On the Ability to Tune a Guitar.

Abstract

The purpose of this study was to develop a tool to compare the ability of individuals to tune a guitar as accurately as possible. This was done by recording the sound waves produced (by the guitar), taken alongside a software called “Audacity”, and manually analysing them using statistical analysis. The resulting tool was astoundingly accurate, however it was too tedious to use and therefore this requires further improvement, perhaps through automation.

Introduction

There are many people throughout the world that want to know how good their musical abilities are and feel the need to check for themselves. Those who are more musically inclined are able to: reproduce a melody, range tones in increasing frequency, maintain a pitch while singing and/or tune an instrument more precisely than those who are not musically inclined.
Instruments can, and in some cases must, be tuned in a variety of different ways depending on the purpose of the instrument. Some musical instruments like pianos or organs are generally tuned by professional tuners, however many instruments can be tuned by the performers themselves, for example guitars and trumpets. In professional settings, musical instruments are tuned to a fixed source when used in a group or orchestra, but this requirement can be dropped when used solo.
In music, each note has its own specific pitch or tone. 12 of these notes, 1 semitone apart from each other, form an octave, and over one octave, a note’s frequency doubles.
The distance between two notes, each of which has its own frequency (sometimes called the pitch in music), and , are expressed as a ratio rather than a simple arithmetic difference or percentage difference:

Where and are the frequencies of the notes.
A unit of distance commonly used for tuning instruments is called a ‘cent’. There are 100 cents between two adjacent notes. In the book Music a Mathematical Offering ^[1], the formula given to calculate the distance in cents is as follows:

• The reason for logarithms in the formula is to account for frequency being a logarithmic unit, which means that instead of scaling linearly, it scales logarithmically.
• It is common knowledge among musicians that an acceptable tuning is within a 5 cent margin of error.

Hypothesis
The more experienced the person is in the musical field, the better their ability to recognise tones will be. It was reasoned that those who have worked with music have got used to the sound of the “correct” pitch, which is standard given the music that they work with, and thus they are able to identify it easier. Naturally talented people will tune the instrument better, perhaps after slight instruction so that they understand what the “correct” pitch would sound like.

Method

First, a stability study of every string was conducted to assess how well the string maintains its frequency over a short playing time (2 seconds) and also over a 15-minute time interval. The guitar was pretuned using a guitar tuner (either a program or a physical device) as a reference to check that all the strings were tuned correctly. To adjust the frequency of a string, its tuning peg was turned.
In this study, 5 participants with different musical backgrounds (musicians and non-musicians) were handed a guitar, the base string of which was pre-tuned. They were then asked to tune the remaining 5 strings of the guitar according to the sequence in the guitar textbook.^[2]
All strings, except for the A string (the second-thickest one), were then mis-tuned by turning the tuning peg of a string by a random amount. The guitar was then handed the guitar to the participant, and they were asked to tune the 5 other strings. After a short break, the task was repeated by the participant.
As soon as the person had completed the second tuning, they were asked to play one string at a time and record the sound via the program ‘Audacity’:
(Below) Figure1: Audacity program recording sound
We then computed the frequencies of individual strings via the formula:

In this study, we typically used about 80 oscillations, however, that number was subject to change depending on the stability of the string.

Figure 2: Calculating frequency in Audacity

Since in music relative tuning is important, mistuning was measured as a ratio:

Where A4 is the base string, relative to which other strings are tuned, f_{string_actual} is the frequency of the string after the tuning, f_{string_nominal} is the ideal frequency of the string, f_{A4_nominal} is the ideal frequency of the A4 string and f_{A4_actual} is the actual frequency of the A4 string during the experiment. A4 is assumed to have no mistune.
Mistuning was measured in cents and the mistuning for the five remaining strings was calculated, and presented in a table for each participant.
It was recognised that it was necessary to produce a single number that sums up the mistunings of a participant. A commonly used measure to compile data into a single number is known as standard deviation (σ), and is expressed as:

Where mist is the mistune of a string, and the subscripts EL, D, G, B, and EH are the names of the five strings and n is the number of strings (5). In this scenario, standard deviation expresses a summary of a person’s mistunings. This allowed the participants’ results to be compared.

• Experimental Variables and Constants
• The main constants of the experiment were:
• •The guitar
• •The environment
• •The base string (A)
• •Tuning instructions and sequence, as described in the book ^[2]
• The independent variable is the hearing of the participant, while the dependent variable is the outcome of the tuning, and all the tools as well as the surrounding environment, are all controlled variables.

Results

Figures 3-7 show the mistunings of each of the five participants, while Figure 8 provides a summary of results. The most important number in the tables is the last number, called Sigma (standard deviation):
Figure 3: Raw data for Participant 1

Participant 1 was a musical student for 3 years at the time this study was conducted. It is worth noting that the G string was particularly unstable in its frequency (see Figure 9) due to how it was tuned, so 3 different sections of its sound were analysed, and the mean was used. The sigma of about 9 cents shows that while some capabilities are present, they are not enough to properly tune the instrument.
Figure 4: Raw data for Participant 2

Participant 2 was a musical student for 8 years at the time of this study. It is worth noting, in particular, that the higher frequency strings had higher mistunes, likely because the proper method of tuning strings relies on using the adjacent string that was last tuned, which can lead to the mistunings accumulating. The sigma here was about 24 cents, which is higher than that of subject 1.
Figure 5: Raw data for Participant 3

Participant 3 was a professional musician, who had been performing consistently for 40 years up to that point. As such, this subject had a great deal more experience than any of the other participants in the study, and a much better understanding of the nuances of how to properly tune a guitar. The sigma of about 4 cents means that this guitar would be considered properly tuned.
Figure 6: Raw data for Participant 4

Participant 4 had no previous experience with using a guitar, and notably had the highest mistuning of all the subjects in this experiment. It is also worth noting that many of the strings, particularly the high E string, the B string, and the G string, were extremely unstable and difficult to count, which resulted in less oscillations being used to calculate the frequency. The sigma in this case was about 363 cents.
Figure 7: Raw data for Participant 5

Participant 5 also had no musical experience, and all of the strings were rather stable, which allowed us to easily and accurately count 80 oscillations. However, it is worth noting that this is the only subject who had a prominent outlier, and despite string D having a high mistune and being used to tune string G, the G string was rather accurately tuned. Excluding the outlier, the sigma was about 18 cents.
Figure 8: Summary of results

Participant	Experience	Mistune (cents)	Notes
1	3 years musical student	9
2	8 years musical student	24
3	40 years performing	4	Professional Musician
4	None	363
5	None	18	An outlier removed
		(besides Subject 4) Average = 13.75
		(Including outlier) Average = 83.6

Discussion

• The developed tool allows one to measure the ability of a person to tune an instrument with good accuracy. We came to this conclusion by checking the data using a fourier-transform, which is a process that analyses waves and breaks them down into their sine wave components, the strongest of which is the primary frequency.
• The professional musician had a much greater ability to do this than any other person. This is clearly shown in the data with a standard deviation of only 4 cents.
• Musical students are anywhere from 2-6 times worse than the professional, but vary widely in their abilities. This is also shown in the data.
• Untrained people vary in their abilities. We concluded this because whilst one untrained person had a sigma of only 18 cents, the other had a sigma (standard deviation) of 363 cents.

Uncertainties and Further Analysis

• We typically used 80 periods (full oscillations) to calculate frequency. We would like to use more oscillations to be able to account for small changes that could become significant over time, as well as to get a more accurate result for the frequency, however that requires tedious monitoring of oscillations.

Figure 9: A situation where counting periods is difficult
Another potential cause of error was the fact that a string may get mistuned by itself during the 20-minute period when the person is tuning, and that the string changes its pitch by itself during the two-second playback.

• Also, determining the exact position of the two peaks (first and last) while measuring the duration of the 80 oscillations was a challenge, as shown in the picture above, where it is unclear where a single oscillation starts and ends.
• However, we assessed that these errors were minimal and only altered the results by about 0.0005 seconds per error.
• Throughout this study, a common theme was present when evaluating the data; How accurate are the tool’s measurements, relative to human error and any other such errors that could have altered the data? As mentioned earlier, most of the technical errors, such as making a minor error in the exact positioning of a peak, were minimal in nature, however those errors are relatively minor when compared to any errors caused by external circumstances. For example, if one of the subjects had experienced a lot of stress before the study, that could have an impact on their tuning capabilities.
• We hypothesised that this is a possible reason for Subject 2’s higher mistune than Subject 1’s, despite the fact that Subject 2 had much more experience in musical studies than Subject 1. However, another possibility is that Subject 1 simply had more natural talent than Subject 2, which could have also led to such an occurrence.
• Subject 4 was also the subject of much debate. As this subject stands out as a noted outlier, compared to the others, we discussed whether or not this subject should be considered as valid data for use in the conclusion. In the end, we decided that since people around the world may have very different capabilities, while Subject 4 was an outlier, they still served as a reminder in this study that not everybody is musically talented.
• Thought the study, we ensured that the environment was stable and consistent by performing the measurements in the same room, and all at roughly the same time. While smaller scale changes in the environment could have affected how the strings vibrated, this potential impact would be rather small, and practically negligible in the data.
• However, despite the accuracy of the data, it is still advised that the results of this study are taken with caution, due to the small sample size.

Applications

• Musicians can use this method for self-evaluation of their ear for music as it is objective and not person-dependent. The tool informs them that they have reached the limit of their tuning ability when the mistune is only a few cents (this applies to simple music without any complex chords).^[1] When this happens, there is no need to improve their ear for music any longer (also known as their capability to distinguish frequencies) and they can concentrate on other skills. A potential application would also be to monitor a person’s rate of improvement over time.
• This study can also be used to watch an instrument become mistuned by itself. The most common reasons being: intensive usage, aging and changes in the environment like temperature and humidity, all of which could affect how the strings vibrate, but may affect each string differently. This may be of use to companies making these instruments, this is so they can better understand how their quality can change based off of their age and environment. It may also be useful for traveling musicians who may move between climates regularly and require the knowledge of how the different locations they perform in, which could potentially affect their music.

Further Research
With regards to further research ideas, we can automate the period-counting process by using programming methods and Fourier-transformation, since manual counting is tedious to do and it is prone to the uncertainties listed above. This study can also be run with multiple other instruments such as violins and pianos to see if a person’s improvement rate is dependent on the instrument.
In his book^[1], Music: A Mathematical Offering, David J. Benson mentions that mistunings of pitch are heard stronger in chords. We would also like to find the limit of allowed mistune before it begins to be audibly heard, this would be another piece of information many performers would find critical to their performances.

Conclusion

Ultimately, it can be concluded from the data that while a person’s experience with music can help improve their abilities, natural talent is still a very large factor. While the best tuner was a professional who had worked in the field, one of the subjects with no musical expertise tuned better than a student who has worked with music for 8 years, despite their disadvantage from lack of experience. As such, a person’s capabilities do not seem to solely rely on experience.
References:
1.Benson, Music a Mathematical Offering. n.p.: Cambridge University Press, 2007.
2.Schmid, Will and Hal L. Pub. Co., 1977.
3.Kostka, Stefan and et al. Tonal Harmony: with an Introduction to Twentieth-Century Music. n.p.: McGraw-Hill, 2013.

About the Author

David Volchonok
David is a person who likes intellectual work: puzzles, math, research, and basically a bit of everything except for French and Astronomy. If there isn\’t anything to do, then he\’ll find some work or make up some project that is almost definitely going to take ages to complete.