Why Glass harp?
What is a musical instrument? When I ask you this question, I assume 99% of you will think of the guitar, violin, piano, or other common instruments. But what actually can be defined as a musical instrument? In my perspective, anything that can be utilized to play music can be regarded as a musical instrument.
Introduction
When I was having dinner with my little sister, she played around with the dining utensils and used her chopsticks to hit the glass cup in front of her. I was attracted suddenly because this generated a pitch. I started to think about why this happens and how it will be like if I use glasses as musical instruments to produce music. Based on this idea, we started to explore the fancy effects of glass harmonics.
We started to explore how to turn this idea into reality. We looked at a few performances by European Glass harp artists, and we looked at a few tutorials about how to make glass harps by ourselves. In this project, we not only want to explore the theoretical aspects of the glass harp — we also want to make the glass harp by ourselves such that we can use it to play music or add it into song productions. We believe that exploring the theories can make us get better use of the tune and the pitch of this instrument, and adding it to music production will definitely enrich the songs, making them creative as well as attractive.
Demo Video
Professional Performances
Theoretical Background
Based on the fact that rubbing your finger around the fringe is very similar to the movement of a bow on the violin strings, and we started our research for glass harp with the Schelleng Diagram — the graphical expression for the effect on violin sound tones from pressure on the string and the distance from the bridge. (Figure is shown below.)
We notice that just as reflected from the Schelleng Diagram, in the “musical glass” performances, the pressure on the fringe of glass from the performer has a significant effect on the sound emitting from the glasses. This partially explains why those performers need to keep their hands wet from time to time — there is too much friction between dry fingers and the rim of the glass. Glass won’t generate proper sounds unless the performer’s finger is having a “stick-slip” motion with the brim of the glass.
We also searched for glass harp related research reports from Google Scholar, and we found two very interesting reports:
The first one focuses on the qualitative measurement of the mode frequencies and mode shapes of a glass cup used for the glass harp. The researchers calculated out the individual aspects of glass cups, made changes to the shapes of cups(e.g stem length, the bulge of the cup), and used FFT for calculating mode frequencies for the cup. It turns out that the cup vibration only depends on the cup itself (unrelated to the stem length), thus timbre and pitch can be changed by just changing the bulge of the glass cup.
The second research is more quantitative-based. Those researchers followed “first-order perturbation theory” and assumed that the presence of liquid is having an insignificant effect on the fundamental frequency of the singing glasses. By choosing a moderate assumption on the shape of the glasses, the researchers gave out the following equation:
*’ω0′ is the mode frequency for the empty glass, ‘α’ is a constant proportional to the liquid density and also depending upon glass shape and wall thickness, and ‘h’ is the depth of the liquid in the glass
*We also have a value ‘β’ which is a constant depending upon the glass dimensions; meanwhile, n = 2β+ 1 for a cylinder, n= 2β+ 2 for a hemisphere, and n = 2β+ 3 for a cone. As a result, the ‘n’ value here in the equation is based on the shape of our glass.
This result is very related to our topic. However, the researchers are omitting the important factor: the amount of liquid in the glass. They later wrote in the report that the existence of the liquid will lower the resonance frequency based on the formula. Unfortunately, due to the complexity of the variable, they didn’t work out an equation for the resonant frequency of a liquid holding cup.
Experiment Expectations
- The fundamental frequency of a glass cup depends on its bulge only and is unrelated to the cup’s stem.
- Increase either liquid depth, liquid density, or cup bulge will make the glass cup generate more low frequencies.
Experiment setup
We generate the sound by rubbing the fringe of the glass cup with the wet right-hand index finger. Though both tapping the glass cup and rubbing the glass cup can generate similar frequency modes, the tapping method may generate excessive noise that interferes with our experimental result.
We use an Audio-Technica AT-2020 condenser microphone to record the frequency response of our glass cup. The mic is set 5 centimeters above the glass cup so that it provides the cup with enough distance to ring. The bottom of the glass cup is fixed with tape so that the cup won’t slip and thus will not affect the experimental result.
Each experimental set-up is recorded 3 times. We take the average value and let Audacity to give out the spectrum of the glass cup. Note that the highest peak is our fundamental frequency f0. There are also some tiny peaks coming after the highest peak. This can either be overtones or, more likely, the high pitch sound generated when we rub the glass fringe. Since the equation we care about only focuses on the fundamental frequency and frequencies below(w must be < w0), the high frequencies can be omitted.
Experiment Analysis
Group A:
This experiment set gives out the fundamental frequency of the two types of cups we use:
Balloon(Fat) Glass Cup: 8.5cm in diameter 19.5oz in volume
Universal(Thin) Glass Cup: 7cm in diameter 18.5oz in volume
Balloon(Fat) Glass Cup: 8.5cm in diameter 19.5oz in volume stemless
The frequency response of the universal glass cup
The frequency response of the balloon glass cup
The frequency response of the stemless balloon glass cup
The experimental result that stemless balloon glass cup has the same fundamental frequency as the balloon glass cup verifies the claim from the first research report that being stemmed or stemless will not affect the resonant frequency.
However, we notice that in our experiment, the bulge is not the only changing variable. In order to make a bulging cup, the manufacture increases the diameter of the cup as well. The surface area of the cup may also have an effect on the fundamental frequency. So instead of modeling fundamental frequency purely out of the bulge, we think it may be better to model the fundamental frequency of the glass cup using a similar way to the Helmholtz resonator.
Despite the fact that there is no coefficient of ‘L’ for our glass cup, the A/V factor can still be used to explain the fundamental frequency for the glass cup. The more bulging balloon glass cup has a lower fundamental frequency since the ‘V'(volume) factor in the denominator is larger for a more bulging glass cup. However, though very different in the bulge, the two cups turn out to have close fundamental frequencies. It is because the ‘A’ factor(surface area, which is determined by the cup diameter) is in the nominator — the opening area and the volume of the glass cup counterbalances. So despite having a much larger volume, the large diameter makes the balloon glass cup only a little lower in f0 than the small-diameter small-volume universal glass cup. We think this pseudo-Helmholtz resonator can better model our glass cup than the bulge factor only.
Group B:
Each cup contains 200-milliliter water liquid. However, the cups’ bulges are different:
The frequency response of the universal glass cup with 200ml water
The frequency response of the balloon glass cup with 200ml water
From the result, we see the large bulge glass cup has an overall lower frequency generated. This makes sense because a larger bulge leads to a larger ‘n’ value in ‘h^n’ in the denominator. Meanwhile, the nominator part ‘w0’ is smaller for the large bulge cup in our case. So it is foreseeable that the balloon glass cup has a lower ‘w’ value triggered.
Group C:
Each cup contains the same liquid: water. However, the volumes are different:
The frequency response of glass cup with No Water
The frequency response of glass cup with Quarter Water
The frequency response of glass cup with Half Water
According to our pseudo-Helmholtz resonator theory: as water takes up the volume for the air in the glass cup, the ‘V’ coefficient goes down and thus should lead to a higher resonant frequency. That can explain why the fundamental frequency goes up a tiny bit for the quarter water case. However, as even more water pours in, the frequency decreases eventually. This potentially verifies the equation given by the second research report: since as the ‘h’ coefficient(depth of liquid) increases, ‘h^n’ in the denominator goes up exponentially. So even if we have a larger ‘w0’(f0) value in the nominator, overall, the ‘w’ value(frequency generated) goes down. We can see that there is no response from the 200-400 Hz frequency band when the glass cup is empty. However, with more water filling in, this low-frequency range is triggered as well.
Group C:
Each cup contains 200-millimeter liquid. However, the liquids are of different densities:
Oil(0.93g/cm^3) has the least density
Water(1g/cm^3) medium
Milk(1.023g/cm^3) has the highest density
The frequency response of glass cup with 200ml oil
The frequency response of glass cup with 200ml water
The frequency response of glass cup with 200ml milk
From the result above, we see that the fundamental frequency is almost the same for different liquids. This is expectable because the liquids contained are of equal volume. However, it is not obvious that with a larger density of liquid containing, we can see more responses resonate from the low frequency. We think it is because comparing with ‘h^n’, which is a large coefficient in value, ‘a’(coefficient proportional with density) is smaller in value and thus is of smaller effect. We can hardly tell the difference between these three experiments.
Points of Improvements
- All of the experiments are done in the dormitory and thus our experimental results are potentially contaminated by the noise floor. Can achieve more accurate results if we can do experiments in an anechoic chamber.
- The balloon cup and universal cup are close but still a little bit off in volume. This set of cups are the closest we can get from Amazon.
- Unable to verify the influence from the surface area since marketable glass cups of the different bulge, even if equivalent in volume, are different in radius. It is hard to only change the bulge of the cup.
Summary
- The performance of the glass harp mimics the violin. Schelleng diagram can hold true for the glass harp. Pressure and friction are the keys to glass harp performance.
- Bulge may not be the only factor deciding the fundamental frequency of a glass cup. Area of the cup / Volume of the cup can potentially explain how to determine the fundamental frequency.
- Function w2 = w02 / (1 + ahn ) is very likely to be reliable when calculating the frequency generated given the fundamental frequency of the glass cup, the shape of the glass cup, the depth of the liquid, and the density of liquid contained.
References
https://www.acs.psu.edu/drussell/Demos/BeerBottle/beerbottle.html
https://arxiv.org/pdf/1801.07514.pdf
https://asa.scitation.org/doi/pdf/10.1121/1.408458
https://asa.scitation.org/doi/pdf/10.1121/1.2198183
https://www.s2ki.com/forums/s2000-modifications-parts-193/helmholtz-resonator-design-1099237/