In instrument making, the scale is basically a measure. denotes which is decisive for the later sound characteristics. In the case of string instruments, this can be the case height, body volume, material thickness etc. but primarily the length of the vibrating string between saddle and bridge and its division, which is important for the fret purity. But other instruments also have important lengths for the sound. For example a wind instrument, which also includes organ pipes, and percussion instruments. Actually on every instrument. It is precisely this circumstance that resulted in a wide variety of stringed instruments and still inspires today to ingenious but also curious or even absurd variants. In any case, whether 3 strings in the strumstick or up to 47 in the harp or up to 220 strings in the piano (grand piano) or even more, there is something for everyone and everything. In the case of stringed instruments that are gripped on a " fingerboard" generate their tones, a value has resulted as a constant for the classification. 17.817 mm The scale consists of 7 whole tones and ends with the 8th note in the octave.
Scale Length : lat. = measure up
From these values, the "Old Masters" the mathematical halftone value 12th root of 2 = 1.059463. This has been confirmed several times in various universities through measurements with modern devices. What is determined today by means of a scale calculator was calculated in a laborious and tedious manner back then. Those who did not know the fixed values had a long way to go. So today you should at least know where they come from and what they mean.
12th root of 2.
(Excerpt from the literature)
The 2 stands for the vibration ratio of the octave 2/1 - i.e. 2.
The 12th power of the 12th root of 2 is 2, so that you can use the " Formula " the 12th root of 2 for a semitone, the 11th root of 2 for the whole tone, the 10th root of 2 for 3 semitones, etc. © Jens Leger 2003
Here is some more math: http://www.math.uni-magdeburg.de/reports/2002/musik.pdf
2^(1/12) / (2^(1/12)-1) = constant
A constant results from these calculations:
1.059463 / 0.059463 = 17,817
A scale calculation, more precisely determination of the individual fret points shows the correctness of the values.
Let's take the typical Gibson scale: 628.65 mm
Scale length / 2^(1/12) = 1. fret point measured from the saddle = 2nd scale length for calculation
2. Scale length / 2^(1/12) = 2. fret point measured from the saddle = 3. Scale length
Point 1 628,65 / 1,059463 = 593,366 (or 35,284 1.Fret)
Point 2 593,366 / 1,059463 = 560,062 (or 33,303 2.Fret)
Calculation with constant
Point 1 628,65 / 17,817 = 35,283 (628,65 – 35,283 =593,367)
Point 2 593,367 / 17,817 = 33,303
As we can see, the old masters had done a great job.
Now you can also "graphically" determine. Due to the accuracy, this should only be determined with a drawing board and only used on instruments such as strumstick etc. But also here you need our constant.
Drawing determination of the fret points
The scale length is drawn as a straight line. Divide this by the constant and get "H" as the first radius. The two end points are connected with an auxiliary line. After the first radius has been struck with the compass from the auxiliary line to the scale line, "H2" results as the radius for the 2nd fret. Continued this way, all other fret points result.
It should be noted that you can work as precisely as possible, a modern electronic caliper and a pocket calculator or a scale calculator is much more accurate!