science

Fingerboard Geometry Part 2: Scoop vs. Radius

As seen in our survey results, scoop gaps along string paths and the cross profile radius were regarded as highest priority.

The model I am using to study their interaction is a grid of 3×3 points. Think of it as 3 smaller radii crosswise (usually around 42mm), and 3 giant arcs lengthwise (representing scoop gaps under G, E, and down the centerline).

FB radius v scoop gap-01

Each of the 9 points are connected to a cross radius and the lengthwise scoop radii, so trying to force definitions on all 6 radii doesn’t always work. That’s too many constraints, like saying, “I want a right triangle with sides of 2, 3, and 4.”

Here, I’ll define 4 of the radii completely, and show the relationship between the remaining two. All solutions along the graphed lines are geometrically compatible with the constraining dimensions specified in the subtitles (which are based on median survey results).

 

Middle crosswise radius vs Center-line scoop gap

FB radius v scoop gap-02

Option 1 (one radius all the way) from our survey is the only one where all crosswise radii are defined. But one of the three scoop gaps can’t be set to our target without changing the middle radius. In this case, with G scoop set to 0.75mm and E set to 0.25mm, the center-line gap comes out to 0.68mm, or 36% over our target of 0.50mm. 

If we instead prioritize our scoop gaps, setting them to 0.75mm, 0.50mm, and 0.25mm, we can calculate the middle radius out to 38mm, which matches our survey’s Option 3 (tighter radius in the middle).

Some survey participants have observed this phenomenon:

I’ve made fingerboards for respected [players] that prefer an even radius along the whole board, but there will wind up being more scoop on the center strings, which makes the string heights higher than the g string in the middle of the board.

Note that our target scoop gaps decrease linearly from G to E. Some makers intentionally skew the gradation of scoop gaps, reasoning that players may not perceive the larger gap due to similar tensions and diameters of the G and D strings. It is difficult to draw conclusions about scoop gap skewing trends from the survey, because responses were given to the nearest 0.25mm.

Below are a few other gap size permutations from our survey.

FB radius v scoop gap-03

Note that in every case with nut & bridge radii set to 42mm, for the centerline scoop size to be halfway between G and E, the middle radius always comes out 38mm. As the middle radius flattens from 38mm, the centerline gap skews toward G. The tighter the radius (<38mm), the more it skews toward E.

 

Crosswise radius of the nut vs middle

FB radius v scoop gap-04

Those who selected Option 4 (tighter radius at the nut) describe it as a section of a cone, with radius changing proportionally to the width of the fingerboard.

Option 2 (flatter radius at the nut) respondents did not give specific numbers about how much flatter. It turns out that gradually increasing the radius toward the nut as I had described in the survey does not produce a plausible solution for our given gap sizes. In general, for a flatter nut radius, the middle radius will still tighten relative to the bridge end.

As shown in Fig 2b, when a centerline gap size is halfway between G and E, the relationships between the crosswise radii are practically identical. Here is another way of looking at that:

FB radius v scoop gap-05

 

tl;dr

Here are some geometrically plausible dimensions:

options1-4

 

 

The differences we are talking about are tiny (though maybe not to a player). Here’s what they’d look like under your 42mm radius template:

templates

Have fun!!!!

 

Coming soon…

I will tell you how to go all Traité de Luthérie so you can resolve your own favorite dimensions! Or if you’re too lazy, I’ll give you an Onshape CAD file to play with.

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Fingerboard Geometry Part 1: Survey dimensions

Here are the results of the survey, with 67 responses collected between 12/19 – 12/25.

Participants were asked pick answers that best described their mental model of a typical violin fingerboard. If their concept of fingerboard dimensions were not well-described by the given answers, they were asked to pick the closest answer and elaborate about discrepancies in the comments section.

 

Fingerboard radius

 

48% One radius all the way; 22% One radius at bridge, flatter at nut; 22% One radius at nut & bridge, flatter in middle; 7% One radius at bridge, tighter at nut

Remarks from participants who selected…

Option one (one radius all the way):

I initially plane a consistent radius but this is slightly altered in the middle portion as a result of asymmetrical scooping. –Mitch McCarthy

Essentially an adjustment to the position relative to the centerline in combination with an adjustment to its height of said radius along the length gives the resultant change in width and edge heights.

Option two (flatter curve at the nut end):

Roundness at the bridge end keeps the middle strings from being in a hole in the upper positions (the string doesn’t need to be depressed as far) making double stopping and bowing easier. Flatter at the nut means the left hand doesn’t have to climb over the strings and chords are easier to play cleanly.

… more flat closer to the nut to give a “security sensation” to the musician, I meant for their finger to press the string, I think that my base of my concept the relation with the bridge radius. –Martin Cruz Aragon

Option three (one radius at nut & bridge ends, tighter curve in between):

I’ve made fingerboards for respected [players] that prefer an even radius along the whole board, but there will wind up being more scoop on the center strings, which makes the string heights higher than the g string in the middle of the board.

I actually make the board 42 at the bridge end slightly tighter in the middle and slightly flatter at the nut. –Nathan Slobodkin

Option four (narrower curve at nut end):

The radius is tighter at the nut, and then flattens out as it widens toward the bridge, allowing dead purchase for double stops and such in high positions. –Chris Jacoby

I am limiting this analysis to arcs, but some participants noted that their cross-sectional profiles were not arcs (parabolic, “egg-shaped,” etc).

 

String scoop gap size

String scoop gap size - mean: G 0.72mm, D/A 0.57mm, E 0.42mm; min: G/D/A/E 0; max: G >1.00mm, D/A >1.00mm, E 1.00mm; mode (27%): G 0.75mm, D/A 0.50mm, E 0.50mm

 

survey results-03

Many players commented that string scoop was dependent upon various factors: string heights, string diameter and properties, player preference & style of playing, etc.

 

Edge scoop gap size

edge scoop gap size smallest to largest

 

 

The treble and bass scoop gaps were symmetrical for the majority of responses.

For asymmetrical edge scoops, bass side was deeper than treble for all but one participant, who explains that:

… reaching over the strings comes from the treble side of the board so I put a little more scoop on this side to facilitate this action.

 

Apparent edge thickness

One participant remarks:

I make the edge thickness the same instead of measuring side scoop.

Those who use edge scoop to achieve uniform edge thickness would need a deeper edge scoop for deeper string scoops, explaining why most asymmetrical responses have heavier gaps on the bass side.

This is also observable in general by correlating string scoop and edge scoop:

Total string scoop vs edge scoop

 

survey results-06

Lifting the floor of the bridge end of the fingerboard is another method for achieving a more uniform edge thickness.

 

String scoop symmetry

survey results-07

Remarks:

I try to get a consistent radius curve throughout the scooping. That means that, for me, scoop centering is irrelevant.

… generally centered on fb, if anything slightly towards nut with the thought that the further away from the bridge the fingering point is, the smaller the angle between the string and fingerboard, requiring more clearance. –Mitch McCarthy

With a bit of thought, it’s apparent that scoop is only important close to nut. –Jim Biggs

I did not receive comments about scoop centering from participants who bias the depth toward the bridge.

 

Priority rankings

survey results-08

Many participants cited player style and player preference as the main determining factors for the dimensions used. The overall highest ranked factors are string scoop size and radius fidelity.

 

Coming soon…

The geometric interactions between scoop gap and the radius at nut, middle, and end will be discussed in greater depth in my next post.

Thanks to everyone involved in this survey.

The Ultimate Glue Test

I have a backlog of stuff I should post about. Today, no comic, but here’s something I did in April, explained in excessive detail.

The short version:

We tested different ratios of glue to water, glued a bunch of blocks together, and broke ’em the next day. Some broke cleanly on the joint, some vaguely on the joint, and some broke elsewhere.

The long version:

Questions:

In what proportions should we mix hide glue? Most will say it depends on what you’re gluing. For gluing a top on, some folks go for something a little weaker, because that top should come off cleanly for repairs. But for a center joint, no one wants that to come apart ever.

For a joint to come apart “cleanly” implies the joint is weaker than the wood. If surrounding wood breaks before the joint does when evenly stressed, we can assume the glue joint is stronger than the wood.

At school we are taught to cover the glue crystals twice over with water. Some people say add water until the crystals are just covered. Others give ratios by mass. What kind of results do these guidelines produce, and what are the percentage thresholds that give clean breaks vs. partial breaks vs. breaks next to the joint?

Is higher concentration always stronger? Does thicker glue have workability issues that adversely affect the strength of the final joint?

Procedure:

Glue: Milligan & Higgins, 192 gram strength. It’s the rather murky stuff that the Chicago School has been using for a while now.

Mixing: The percentages I chose are supposed to approximate visual ratios by volume, because we are used to just eyeballing it at school. We’ve been taught to add water to cover the glue flakes twice over (in other words, 1:2 visual ratio), so for this test, I used a visual ratio of 1:1 for the upper limit (weighing out at 39% glue) and 1:3 for the lower limit (at 11%, that is definitely too watery).

sample number visual ratio (volume) % by mass glue water total
1 ~ 1:1 39% 5.0g 7.8g 12.8g
2 35% 5.0g 9.3g 14.3g
3 31% 5.0g 11.1g 16.1g
4 27% 5.0g 13.5g 18.5g
5 23% 5.0g 16.7g 21.7g
6 ~ 1:2 19% 5.0g 21.3g 26.3g
7 15% 5.0g 28.3g 33.3g
8 ~ 1:3 11% 5.0g 40.5g 45.5g

I put 5g of dry hide glue in each jar, then added water to make the desired ratios for testing. I waited about half an hour before putting the jars in the hot water (~140ºF).

8 jars of hide glue with different water:glue ratios

8 jars of hide glue flakes dissolving in water. Jar 1 is 39% glue (by mass), jar 8 is 11% (ridiculously watery…)

Joint samples: I thicknessed a board of some riftsawn softwood (pine?) to about 12mm. I ripped the board in half, and planed the two halves side by side with a strong hollow. I labeled and sawed the boards into 8 pairs to glue back together.

For each pair, I applied the appropriate glue sample on both jointing surfaces together, closed them together, placed them on glass, and drove a wedge between one side and a fence to provide some clamping force. The time elapsed between applying glue and clamping was around 10 seconds each.

wood samples joined together with different ratios of hide glue

All 8 samples came from one board, which was ripped lengthwise, planed side by side, then chopped into 8 pairs, and reglued.

I left them to dry overnight before handling (~16 hours). I planed them to the same thickness and cut them to the same height, hopefully eliminating any bias from unevenness. All the joints looked good – no glue lines, no gaps.

Final dimensions of each sample before breaking: 12mm x 140mm x 50mm

The Break Test: We broke each piece by increasing pressure in the vise with a jig. The jig had to be modified a few times because I didn’t anticipate how much the wood could deform before breaking.

3 breaking 128

Our breaking jig got modified 3 times because we didn’t anticipate just how flexible those pieces of wood were.

Everyone played with each jar of glue to hypothesize how each consistency would behave before we broke them. A big point of the test was to see how our perception of glue would line up with the results.

Results:

Jar 1 (39%) and Jar 2 (35%) produced breaks away from the joint.

Jar 3 (31%), Jar 4 (27%) and Jar 5 (23%) gave partial breaks.

We consider Jar 6 (19%) to be a clean break, though a few pieces at the edge of the joint did stick.

Jar 7 (15%) and Jar 8 (11%) resulted in clean breaks very quickly.

5 data sheet

> 35% broke away from the joint (suggesting the joint was stronger than the wood) 23% – 31% gave a partial break on the joint < 19% gave a clean break

Analysis: The results were more or less in line with students’ hypotheses. Most expected Jar 3 to produce a stronger joint than it did, suggesting that we can mix glue just a little bit thicker than we thought for applications where we need a strong joint. Some expected Jar 1 to produce a partial break, citing the problem of thick glue gelling up. This did not pose a problem for this particular gluing application, where the working time was about 10 seconds.

Your results may vary depending on many many factors. Your hide glue might be different. Fresh glue and glue a few days old behave differently. Working time is a big factor. The list goes on. Best thing for you would be to try it yourself!

Suggestions for follow-up tests, criticism of this test, and questions are very welcome! Comment below or on Youtube.