma pythagorician than their diésées enharmonic notes (for example RÉ♭ and DO♯). Attention: this is worth only for the range of Pythagore.

the alchemy of the figures in the range pythagorician

By taking again the range built previously and by making an inventory of the variations, one had obtained the following table

The preceding observations make it possible to affirm that

• a apotome - one filed = a coma pythagorician
• a tone = a apotome + one filed
• a third (for example the C-semi interval) = 2 apotomes + 2 filed = 2 tons
• a quad (for example the interval C-F) = 2 apotomes + 3 filed = a third + one filed
• a fifth = 3 apotomes + 4 filed
• an octave = 5 apotomes + 7 filed

one then finds the relation between 7 octaves and 12 fifths:

12 fifths - 7 octaves = 36 apotomes + 48 filed - 35 apotomes - 49 filed

12 fifths - 7 octaves = a apotome - one filed = a coma

One conceives without sorrow that the mathematicians of Antiquity, within sight of these somewhat magic relations, lent to the music a divine origin. More close to us, Rameau had even in the idea that the music was the base of mathematics.

For those which would reject (one includes/understands them) calculations of powers of 2 and 3, the diagram below summarizes the relations between the various intervals of the range of Pythagore (to notice the shift of MI♭ and SI♭) :

Properties of the intervals

A range pythagorician is all range (or scale) musical founded only on intervals of octave S and acoustically pure Quinte S (except one) - the Quarte S, inversion of the fifths, are it then too. An important property - and even founder - of such a range are that twelve fifths are equivalent “almost” to seven octaves: one will consider that these intervals are equivalent. However there is a residual variation which one calls “Comma pythagorician” or “ditonic”.

A property of the “range pythagorician” is the interval of “diton” (two tons “pure” successive: 9/8x9/8) lower than a quad. The diton form the interval called “third pythagorician”. It differs from a quad of the interval from “filed” (litt. “the remainder”). It is at Plato ( the Republic ) that we find the terms of the report/ratio of “filed” (256/243) any text of Pythagore did not reach us. The third pythagorician, from report/ratio 81/64 differs slightly from the “pure” third, report/ratio 5/4 or 80/64. There is between the two a variation which one calls “syntonic Comma” and to which the value is rather close to the coma pythagorician.

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

Table of synthesis: notes, intervals, frequencies

Is a tone worth 9 comas?

The table below presents the instantaneous frequency deviations (in approximate value) according to the number of comas

Knowing that a tone is worth 9/8 exactly is 1,125, it is seen that there is between 8 and 9 comas in a tone (9 comas will thus be taken)

Knowing that a apotome is worth 3⁷/2¹¹ exactly is approximately 1,068, it is seen that one can reasonably say that there are 5 comas in a apotome.

Lastly, knowing that one filed is worth 2⁸/3⁵ exactly is approximately 1,053, it is seen that one can consider that one filed is worth 4 comas.

This is why it is often considered that the octave is worth 53 comas (either 7x4 + 5x5): this equality is however only one approximation (because to raise the 3¹²/2¹⁹ number with power 53 cannot obviously give 2 for exact result).

This approximation is at the base of the general assertion according to which:

1. a semitone diatonic is worth 4 comas;
2. a chromatic semitone is worth 5 comas;
3. a tone is worth 9 comas.

It will have been understood that implicitly, the coma of which it is question is the coma pythagorician and which it is not a question of exact values.

One defines the “coma of Holder” like dividing exactly 53 times the octave. This coma, strong near to the coma pythagorician is at the base of a Tempérament by multiple division.

See too

Articles in relation

• Ranges and temperaments

• Coma
• natural Range
• moderate Range
• Temperament
• Temperament mesotonic
• unequal Temperament
• Temperament by multiple division
• Its (physics)
• Cycle of the fifths

Other articles

• theoretical and technical Glossary of the Pharaonic Western music

• Range

Bibliography and sources

• Pierre-Yves Asselin: Music and temperaments (Quebec), Jobert Editions, 2000 - ISBN 2-905-335-00-9
• Deviates Dominique, the temperament musical, philosophy, history, theory and practical , International Musical Librairie, Marseilles (second edition 2004).
• Patrice Bailhache: a history of the musical acoustics - CNRS Editions Paris 2001 - ISBN 2-271-05840-6
• 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 musical the , Séminaire, 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.
• http://www.univosite.com/gammepythagore.html

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