Scale length & fret spacing comparisons from soprano to baritone

Sporky

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[EDIT: Scroll down for comparative images of scales]


Hola, earlier I made this plot relating scale length to fret spacing all the way up the fretboard, to compare the playability of different ukulele sizes. Basically it tells me that capo'ing a DGBE 20 inch baritone scale at the 5th fret leaves almost the same GCEA fret spacing as a 17 inch tenor. Enjoy this masterpiece!

fretspacing.jpg
 
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merlin666

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This looks scientifically cool. To get this in cm I think we have to multiply with 2.54.
 

ripock

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this is definitely a case of TMI for me but I really like the graph itself. It reminds me of the spirograph I used to have as a kid. I liked sitting at the kitchen table and making those designs.
 

Ed1

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This is a great chart, and yes, it is a masterpiece. With some hand problems, my interest in it is at the lower end; that is the first five frets. I find that certain positions are easier for me with 13.5 inch soprano than a 14 inch soprano length. Yes, that small change made a difference in comfort.

If you have the time could you post an enlargement with the fret number x-axis just going up to about 6 which should put the y-axis distance from nut to about 6. (Or, could you pm me with the data and program you used to create it if that's easier)

Yes, I could always measure the frets on my ukes, but seeing/plotting the graph and having the info for future use would be of interest to me. Thanks again.
 

Sporky

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This is a great chart, and yes, it is a masterpiece. With some hand problems, my interest in it is at the lower end; that is the first five frets. I find that certain positions are easier for me with 13.5 inch soprano than a 14 inch soprano length. Yes, that small change made a difference in comfort.

If you have the time could you post an enlargement with the fret number x-axis just going up to about 6 which should put the y-axis distance from nut to about 6. (Or, could you pm me with the data and program you used to create it if that's easier)

Yes, I could always measure the frets on my ukes, but seeing/plotting the graph and having the info for future use would be of interest to me. Thanks again.
Sure thing 👍 which scale lengths do you want? 13 through 20 or less?
By the way I used the StewMac calculator to get the measurements. It was a bit laborious but less overkill than calculating the spaces myself it seemed. However @Bill1 is making me want to do that if I ever have too much time on my hands.
 
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Ed1

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Sure thing 👍 which scale lengths do you want? 13 through 20 or less?
By the way I used the StewMac calculator to get the measurements. It was a bit laborious but less overkill than calculating the spaces myself it seemed. However @Bill1 is making me want to do that if I ever have too much time on my hands.
I'm most interested in 13 through 16. If others are interested, then perhaps going through 18 is a good idea. Thanks again.
 

Ed1

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Its just a calculation. Once you set up the equation in a spreadsheet platform, you can make the computer do all the work. The Equal Temperament equation is a relatively simple exponential equation. It has a factor of 1/12 and 2, 1/12 to divide the scale into 12 equally spaced notes, and 2 for the octave interval ratio of 2:1. If you understand that, then the equation becomes relatively simple exponential algebra.

The Rule of 18 (or 17.82) is even easier to set up in a spreadsheet. (https://artsintegration.com/2017/05/01/rule-18-nothing-fret/ ).

So many makers get flummoxed when they see an exponential equation or maths. They see the math and not the relatively simple concept of a Temperament. If you show them how to make the table, and how to create the spread sheet, you will be helping them a lot more than you would by just giving them the tables. The rule of 18 math is very simple, its not hard to make a table in a note book using a calculator, some makers just need to be trained in how to do it. The equations in the Equal Temperament look daunting, but with a modern spreadsheet platform, they are not much harder than using a calculator. Once they have done it a few times, they will realise that it is not hard. For the hardcore old school makers who refuse to use computers, show them how to make the table in their notebook using a slide rule, thats not all that hard either.
Thanks for this Bill1. Although Sporky's graphs are helpful and intersting, I think I'll use your info to set up a chart to help me figure out the exact distances between frets for various soprano overall string lengths.
 

Sporky

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Ahh! I'm just happily surprised that someone was interested, and pleased to have learned something myself.
 

Sporky

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I decided to experiment further! This time I calculated the fret distances from the equal temperament exponential and produced scaled images of scales 🙃
If you want to print them out, you can calibrate using the "5" and "10" labels which are inches from the nut; also, the 12th fret is right in the middle of the scale so for example in baritone it's at 10 inches.
The bold lines are 5th and 10th frets. I'm including all of the individual scale images too.

Each scale is 30mm tall to represent string spacing at the nut (rather than fretboard width) but that doesn't really mean anything since there's no widening toward the saddle.


scales.png
 

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Ed1

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Thanks for this Sporky. Visualizing the relative distances of different string lengths at different frets is interesting.
 

ubulele

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It would be more correct to say that capoing a bari at the 5th fret yields about the same fret spacing as on a concert (starting from the nut). It's equivalent to tenor spacing at the 3nd fret. I first proved this (years ago, to my satisfaction) through a though experiment, and just now I double-checked this particular case by holding up the necks of actual ukes side by side.

For comparing scales, it's easier to think by comparing to where the various scales come on a full-sized guitar (if you want to work various sizes of guitars into the mix as well).

guitar: nut (0th fret)
20" baritone: 4rd fret
19" baritone: 5th fret
17" tenor: 7th fret
15" concert: 9th fret
13" soprano: 11th fret

No matter where you start on an instrument, for each successive fret (half-step up) the spacing will reduce by the same multiplier: the twelfth root of 2. Therefore, from 20" bari to tenor scale there's a three-fret difference, and the spacing upward from the 3rd fret of the bari will almost exactly match that of tenor scale starting at the nut, fret for fret, till you run out of fretboard. Similarly, at the tenor 2nd fret, the spacing is like at the nut of a concert, and at the 4th fret, the spacing is like at the nut on a soprano. Basically, just take the difference in fret numbers from the above chart, and you'll find the same spacing that many frets apart on the sizes you're comparing. This is simple enough to carry around in your head. If you want to ignore guitars and make it simpler still, it looks like this:

20" bari: 0
19" bari: 1
17" tenor: 3
15" concert: 5
13" soprano: 7

This is also the model I use for predicting what the "optimal" tuning will be on a uke when you install a string set designed for a different scale. For instance, if you want Bb tuning on a tenor, try a set of concert strings, since you'd get the same C tuning pitches at the tenor's 2nd fret, which corresponds to the nut on a concert, and thus the string tension would remain constant when you tune to Bb on a tenor. (For this application, you should think in opposite directions: for a lower tuning, try strings for a smaller size; for a higher tuning, try strings for a lower size.) With guitar strings, you can get lower tunings by using the middle four strings rather than the highest four; this roughly lowers the appropriate tuning by a fourth (5 semitones or "frets").