Fundamentals                  

       Madroños (F. Moreno Torroba) (interprete Rene Mora).

 

 

Up to Euler and dÁlambert, music and mathematics were studied together, having had as a pattern,the union of ideas of proportions, Pythagoras, Ptolomy, Descartes ( Musical Compendium ), Galileo ( Discource ). Nonetheless, the new theoríes of harmony with which Euler worked, were too mathematical for musicians.

On the other hand music progressed to unimaginable limits, ( Bach, Haendel, Mozart, Beethoven ). The result was that mathematics was too complex for musicians and music became too musical for mathematicians.

This separation is apparent since composers and musicians have continued using the idea of symmetry, musical progression or criteria of proportions, for example, that represented by the number of Fibonacci ( Leonardo de Pisa), and Chopin ( 1810-1849 ) stated that the fugue was pure logic.

 http://www.anarkasis.com/pitagoras/052_matematica/

 

 http://www.anarkasis.com/pitagoras/061_modulo/

It seems as though until very recently that musicians kept to simple mathematical ideas without arriving at new mathematical development, possibly conditioned by the general public demand.

Since the fundamentals of the System are basically mathematical, let us capitulate on the union between music and mathematics, precisely in its construction of musical instruments, but done only in a conceptual way.

 The first thing is to understand and prove something I beleive everybody knows, that is how the shape of an instrument affects the sound. Let´s reflect on some simple examples.

Take a tube and a bell, both constructed with the same metal and the same thickness in body. It´s easy to hear how differently they sound; whatsmore, if you close the tube at either extreme the sound changes

If furthermore, we adjust its díameter and curves with the harmonic tuning frequencies of the bell, we would obtain one tuned to a determined pitch, with at least two other octave harmonics, one low and the other corresponding to the higher octave, and consequently achieving greater sound duration because of the inter-feedback of the harmonics, therefore, a better and larger sound and of course greater projection.

Si además, ajustamos sus diámetros y sus curvas a las frecuencias armónicas de afinación en la campana, tendremos una afinada a una determinada nota, con al menos dos armónicos de octava, uno grave y el otro el correspondiente a la octava aguda, consiguiéndose además, una mayor duración del sonido, por una auto-alimentación de los armónicos y por tanto, una mejor y mayor sonoridad y por supuesto con una mayor proyección.

Another simple observation is that of the glass more or less filled with water. The emitted frequency ( the musical note ) is different depending on the quantity of water we add; therefore, the length of the vibrating side and empty space condition the note.

Another very familiar example is when we fill a bottle or a tube. As we fill it the sound changes in continuous “glissando” up to the higher tones. The sound effect there will be frequencies adjusted to notes that fit with a tuning reference and others that are intermediate; that is to say, there are adjusted volumes and other intermediate ones that are not.

Helmholtz related the sound with the volume of the resonator and also with wind instruments with length and area of mouth piece, though with possible solutions nowadays.

The first question that arises is whether or not the volume of the guitar is adjusted.

 

Eduardo Thenon presents an interesting article in which he questions the acoustic identity of the guitar in relation to the response of different tuning pitches.

                 http://guitarreando.iespana.es/identidad.htm

  …..Torres may have designed a body in such away as to amplify the generating harmonics tuned to the pitch of 404 cycles………,

 …… modern guitars sound on the basis of old pitch, which means that the instruments are not modernised. This is a seriuos conclusion, which having risen from analysis we beleive it to be unavoidable…………..

……the new acoustic guitar identity will consist of coinciding the “ A “ of resonance or harmonic constructión with standardly used “A “. This way the resonance activity will be able to create harmonic richness , without sacrificing the conditions of a modern instrument.

Apart from the harmonic clashes present in the guitar today……  The solution of eliminating the instruments resonance as a way to solve undesirable reverberations, can also be avoided………………

 

In general the present day guitar is designed with the idea of proportions and is done based on the 650 mm string length, the most required; and so the body is ¾ of that of the length, the large waist is ¾ that of the body and the small waist is ¾ that of the large waist. Therefore we need to prove whether or not the vibrating surface is adjusted, in other words-is the frequency 440 Hz the most adequate for the most required length ( 650 mm ).

The author of the System presented a publication titled “Orchestral tuning and the guitar length; two unresolved matters

                http://opusmusica.com/003/guitarra.html

  Andres Segovia always defended the 660 mm ( 66 cm) length and even longer. Many other classical and flamenco guitarists coincided. On the contrary, another group of guitarists required 650 mm ( 65cm ), even less, owing to anatomical reasons.

 In another section, the author continues that the relation between the nodal distances and emitted frequencies are inversely proportional with the constant of harmonic and nodal progression.

……….a nylon string constitutes a different module of elasticity to that of steel; nonetheless, both are adjusted to a determined tuning frequency, both have the same nodal progression ( frets ) and the same musical progression ( frequencies ), therefore, the frequency progression and the nodal distance progression is independent of the material used,…………...

In other words the musical progression is a function of λ and to pass from one semitone to another we divide or multiply by λ, the same thing happens with distribution of frets or nodal distances which we multiply or divide depending on the distance we want to obtain.

 2 -1/12 = λ = 1.0594630943593

 The solution is obtained by seeing both sides right. Those who defended the 660 mm length, because it is adjusted to the tuning frequency of 440 hz (London Convention, 1939) (659,25 mm). Those who required the 650 mm length are also right because they are unconsciously defending Universal Harmonics (645mm).

 The Schiller Institute has defended singers ( tenors, baritones, sopranos ) in relation to justification of Universal Harmonics, which adjust to the tuning used by Verdi and defended by the Italian government in that epoch.

            http://schillerinstitute.org/newspanish/InstitutoSchiller/Arte/afinacionantigua.html

……….  the most outstanding scientist in this area have newly argued that the coherence of “A“=432 (corresponding with C5=256) with the laws that govern the physical universe from the functioning of the solar system , to the production of the human voice. From a more advanced scientific approach they have demonstrated that any other tuning based on “A“=440,443 OR 450 not only does not reflect the natural laws but even violates them and can actually produce injuries to the voice .

One small clarification; “A“= 432 adjusted to the Universal Harmonics is “A“ =430.54 Hz.  To follow up this issue consult he following address:

Following the issue of orchestral tuning, please consult:

              http://es.wikipedia.org/wiki/Afinaci%C3%B3n

Show us the different tunings that have been used.

It shows that “ A” at 440 hz is a rounded off medium of all the tunings that have existed.

 Another basis of the research for the development of the System was to reflect on the evolution of tastes and musical requirements of each epoch, as well as technological means and available building materials, etc.

 For instance the characteristics of the materials used for strings (gut are completely different from todays nylon and/or steel is a convenient use of technological development, seeing that many strings used today are made for applications other than music.

In the near future, the mathematical-physical criteria will be used for the manufacturing specifically for musical applications.

 A both simple, and simplified example which appears in any musical treatise; two functions which can allow us to illustrate on two important questions. Firstly, why harmonics reinforce the principal note; and secondly, how different measurements are related with the frequencies that musical instruments emit ( length, size and shape of the sound-board, volume of the body etc.) these simple examples of trigonometry functions allow us to see once more the relation between music and mathematics.

The axis are:

X lengths

Y units of sound strength

 

 

When the waves interfere with the axis x the carrying energy is 0, therefore it is the point where for instance, the ends of the soundboard should be situated..

This graph marks the distance and its multiples, where the nodal breaking points should be, that is to say, the end of the soundboard, since the point in which the two wave lines meet, distances ,in the case of sen 4x, at twice the wave length of this function, and in the case of sen6x, at three times.

There is another secondary mode at 1 of the wave length and in the function sen 4x and at 1 plus ½ in the function sen 6x.

The question remains of how we can calculate the different frequencies. The answer is that all the frequencies emitted by an instrument are multiples of the constant of the progression λ and of the frequency of reference, nowadays 440 Hz.

These reflections take us on to the interior structure of the instrument.

 In all string instruments, the two fundamental parameters which should be studied are: the distribution of forces generated by tensing the strings and their characteristics of vibration as generator of sound.

Until recently, the element most considered was the sound board, and to avoid any deformations against the generated force caused by tense strings. The correct wave propagation is precisely the vibrating element, what generates sound, that was not considered.

On the guitar the direction of the dominant force is traction. Nonetheless, when the strings are tied to the bridge, on the soundboard and just under the point below the bone, there is a component of negative stress on the front part.. That is the bridge presses inwards with another positive deformation (raising) on the back part where the strings are tied. Therefore, if we add to the effect of traction we have flexo-rotation, which rightly justifies our hitherto know bracing structures.

 The systems used before Antonio Torres Jurado, used two straight struts in front of and behind the bridge. His innovation was using a fan bracing system to compensate for these deformations proving that all tensions are distributed throughout the soundboard, he even built a guitar with cardboard waists and back. This system evolved with numerous modifications, nonetheless, it is essentially the same that is used up to today.

 Recently, the net system has appeared, some regular and others irregular shaped which proposed a mix of the old baroque style system with the fan brace system.

 In other cases hybrid materials (natural-carbon, fibres), undoubtedly question those studies which attribute the good sound of an instrument to aged woods.

The final evaluation is summed up as simple as better or worse sounding, which is how we can all accept it as something subjective. Nowadays, we we have techniques which allow us to evaluate and quantify the quality of sound, in clarity and volume. Two instruments recorded in the same conditions give two different curve spectrums.

In any case the common denominator of these designs is to compensate those two effects of soundboard deformation: traction and flexo-rotation.

Nowadays, thanks to the capacity of calculation that computers offer it has been possible to create a different design, making compatible the force distribution that the soundboard bears with the effects of vibration and the nature of propagation through the board (wave movement ) which are the sound generating agents. 

Sound in the air is propagated in a spherical shape, on the other hand in the soundboard the way of propagation is by an abstraction ,approximately elipsis and flat. It is obvious that it is flat, and that it is not spherical, seeing that the waves travel master in the direction of the wood grain, as the elipsis has two centres ( focuses ), unless correcting this form, sound emissions that generate destruction of sound can occur. This is why the curved bracing perpendicular to the grain of the wood was invented.

The interest to better the sound of the guitar has been permanent as shown by numerous patents carried out by important makers and musicians; some of which are:

 Narciso Yepes                                                1964

Michael Kasha                                                1967

Melchor Rodríguez                                          1973

Melchor Rodriguez                                          1980

Jose Ramirez Martinez                                     1984

Manuel González Contreras                             1988

Antonio Losada Ordoñez                                 1989

Santos Bayón Ruiz                                           1994

Mariano y Felipe Conde Cavia                         1998

 What is the desired sound of the guitar? Everybody coincides in sufficient sound volume. Nonetheless, if we delve into tone colour, each guitarist has a different appreciation of sound. One fact is that on raising the volume, we lose part of the magic of the characteristic delicateness of the guitar. Therefore, the dilemma of whether or not we maintain the timbre and fragile sound or do we increase the volume. one commonly heard expression by guitarists is ”it sounds strong but unlike a guitar“. This is why we have opted for raising the volume a little, but mostly having more clarity which gives better projection through other voices. An example is when in a full room with mixed conversation it is the high and clearest voice that project and not necessarily the one with greater volume.

 This projection has been displayed in concert in different auditoriums, theatres, concert halls etc. Sound recordings in different parts of the hall show the little loss of sound with regards to the micro pone placed on stage.

Greater clarity and naturalness is noticed in recordings without the need for equalization in such a way that the sound difference is noticed.

 The background music throughout the whole web are samples from a direct recording without equalisation nor corrections, needless to say, the dome effect disappears, and, on the other hand clarity and definition is gained giving a realistic sense of presence.