J. Murray Barbour brought new attention to Stråhle's construction along with Faggot's treatment of it in the 20th century. Introduced in the context of Marpurg, he included an overview of it alongside the more famous methods of determining string lengths in his 1951 book
Tuning and Temperament where he characterized the tuning as an "approximation for equal temperament". He also demonstrated how close Stråhle's construction was to the best approximation the method could provide, which reduces the maximum errors in major thirds and fifths by about half a cent and is accomplished by substituting 7.028 for the length of
QP. Barbour presented a more complete analysis of the construction in "A Geometrical Approximation to the Roots of Numbers" published six years later in
American Mathematical Monthly. He reviewed Faggot's error and its consequences, and then derived Stråhle's construction algebraically using
similar triangles. This takes the generalized form :N^m \doteq \frac{OA + BA + (OA - 3BA) \times{m}}{OA +BA - 2BA\times{m}} Using the values from Stråhle's instructions this becomes :\frac{24 + 10m}{24 - 7m} Letting \scriptstyle OA-BA=1 so that \scriptstyle OA+BA=\sqrt{N} leads to a form of the first formula that is more useful for calculation :N^m \doteq \frac{Nm+\sqrt{N}(1-m)}{m+\sqrt{N}(1-m)} Barbour then described a generalized construction using the easily obtained
mean proportional for the length of
MB that avoids most of the specific angles and lengths required in the original. For musical applications it is simpler and its results are slightly more uniform than Stråhle's, and it has the advantage of producing the desired string lengths without additional scaling. He instructed to first draw the line
MR corresponding to the larger of the two numbers with
MP the smaller, and to construct their mean proportional at
MB. The line that will carry the divisions is drawn from
R at any acute angle to
MR, and perpendicular to it a line is drawn through
B, which intersects the line to be divided at
A, and
RA is extended to
Q such that
RA=
AQ. A line is drawn from
Q through
P, intersecting the line through
BA at
O, and a line drawn from
O to
R. The construction is completed by dividing
QR and drawing rays from
O through each of the divisions. Barbour concluded with a discussion of the pattern and magnitude of the errors produced by the generalized construction when used to approximate exponentials of different roots, stating that his method "is simple and works exceedingly well for small numbers". For roots from 1 to 2 the error is less than 0.13%—about 2 cents when
N=2— with maxima around
m=0.21 and
m=0.79. The error curve appears roughly sinusoidal and for this range of
N can be approximated by about 99% by fitting the curve obtained for
N=1, f(m)=m(1-m)(1-2m). The error increases rapidly for larger roots, for which Barbour considered the method inappropriate; the error curve resembles the form \scriptstyle f(x)=x(1-x^{2a}) with maxima moving closer to
m= 0 and
m=1 as
N increases.
Schoenberg's refinements of Barbour's methods The paper was published with two notes added by its referee,
Isaac Jacob Schoenberg. He observed that the formula derived by Barbour was a fractional linear transformation and so called for a perspectivity, and that since three pairs of corresponding points on the two lines uniquely determined a projective correspondence Barbour's condition that
OA be perpendicular to
QR was irrelevant. The omission of this step allows a more convenient selection of length for
QR, and reduces the number of operations. Schoenberg also noted that Barbour's equation could be viewed as an interpolation of the exponential curve through the three points
m=0,
m=1/2, and
m=1, which he expanded upon in a short paper titled "On the Location of the Frets on the Guitar" published in
American Mathematical Monthly in 1976. This article concluded with a brief discussion of Stråhle's fortuitous use of \scriptstyle \frac {41}{29} for the half-octave, which is one of the convergents of the
continued fraction expansion of the \scriptstyle \sqrt{2}, and the best rational approximation of it for the size of the denominator.
Stewart and continued fractions The use of fractional approximations of \scriptstyle \sqrt{2} in Stråhle's construction was expanded upon by Ian Stewart, who wrote about the construction in "A Well Tempered Calculator" in his 1992 book ''Another Fine Math You've Got Me Into...'' as well as "Faggot's Fretful Fiasco" included in
Music and Mathematics published in 2006. Stewart considered the construction from the standpoint of
projective geometry, and derived the same formulas as Barbour by treating it from the start as a fractional linear function, of the form \scriptstyle y= \frac{(ax+b)}{(cx+d)}, and he pointed out that the approximation for \scriptstyle \sqrt{2} implicit in the construction is \scriptstyle \frac {17}{12}, which is the next lower convergent from the half octave it produces. This is the consequence of the function simplifying to \scriptstyle \frac{p+2q}{p+q} for
m=0.5 where \scriptstyle \frac{p}{q} is the generating approximation. ==Similar methods applied to musical instruments==