Strähle construction
Encyclopedia
Strähle's construction is a geometric
method for determining the lengths for a series of vibrating string
s with uniform diameters and tensions to sound pitches
in a specific rational tempered
musical tuning
. It was first published in the 1743 Proceedings of the Royal Swedish Academy of Sciences
by Swedish master organ
maker Daniel Stråhle (1700–1746). The Academy's secretary Jacob Faggot appended a miscalculated set of pitches to the article, and these figures were reproduced by Friedrich Wilhelm Marpurg
in Versuch über die musikalische Temperatur in 1776. Several German textbooks published about 1800 reported that the mistake was first identified by Christlieb Benedikt Funk in 1779, but the construction itself appears to have received little notice until the middle of the twentieth century when tuning theorist J. Murray Barbour presented it as a good method for approximating equal temperament
and similar exponentials of small roots, and generalized its underlying mathematical principles. It has become known as a device for building fretted musical instruments through articles by mathematicians Ian Stewart
and Isaac Jacob Schoenberg
. The name "Strähle" used in recent English language works appears to be due to a transcription error in Marpurg's text, where the old-fashioned diacritic raised "e" was substituted for the raised ring.
; he is also known to have made clavichord
s, and a notable example with an unusual string scale and construction signed by him and dated 1738 is owned by the Stockholm Music Museum. His apprentices included his nephew Petter Stråhle and Jonas Gren, partners in the famous Stockholm organ builders Gren & Stråhle, and according to Abraham Abrahamsson Hülphers in his book Historisk Afhandling om Musik och Instrumenter published in 1773, Stråhle himself had studied mechanics (which has been assumed to have included mathematics) with Swedish Academy of Science founding member Christopher Polhem
. He died in 1746 at Lövstabruk in northern Uppland.
Stråhle published his construction as a "new invention, to determine the Temperament in tuning, for the pitches of the clavichord and similar instruments" in an article that appeared in the fourth volume of the proceedings of the newly formed Royal Swedish Academy of Sciences, which included articles by prominent scholars and Academy members Polhem, Carl Linnaeus, Carl Fredrik Mennander, Augustin Ehrensvärd
, and Samuel Klingenstierna
. According to organologist Eva Helenius musical tuning was a subject of intense debate in the Academy during the 1740s, and though Stråhle himself was not a member his was the third article on practical musical topics published by the Academy—the first two were by amateur musical instrument maker, minister, and Academy member Nils Brelin which related inventions applicable to harpsichords and clavichords.
Stråhle wrote in his article that he had developed the method with "some thought and a great number of attempts" for the purpose of creating a gauge for the lengths of the strings in the temperament which he described as that which made the tempering ("sväfningar") mildest for the ear, as well comprising as the most useful and even arrangement of the pitches. His instructions produce an irregular tuning with a range of tempered intervals similar to better known tunings published during the same period, but he provided no further comments or description about the tuning itself; today it is generally considered to be an approximation of equal temperament
. He also did not elaborate upon any advantages of his construction, which can produce accurate and repeatable results without calculations or measurement with only a straightedge and dividers; he described the construction in only five steps, and it is less iterative than arithmetic methods described by Dom Bédos de Celles
method for determining organ pipe lengths in just intonation or Vincenzo Galilei
for determining string fret positions in approximate equal temperament, and geometrical methods such as those described by Gioseffo Zarlino
and Marin Mersenne
—all of which are much better known than Stråhle's. Stråhle concluded by stating that he had applied the system to a clavichord, although the tuning as well as the method of determining a set of sounding lengths can be used for many other musical instruments, but there is little evidence showing whether it was put into more widespread practice other than the two examples described in the article, and whose whereabouts today are unknown.
Stråhle wrote that he had named the line PR "Linea Musica", which Helenius noted was a term Polhem had used in an undated but earlier manuscript at the Linköping Stifts- och Landsbibliotek and which is accompanied by notes from composer and geometer Harald Vallerius (1646–1716) and Stråhle's former employer J. N. Cahman.
Stråhle also showed line segments parallel to MR through points NHS, LYT, and KZV in order to illustrate how once created the construction could be scaled to accommodate different starting pitches.
Stråhle stated at the conclusion of the article that he had implemented the string scale in the highest three octaves of a clavichord, although it is unclear whether this section would have been strung all with the same gauge wire under equal tension like the monochord which he wrote it resembled, and whose construction he described in more detail. He only described an indirect method of setting its tuning, however, requiring that he first establish reference pitches by transferring the corresponding string lengths to the movable bridges on a keyed thirteen string monochord whose open strings had been previously tuned in unison.
In his analysis of Stråhle's article Faggot outlined the trigonometric steps he had used to calculate the sounding lengths of the individual pitches, for the purpose of comparing the new tuning produced by Stråhle's method, against a tuning with pure thirds, fourths and fifths (labeled "N.1." in the table), and equal temperament, which he called only "an older temperament and [which] is introduced in Mr. Mattheson's
Critica Musica" ("N.2."), He intended the resulting set of figures to show whether "the tuning of the pitches, following the previously described invention, satisfies the ear with pleasant sounds and with better evenness, in the Musical pitches on a keyboard instrument, and therefore teaches understanding better can judge than the old and previously known manner of tuning, when the eye can see what the ear hears."
Both articles were reproduced in a German edition of the Academy's proceedings published in 1751, and a table of Faggot's calculated string lengths was subsequently included by Marpurg
on his 1776 Versuch über die musikalische Temperatur, who wrote that he accepted their accuracy but that rather than accomplishing "Strähle"'s stated goal, the tuning represented an unequal temperament "not even of the tolerable type."
The sounding lengths calculated by Faggot are substantially different from what would be produced according to Stråhle's instructions, a fact which appears to have been first published by Christlieb Benedict Funk in Dissertatio de Sono et Tono in 1779, and the tuning he created includes intervals tuned outside of the range conventionally used in Western art music. Funk is credited the observation of this discrepancy in Gehler's Physikalisches Wörterbuch in 1791, and Fischer's Physikalisches Wörterbuch in 1804, and the error was pointed out by Ernst Chladni
in Die Akustik in 1830. No similar comments appear to have been published in Sweden during the same period.
These works report Faggot's mistake as the result of having used a value from the tangent instead of the sine column from the logarithmic tables. The error itself consisted of making the angle of RP about seven degrees too great, which caused the effective length of QP to increase to 8.605. This greatly exaggerated the errors of the temperament compared to the tunings he presented alongside of it, although it is not clear whether Faggot observed these apparent defects as he made no further comments about Stråhle's construction or temperament in the article.
fifth to two cents sharp of just
3:2; the range of major thirds is from 396 cents to 404 cents, or ten cents sharp of just 5/4 to three cents flat of Pythagorean
81/64. These intervals fall within what is considered to have been acceptable but there is no distribution of better thirds to more frequently used keys that characterize what are today the most popular of the tunings published in the seventeenth and eighteenth centuries, which are known as well temperament
s. The best fifth is pure in the key of F♯—or the pitch given by MB—which has a 398 cent third, and the best third is in the key E, which has a 697 cent fifth; the best combination of the two intervals is in the key of F and the worst combination is in the key of B♭.
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
Using the values from Stråhle's instructions this becomes
Letting
so that 
leads to a form of the first formula that is more useful for calculation
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,
. The error increases rapidly for larger roots, for which Barbour considered the method inappropriate; the error curve resembles the form 
with maxima moving closer to m= 0 and m=1 as N increases.
. 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
for the half-octave, which is one of the convergents of the continued fraction expansion of the 
, and the best rational approximation of it for the size of the denominator.
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 
, and he pointed out that the approximation for 
implicit in the construction is 
, which is the next lower convergent from the half octave it produces. This is the consequence of the function simplifying to 
for m=0.5 where 
is the generating approximation.
To construct the twelve note scale used as the basis for the upper half of a piano Sievers specified to extend a line segment qs-192mm long for the sounding length for note A5 (a′′)-at 45 degrees to the line qr, and from a point t, located midway between q and a, to extend a line perpendicular to qr intersecting it at u, then to divide tu into 12 equal parts. A point v on qr is located by transferring the lengths of qt, tu, and ut from q towards r, and rays extended from v through the points dividing tu and intersecting qs to locate the different endpoints of the string lengths from s.
This arrangement is equivalent to using the right triangle qtu to generate the exact mean proportional of 192 and 92.
Wolfenden explicitly advocated equalizing the tension of the plain strings which he proposed to accomplish in the upper range by combining a 9:17 ratio between octave lengths with a uniform change in string diameters (achieving slightly more consistent results over the otherwise similar system published by Siegfried Hansing in 1888), in contrast to Sievers scale whose stringing schedule results in higher tension for the thicker, lower sounding pitches.
Like Sievers, Wolfenden constructed all of the sounding lengths on a single segment at 45 degrees from the base lines for the rays, starting with points located for each C in the range designed at 54, 102, 192.5, 364 and 688mm from the upper point. The four vertices for the rays are then located by the intersections of the horizontal base lines extended from the lower C in each octave with a second line angled from the upper starting point for the string line, however, which he specified should both be at 51.5 degrees to the base lines and that the base lines have a 35:13 ratio with the difference between the two octave lengths.
Wolfenden's method approximates
with roughly 1.3775, and is equivalent to 
in Barbour's form. Compensating for its smaller octaves this produces 596 cent half octaves, an error of about 1mm at note F4 (f′) compared with his calculated figures.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
method for determining the lengths for a series of vibrating string
Vibrating string
A vibration in a string is a wave. Usually a vibrating string produces a sound whose frequency in most cases is constant. Therefore, since frequency characterizes the pitch, the sound produced is a constant note....
s with uniform diameters and tensions to sound pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
in a specific rational tempered
Musical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
musical tuning
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
. It was first published in the 1743 Proceedings of the Royal Swedish Academy of Sciences
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...
by Swedish master organ
Organ (music)
The organ , is a keyboard instrument of one or more divisions, each played with its own keyboard operated either with the hands or with the feet. The organ is a relatively old musical instrument in the Western musical tradition, dating from the time of Ctesibius of Alexandria who is credited with...
maker Daniel Stråhle (1700–1746). The Academy's secretary Jacob Faggot appended a miscalculated set of pitches to the article, and these figures were reproduced by Friedrich Wilhelm Marpurg
Friedrich Wilhelm Marpurg
Friedrich Wilhelm Marpurg was a German music critic, music-theorist and composer. He was friendly and active with many figures of the Enlightenment of the 18th century.-Life:...
in Versuch über die musikalische Temperatur in 1776. Several German textbooks published about 1800 reported that the mistake was first identified by Christlieb Benedikt Funk in 1779, but the construction itself appears to have received little notice until the middle of the twentieth century when tuning theorist J. Murray Barbour presented it as a good method for approximating equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
and similar exponentials of small roots, and generalized its underlying mathematical principles. It has become known as a device for building fretted musical instruments through articles by mathematicians Ian Stewart
Ian Stewart (mathematician)
Ian Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...
and Isaac Jacob Schoenberg
Isaac Jacob Schoenberg
Isaac Jacob Schoenberg was a Romanian mathematician, known for his discovery of splines.He studied at the University of Iaşi, receiving his M.A. in 1922. From 1922 to 1925 he studied at the Universities of Berlin and Göttingen, working on a topic in analytic number theory suggested by Issai Schur...
. The name "Strähle" used in recent English language works appears to be due to a transcription error in Marpurg's text, where the old-fashioned diacritic raised "e" was substituted for the raised ring.
Background
Daniel P. Stråhle was active as an organ builder in central Sweden in the second quarter of the eighteenth century. He had worked as a journeyman for the important Stockholm organ builder Johan Niclas Cahman, and in 1741, four years after Cahman's death, Stråhle was granted his privilege for organ making. According to the system in force in Sweden at the time a privilege, which was held by only a few of the most established makers of each type of musical instruments, gave him the legal right to build and repair organs, as well as to train and examine workers, and it also served as a guarantee of the quality of the work and education of the maker. An organ by him from 1743 is preserved in its original condition at the chapel at Strömsholm PalaceStrömsholm Palace
Strömsholm Palace, sometimes called Strömsholm Castle , is a Swedish royal palace. The baroque palace is built on the site of a fortress from the 1550s, located on an island in the Kolbäcksån river at the west end of Lake Mälaren...
; he is also known to have made clavichord
Clavichord
The clavichord is a European stringed keyboard instrument known from the late Medieval, through the Renaissance, Baroque and Classical eras. Historically, it was widely used as a practice instrument and as an aid to composition, not being loud enough for larger performances. The clavichord produces...
s, and a notable example with an unusual string scale and construction signed by him and dated 1738 is owned by the Stockholm Music Museum. His apprentices included his nephew Petter Stråhle and Jonas Gren, partners in the famous Stockholm organ builders Gren & Stråhle, and according to Abraham Abrahamsson Hülphers in his book Historisk Afhandling om Musik och Instrumenter published in 1773, Stråhle himself had studied mechanics (which has been assumed to have included mathematics) with Swedish Academy of Science founding member Christopher Polhem
Christopher Polhem
Christopher Polhammar , better known as , which he took after his ennoblement, was a Swedish scientist, inventor and industrialist. He made significant contributions to the economic and industrial development of Sweden, particularly mining.-Biography:Polhem was born on the island of Gotland...
. He died in 1746 at Lövstabruk in northern Uppland.
Stråhle published his construction as a "new invention, to determine the Temperament in tuning, for the pitches of the clavichord and similar instruments" in an article that appeared in the fourth volume of the proceedings of the newly formed Royal Swedish Academy of Sciences, which included articles by prominent scholars and Academy members Polhem, Carl Linnaeus, Carl Fredrik Mennander, Augustin Ehrensvärd
Augustin Ehrensvärd
Field marshal count Augustin Ehrensvärd was a Swedish military officer, military architect, artist, creator of the Sveaborg fortress and the Swedish Archipelago fleet,...
, and Samuel Klingenstierna
Samuel Klingenstierna
Samuel Klingenstierna was a very renowned Swedish mathematician and scientist. He started his career as a lawyer but soon moved to natural philosophy. He was the first to enunciate errors in Newton's theories of refraction, geometrical notes that were used by John Dollond in his experiments...
. According to organologist Eva Helenius musical tuning was a subject of intense debate in the Academy during the 1740s, and though Stråhle himself was not a member his was the third article on practical musical topics published by the Academy—the first two were by amateur musical instrument maker, minister, and Academy member Nils Brelin which related inventions applicable to harpsichords and clavichords.
Stråhle wrote in his article that he had developed the method with "some thought and a great number of attempts" for the purpose of creating a gauge for the lengths of the strings in the temperament which he described as that which made the tempering ("sväfningar") mildest for the ear, as well comprising as the most useful and even arrangement of the pitches. His instructions produce an irregular tuning with a range of tempered intervals similar to better known tunings published during the same period, but he provided no further comments or description about the tuning itself; today it is generally considered to be an approximation of equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
. He also did not elaborate upon any advantages of his construction, which can produce accurate and repeatable results without calculations or measurement with only a straightedge and dividers; he described the construction in only five steps, and it is less iterative than arithmetic methods described by Dom Bédos de Celles
Dom Bédos de Celles
François Lamathe Bédos de Celles de Salelles, known as Dom Bédos de Celles, was a Benedictine monk best known for being a master pipe organ builder.He was born in Caux, Hérault, near Béziers, France...
method for determining organ pipe lengths in just intonation or Vincenzo Galilei
Vincenzo Galilei
Vincenzo Galilei was an Italian lutenist, composer, and music theorist, and the father of the famous astronomer and physicist Galileo Galilei and of the lute virtuoso and composer Michelagnolo Galilei...
for determining string fret positions in approximate equal temperament, and geometrical methods such as those described by Gioseffo Zarlino
Gioseffo Zarlino
Gioseffo Zarlino was an Italian music theorist and composer of the Renaissance. He was possibly the most famous music theorist between Aristoxenus and Rameau, and made a large contribution to the theory of counterpoint as well as to musical tuning.-Life:Zarlino was born in Chioggia, near Venice...
and Marin Mersenne
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...
—all of which are much better known than Stråhle's. Stråhle concluded by stating that he had applied the system to a clavichord, although the tuning as well as the method of determining a set of sounding lengths can be used for many other musical instruments, but there is little evidence showing whether it was put into more widespread practice other than the two examples described in the article, and whose whereabouts today are unknown.
Construction
Stråhle instructed first to draw a line segment QR of a convenient length divided in twelve equal parts, with points labeled I through XIII. QR is then used as the base of an isosceles triangle with sides OQ and OR twice as long as QR, and rays drawn from vertex O through each of the numbered points on the base. Finally a line is drawn from vertex R at an angle through a point P on the opposite leg of the triangle seven units from Q to a point M, located at twice the distance from R as P. The length of MR gives the length of the lowest sounding pitch, and the length of MP the highest of the string lengths generated by the construction, and the sounding lengths between them are determined by the distances from M to the intersections of MR with lines O I through O XII, at points labeled 1 through 12.Stråhle wrote that he had named the line PR "Linea Musica", which Helenius noted was a term Polhem had used in an undated but earlier manuscript at the Linköping Stifts- och Landsbibliotek and which is accompanied by notes from composer and geometer Harald Vallerius (1646–1716) and Stråhle's former employer J. N. Cahman.
Stråhle also showed line segments parallel to MR through points NHS, LYT, and KZV in order to illustrate how once created the construction could be scaled to accommodate different starting pitches.
Stråhle stated at the conclusion of the article that he had implemented the string scale in the highest three octaves of a clavichord, although it is unclear whether this section would have been strung all with the same gauge wire under equal tension like the monochord which he wrote it resembled, and whose construction he described in more detail. He only described an indirect method of setting its tuning, however, requiring that he first establish reference pitches by transferring the corresponding string lengths to the movable bridges on a keyed thirteen string monochord whose open strings had been previously tuned in unison.
Faggot's numerical representation
The article following Stråhle's was a mathematical treatment of it by Jacob Faggot (1699–1777), then secretary of the Academy of Sciences and future director of the Surveying Office, who in the same volume also contributed articles on a weight measure for lye and methods for calculating the volume of barrels. Faggot was one of the first members of the Academy, and had also been member of a special commission on weights and measures. He apparently was not a musician, though Helenius described he was interested in musical topics from a mathematical perspective and documented that he periodically came in contact with musical instrument makers through the Academy. Helenius also presented a theory that Faggot had a more active, if indirect and posthumous influence on the construction of musical instruments in Sweden, claiming that he may have suggested the long tenor strings used in two experimental instruments built by Johan Broman in 1756 which she proposed influenced the type of clavichord built in Sweden in the late eighteenth and early nineteenth centuries.| note | Faggot | Stråhle | |||
|---|---|---|---|---|---|
| note | length | cents | length | cents | ratio |
| C | 10000 | 0 | 10000 | 0 | 1/1 |
| c ♯ | 9379 | 111 | 9432 | 101 | 211/199 |
| d | 8811 | 219 | 8899 | 202 | 109/97 |
| d ♯ | 8290 | 325 | 8400 | 302 | 25/21 |
| e | 7809 | 428 | 7930 | 401 | 29/23 |
| f | 7365 | 529 | 7490 | 500 | 239/179 |
| f ♯ | 6953 | 629 | 7073 | 600 | 41/29 |
| g | 6570 | 727 | 6676 | 699 | 253/169 |
| g ♯ | 6213 | 824 | 6308 | 798 | 65/41 |
| a | 5881 | 919 | 5955 | 897 | 89/53 |
| a ♯ | 5568 | 1014 | 5621 | 997 | 137/77 |
| b | 5274 | 1108 | 5303 | 1098 | 281/149 |
| c | 5000 | 1200 | 5000 | 1200 | 2/1 |
In his analysis of Stråhle's article Faggot outlined the trigonometric steps he had used to calculate the sounding lengths of the individual pitches, for the purpose of comparing the new tuning produced by Stråhle's method, against a tuning with pure thirds, fourths and fifths (labeled "N.1." in the table), and equal temperament, which he called only "an older temperament and [which] is introduced in Mr. Mattheson's
Johann Mattheson
Johann Mattheson was a German composer, writer, lexicographer, diplomat and music theorist.Mattheson was born and died in Hamburg. He was a close friend of George Frideric Handel, although he nearly killed him in a sudden quarrel, during a performance of Mattheson's opera Cleopatra in 1704...
Critica Musica" ("N.2."), He intended the resulting set of figures to show whether "the tuning of the pitches, following the previously described invention, satisfies the ear with pleasant sounds and with better evenness, in the Musical pitches on a keyboard instrument, and therefore teaches understanding better can judge than the old and previously known manner of tuning, when the eye can see what the ear hears."
Both articles were reproduced in a German edition of the Academy's proceedings published in 1751, and a table of Faggot's calculated string lengths was subsequently included by Marpurg
Friedrich Wilhelm Marpurg
Friedrich Wilhelm Marpurg was a German music critic, music-theorist and composer. He was friendly and active with many figures of the Enlightenment of the 18th century.-Life:...
on his 1776 Versuch über die musikalische Temperatur, who wrote that he accepted their accuracy but that rather than accomplishing "Strähle"'s stated goal, the tuning represented an unequal temperament "not even of the tolerable type."
The sounding lengths calculated by Faggot are substantially different from what would be produced according to Stråhle's instructions, a fact which appears to have been first published by Christlieb Benedict Funk in Dissertatio de Sono et Tono in 1779, and the tuning he created includes intervals tuned outside of the range conventionally used in Western art music. Funk is credited the observation of this discrepancy in Gehler's Physikalisches Wörterbuch in 1791, and Fischer's Physikalisches Wörterbuch in 1804, and the error was pointed out by Ernst Chladni
Ernst Chladni
Ernst Florens Friedrich Chladni was a German physicist and musician. His important works include research on vibrating plates and the calculation of the speed of sound for different gases. For this some call him the "Father of Acoustics"...
in Die Akustik in 1830. No similar comments appear to have been published in Sweden during the same period.
These works report Faggot's mistake as the result of having used a value from the tangent instead of the sine column from the logarithmic tables. The error itself consisted of making the angle of RP about seven degrees too great, which caused the effective length of QP to increase to 8.605. This greatly exaggerated the errors of the temperament compared to the tunings he presented alongside of it, although it is not clear whether Faggot observed these apparent defects as he made no further comments about Stråhle's construction or temperament in the article.
The tuning
The tuning produced following Stråhle's instructions is a rational temperament with a range of fifths from 696 to 704 cents, which is from about one cent flatter than a meantoneQuarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
fifth to two cents sharp of just
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
3:2; the range of major thirds is from 396 cents to 404 cents, or ten cents sharp of just 5/4 to three cents flat of Pythagorean
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
81/64. These intervals fall within what is considered to have been acceptable but there is no distribution of better thirds to more frequently used keys that characterize what are today the most popular of the tunings published in the seventeenth and eighteenth centuries, which are known as well temperament
Well temperament
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
s. The best fifth is pure in the key of F♯—or the pitch given by MB—which has a 398 cent third, and the best third is in the key E, which has a 697 cent fifth; the best combination of the two intervals is in the key of F and the worst combination is in the key of B♭.
Barbour's algebraic representation and geometric construction
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
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
. This takes the generalized form
Using the values from Stråhle's instructions this becomes
Letting
so that leads to a form of the first formula that is more useful for calculationBarbour then described a generalized construction using the easily obtained mean proportional
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
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,
. The error increases rapidly for larger roots, for which Barbour considered the method inappropriate; the error curve resembles the form 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 SchoenbergIsaac Jacob Schoenberg
Isaac Jacob Schoenberg was a Romanian mathematician, known for his discovery of splines.He studied at the University of Iaşi, receiving his M.A. in 1922. From 1922 to 1925 he studied at the Universities of Berlin and Göttingen, working on a topic in analytic number theory suggested by Issai Schur...
. 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
for the half-octave, which is one of the convergents of the continued fraction expansion of the , and the best rational approximation of it for the size of the denominator.Stewart and continued fractions
The use of fractional approximations of 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 , and he pointed out that the approximation for implicit in the construction is , which is the next lower convergent from the half octave it produces. This is the consequence of the function simplifying to for m=0.5 where is the generating approximation.Similar methods applied to musical instruments
The geometric and arithmetic methods for dividing monochords as well as musical instrument fretboards compiled by Barbour were for the stated purpose of illustrating the different tunings each represents or implies, and Schoenberg's and Stewart's works retained similar focus and references. Two textbooks on piano building that are not included by them show similar constructions to Stråhle's for designing new instruments but treat the tuning of their pitches independently; both employ a non-perpendicular form as suggested by Schoenberg's observation in Barbour's "A Geometrical approximation to the Roots of Numbers", and one demonstrates an application of the construction to a root other than 2.Sievers
Giacomo Sievers, a Russian-born piano maker working in Naples in the middle of the nineteenth century, illustrated a construction in his book Il Pianoforte, published there in 1868, that he claimed was the best practical method for determining sounding lengths of strings in a piano. He did not describe its source or extent of its use, and did not explain any theory behind it. He also did not suggest it had any use for tuning, although the book contains a chapter outlining four aural tunings including equal temperament and instructions in the use of Scheibler's then commercially available thirteen fork tonometer.To construct the twelve note scale used as the basis for the upper half of a piano Sievers specified to extend a line segment qs-192mm long for the sounding length for note A5 (a′′)-at 45 degrees to the line qr, and from a point t, located midway between q and a, to extend a line perpendicular to qr intersecting it at u, then to divide tu into 12 equal parts. A point v on qr is located by transferring the lengths of qt, tu, and ut from q towards r, and rays extended from v through the points dividing tu and intersecting qs to locate the different endpoints of the string lengths from s.
This arrangement is equivalent to using the right triangle qtu to generate the exact mean proportional of 192 and 92.
Wolfenden
English piano maker Samuel Wolfenden presented a construction for determining all but the lowest sounding plain string lengths in a piano in A Treatise on the Art of Pianoforte Construction published in 1916; like Sievers, he did not explain whether this was an original procedure or one in common use, commenting only that is was "a very practical method of determining string lengths, and in past years I used it altogether". He added that at the time of writing he found calculating the lengths directly "somewhat easier" and had preceded the description with a table of computed lengths for the top five octaves of a piano. He included frequencies in equal temperament, but only published aural tuning instructions in his 1927 supplement.Wolfenden explicitly advocated equalizing the tension of the plain strings which he proposed to accomplish in the upper range by combining a 9:17 ratio between octave lengths with a uniform change in string diameters (achieving slightly more consistent results over the otherwise similar system published by Siegfried Hansing in 1888), in contrast to Sievers scale whose stringing schedule results in higher tension for the thicker, lower sounding pitches.
Like Sievers, Wolfenden constructed all of the sounding lengths on a single segment at 45 degrees from the base lines for the rays, starting with points located for each C in the range designed at 54, 102, 192.5, 364 and 688mm from the upper point. The four vertices for the rays are then located by the intersections of the horizontal base lines extended from the lower C in each octave with a second line angled from the upper starting point for the string line, however, which he specified should both be at 51.5 degrees to the base lines and that the base lines have a 35:13 ratio with the difference between the two octave lengths.
Wolfenden's method approximates
with roughly 1.3775, and is equivalent to in Barbour's form. Compensating for its smaller octaves this produces 596 cent half octaves, an error of about 1mm at note F4 (f′) compared with his calculated figures.