Fender Stratocaster Analysis of Amplitude and Time Decay

Fender Stratocaster Analysis of Amplitude and Time Decay
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Josh Penas
In this project, the amplitudes and time decay of the harmonics of guitar strings were measured.
The open low E string and the G string were played with each of the 5 different pickups. Each time, the
string was picked at about the middle pickup area but slightly toward the neck. After the sounds were
recorded, the harmonics of each of the notes with the different pickups were analyzed using a computer
program. The guitar was a Fender Stratocaster made in Mexico with a hum bucker pickup by the bridge.
Both tone nobs were kept at the same level for each note.
One of the main properties that was looked at was the amplitudes of each of the different
harmonics for each played string. A 3D graph was looked at for each combination of string and pickup.
This displayed the harmonic or mode on one axis, time on the one perpendicular to it, and the
amplitude on the vertical axis. This showed how loud each of the frequencies were and for how long.
What was consistently found was that the higher number harmonics of the notes played on the G string
had higher amplitudes than higher modes of the notes of the E string no matter what the guitar pickup
was at. This means that in general, the higher harmonics on the G string can be heard better than the
higher harmonics on the E string. This makes sense when hearing the strings played. The notes on the E
string have a deeper and soft sound while the higher strings have a higher and sharper sound. They are
lower not only because the string is thicker and the frequencies themselves are lower but the same note
played on a higher string would sound different than played on a lower string. For example, when
tuning the guitar, the notes are supposed to be the same but they still sound different because the
strings themselves are different. On the following page are some comparisons of the 3D graphs of the E
and G strings with the same pickups with E on the left and G on the right. The one right below is a
comparison of the strings played with the neck pickups.

Neck PU Comparison: Open low E vs. Open G
Neck‐Mid PU Comparison: Open low E vs. Open G
The differences between the amplitudes of the different harmonics for the different pickups
were also looked at. These differences were not as noticeable but the higher harmonics of the notes
played with the neck‐mid pickups and the mid‐bridge pickups had lower amplitudes the higher
harmonics of the other three. This is because the second and fourth pickups have a deeper and soft
tone while the other ones have a sharper tone. The bridge pickup has the sharpest tone and the higher
modes of the notes played using bridge pickups were louder than any of the other pickups. The tone is
sharp because many of the higher harmonics can be heard. Below are the 3D graphs of the low E string
at the third, fourth, and firth pickups up to harmonics with frequencies of 1000Hz.
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The mid, fourth (or mid‐bridge), and bridge pickups
are shown to the right with the mid on the top and
the bridge on the bottom. It is hard to tell much of a
difference and the graphs look very similar to each
other, but if you look closely, you can see that the
amplitudes of the higher frequencies of the fourth
pickup seem a little bit higher. This also makes sense
since the sound of the middle pick up is sharper than
the second and fourth not that sharp and is still
somewhat soft at the same time. The higher
harmonics on the E string played with the bridge
pickup have visibly higher amplitudes though. This
also makes sense since the bridge pickup has the
sharpest tone and would be the easiest to
distinguish out of the three.
All these results seem to show that higher pitched/
sharper sounds are richer in the higher harmonics.
To make a sound sharp sounding, add higher
harmonics; and to make a deeper and soft sound,
add less of the higher harmonics. The amplitudes
of harmonics seem to influence the tone of the
sound and the pickups are probably designed to
cause this effect on the amplitude of the harmonics.
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Another property of the harmonics that was looked at was the time decay, how long each of the
harmonics lasted. The harmonics not only have different amplitudes by also different durations for
which they sound.
Looking at graphs of the sounds as a whole, the pickups do not seem to impact the time the
sound is heard that much but the string seems to be an important factor. The sound signals for the
notes on the E string seem to last longer while the sounds on the G string seem to get much quieter
more quickly. The notes on the E string also become quieter at a fairly fast rate but not as much as the
G strings. The open E notes also seem to have a consistently higher signal. The graphs that measured
this showed the signal on the vertical axis and time on the horizontal. Below are graphs of the signals of
the open E and G strings with the first, second, and fifth pickups.
Open Low E
Open G
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The time decays for each of the individual harmonics were also looked at. The signal of each of
the first sixteen harmonics for all of the notes were measured over a period of about 3 seconds and a
least squares exponential fit was made for each note’s harmonics since sound vibration is supposed to
decay exponentiallyThe decay time constant was calculated for each of the harmonics. The analysis of
each of the harmonics was also split into two parts. One was the signal from 0.1 to 0.4 seconds and the
other was from 0.5 to 3 seconds. The reason that this was done is because the signals for all of the
harmonics actually increase from 0.1 to 0.4 seconds due to the transient response of the pickup. The
Fender Stratocaster that was used was not an easy guitar to analyze because there are many things
about it that complicate the measurements – coupling of string vibrations due to the tremolo bridge.
Below are some of the graphs of the individual harmonics decaying, with four at a time.
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The two graphs on the top are of the first four harmonics of one of the notes, the bottom left is a graph
of the least squares line for one of the notes and the bottom right one is a graph of the least squares line
for the next eight harmonics of the same note as on the graph next to it. Hopefully, the little increase at
the beginning is noticeable, at least on the top graphs. It is also probably noticeable how the line
representing the fourth harmonic on the second and third graphs is significantly lower and the lines on
the fourth graph are much squigglier. This is because the higher harmonics were harder to measure.
This was also the first time the program was used on a guitar and as mentioned before, the particular
guitar caused many complications.
Least squares exponential fits were calculated, a graph was made as mentioned earlier. This
graph displayed the time constants of the different frequencies both during the first half of a second and
the rest of the three seconds. The time constants of the first half a second are blue and the time
constants for the rest of the three seconds are pink. Since the first half second measured was
somewhat irregular, the time constants of the harmonics for this time frame were not really looked at.
Unfortunately, I was not able to find many patterns for the time decays of the pink points either. It did
seem like generally, the lower harmonics took longer to decay which makes sense. It also seems like
after a certain harmonic, the time constants stayed roughly the same, for some of the graphs. For
others, the time constants are all over the place and sometimes they increase as the harmonics get
higher. The time constants for the harmonics during the first half of a second are mostly toward the
bottom of the graph but in of them, they seem to correspond with the pink time constant of the same
harmonic and follow a similar pattern/relative position to the other points. Another thing that was
consistent was that the second harmonic on the open E string was always higher than the first and the
highest. This is probably due to the specific guitar being a Fender Stratocaster on not for all E string in
general. Here are a few of the graphs were a pattern is somewhat present below. The graph on the
left is of the open E with the neck pickup and the right one is E with the fourth pickup.
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Some other things that I noticed were the points for the low harmonics on the E string were
missing for a lot of the pickups and this was not the case for G. This could mean that the harmonics on
the E string, or at least the low ones, are harder to measure because of the way the string is made. I did
not notice any significant patterns between the time decays for different pickups for both the E and the
G string when looking at both the graphs and the data. The time constants seem to be longer on the
second and fourth pickups as a whole from looking at the numbers and glancing at the first two digits of
each number in each column but only for the G string and not by much, and the numbers for the G
string and the fourth pickup did not make much sense at all. When looking at the numbers, I could also
see the pattern of the first few harmonics being significantly higher while the rest being about at the
same level. The time constants for the bridge pickups seemed shorter overall than the rest of the
pickups which is what I expected. I did not notice many differences between the harmonics individually
(if the harmonics were higher, then they were higher as a whole, the pattern of them decreasing did not
seem to change, when it did follow the decreasing pattern). It seemed like the harmonics for the E
string lasted longer than the harmonics on the G string but that was seen on the graphs of the signal as a
whole shown earlier.
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The results from the time decay analysis seem to show that if anything that harmonics ring
longer than the higher ones. This means that the note would become gradually less rich in harmonics as
time progressed if it was true. Harmonics on the low E string also seem to ring longer than the G string
as a whole and this seems certain. Also, among the first sixteen harmonics, there seems to be a point
where the time constant does not decrease much or at all after a certain point, and that the lower
harmonics last significantly longer than the higher ones. This may not be true but if it was, it would
mean that the notes are not very rich in harmonics after a certain point in time once all the higher
harmonics stop ringing. The pickups did not seem to affect how long the harmonics lasted that much
but if they did, then the harmonics would probably last the longest on the second and fourth pickups
and the shortest on the fifth or bridge pickup. This would make sense since the firth pickup has the
sharpest sound and the second and fourth ones have softer sounds. The fifth pickup is also located near
the bridge so the string is not vibrating as much. As mentioned earlier, the program was used on a
guitar for the first time, and this guitar was not a very good example for the first time so much of the
data is hard to makes sense of but there are some things that seem to be consistent and many
possibilities.
Overall, the pickups seem to have a bigger impact on the amplitudes of the harmonics than the
time the last for. The different tones is most likely due more to the fact that more of the higher
harmonics can be heard on some than others, though the time each of the harmonics last for may be a
contributing factor. This would mean that the louder amplitudes do not last any longer than the quiet
ones. However, the amplitudes of harmonics on guitar strings has been measured before so the method
and program or doing it was much more developed and the time decay had never been measured
before. Maybe later, and when a simpler guitar is being measured, more conclusions will be made, but
there are still some findings, which shows that it will definitely be possible to find more in the future. As
for the different strings, there seems to be differences among the harmonics as well. As a whole, they
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seem to last longer on the low E string but the higher harmonics on the G string seem to be louder.
Between playing many notes on the different strings while using different pickups and changing many
other things on the guitar, there are many sounds that can be made, and much more that can be
analyzed about them, and even more that can be found out in the future.
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