Detailed and clear instructions for the rational calculation

The treatise Detailed and Clear Instructions for Rational Calculation is a music-theoretical work by Georg Andreas Sorge (1703–1778), which he published in 1749 in Lobenstein. The full title reads: Detailed and clear instructions on the rational calculation, and the associated measurement and division of the monochord, conveying which one The musical temperature, as it is required by today's practice, which all, both with music, as well as with To avoid making organs and instruments, to understand them as necessary as useful, as precisely as the ear can grasp, not only calculate in the most varied of ways, but also measure down to a hair, and consequently bring them to organs and all sorts of other instruments . Along with a detailed message from the new Telemannic interval system. To promote pure harmony, for the sake of clarity, put in a conversation-wise manner and brought to light by Georg Andreas Sorgen / Gräfl. Reuss-Plauischen court and city organists in Lobenstein, and members of the corresponding society of musical sciences in Germany. Lobenstein, published by the author. 1749

meaning

At a time when the music-theoretical specialist discussion was more and more shifting from the mathematical considerations of the tone system and the interval relationships to the questions of a music theory that considered mathematical calculations to be dispensable, Sorge set a music-theoretical counterpoint with a substantial and complex work that was perceived as not very modern . According to his publications ( Instructions for Mood and Temperature , 1744; Vorgemach der Musicalischen Composition , 1745 and Conversation between a Musico Theoretico and a Studioso musices , 1748), which had devoted themselves to a similar subject, the Rational Calculation was the most important concern . The writings mentioned are to be seen in connection with the discussions within the Corresponding Societät der Musicalischen Wissenschaften of the Bach student Lorenz Christoph Mizler . After Johann Sebastian Bach in 1747, Sorge was accepted into this society as the 15th member and was actively involved in the dispute over the New Musical System presented by Georg Philipp Telemann in 1743 . The famous Hamburg composer, who was also a member of Mizler's Societät, proposed a tone system in which the octave should be divided into 55 small intervals. Telemann had already suggested 55 different tone names for the tones created in this way: The tone c, for example, could be raised a maximum of three times with crosses and, as an unsigned prime, led to the smallest prime, followed by the small prime with a cross (ces), the large prime with two crosses (cex) and the largest prime with three crosses (cexes).

Telemann, who had been asked by members of the Society to provide a mathematical description of his microtonal system, did not want to face this task, as he considered a more precise specification in the form of exact mathematical descriptions to be superfluous. There were members of the society, for example Christoph Gottlieb Schröter, who were skeptical of Telemann's system. On the other hand, Johann Adolph Scheibe , who was remote from the law firm , supported this proposal. Sorge got in touch with the now almost unknown mathematician Johann Christoph Breitfeld, with whose help it was possible to describe Telemann's system mathematically to the Society. Breitfeld had calculated the size of the Telemann comma as 405073: 400000 and Sorge had sent a table for Telemann's system based on a logarithmic basis. Starting from log (½) = 0.3010299.9565 he gave the 55th part of this decadic logarithm as 0.00547327265 (p. 223). Around this central statement of the book there is a dialogue between teacher and pupil, based on fundamental considerations of the rational calculation of the tone system. Breitfeld had so convinced Sorge of the use of the logarithmic method in music-theoretical questions that the music theorist devoted the tenth lesson of his rational calculation to this topic. He published a logarithmic table from 1 to 1000 and added exercises on this type of calculation. On the basis of this table it was not only possible to specify the 55th part of the octave, but also the exact mathematical description of the division of the octave into twelve semitones could now be given with the logarithmic value 2508 7/12. Andreas Werckmeister , Georg Heinrich Bümler and Johann Georg Neidhardt had already requested the same temperature, but Sorge had now succeeded in determining the size of this semitone.

construction

Introduction page 1.
I. Lesson on the relationships between the intervals, their sexes and names. 20.
II. Lection, from the Monochord. 63.
III. Lesson on the addition of proportions. 70.
IV. Lesson on the subtraction of proportions. 83.
V. Lesson on the Division of Relations. 118.
VI. Lesson from the multiplication of the ratio. 126.
VII. Lesson from the calculation and measurement of a temperature. 131.
VIII. Lesson on the actual exercise of the theory of temperature, which was obtained through the art of arithmetic and measuring. 196.
IX. Lesson on the interval system of Herr Capellmeister Telemann 200.
X. Lesson on logarithmic arithmetic 238.
XI. Lesson of the rationally equal temperature according to the logarithmic art of arithmetic, together with instructions for the extraction of the square and cubic roots, and instructions on how to use such in the calculation of the rationally equal temperature. 263.
Mr. Cämmerer Breitfeld's calculation of the rational-equal temperature per numeros logarithmicos. 294. The
same calculation of a temperature that is almost rationally the same by combining the circles of fifths and fourths. 296. The
same geometric distribution of the commatis ditoniei in 12th, the same of the diesis in 3rd and the excess 648: 625 in 4th, the same proportions, together with clear instructions to be used as such according to Herr Neidhardt's method when calculating the rationally equal temperature. 299. seqq.

reception

Mizler, who presumably had first become aware of Sorge's Rational bill in the early 1750s , told Meinrad Spieß the - but then not implemented - plan to review Sorges Rational bill in the first part of the fourth volume of the musical library . The postponement of the review could be due to Mizler's appreciation, because a critical review, written quickly, would not have been honest in view of the thoroughness of Sorge's work. The rational calculation required a thorough study, for which Mizler could not spare any time during these years. In any case, Mizler was very interested in the discussion about the topic introduced by Telemann and discussed it directly with the Hamburg conductor. The postponement of the review could also be related to the power struggles within the society between Mizler and Sorge. The correspondence between Mizler and Spieß shows that he had also dealt with Sorge's rational calculation . At least Sorge had sent him four copies. The Capellmeister Telemann, who was particularly affected by this music-theoretical work, and presumably all the other members of the society, probably Johann Sebastian Bach as well , had taken note of Sorge's treatise. The fact that Bach was disinterested in such mathematical questions was asserted several times by his second eldest son, but there is no evidence from the first source for this assumption. As Telemann's godson, Carl Philipp Emanuel Bach was very close to the views of the "modern" Hamburg composer, but some of his father's "dry" counterpoints were alien to the son and friends of the gallant style.

Internet

Detailed and clear instructions for the rational calculation , Lobenstein 1749. Source online .

literature

Lutz Felbick : Lorenz Christoph Mizler de Kolof - student of Bach and Pythagorean "Apostle of Wolffian Philosophy" (= University of Music and Theater "Felix Mendelssohn Bartholdy" Leipzig - writings. Volume 5). Georg-Olms-Verlag, Hildesheim 2012, ISBN 978-3-487-14675-1 .

Individual evidence

↑ Felbick 2012, pp. 177, 278, 280, 297, 300-308, 328, 329, 334, 335, 340, 341 and others. 349. See Wikipedia article on Georg Andreas Sorge and Musikalische Bibliothek , III.4 [1752], pp. 713–719 and 720–726; cf., Last Employment / Georg Philipp Telemanns, in the 86th year of life, consisting / in a musical / sound and interval table , Hamburg 1767.
↑ Telemann's reasoning was published in the journal of the Societät: “But since I have hitherto considered mathematics and the art of measurement to be dispensable in composing, u. I only looked around at the top ”( Musikalische Bibliothek , III.4 [1752], p. 717).
↑ The title of both of Sorge's other writings also suggests these disputes: Georg Andreas Sorge, Thorough investigation into whether the Schroeter piano temperatures in the third part of the third volume of the Mizlerische musicalische Bibliothek, pp. 457 and 580, can pass for the same balance or not , Lobenstein 1754 and Georg Andreas Sorge, Reliable instruction on how to temper and tune pianos and organs appropriately, along with a copper, which represents the measurement and calculation of the temperature, as well as the Telemannic system; at the instigation of Mr. B. Barthold Fritzens [...] to tune the mechanical type published, and designed to defend against the same attack , Leipzig and Lobenstein 1758.
^ Mizler to Spieß, March 1, 1752 (Hans Rudolf Jung and Hans-Eberhard Dentler: Letters from Lorenz Mizler and contemporaries to Meinrad Spieß, in: Studi musicali 32 (2003), p. 141). Spieß may have told him about this book or sent him a copy, because Sorge had sent four copies to Spieß on September 7, 1750 (ibid., P. 123).
^ Mizler to Telemann, March 16, 1744 (Georg Philipp Telemann. Correspondence. All available letters from and to Telemann, edited by Hans Große and Hans Rudolf Jung, Leipzig 1972, p. 323. In this letter, Mizler referred to Telemann's letter of February 12, 1744, which is considered lost); see. Mizler to Telemann, October 31, 1753 (ibid., P. 326).
↑ Felbick 2012, pp. 426–446.

[1] Felbick 2012, pp. 177, 278, 280, 297, 300-308, 328, 329, 334, 335, 340, 341 and others. 349. See Wikipedia article on Georg Andreas Sorge and Musikalische Bibliothek , III.4 [1752], pp. 713–719 and 720–726; cf., Last Employment / Georg Philipp Telemanns, in the 86th year of life, consisting / in a musical / sound and interval table , Hamburg 1767.

[2] Telemann's reasoning was published in the journal of the Societät: “But since I have hitherto considered mathematics and the art of measurement to be dispensable in composing, u. I only looked around at the top ”( Musikalische Bibliothek , III.4 [1752], p. 717).

[3] The title of both of Sorge's other writings also suggests these disputes: Georg Andreas Sorge, Thorough investigation into whether the Schroeter piano temperatures in the third part of the third volume of the Mizlerische musicalische Bibliothek, pp. 457 and 580, can pass for the same balance or not , Lobenstein 1754 and Georg Andreas Sorge, Reliable instruction on how to temper and tune pianos and organs appropriately, along with a copper, which represents the measurement and calculation of the temperature, as well as the Telemannic system; at the instigation of Mr. B. Barthold Fritzens [...] to tune the mechanical type published, and designed to defend against the same attack , Leipzig and Lobenstein 1758.

[4] Mizler to Spieß, March 1, 1752 (Hans Rudolf Jung and Hans-Eberhard Dentler: Letters from Lorenz Mizler and contemporaries to Meinrad Spieß, in: Studi musicali 32 (2003), p. 141). Spieß may have told him about this book or sent him a copy, because Sorge had sent four copies to Spieß on September 7, 1750 (ibid., P. 123).

[5] Mizler to Telemann, March 16, 1744 (Georg Philipp Telemann. Correspondence. All available letters from and to Telemann, edited by Hans Große and Hans Rudolf Jung, Leipzig 1972, p. 323. In this letter, Mizler referred to Telemann's letter of February 12, 1744, which is considered lost); see. Mizler to Telemann, October 31, 1753 (ibid., P. 326).

[6] Felbick 2012, pp. 426–446.