Analysing the pan flute: a strategy for understanding sound

This article analyses the application of a didactic strategy through the inquiry methodology, managing to relate topics of interest to students such as music and the physical concepts dealt with in the theme of waves and their characteristics, specifically sound. The main objective is the understanding of terms related to sound such as: frequency, tone, speed and air columns from the construction of a musical instrument, specifically the pan flute. An inquiry methodology was used, for Camacho, Casilla & Franco [1] “inquiry is defined as those activities that lead students to make observations; ask questions; examine books and other sources of information; thinking investigations; review what is known in light of experimental or experiential evidence, collect, analyse and interpret data; propose questions, explanations, predictions, communicate and socialize the results product of the systematic processes developed.” Following this methodology, the students carried out research on how to make a pan flute, encouraging their curiosity and reaching viable solutions that would allow pleasant sounds. The construction of the prototype occurred with easy-to-acquire materials, including the necessary attachments to obtain the desired sound. Finally, it is determined what is the frequency achieved with flutes made of different materials, theoretically and through an App, managing to establish relationships between physical concepts and those referring to music to determine the best way to build a pan flute for educational purposes.


1.Introduction
Our ancestors quite a relation with objects within their reach such as animal bones and the stems of some plants with music.If we go back to Homo Sapiens, the vestiges indicate that they already have the bases of a whistle.
The improvement of these objects of nature managed to make the first pan flutes; perhaps they used them as a media individually or in groups.During the Upper Paleolithic (30,000-10,000 BC) Kostenki mammoth bone and bird bone flutes were made, later in the Neolithic (7000-3600 BC) the evidence shows that their manufacture occurred with bird bones such as vultures and eagles.One of the most interesting precedents is related to the Etruscans, who were music lovers influenced by the Greek civilization.They designed various musical instruments, including the pan flute, of which there are traces of bronze reliefs of the god Pan, playing the flute.[2] Later in the Late Pre-Ceramic (3000 -1800 BC) there are findings of pan flutes made of reeds.During the Middle Ages, carved wooden instruments were made found in archaeological sites, being used as an advertisement for sharpeners or workers dedicated to trades such as pig castrators.In various sources we can find fascinating stories about flutes, and we will be surprised by the number of instruments that exist depending on the region, time and materials.[3] Figure 1.This "God Pan playing the flute" has been obtained by the author from the Wikimedia website, where it is stated to have been released into the public domain.It is included within this article on that basis.

2.Music and the relation with physics and mathematics
Physics is also found in music, especially when we refer to musical concepts such as tone, timbre or frequency, generated vibrations and the musical scale.This musical scale was discovered by Pythagoras more than 20 centuries ago, being the first to relate music to mathematics, describing the relationship between harmonic sounds and whole numbers.Pythagoras, experimenting with the monochord, a musical instrument consisting of a string with variable length and proportional to twelve, managed to establish that pleasant sounds can be produced at certain lengths.He discovered that dividing a string in half would produce an octave higher than the original, using ratios of 1:2, 2:3, and 3:4.[4] The property that relates the length of a vibrating object to a musical note determined is verified for any class of object, as is the case of the pan flute which is based on tubes as opposed to the monochord which is based on strings.In this way Pythagoras will determine the first and the last musical note with the ratio 1:2 to later find the rest of the ratios.In this case, the monochord is a string instrument as is the violin or the guitar, however, the same musical note can sound different from one instrument to another, a C on the piano will sound different from a C on a flute.This characteristic is called quality or timbre.

2.1.The Pan flute
The pan flute is a musical instrument that is made up of several tubes arranged in a vertical formation closed at one end, to generate the sound it is only necessary to blow through the open end.The emitted sound varies depending on the length of the tube.When the air enters through the closed tube, a belly is generated from the open end and a node is generated at the closed end, producing standing waves that corresponds to the first harmonic L= λ/4, (Fundamental tone corresponding to the ratio 1:1 and to the musical note C) [5] , in other words, when the air produces a sound wave, it generates a pressure difference ,this variation propagates through air with a specific speed c the value of C is 345 ± 2 / . .The relation obtained for the calculation of the frequency for a closed tube considering the length of the tube and the harmonic is the following: = frequency emitted by the tube  = 1,2,3,4,...  = speed of the sound 345 ±2 m/s  = tube length One of the main physical concepts present in the operation of the pan flute is resonance, which is a phenomenon that occurs when a body capable of vibrating is subjected to the action of a periodic external force.In this case the frequencies obtained are called the resonant frequencies of the tube.
Various musical instruments are modelled with resonant cavities to produce a variety of sounds; coupling a resonant cavity generates a wave that is perceived by our ear [6].

3.Methodology
The inquiry method is a student centered teaching that is related in how knowledge is acquired encouraging the curiosity about something.In this case the role o the teacher is to be a mentor, providing students with topics that support their cognitive thinking [7].
Students begin the inquiry methodology by following these steps: problem statement, research in various sources, review of experiences obtained by other students or people interested in the subject, making predictions, construction or experimentation, analysis of results and discussion of the results.

Steps Activities Statement of the problem
How to build a pan flute?Objective Establish the relation between terms related to sound such as: frequency, tone, speed and air columns from the construction of a pan flute.

Research in various sources
Theoretical fundament of the pan flute.How to build a pan flute?Characteristic of the sounds emitted with a pan flute.Suitable materials to obtain a pleasant sound Review experiences obtained by other students or people interested in the subject.
Review of sites that explain how to build a pan flute.Videos showing the results obtained in the construction of a pan flute.

Make predictions
Getting better sound quality and notes with a given size and type of material Construction or experimentation Construction of the pan flute with the chosen materials.

Analysis of results
Determination of the materials is best for the construction of the pan flute, as well as the size of the tubes.

Discussion of the results
In groups, the students will show the results obtained with the flute they made and the comparison of materials, size, as well as the best option in the elaboration of the flute.
Research question: How does the inquiry methodology contribute to learning the phenomenon of sound?Hypothesis: It is hypothesized that the implementation of the inquiry methodology in a group of students will significantly increase their ability to develop critical thinking skills, foster greater curiosity about learning, and improve their knowledge retention compared to more traditional teaching methods.
In relation to the activity proposed by the students: Central problem: What material provides a better sound in the construction of a pan flute and what frequencies will be obtained?Hypothesis: Materials that are more rigid will have a better sound resonance, which will generate better sound quality.The plugs that close the ends of the tube must have the same characteristics to prevent air from escaping and interfering with resonance.The frequencies obtained by the tubes will correspond to certain musical notes, the theoretical calculation as well as the experimental one will coincide with a minimum of difference.

Development:
In teams, students develop the inquiry methodology in which they obtain information about the materials and the construction procedure of the flute.They chose the materials that they considered most accessible and efficient for its construction.

Tubes material
Plugs PVC pipes maximum one meter long and approximately half an inch in diameter, from 5 to 8 tubes.

Table 2. Materials for the construction of pan flutes
The arrangement of the tubes would be staggered, the measurements may vary depending on the research source, however, a difference of 2 cm is defined for each tube.The number of tubes depends on the material that has been chosen, such as pen barrels, as they have a short length, it does not allow a greater number of tubes compared to PVC tubes, in which there is a possibility that the longest tube has a considerable length allowing for more tubes.

4.Results
All the elaborate flutes emitted pleasant sounds in different intensity and frequencies.Each one of the teams showed their flute before the group presenting difficulties to play it since it must have a specific technique.
Straws: The flutes made with straws obtained sounds more treble and with a low intensity, those made with rigid straws had a better sound quality, however, plasticine was not a good option.
Pen barrels: The feather barrels are rigid, and their difficulty lies in the method to cut them since they can be completely damaged, due to their size the number of tubes was less, the sound obtained was of good quality tending to the treble, the best option to seal the tubes was silicone.
PVC pipes: The pvc tubes were one of the best options since the sound quality is excellent due to its characteristics that allow better resonance, the plugs to seal the end of the tubes are made of the same material.Due to its length, it was possible to have a greater number of tubes and the sound intensity was greater than those of the other materials.
Reed: The reed tubes were the second-best option, even though the pan flutes are made with similar materials, because they had to find reeds that had the natural knot that would play the role of stopper, since all the tubes did not have said possibility is used to use cork which reduced the quality of sound.
Frequencies: The emitted frequencies were calculated for each of the flutes by teams using the relation for closed tubes, only one of the cases is shown.

Tube
Frequency grave  3. Calculation of frequencies for a pan flute.
According to the frequencies obtained, we can observe that, for example, the frequency of 616 Hz corresponds approximately to the musical note D#, the frequency of 862 Hz corresponds approximately to the musical note A [8] Frequency measurement through an App: Another method by which students can know the value of the frequency emitted by each of the tubes of the flute is through an App, for example: Sound Frequency Analyzer, Sound Spectrum Analyzer or Spectroid, in which they can display the frequency in Hz (Hertz) of the emitted sounds.

5.Analysis and discussion of results
At the end of the construction of the flutes, it was observed that the ones that obtained a better sound and a better presentation were the flutes made with PVC pipes, obtaining a greater frequency range than those made with other materials.The construction of the pan flute allowed the students to carry out inquiry, investigation and application of the concepts dealt with in the theme of waves and their characteristics.Motivation plays an important role in carrying out such activities.It is important to mention to the students that the instrument they will elaborate is for didactic purposes.
The limitations of this didactic strategy are the following: • The frequencies obtained do not always coincide with the frequencies of some musical notes.
• The pan flute is an instrument that due to its characteristics can be modified either by adding or removing tubes, which does not happen with other types of instruments, which does not guarantee that the flute is tuned to a certain musical note.
• For the pan flute there is no specific tuning note since its immense variety allows it to be tuned in different notes.
• If students want to build a pan flute that emits the frequencies corresponding to musical notes in the first octave, its dimensions would be approximately one meter.

5.1.Results of the applied methodology
With the use of the inquiry methodology, interaction between teacher and students is allowed in the construction, deconstruction and reconstruction of the knowledge concretized in each of the sessions to discuss, give opinions or generate ideas, contributing to active learning.
The methodology encourages the development of critical and reflective processes, generating assumptions and presuppositions, analysis, synthesis as well as a relationship of concepts and findings.

6.Conclusions
The application of the didactic strategy allows students to understand the characteristics of the waves, specially the sound waves.The concepts of sound as a longitudinal wave will be consolidated, identifying the tone of the emitted sound, which is related to the frequency, as well as the presence of the resonance phenomenon.By applying the inquiry methodology, students acquire skills for the development of future research, putting into practice the didactic approach of the subject.

Figure 2 .
Figure 2. Pythagoras Diatonic Scale showing the musical notes and their ratios.(Personal source)

Figure 3 .
Figure 3. Column of air in a closed tube.a) first harmonic b) third harmonic.(Personal source)

Figure 4 .
Figure 4. Example of the arrangement of the tubes and their size (Personal Source)