Report :: Simulation of Fingerboard Scoop and String Buz
- Hwang, Il-Seok
- Aug 5
- 25 min read
Using computer simulation, this study looks at the scoop that is cut into the fingerboard to prevent buzzing, and examines how the shape and depth of the scoop curve relate to the fingerboard clearance and what effect they have.
1. Overview
The fingerboard of a bowed string instrument is slightly hollow in the middle along its length (the scoop). This is a device for preventing the string from touching the fingerboard while it vibrates (buzzing), and its shape has a large influence on buzzing.
In this study I want to see several things related to this fingerboard scoop with my own eyes, through computer simulation.
We will look at the position and depth of the scoop vertex, the shape of the scoop curve, the place where buzzing occurs, and the relationship between the fingerboard clearance and the depth of the scoop vertex.
2. Basics
2.1. Sections of the vibrating string
A vibrating string is divided into an open string and a stopped string. The stopped string is further divided into the main vibrating section (from the stopping point to the bridge) and the secondary vibrating section (from the nut to the stopping point). On an open string the nut takes the role of the stopping point, so only the main vibrating section exists.
2.2. The path of the string and how it really moves
In the main vibrating section the bow bends the string into a "V" shape, and the corner of the V travels along a parabola. This is Helmholtz motion. The secondary vibrating section, however, vibrates in a fine, smooth sine-curve shape, driven by the weak vibration passed on from the main vibrating section and by resonance. - Mov.1 -
* In the video below the amplitude of the secondary vibrating section has been set large so that its path can be observed easily.
2.3. Impact zone and maximum impact point
First, the related ideas and terms.
Impact Zone: the area of the fingerboard that the string touches when buzzing occurs
Penetration Depth: the depth to which the path of the string, in the virtual model, passes below the top line of the fingerboard and enters the fingerboard
Maximum Impact Point: the point within the impact zone where the impact of the string is strongest, that is, the point with the greatest penetration depth
Following the ideas above, when one particular point on the fingerboard is stopped, or when the string is open, I will use:
Instantaneous Impact Zone
Instantaneous Penetration Depth
Instantaneous Maximum Impact Point
And when the instantaneous values for the whole area of the fingerboard are added up, I will use:
Cumulative Impact Zone
Cumulative Penetration Depth
Cumulative Maximum Impact Point
The 'cumulative impact zone' is the union of all instantaneous impact zones.
The 'cumulative penetration depth' is the accumulated value of the instantaneous penetration depths of all stopping points.
The 'cumulative maximum impact point' is the point on the fingerboard where the cumulative penetration depth is largest.
When all other conditions are the same, the instantaneous and cumulative impact zones become wider as the amplitude of the string grows, and when the position of the stopping point changes, the instantaneous impact zone and the position of the instantaneous maximum impact point change as well.
The impact zone depends on the path of the string (parabola, sine wave, and so on).
In Helmholtz motion the path of the corner of the string is basically a parabola. Therefore the impact zone and the position of the maximum impact point stay the same whether the whole string vibrates in a parabolic shape or the string performs Helmholtz motion.
In Helmholtz motion, down-bowing and up-bowing only reverse the direction of the string's movement. They do not affect the shape of the vibration path, nor the impact zone and the position of the maximum impact point that follow from it.
In this simulation I apply Helmholtz motion and a down-bow condition to the string vibration of the main vibrating section, and a sine-wave vibration to the secondary vibrating section.
* For an open string only the 'instantaneous' items exist. For a stopped string, unless stated otherwise, all items are the 'cumulative' ones.
2.4. Maximum amplitude of the string
If the amplitude of the string is large, the scoop of the fingerboard must be deep to match it; if the amplitude is small, the scoop must be shallow. The same principle applies to the gap between the fingerboard and the string.
Knowing the amplitude of the string is therefore very important. However, the real amplitude of a string is hard to determine exactly by calculation alone. This is because the many physical factors that decide the amplitude - bow speed, bow position, pressure, string material, damping behaviour, and so on - are hard to generalize.
But if we assume that all physical parameters are the same, the relative ratio of amplitude between the strings can be calculated.
For example, assume the maximum amplitude of the E string is 2 mm, and assume that the bow speed, the bow contact position, the applied pressure and all other physical conditions are the same on the E string and the G string, and that both strings are speaking normally. In Helmholtz motion the amplitude (A) is inversely proportional to the fundamental frequency (F), so the amplitude ratio of the E and G strings can be calculated simply from the fundamental frequency of each string:
If E string = 659.25 Hz and G string = 196.00 Hz,
Ag / Ae = Fe / Fg = 659.25 / 196.00 ≈ 3.36 (1)
* Ag, Ae: amplitude of the G and E strings, Fg, Fe: frequency of the G and E strings
So in theory the amplitude of the G string is calculated to be about 3.36 times larger than that of the E string.
Thinking simply, this would mean that the scoop depth and the fingerboard-string gap should also be about three times or more larger on the G string side. In a real instrument, however, the amplitude ratio between the E and G strings is different from the theoretical value, and fingerboards are not made to such a ratio either.
For example, if the fingerboard-string gap of the E string is 3.5 mm, applying the ratio above would give about 11.7 mm for the G string, which would make the instrument unplayable.
According to the theory of John C. Schelleng, for a violin string to produce normal Helmholtz vibration, a proper 'range of bow speed and pressure' must be kept for each string. For example, if the G string is bowed with the light pressure suited to the E string, the string produces a surface, flautando-like sound or noise; conversely, if the strong pressure meant for the G string is applied to the E string, a scratching sound appears and the correct vibration breaks down.
In addition, because of physical limits such as bending stiffness and torsional waves, the real maximum amplitude of the G string turns out smaller than the theoretical value (3.36 times). On top of this, players unconsciously reduce the amplitude as a playing habit, so the real amplitude ratio of the G string to the E string becomes much smaller.
For this reason, in this simulation I estimate the maximum amplitude of each string from values commonly used in making and setting up instruments, and I assume that the amplitude of a string is inversely proportional to the length of the string.
I also divide the maximum amplitude into the 'maximum amplitude of the open string' and the 'maximum amplitude of the stopped string'. I define the maximum amplitude at which the string does not touch the fingerboard when any point over the whole fingerboard is stopped as the 'maximum amplitude of the stopped string'; the 'maximum amplitude of the open string' therefore applies only to the open-string situation. On a stopped string the string touches the fingerboard at the stopping point, whereas on an open string the nut holds the string away from the fingerboard, so the maximum amplitude of the open string is always larger than that of the stopped string.
3. Preparing the simulation
3.1. About the simulator
This is a description of the simulator used in this study.
* Simulator: made by me, Python 3.10
A : the string and fingerboard, magnified in the vertical direction
B : the string and fingerboard in their real proportions
C : string - main vibrating section
D : string - secondary vibrating section
E : nut
F : stopping point (finger position)
G : bridge
H : path of the string (envelope, at maximum amplitude)
I : fingerboard (top line)
J : position of the deepest point of the fingerboard scoop (distance from the nut)
K : depth of the deepest point of the fingerboard scoop
L : impact zone
M : maximum impact point of the main vibrating section (distance from the nut)
N : maximum impact point of the secondary vibrating section (distance from the nut)
O : combined maximum impact point of main + secondary vibrating sections (distance from the nut)
P : gap between the string and the fingerboard (nut clearance)
Q : gap between the string and the fingerboard (fingerboard clearance)
R : nut-side end of the impact zone (distance from the nut)
S : bridge-side end of the impact zone (distance from the nut)
T : maximum amplitude of the string
U : open / stopped string selection
V : selection of the curve shape of the string's vibration path
W : selection of the curve shape of the fingerboard scoop
X : other parameter settings
* From here on, unless stated otherwise, "clearance" means the 'fingerboard clearance'.
3.2. Reference conditions
Some parameters are used as fixed values.
Finger Space : 5 mm
FB Length : 270 mm
ST Length : 330 mm
* Sub DR : 5%
* Excl Zone : ±10 mm
* Phs Step : 0.05
Fingerboard scoop curve : parabola
* Sub DR : the ratio of the amplitude of the secondary vibrating section to that of the main vibrating section
* Excl Zone : when a string is stopped, the string is pressed flat against the fingerboard over the area of the finger (the stopping zone). Right next to the stopping zone the gap between string and fingerboard is very small, so buzzing can occur even with a tiny vibration. But buzzing in this area is too slight for a person to notice, so it is reasonable to exclude it from the simulation calculation.
I therefore set the stopping zone plus part of the area next to it as the 'exclusion zone', and treat impacts there as not having occurred. Note that this value is an intuitive estimate.
* Phs Step : the angular increment used to check for impact on the fingerboard. One vibration cycle of the string is divided finely into equal angles and impact is checked at each step; this value sets how finely it is divided.
A simulation needs some starting point (a reference). I call this the 'Reference conditions', and I used setting values that are commonly used when making a violin.
Violin | 1st String (mm) | 4th String (mm) | Ratio - 1st : 4th |
Clearance-Nut | 0.30 | 0.50 | 1 : 1.67 |
a. Clearance-FB | 3.50 | 5.50 | 1 : 1.57 |
b. Vertex Depth | 0.30 | 0.70 | 1 : 2.33 |
Vertex Position | 135.00* | 135.00* | - |
Max Amp (Open) | 2.03 | 3.48 | 1 : 1.71 |
Max Amp (Stopped) | 1.45 | 2.56 | 1 : 1.77 |
Ratio - b : a | 1 : 11.67 | 1 : 7.86 | - |
Table 1: Reference conditions for the 1st and 4th violin strings used in the simulation. (Denotes distance from the nut)*
Table 1 above shows the reference conditions for the 1st and 4th strings. The orange part is the reference conditions, and the green part is the result of analysing them. The two maximum amplitudes are the maximum amplitudes at which no buzzing occurs under those conditions, and I found them through simulation. (The maximum amplitude can change slightly depending on the setting of the exclusion zone.)
The point to note in the analysis is this:
Under the reference conditions, the maximum amplitude of the 4th string is about 1.7 to 1.8 times larger than that of the 1st string. (both for open and stopped strings)
This value changes according to how the reference conditions are set, but given that the reference conditions are setting values commonly used on violins, we can infer that the real maximum amplitude ratio between the 1st and 4th strings of a violin is about 1.7 to 1.8. This value can be an important reference when deciding the clearance and the depth of the scoop.
Mov.2 below is a short video showing how the maximum amplitude of the 1st string (open and stopped) under the reference conditions is found.
4. Simulation
We will go through the subjects in order, starting with what can be seen as the relational characteristics of scoop and buzzing.
4.1. What causes the buzzing
Let us look at whether buzzing happens in the main vibrating section, in the secondary vibrating section, or in both.
(The secondary vibrating section exists only on a stopped string, so this examination deals with the stopped string.)
In the main vibrating section the friction of the bow makes the string vibrate directly, but the secondary vibrating section vibrates from the vibration that passes through the stopping point and from its resonant amplification. Data that directly measures how much vibration is transmitted is hard to find, but in terms of amplitude it is estimated at about 1% of the main vibrating section.*
* This is a theoretical estimate. It treats the stopping finger as a mass hanging on the string, and was calculated for A-string conditions (tension 50 N, linear density 0.6 g/m, 440 Hz) with an assumed effective finger mass of 10 g. In reality the finger presses the string onto the fingerboard, so the fingerboard surface acts as a termination and less vibration is expected to pass through than this model suggests.
Resonance occurs when the natural frequency of the secondary vibrating section, or an integer multiple of it, matches the note being played. Stopping near the nut transmits a lot of vibration, but resonance is only possible with very high-order harmonics, so the amplitude is small; stopping at the end of the fingerboard gives a high played note, so the transmitted amount itself is small. Both extremes are safe for different reasons, so the middle area, where the two sections are of similar length, is the most unfavourable.
Among these, the points where the length ratio is 1:1 or 1:2 are where the natural note of the secondary vibrating section matches the played note and its second harmonic exactly, so resonance is established most clearly. However, because the finger is an imperfect termination, the actual amplification factor is hard to predict.
Therefore, instead of predicting the amplitude of the secondary vibrating section, I did the reverse and observed at what percentage of the main vibrating section's amplitude the secondary vibrating section starts to buzz under the reference conditions. - Mov.3 -
Mov.3 is the case of the 4th string, and I confirmed that impact occurs in the secondary vibrating section only when its amplitude is 34.37% or more of the amplitude of the main vibrating section. (For the 1st string it is 30.77%.)
Considering that the transmitted amount estimated earlier is on the order of 1% in terms of amplitude, it is practically impossible for the amplitude of the secondary vibrating section to reach 30% of the main vibrating section, even if the resonance conditions are met.
Therefore,
Under the reference conditions, there is almost no possibility of buzzing in the secondary vibrating section of a stopped string.
is the conclusion we can draw. Buzzing of the secondary vibrating section thus need not be taken into account, but in order to allow for extreme situations, in the following simulations I set the amplitude of the secondary vibrating section to 5% of the main vibrating section.
4.2. Effect of the fingerboard-string gap on the impact zone and the position of the maximum impact point
If the gap between the fingerboard and the string becomes smaller, the impact zone and the maximum impact point are expected to move that way. To check this, I ran simulations while changing the fingerboard-string gap.
Below are the results of changing the nut clearance (CLR.Nut) to 0.5, 0.3, 0.1 and 0.01 under the reference conditions of the stopped 4th string. (damping ratio 5%)
In Mov.4 there is no impact at all, even when the nut clearance is reduced to 0.01.
What deserves attention here is the shape of the string's vibration path and of the fingerboard scoop curve. The vibration path produced by Helmholtz motion is a parabola, and the scoop curve of the reference conditions is also a parabola, so the two curves have the same shape. Under this condition, therefore, no impact occurs in theory no matter how far the nut clearance is reduced. If the shape of the scoop curve changes, however, the result changes too, and this will be examined in the next chapter.
Next is the open 4th string. As with the stopped string, I changed the nut clearance to 0.5, 0.3, 0.1 and 0.01. (The exclusion zone applies to open strings as well.) - Mov.5 -
The maximum amplitude of the open string under the reference conditions was the maximum amplitude at which there is no impact while the string is held a certain distance away from the fingerboard on the nut side, so reducing this distance naturally produces impact.
Stopped String | Open String | |||
Impact Zone | Max Impact Point | Impact Zone | Max Impact Point | |
0.50 | No Impact | No Impact | No Impact | |
0.30 | 30.60 ~ 109.80 | 70.20 | ||
0.10 | 10.10 ~ 123.00 | 66.10 | ||
0.01 | 10.10 ~ 127.50 | 64.20 | ||
Table 2: Effect of Clearance-Nut on the Impact Zone and Maximum Impact Point (Stopped/Open 4th string)
Table 2 summarises how the impact zone and the position of the maximum impact point change with nut clearance, for the open and stopped 4th string.
* The minimum value of the impact zone on the open string comes out as 10.10 because the exclusion zone is set to 10.
Under the reference conditions, when the nut clearance is reduced, the impact zone of the open string becomes wider and the maximum impact point moves towards the nut, while the stopped string shows no impact at all.
Next we will look at the effect of the fingerboard clearance on the impact zone and the position of the maximum impact point.
Mov.6 and Table 3 show the results of changing the clearance (CLR.FB) to 5.5, 4.5, 3.5 and 2.5 under the reference conditions of the 4th string.
Stopped String | Open String | |||
Clearace-FB | Impact Zone | Max. Impact Point | Impact Zone | Max. Impact Point |
5.5 | No Impact | No Impact | No Impact | No Impact |
4.5 | 15.10 ~ 126.80 | 42.70 | 36.40 ~ 153.70 | 95.00 |
3.5 | 15.10 ~ 196.20 | 75.70 | 27.40 ~ 204.10 | 115.70 |
2.5 | 15.10 ~ 250.80 | 108.60 | 22.40 ~ 250.60 | 136.50 |
Table 3: Effect of Clearance-FB on the Impact Zone and Max Impact Point - Stopped/Open 4th string -
What was confirmed in the simulation above can be summarised as follows.
Under the reference conditions, when the fingerboard clearance is reduced, the impact zone becomes wider on both the open and the stopped string (it widens more towards the bridge) and the maximum impact point also moves towards the bridge. The impact zone and the maximum impact point are closer to the bridge on the open string.
4.3. Effect of the scoop curve shape on the impact zone, the position of the maximum impact point and the maximum amplitude
By checking what happens when the scoop curve of the fingerboard is changed to another shape, we will find out which scoop curve is the most suitable.
The candidate curves are the parabola (the reference conditions), the sine curve and the elliptical arc.
An elliptical arc cannot be drawn from the width and depth of the curve alone, so I constructed it with an additional variable: the area of the closed figure formed by the sine curve and the parabola. - Fig. 3 -
Given the width and depth of the curve, and calling the area of the closed figure formed by the sine curve and the parabola (Fig.3-b) K, I draw the elliptical arc so that the area of the closed figure formed by the parabola and the elliptical arc (Fig.3-c) is also K. In other words, the elliptical arc is closest to a U shape and the sine curve is closest to a V shape (the parabola lies in between).
Table 4 below shows the results of the test in which the scoop curve of the fingerboard was changed under the reference conditions of the 4th string. With the elliptical arc the maximum amplitude increased for both the open and the stopped string, and with the sine curve it decreased. Under the reference conditions the impact points for the sine curve are concentrated near the nut. This means that the place where impact is most likely to occur is near the nut, and this result appears because the sine curve is shallower there than the parabola while the elliptical arc is deeper. That is, if we focus on the maximum amplitude, the elliptical arc, which is deepest near the nut, is the most favourable.
Stopped String | Open String | |||||
Scoop Shape | Impact Zone | Max Impact Point | Max Amp | Impact Zone | Max Impact Point | Max Amp |
Sine wave | No Impact | No Impact | 2.43 | No Impact | No Impact | 3.44 |
15.10 ~ 35.40 | 18.70 | 2.56 | 56.00 ~ 86.20 | 71.00 | 3.48 | |
Parabola | No Impact | No Impact | 2.56 | No Impact | No Impact | 3.48 |
Ellipse | No Impact | No Impact | 2.62 | No Impact | No Impact | 3.51 |
Table 4: Effect of Scoop Shape on the Maximum Amplitude
Next are the results obtained after changing all maximum amplitudes to the maximum amplitude of the elliptical arc. - Table 5 -
Stopped String | Open String | |||||
Scoop Shape | Impact Zone | Max Impact Point | Max Amp | Impact Zone | Max Impact Point | Max Amp |
Sine wave | 15.10 ~ 45.70 | 22.50 | 2.62 | 51.30 ~ 93.40 | 72.10 | 3.51 |
Parabola | 15.10 ~ 40.90 | 15.80 | 2.62 | 62.40 ~ 88.40 | 75.40 | 3.51 |
Ellipse | No Impact | No Impact | 2.62 | No Impact | No Impact | 3.51 |
Table 5: Effect of Scoop Shape on the Impact Zone and Maximum Impact Point
When the maximum amplitude is fixed at that of the elliptical arc and the curve shape is then changed, both the sine curve and the parabola produce impact on the open string as well as on the stopped string, but the impact position moves in opposite directions for the open and the stopped string. As the curve shape gets closer to a V, the impact position moves towards the bridge on the open string and towards the nut on the stopped string.
The above can be summarised as follows.
The closer the scoop curve is to a U shape, the larger the maximum amplitude; the closer it is to a V shape, the wider the impact zone becomes and the maximum impact point moves in different directions depending on whether the string is open or stopped.
From all this, we can say that the best fingerboard scoop curve is the elliptical arc. But we cannot guarantee that the elliptical arc, which is the best numerically, is also the best in actual playing. This is because the elliptical arc closest to a U shape has the steepest slope at both ends of the fingerboard.
Table 6 below shows the maximum amplitude of the three curves under the reference conditions, together with the fingerboard-string gap at five points on the fingerboard.
Scoop Curve | Max Amp Stopped String | Max Amp Open String | Gap 1/6 | Gap 2/6 | Gap 3/6 | Gap 4/6 | Gap 5/6 |
Sine Curve | 2.43 | 3.44 | 1.68 | 2.77 | 3.70 | 4.44 | 5.02 |
Parabola | 2.56 | 3.48 | 1.72 | 2.79 | 3.70 | 4.46 | 5.06 |
Ellipse | 2.62 | 3.51 | 1.76 | 2.80 | 3.70 | 4.47 | 5.09 |
Table 6: String-FB Gap along Scoop Curve (CLR.Nut=0.5, CLR.FB=5.5, 4th String)
The Gap items are the gaps between the fingerboard and the string at the 1/6 to 5/6 points along the length of the fingerboard. The 3/6 point is the lowest point of the scoop curve, so it is the same for all curves, and both ends (CLR.Nut, CLR.FB) are also the same because they are the reference conditions.
Comparing the sine curve and the elliptical arc, there are differences of 0.08, 0.03, 0.03 and 0.08 at the 1/6, 2/6, 4/6 and 5/6 points, while the maximum amplitude differs by 0.19 on the stopped string and 0.07 on the open string, so a large gain can be obtained on the stopped string.
From this it can be assumed that the slope at both ends of the fingerboard mentioned above will not be a serious problem. Of course, if a more extreme elliptical arc were used, it could become a problem.
The above can therefore be summarised as follows.
If an elliptical arc, slightly closer to a U shape than a parabola, is used as the scoop curve, the clearances (nut and fingerboard) and the maximum amplitude of the open string increase slightly, while the maximum amplitude of the stopped string increases more than twice as much. Using an elliptical arc as the scoop curve is therefore worth considering.
4.4. Position of the scoop vertex
This time we will look at the subject of the 'position of the scoop vertex', that is, which part of the fingerboard should be made deepest.
In the present reference conditions I set the position of the scoop vertex at the centre of the fingerboard, but whether that is the most suitable position is not yet known.
If the position of the fingerboard scoop vertex is moved left and right under the reference conditions, there will be a certain range of vertex positions in which no impact occurs (the Valid Vertex Zone), and the midpoint of that range will be the position safest from impact (the Safest Position).
One point to be careful about: the present reference conditions assume the maximum amplitude obtained when the vertex position is set at the centre of the fingerboard, so if the vertex position changes, the maximum amplitude changes too. In other words, the maximum amplitude inevitably depends on the vertex position.
Fig.4 below shows the maximum amplitude and the valid vertex zone calculated with the clearance and the vertex depth specified but the vertex position left unspecified, and Fig.5 shows the result when the vertex position is specified as well (the centre of the fingerboard). (4th string, reference conditions)
I designed the program so that, when the vertex position is not specified, it finds the maximum amplitude that the given geometry (clearance and vertex depth) can handle together with the valid vertex zone; and when the vertex position is specified, it finds the maximum amplitude that can be handled at that vertex position and shows the valid vertex zone over which that maximum amplitude can be maintained.
To add a word about the valid vertex zone: within the valid vertex zone the maximum amplitude is the same wherever the vertex position is set. So for the stopped string with the vertex position set to 135 (the lower picture of Fig.5), the valid vertex zone is 89 to 135, which means the maximum amplitude is always 2.56 wherever the vertex position is set within the range 89 to 135.
The valid vertex zone is therefore different when the vertex position is not specified and when it is specified. I define the former as the 'Inherent Valid Vertex Zone' and the latter as the 'Conditional Valid Vertex Zone'. In the same way, I define the maximum amplitude as the 'Inherent Maximum Amplitude' when the vertex position is not specified and as the 'Conditional Maximum Amplitude' when it is specified.
Looking at the results above, the inherent maximum amplitude, obtained without specifying the vertex position, is larger, and the inherent valid vertex zone is concentrated towards the nut. If the vertex position is specified anywhere outside the inherent valid vertex zone, the maximum amplitude will always be reduced.
A feature of the conditional valid vertex zone is that its range leans towards the inherent valid vertex zone. If the vertex position is set to 135, this number lies outside the inherent valid vertex zone, so the conditional valid vertex zone includes 135 while leaning as far as possible towards the inherent valid vertex zone.
Fig.6 below summarises the maximum amplitude as the vertex position changes under the reference conditions, and Fig.7 summarises the change of the valid vertex zone with vertex position.
The conditional maximum amplitudes are all smaller than the inherent maximum amplitude, and the value becomes smaller and smaller as the vertex position moves towards the bridge.
That is, from the point of view of maximum amplitude, the closer the vertex position is to the inherent valid vertex zone the better, and the structure is such that moving it towards the bridge costs maximum amplitude.
The valid vertex zone becomes wider and wider as the vertex position is moved towards the bridge. Since the maximum amplitude decreases as the vertex position moves towards the bridge, this is a natural result. The reason the conditional valid vertex zone of the 4th string is narrower than that of the 1st string is that the ratio of the string gaps at the two ends of the fingerboard is different. In addition, in the conditional case the safest position has no meaning, because the vertex position has already been specified.
The above can be summarised as follows.
The maximum amplitude is largest when the position of the scoop vertex is around 1/3 of the fingerboard; as the vertex position is moved towards the bridge, the maximum amplitude becomes smaller and the width of the valid vertex zone becomes larger.
From the point of view of maximum amplitude, then, it is good to move the position of the scoop vertex from the centre of the fingerboard towards the nut, and moving it towards the bridge is not right. However, the further the vertex position moves towards the nut, the steeper the fingerboard slope near the nut becomes, and if further measures such as using an elliptical arc for the scoop curve are added, the fingerboard slope will become too extreme. Placing the vertex position at the centre of the fingerboard can therefore be seen as a compromise.
If one wants a little more maximum amplitude, one could also try an asymmetric curve: keep the vertex position at the centre of the fingerboard, but make the nut-side half of the scoop curve an elliptical arc and the bridge-side half a sine curve or a parabola.
4.5. Relationship between the scoop vertex depth and the fingerboard-string gap
In principle, to lower the probability of buzzing the maximum amplitude must be increased, which means either the scoop vertex depth or the clearance must be increased. So if the vertex position and the maximum amplitude are fixed, increasing the vertex depth requires reducing the clearance, and conversely increasing the clearance requires reducing the vertex depth.
Since the two are in inverse proportion in this way, there will be several combinations (of vertex depth and clearance) that give the same maximum amplitude. For example, if there are combinations such as 2.55/5.10/0.80 (amplitude/gap/depth) and 2.55/5.50/0.70, which one should be chosen?
For this question we have to look at the gap between the fingerboard and the string over the whole fingerboard. If the gap of one of them is smaller over the whole fingerboard, then, with no other reason to decide otherwise, that combination should be chosen.
In this chapter we will look for other combinations whose maximum amplitude is similar to that of the reference conditions, examine the fingerboard-string gap of each combination over the whole fingerboard, and see whether there is a combination that has the same maximum amplitude as the reference conditions but is easier to play.
The simulation below shows the maximum amplitude calculated while changing the vertex depth from 0.1 to 1.0 and the clearance from 0.1 to 1.0, under the reference conditions of the stopped 4th string. - Fig.8 -
There are three combinations within ±0.01 of the reference-condition maximum amplitude of 2.56. Now let us look at the gap between the string and the fingerboard over the whole fingerboard for these four combinations.
Fig.9 compares the gap between the fingerboard and the string at the six division points of the fingerboard for the three combinations whose maximum amplitude is close to that of the reference conditions. The one with the smallest overall gap is the 4.80/0.90 (gap/depth) combination, and the one with the largest gap is the 5.90/0.60 combination.
* To check the gap over the whole fingerboard quickly, one can also look at the Area value (the area of the closed figure formed by the string and the fingerboard) or the CLR+Depth value in Fig.8.
What the result above tells us is that the gap over the whole fingerboard is decided by the fingerboard clearance. The 5.90/0.60 combination, which has the largest clearance, has the largest fingerboard-string gap over the whole fingerboard, and the 4.80/0.90 combination, which has the smallest clearance, has the smallest fingerboard-string gap over the whole fingerboard.
Therefore, if you want to reduce the fingerboard-string gap over the whole fingerboard, you must reduce the fingerboard clearance.
However, the combination with the smallest gap over the whole fingerboard is not automatically the best. A small gap means a deep vertex, which means the fingerboard is strongly curved. A proper balance must therefore be found between the fingerboard-string gap and the curvature of the fingerboard. An agreement must also be found that suits the open-string situation and the situation of the other strings.
The CLR/Depth value in Fig.8 is the clearance value divided by the vertex depth value, that is, the ratio of the two parameters. Referring to this value when comparing with the other strings should make it easier to find an agreement.
Now the open-string case. - Fig.10, 11 -
The open string shows a similar tendency to the stopped string.
Overall, the 5.10~5.20/0.80 combination (CLR/Depth = 6.37~6.50) seems better than the present reference-condition combination of 5.50/0.70 (CLR/Depth = 7.86). Let us go on and check the 1st string as well.
For the stopped 1st string many combinations were found, so I excluded those whose CLR/Depth value differed greatly from the reference conditions and examined the overall fingerboard-string gap again for six combinations (including the reference conditions). (For the 1st string I found no other combination with a maximum-amplitude error of ±0.01, so I increased the search precision.)
For the open string too, I excluded combinations with a large difference in CLR/Depth value and examined a total of six combinations.
For the 1st string, the 3.30~3.35/0.35 combination (CLR/Depth = 9.43~9.57) stands out.
For both the 1st and the 4th string, increasing the depth of the scoop vertex just a little allows the fingerboard clearance to be reduced considerably, and as a result the distance to the string over the whole fingerboard can be made smaller. But increasing the vertex depth means increasing how much the fingerboard is curved, so care is needed. It must also be kept in mind that reducing the fingerboard clearance changes the whole geometry of the instrument.
When the size of the instrument changes, using the CLR/Depth value should help in deciding the vertex depth and the clearance.
The simulation above examined combinations with the same maximum amplitude as the reference conditions, so if the reference conditions change, the candidate combinations change too - and the starting point is always the maximum amplitude.
5. Conclusion
The starting point of fingerboard design is the 'maximum amplitude of the string'. But since it cannot be known by calculation, I introduced setting values commonly used when setting up a violin as the reference conditions and estimated the maximum amplitude under those conditions.
The estimated maximum amplitude of the string is about 1.7 to 1.8 times larger on the 4th string than on the 1st string. This ratio becomes the starting point of fingerboard design when the size of the instrument changes, and it plays an important role in deciding the clearance and the scoop vertex depth of the 1st and 4th strings.
The vibration of the string on a stopped string can be divided into a main vibrating section and a secondary vibrating section, and the possibility of buzzing occurring in the secondary vibrating section is almost nil.
The string and the fingerboard are not parallel to each other; they have a tilted relationship, close together at the nut side and far apart at the bridge side. Because of this relationship, buzzing mainly occurs near the nut, and the position of the buzzing moves in the direction in which the fingerboard tilts.
The scoop curve should basically follow the vibration path of the string. The path of the string in the main vibrating section is a parabola, so a parabolic scoop curve is the common choice. But the closer the scoop curve is to a U shape, the larger the maximum amplitude becomes, so in some cases a curve such as an elliptical arc could be used. In that case, however, the position of the scoop vertex must be considered together with it, so that the fingerboard slope near the nut does not become excessive.
The maximum amplitude of the string is largest when the position of the scoop vertex is around 1/3 of the fingerboard, and the further the vertex position is from this area, the smaller the maximum amplitude always becomes. In other words, moving the vertex position from the centre of the fingerboard further towards the bridge is a great loss from the point of view of maximum amplitude.
There can be several pairs of fingerboard clearance and scoop vertex depth that give the same maximum amplitude. Such a combination can be defined by the ratio of the two parameters, and the most suitable ratio - that is, the most suitable combination - can only be decided through the accumulation of much experience.
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