Virtual Analysis of Mayan Whistles

Experimental clay model

Roberto Velázquez Cabrera
Virtual Research Institute Tlapitzcalzin

First version March 26, 2001. Last actualization December 29, 2002

One Spanish version of this paper was presented in the 8º Mexican Congress of Acoustics, Oaxtepec, Morelos, México, November 14-16, 2001, organized by the Mexican Institute of Acoustics (IMA) and The National Center of Research and Technological Development (CNIDT).

The main object of the study is to make a virtual analysis of an important group of old Mayan whistles, using a official study (3) and an experimental Mayan clay model. The analysis is to respond in part to some commentaries emitted in a conference (1) by a teacher on organology Felipe Flores, the co-author of the analyzed study. I hope that the following exercises and commentaries are of utility to enrich his future works on the Mexican clay singers.

During the works of bibliographical search of a MS thesis on Mexican aerophones (2) the study of reference (3) was analyzed and included, since it is remarkable by several causes: A) it is the only case known in which there has been access to a wide set of aerophones of a specific ancient Mexican civilization, 356 Mayan whistles; B) it is the only known study in which they inform to have used computers to analyze old Mexican aerophones and: C) the study was published by the National Institute of Anthropology and History (INAH) in the Special Serie of Scientific Investigation of Archaeology.

It mentions that works were developed to capture the data of the whistles in formats (cédulas), classified and processed them in computer, took x-rays and made drawings of some of them and they used the analysis of the clay to obtain indicators of the possible precise origin, since that data is not known for many of the whistles. It is said that they selected them, because they had been archeologicaly studied already. The best content of the analyzed study is an X-ray of the whistle 5-0003 (Figure 1) taken by the Department of Physical Anthropology from INAH.

The excellent thing about the study, from the point of view of the acoustic analysis, is that includes resulting data of an analysis of internal tone-volume of the whistles to try to fit it to a mathematical or graphical function of behavior. The acoustic analysis of the whistles is very pertinent, because the substantive function of any aerophone is to produce sounds. However, several errors and limitations for the possible technical advantage of the information included in the document of the whistles are observed, of which the following ones may be emphasized:

The result of its measurements of the sounds includes groups of values of volume of the whistles that give the same tone. That result indicates that the parameters adapted according to the assembly of the considered whistles were not well selected, since they are very heterogeneous in shape and dimensions.

The main error of mathematical criterion was the one to try to fit to a function of very high degree to the series of points tone-volume of the whistles. It has , (Graph 2) 13 maximums and minimums and its algebraic description is not provided. It is known that the acoustic function must be one of second degree, logarithmic, hyperbolic or exponential decreasing, if one considers the well-known equation of tubular resonators (of Bernoulli) and the data shown in the study. That deficiency is remarkable, mainly, when seeing that they were advised by experts in computation of the UNAM (Autonomous National University of México) and the IMP (Petroleum Mexican Institute). That error indicates that it was not known how to apply or the elementary techniques of numerical analysis to fit mathematical functions.

A fundamental fault of organology knowledge is that it is not considered that the Mayan whistles under study are not tubular aerophones, reason why the equation of Bernoulli does not apply. It was used in Mexico for the first and only time in our history by the engineer colleague Daniel Castañeda (4), in his studies of Mexica and Purépech flutes (or Aztecs and Tarasc).

The mathematical model of German investigator H. von Helmholtz applied to globular resonators was not taken into account and that considers, in addition to the volume of the resonator and other factors, the geometry and dimensions of the hole of the mouth, to estimate the fundamental frequencies. The model of Helmholtz can be used for the acoustic analysis of some ancient aerophones, because it is possible to be applied to the globular forms of ocarinas and whistles like many of the Mayans. It is not possible to be argued that the studies of Helmholtz were secret, since they were published at the begining of the last century (5). That equation is described in the elementary books that are used in the acoustics courses in the schools of engineering and physics of our country (6). The author has already used and proven the equation of Helmholtz in experimental models of globular Mexican whistles (2).

The lack of knowledge of pertinent methodologies in the schools and professors of some fields related like music, ethnomusicology and anthropology is originated by the use of texts on musical acoustics and organology. For example the one of Tirso de Olazabal (included in the bibliography of the analyzed study) can serve to study the present instruments of music, but it is not useful to study the organology of other great old civilizations that were different and more varied in the field of the aerophones. The musical instruments do not include globular aerophones, like the Mayan whistles and many other sonorous devices of other Mexican civilizations.

An error of the study, frequent in the investigations on the Mexican organology made under colonized approaches, is the one to use criteria of present music to analyze the ancient sonorous devices, since "ancient music" was different (in scales, notes, diapasons, est.) and their sounds had other uses. In addition, using a piano (like metrology equipment) to fit notes of the whistles, is not a realistic form to analyze the sounds of ancient aerophones, since they do not have to agree necessarily with the discreet values of the present musical scales. That also avoids to get the fundamental frequencies between halftones. With that procedure, it is not possible to get the real pitch, because it is not known the scale of the piano, the tuning fork used nor the ranges of the sounds in pitch. It was a serious error to use the numbered keys of the piano instead of the value of the frequency in cycles per second or Hertz. The representations of the musical note sounds are of utility for the musicians, but it does not stop to make mathematical or scientific exercises, because they are not possible to be operated numerically.

It is mentioned that to know the tones of the whistles they used a "stroboconn" (or stroboscope) with electronic technology of the first generation (of 1936) of the National School of Music of the UNAM, but the information of its measurements is not provided. That stroboscope, is the only exact measuring instrument of pitch sounds that existed in the professional schools of music. Teachers of that school, like Felipe Ramirez Gil who knew the study reference, commented that the stroboscope of that school was requested for but the works of the Mayan whistles were not made with it.

To fit the sounds of the whistles to discreet musical notes does not allow the real analysis of its sounds. The aerophones generate a single fundamental (frequency), when the air excitation is produced by a mechanical tool or devise of constant and uniform air flow. They do not mention if the obtained musical notes correspond to the minimum, average or maximum pitch values.

Reducing the acoustic study to the pitch analysis, eliminates other techniques or scientific tools available for the study of sounds, like the analysis in the dominion of the frequencies, using spectrum analysis or Fast Fourier Transform (FFT), and the measurers of frequencies and the sonorous power, that also existed at the time of the study and they had been used to analyze whistles of old cultures of other zones (7). That limiting subsists, because it seems that even they do not have devices for the analysis of the old sounds. Also other techniques of analysis of sonorous signals of certain similarity exist, like sound characterization which was not used.

The study does not mention the predicted organology use of the catalogue of formats automated and their data, in addition to the analysis shown in the document and to the one of internal registry of the museum, since they do not provide descriptive information to the public, obtained of those whistles neither of other old aerophone under its guard, nor other later studies are known on those whistles or other groups of old Mexican aerophones. The data presented in the study gives a good idea of the distinguishing structure of the Mayan whistles, but it is not the descriptive information sufficient to be able to analyze them nor to reproduce them as good experimental models.

They mention that the processing in computer of the formats was made by a psychologist of UNAM (Luis Monzon). They show the data of the format, drawings and a x-ray (without indicating scales) of only a whistle (5-0003). Without a doubt, those works are the most important product of the study, but they are useless if that information stay in archives inaccessible (or died) for the investigators, as the subjects of the study that have been maintained in warehouses. The absence to public availability of the information and the descriptive data of the Mexican organology have prevented that investigators can make studies or contributions to them.

Unfortunately, some contribution or exercise adapted to the equation cannot be done with Helmholtz'equation, not even for that Mayan whistle, since they did not include in the format all the data required for its use, like the thickness of the wall in the mouth. Other fundamental data was not included that they influenced in the pitch of the sounds, like the length of the tone holes or the thickness of the wall. Nevertheless, it is possible to do some exercises, considering some of the data provided in the format of the Mayan whistle 5-0003: The obtained tone of the key of piano 56 = Sol4 (+44 cents) = G4 + 44 cents. If one assumes that they used a standard tuning fork A4 = 440 Hertz and one Tempered Scale the frequency (6) of note G4 = 391.996 Hertz and the superior halftone G can be considered G#4 = 415.305 Hertz = G4 + 50 cents, reference that is slightly superior to the data of G4 + 44 cents.

With the data of the format it is not possible to accurately consider the frequency of the sound, but a calculation can be made [1] to consider the theoretical thickness of the mouth, required to produce a frequency near the measurement and provided in the format. Considered thickness L is of 0.7 cm and corresponds to a frequency of 410.6 Hertz. The considered thickness previously is near the average of the values of the thickness of the environs of the mouth (including inner, resulting the tongue-piece to make the hole of the mouth) that are in the drawings of laminae 6-1 and 6-2 and in the image of x-ray 4 of the Mayan whistle (Figure 1), including in the study. It is advisable to comment that the used equation was applied originally to spherical resonators with circular mouth, reason why is pending the study of his possible more precise adjustment to aerophones of complex and diverse forms, like the Mayan whistles. The equation of Helmholtz [1] and data used in the numerical exercise is:

F = (17000/PI) * ROOT(S/(((L+0.7*D)*V)), [ 1 ] where:

F = frequency, in cycles/second = 410,6 Hertz
V = volume of the resonating chamber of the whistle = 150 cm3
S = area of the mouth = approximated to 1.0 cm*1.3 cm = 1.3 cm2
L = thickness of the mouth in cm. It was not provided. Estimated = 0.7 cm
D = diameter of the mouth = approximated to (1.3 cm +1.0 cm)/2 = 1.15 cm 17000 = speed of sound/2. It is function of the temperature
PI = 3.1416 0.7 = correction factor
ROOT = square root.

It is said that the whistles that produced good sounds were analyzed, but the factors or parameters that were considered are not mentioned. With the calculated L, such data used and an equation similar [2] to the used one, can be considered the factor of "quality" Q of the sound that is used in acoustics, same that is 95.9:

Q = 2*PI * ROOT(V*((L+0.7*D)/S) ^3 ) = 95.9 [2]

Other fault in the design of the formats, derived are observed to adopt systems of other cultures, like the one of classification of applied Sachs-Hornbostel to musical instruments and that already has been commented in detail by other investigators, like Jorge Dájer (9). That system is not adapted to classify the varied Mexican organology. It is enough to mention that for all the Mayan whistles, in spite of being very heterogeneous, the same classification is applied in the cells, since single is included to put an X to indicate if they have holes (42122112) or not (411221311). If that classification to the thousands of whistles extended that exist anywhere in the world, the same key would be assigned to them. The fact to designate as whistles the globular aerophones with pitch holes oppose with the definition and the international custom to call whistles the aerophones without holes (that produce a note).

On the procedures for the construction of the Mayan whistles, something excellent is not mentioned.

On the material, they include in an annexed study on the clay of the whistles, of the Nuclear Training Center of the UNAM, using the method of Mossbauer, to determine its possible origin, but surely they did not obtain its objective, since in the final report they mention that of 87 % of the whistles the precise place of its processing is unknown.

In order to show with clarity that better analyses can be done, it will be made an exercise with the 172 data of tone and volume of the Mayan whistles without tone holes including in its Table IV: First, the musical data of the keys of the piano and notes to values of Hertz become, with the corrections due and the aid of a table of equivalencies of Bart Hopkins (10). In the document the notes are in excess more in relation to the norm accepted in the international scope for the musical tuning fork La4 or A4 = 440 Hz, since it is put as La5. The musical notes are included so that the musicians see equivalence. Second , with a personal computer and a spreadsheet we capture and we rearrange the pairs of data of the series, considering the volume (cm3 or milliliter) like X and Y the frequencies (Hertz). Third, tone-volume of the data of the whistles without holes is obtained a graph (Figure 2) with the relation. It is possible to see that in the average part of the curve there are significant dispersions in the values, originated because not the suitable parameters nor the values of notes in cents or in Herz were considered. The image of the dispersion of the data distorted and amplified much in Graph 2 of the study, because the axis of the tones in its ends was compacted. Quarter, with such data can be made an exercise to fit them to a suitable analytical function, for example to an exponential one. The function and the parameters fit by the method of square minimums, obtained with a good computer program downloaded from Internet (11) and some measures of the adjustment are:

Y = a*x^b [ 3 ], a = 2642,4867 and b = - 0.40115241

x is the volume (milliliter),
and Y is the frequency (Hertz)
Standard error = 84.4
Precision of the calculations = 8 exact digits.

With the information available the sounds of the Mayan whistles nor their characteristics can be known in the space of the frequencies. Using one of the experimental model of the subscribed one of typical Mayan form which is in the photo (Figure 3) , it was possible to produce a short sound (2 seconds, in Wav format), to listen to it, to record it in a computer and with the Gram program (11), based on FFT, to obtain his spectrogram in 2D of time (seconds) against frequency (Hertz) that is in a graph of frequencies(Figure 4). The 2D spectrogram was made using the following parameters:

Sampling fraction of signal = 22 kHz;
Length of the sample = of 124 data kB;
Size of FFT = 2048 (points);
Resolution in frequency = 15.1 Hz;
Band = 11 kHz and; Linear scales of frequency and time.

The first part of the whistle sound is played with the holes covered. It is possible to observe that it appears a single fundamental frequency (without overtones), like most of the globular old aerophones (whistles and ocarinas). That is the reason of the sweet and depth of the timbre of its sounds. The following part of the sonorous phrase is obtained opening and closing one of the holes of the whistle, in bipitch form, which produces a signal similar to a little snake or serpent, image similar to those of the graphic symbols that were used by our ancestors to represent all wavy class of beings and phenomena, indeed like that of sound. It can be a coincidence, but it can not be, since they had iconography with images even of colors to represent diverse types of sounds, as the speech, the song, the sounds of many instruments, etc. In contrast, single present music has the representation in black and white of the discreet notes which are used to write music in the staff. Also, it was verified that its level of relative intensity in dB is low and that the sound takes place in a close rank of frequencies, as it is possible to be observed in the spectrograms in 3D (Figure 5), obtained with the Tuneit program (12).

That model can generate sounds in the rank of 440 Hertz to 494 Hertz (A4 - B4 ) depending on the power and mass of the excitation air, which proves that those aerophones do not produce a single note only. With the measured internal volume of 65 cm3 and formula [3] fitted to the Mayan whistles can be considered to make numerical exercise [4]. Its theoretical frequency = 495 Hertz, almost equal value to the maximum of the real rank.

Y = 2642*65^(-0.4105) = 495 Hertz [4]

The previous numerical and signal analysis exercises serve to formulate several theories on the whistle 5-0003: A) in spite of being a good one, since it was selected like an example for that reason, can not be used like a present musical instrument, because if it is played accompanied by other instruments of greater sonorous power as those of an orchestra or a modern or old band (like one of the Bonampak's murals) hardly could be listened due to its low intensity; B) Its use must have been of another type of "music", without any accompaniment or a different application (like magicians or rituals), necessarily of smaller sonorous power or greater smoothness and refinement, because they did not have electronic amplifiers or synthesizers and; C) if it is played with other similar whistles, it is possible that it can produce beats in the brain that have special effects (for example, healing ones) in the human being. If they used molds for the processing of whistles, as it seems to be, since they have been in the same Mayan zone (8), it is very feasible that they were able to make copies very similar, but hardly identical. It has already been experienced (7) that when playing groups of old whistles subsonic beats are generated, vibrations that can affect the internal control system of the human being and their neurons, generating states of superior conscience.

Whistle 5-0003 cannot be used with efficiency to emit signals for long distances nor in conditions of high level of noise like other ancient and modern whistles of greater sonorous power and impact in the human being and many animals, for their under value of intensity or power, its greater volume and lower frequency, as well as its irregular internal form. The radiated acoustic power of the experimental replica operated in closed mode was estimated in 0.0005 Watts, similar to other Mayan whistles like the Clay Frogs of Yaxchilan (16).

. So for a whistle to be effective in the function to transmit signals between humans it must produce sounds of greater power and/or in the rank of greater auditory sensitivity of human beings (1 kHz to 4 kHz), which requires a smaller resonating volume and a more regular form of the resonator, like the ones used for vigilance, and signaling (police, military, etc.) and sports. The generic designation of whistles, applied to aerophones, does not have organologic relation with its characteristics and properties, nor with the use of its sounds.

The exercise to fit equation [3] is relevant because it constitutes the first known representation or mathematical model of an important group of old whistles. It is necessary to indicate that the parameters of equation 3 are the turn out to diminish the sum of the squares of the errors between the function and the data. Error of adjustment is significant, for whistles specific but not much in relation to the maximum values of the equation, which means that it can represent very well the general tendency of behavior of the data, but does not have to be used to accurately interpolate or extrapolate the frequency of the given volume of a whistle. In Figure 6) is the graph of the fit model and the used data. The data of frequencies also can be used to obtain significant information. The average arithmetic of the data of the frequencies is of 721 Hertz (above of F5), but is not very representative. If the 3 data of the ends are eliminated, the rank of the sounds of 166 data (97%) of the used whistles goes more from 277 Hertz (C#4 ) to 1.480 Hertz (F# 6) or 1.003 Hertz (near two and half musical octaves). And if the ten higher values of tones are eliminated, The number data is reduced to 159 (92%) and the rank is of 721 Hertz (near two musical octaves). That means that they did not require of a wide range of sounds and that they did not like the very acute or narrow fundamental sounds, in that set of Mayan whistles.

Some erroneously consider whistles or ocarinas like flutes or melodic instruments, arriving to despising them because they do not have many pitch holes or wanting to put to them but holes to adapt them to present melodic music. One has already been that until old flutes can be played in different form (12). It seems that they do not realize that its use was different, since they preferred to have ranges and scales reduced but of greater wealth in timbres, special rhythms and effects like those of micropitch continuous variations or when groups of similar whistles were played to produce choirs. That also is consistent with the idea that our ancestors could use the subsonic beats, since these are possible to be produced with near sounds in their fundamental (frequencies). Generally, in harmonic present music the beats are undesirable since they are called disharmonies, but it seems that to our ancestors they enchanted to them. But this subject is matter of other studies and experiments of greater depth.

The result of the exercise with the equation [4] can prove 3 things: A) the adjustment made and the mathematical model [3] of Mayan whistles is acceptable; B the values of tone of the obtained Mayan whistles sounds in the analyzed study could have been the maximums and; C) such clay model that looks like the Mayan little figures, also sings in form similar to the Mayan whistles. The exercises made with the experimental Mayan type model also can be applied to the old whistles, because for the mathematical and computing tools used are no differences between the old aerophones and sounds and their present descendants who some confused ones consider apocryphal.

We can use the data of the clay replica: V = 65 cm3, S = 1.1 cm * 0.6 cm = 0.66 cm2, L = 1 cm y D = (1.1 cm+ 0.66 cm) / 2 = 0.535 cm, and the equations [1] and [2] to get F = 465 Hz (located in the range of the real sounds) and Q = 152 (bigger than the estimated Q for the mayan whistle 5-0003). The variations in the dimensions of the mouth are very sensitive for F and Q values. For example, if we use the minimum value for L = 0.4 cm (it is the smaller thickness of the wall, instead the average = 1 cm) the frequency rises to F = 640 Hz and the quality of the sound lowers to Q = 64.

The estimated Q values are very high, if we consider the data from a recent paper on clay ocarinas (14) provided by Dr. Ray Dessy (an Emeritus Professor of Chemistry and lover of music and simple aerophones). In relation to acoustic sound quality it is very relevant the following information: "typical modern flutes have Qs of 35 - 40". It has more information from Dr. John Coltman (an Expert in Physics and Acoustics of Flutes) on the effects of holes sizes and the pitch of the sound from globular resonator in relation to his acoustic quality: "if Q > 5 is good!"


  1. The analyzed document shows faults of formal analysis and ignorance of the technical tools available, at least, in matters of: organology of old aerophones, acoustics, computation and numerical analysis. The possible advantage of the analyzed document is like an example of what it is not to do when analyzing old aerophones. It shows also that the bureaucratic monopoly in the study of ancient Mexican organology did not work well.

  2. In a wider context, the contents of the analyzed study show with clarity the basic limitations that have prevented to comply suitably, in the field of the organology, the obligations of the law (Articles 3 and 4 from our Constitution) to investigate and to disclose our valuable and rich indigenous and pre-Hispanic plurycultural patrimony. It is observed that the fundamental technical limitations for the suitable investigation of the sonorous archaeological goods are four: personnel with preparation and capacity, methodologies of analysis, equipment of metrology and descriptive information of the ancient sonorous artifacts. There are other non technical limitations like, racism, malinchism, laziness, colonization, dependence, ethnocentrism, etc. If it is not possible to surpass the existing limitations, the law related with the indigenous cultures, the Mexican polycultural patrimony and the disclosure of information from the government will remain only on the paper, as death words.

  3. Dr Rolando Menchaca, professor expert in acoustics, already proposed a formal methodology for the analysis of ancient wind devices (13). The author already has proven other simple techniques with low investments, that can serve to begin systematically to analyze and to disclose our valuable old sonorous patrimony (2), in the previous exercises are included. Among which the ones used.

  4. In order to make correlations, it is necessary to study with depth each relevant aerophone and to obtain all the data that can define their sonorous behavior noun. The first condition to make any mathematical correlation is the one to assure that a good real relation between the selected data exists, which can be facilitated selecting similar aerophones as far as their dimensions of internal structure and sounding mechanism. In addition, they would have to include the missing information and data, like the indicated ones in the commentaries, to be able to make other exercises of analysis.

  5. One analysis has been shown, that even with the inadequate and insufficient data of the published study and without having at sight, at the hand or the ear the Mayan whistles it was possible to make other exercises of analysis, to obtain additional information of its properties and to raise theories on possible uses of their sounds.

  1. Velázquez-Cabrera, Roberto,"aerófono de Piedra Negra" (Black Stone Aerophone), Conference presented in the International Congress of Computation CIC-2000, the 27 of November of 2000, in the IPN, México. A short study of this aerophone, with additional acoustical measurenments was presented in the 143rd Meeting of the Acoustical Society of America, Pittsburgh, Pennsylvania, June 2 - 7, 2002.
  2. Velázquez-Cabrera, Roberto, "Estudio de Aerofonos Mexicanos Usando Técnicas Artesanales y Computacionales. Polifonía Mexicana Virtual", A Ms thesis in Sciences of the Computation, CIC, IPN, Mayo 2000.
  3. Flores-Dorantes, Felipe and Flores-Garcia, Lorenza. "Organología Alicada a Instumentos Pehispánicos. Silbatos Mayas". INAH. MNA. 102 Colección Científica. Instumentos Musicales Prehispánicos. Mexico. 1981.
  4. Castañeda, Daniel. "Las Flautas en la Civilizaciones Azteca y Tarasca. II. Civilización Tarasca.", Revista Mexicana, S. A. Editora de Música Mexicana, No. 9-10. Enero de 1931
  5. Helmholtz, H. von, "On The Sensations of Tones", Trnas. To J. Ellis, Longmans, Green & Co, Inc., Dover, New York, 1954. (Ed. 1, 1912).
  6. Kinsler, Larry, "Fundamentos de Acústica", Limusa, 4ª. Ed. 1995. Pp197-300.
  7. Garret, S. and Statnekov D. K., " Peruvian Whistling Bottles ", The Journal of Acoustical Society of America (JASA), Vol. 62, no. 2, August, 1977, (
  8. Schelle, Linda. "Rostros Ocultos de los Mayas". Impetus Communications". 1997. ISBN 968-7917-00-8. (He includes photographies of Jorge Perez de Lara and introduction of Román Piña Chan)
  9. Dájer, Jorge. "Artefactos Sonoros Precolombinos. Desde su Descubrimiento en Michoacán". FONCA-ELA. Mexico. 1995.
  10. Hopkin, Bart, " Air Columns and Toneholes. For Principles Wind Instrument Design ". EMI "Experimental Instruments Musical comedy), 999, (
  11. Oakdaleengr. DataFit V6.0.10 ( /). It was used in the time of test. They say that the program was proven in its precision by the NIST of the EUA. Is excellent. Its cost is of $749 with license for a network.
  12. Horne, Richard, Spectogram V 5,0,5, Freeware. Kindly, he has authorized to use and to mention his excellent program in my thesis and other studies (
  13. Volkmer, D., Shareware, "TUNE!IT", The program was used during the period allowed for testing. (
  14. Dessy, Ray and Lee, "The clay pot that sings", American Recorder magazine, March 2001, pp 9-14 (
  15. Menchaca-Garcia F. Rolando and Velázquez-Cabrera, Roberto."Análisis de Artefactos Sonoros del México Antiguo". Conference for 7º. Mexican Congress of Acoustics, celebrated in the City of Veracruz, Mexico, 26 and 27 of October of 2000.
  16. Velazquez-Cabrera, Roberto. "Clay Frogs of Yaxchilan". A short version of this study was presented in the First Iberoameriban/Iberian Meeting on Acoustics, Cancun , December 2-6, 2002. A Lay paper is posted in the ASA World Wide Press Room (