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Comma

Comma ( Greek κόμμα - segment) in the theory of music is the general name for micro-intervals of about 1/7 - 1/10 of a whole tone , which arise, as a rule, when comparing intervals of the same type in different musical systems [1] . The most famous are the synthonic (Didim) comma and the Pythagorean (Pythagorean) comma. Artificial (Holder or Arabic) and septimal are also known. (architects) of the comma.

There are comms less than 1/10 of a whole tone, for example, the Mercator comma [2] , which does not contradict the definition of comma as the difference between the mathematical values ​​of two tones, approximately equal in height [3] . Based on this definition, the varieties of coma should be recognized, for example, minor sharpening , more than 1/7 of a whole tone and schism , less than 1/10 of a whole tone .

The usual uniform temperament destroys all varieties of coma, with the exception of rare exceptions [4] . When talking about a comm without specifying its name, we are talking about a syntonic comm.

History

Despite the antiquity of the term (in ancient times it was actively used in the context of rhetorical teachings ), the first evidence of the use of coma as a musical-theoretical term refers only to the 5th century AD. e. It is in Proclus' commentary on Timaeus of Plato (Plato himself does not have the term “comma”). In Latin literature, the first evidence of a comma is in the treatise "Fundamentals of Music" (about 500 years) Boethius . Proclus defines commune (in modern times called “Pythagorean") as the difference of apotoma and limma , but calculates it as the difference of the relations of a whole tone and two limms (this calculation of Proclus, however, contains an arithmetic error). Boethius knows these methods, adding to them also the calculation of the coma as the difference between six whole tones and an octave. Boethius (De inst. Mus III, 10). In his opinion, comma is the smallest (or “most recent”) of what a person’s hearing can perceive (est enim comma, quod ultimum conprehendere possit auditus). Nowadays it is well known that this is not so. Not only the Pythagorean comma [5] , but also its shares are accessible to human hearing.

Performing the usual uniform temperament , for example, requires the ability to hear 1/12 of the Pythagorean coma. It is at such an interval that each natural pure fifth (3: 2) [6] must be reduced in order for the mentioned setting to be completed successfully. This method of fulfillment of temperament [7] was affirmed as a result of the historical development of the so-called “good temperaments” proposed in the time of JS Bach.

Pythagorean Comma

Twelve fifths should add up to seven octaves . However, in the Pythagorean system (in which the ratio of the frequencies of the tones forming the fifth) is 3: 2) there is a difference called the Pythagorean , or Pythagorean comma, equal to about a quarter of a semitone :

(32)1227=3122nineteen=531441524288≈23,46Cent{\ displaystyle {\ frac {\ left ({\ frac {3} {2}} \ right) ^ {12}} {2 ^ {7}}} = {\ frac {3 ^ {12}} {2 ^ {19}}} = {\ frac {531441} {524288}} \ approx 23 {,} 46 \; \ mathrm {Cent}}   [eight]

Synth Comma

The Didim Comma is also called, by the name of Didim the Musician , a scientist of the 1st century BC. e., for the first time describing the 5: 4 thirds in a tetrachord of the diatonic genus (the musical and theoretical teachings of Didim were not preserved; it is known in the account of Ptolemy and Porfiry ). The very phrase “Didim Comma” itself appeared, apparently, in the New Time . In ancient treatises on music (Greek and Latin), the term "Didim Comma" is not.

If you add together four pure fifths (3: 2) and subtract two octaves (2: 1), you get the Pythagorean big third (deaton) :

(32)four22=3four26=8164≈407,82Cent{\ displaystyle {\ frac {\ left ({\ frac {3} {2}} \ right) ^ {4}} {2 ^ {2}}} = {\ frac {3 ^ {4}} {2 ^ {6}}} = {\ frac {81} {64}} \ approx 407 {,} 82 \; \ mathrm {Cent}}  

Deaton is larger than the natural major third [9] (81:64> 5: 4) by synthonic (or didim) comm:

8164:fivefour=8164∗6480=8180≈21,51Cent{\ displaystyle {\ frac {81} {64}}: {\ frac {5} {4}} = {\ frac {81} {64}} * {\ frac {64} {80}} = {\ frac {81} {80}} \ approx 21 {,} 51 ​​\; \ mathrm {Cent}}  

Artificial Comma

The following is known about artificial comm [10] :

Nikolai Mercator , a modest person and a scientist and intelligent mathematician <...> brought out the ingenious invention of the search and application of the smallest general measure of all harmonic intervals, not strictly ideal, but very close to it . Assuming that the commune 1/5th part of the octave <...> he calls this 1/53th artificial artificial comma not exact, but different from the true natural comma by about 1/20 of the comma

Original text
Nicholas Mercator a modest person and a learned and judicious mathematician <...> has deduced an ingenious invention of finding and applying a least common measure to all harmonic intervals, not precisely perfect, but very near to it . Supposing a comma to de 1/53 part of diapason <...> wich 1/53 he calls an artificial comma not exact, but differing from the true natural comma about 1/20 part of a comma
- Golder (quoted from the book of G. Riemann) [11]

In musical theory, artificial commune is also called a holder comma [12] [13] , sometimes an Arab comma [14] ; this micro-interval is between any pair of neighboring heights in a system of 53 equal divisions of an octave (1200 cents) and its value is easily calculated:

120053≈22,641fiveCent{\ displaystyle {\ frac {1200} {53}} \ approx 22 {,} 6415 \; \ mathrm {Cent}}  

An artificial comm is equally suitable and convenient to use instead of the Pythagorean and Didim comm. It allows you to not make distinctions between Didim and Pythagorean comms in the specified musical notation. Only one universal set of signs of alteration to indicate commatic difference [15] is necessary and sufficient. There is no need to observe the mentioned differences for the construction of musical instruments.

Along with an indication of Golder’s report on the significant contribution to the theory of music by the humble Nikolai Mercator, the recognized music theorist of the turn of the 19th-20th centuries Hugo Riemann also published the following statement:

mathematicians have irrefutably proved that for the free use of all tonalities only a system of 53 steps in an octave is better than a conventional system of 12 uniformly tempera

- G. Riemann [16]

Comm Mercator

It was noted above that the Mercator comma is much smaller than the most well-known comms, because it is the difference between the chains of 53 natural fifths and the 31st natural octave with the value:

(32)53231=353284=1938324566768001989679672319342813113834066795298816≈3,6150Cent{\ displaystyle {\ frac {\ left ({\ frac {3} {2}} \ right) ^ {53}} {2 ^ {31}}} = {\ frac {3 ^ {53}} {2 ^ {84}}} = {\ frac {19383245667680019896796723} {19342813113834066795298816}} \ approx 3 {,} 6150 \; \ mathrm {Cent}}   .

Narrowing each natural fifth to an insignificant amount of 1/53 of Mercator’s comme, you get the so-called Mercator’s cycle, which closes the chain of 53 such fifths, which leads to the division of the octave into 53 artificial comms. Like the destruction of the Pythagorean coma in a cycle of 12 evenly tempered fifths, the Mercator cycle destroys the Mercator commune, but the Pythagorean comma is not destroyed, but replaced by almost the same artificial one.

Comma and music

The Comma does not form a separate step in the traditional West European modal frets and in the major-minor key (and, accordingly, is not endowed with a special fret function ), however it is used by musicians (vocalists and performers on instruments with unfixed scales, for example, on the violin ) to give the performance a greater expressiveness.

Contrary to the prevailing opinion that it is possible to exclude the comma from a number of intervals necessary for full-fledged playing music [17] , there are facts in favor of other views:

Adding or subtracting a comma tells ... to both sounds of any interval a completely different dynamic direction ... In the temperament, the coma additives are cut off (instead of a diatonic halftone with a comma, an amorphous tempera grayscale is added) ... The logic of musical thinking is controlled by the relationships and interactions of sounds within the system non-tempered (for us, deterministic) form.

- A. S. Ogolevets [18]

If we take as the smallest interval the value of the Pythagorean coma (24 cents) as an interval freely distinguishable by our hearing (al-Farabi also argued that this interval should be considered one of the main ones in musical theory and practice, and within the octave range should be called typical, the most stable intervals, it is possible to determine almost 30 steps, which are conscious and creatively used in the melodic structures of musical practice of many peoples of the East.

- G. A. Kogut [19]

Exploring Pers. east, Khorasan tanbur, F. [Arabs] calculated the Pythagorean large whole tone (see. Pythagorean system), breaking up into 3 micro-intervals (two limms and a comma). This whole tone was the basis of the 17-stage scale developed by the middle century. theorists of the East.

- O. V. Rusanova [20]

In Azerbaijan, comms are quite deliberately used in traditional music, along with the search for suitable systems for their notation [21] .

 

Modern musical writing of Turkey directly indicates the use of comma in Turkish music. In measures 3..11 of the proposed musical example, it is required to play a si- békar note (Turkish name bûselik), however, in the first two measures it is prescribed to play a si-on-commune-lower note (segâh). The independent names of two notes at a distance of the comma indicate the existence of a commatonic step in the Turkish scale.

One of the features of Nar. melodies - their modal variability (constant short-term deviations from one fret to another). The special “flowering” of the melos is also explained by the increase and decrease in diatonicity. steps on the comm; in T [Uretsk] m [music] <...> there is a special mode system (Turkish theorists believe that this system corresponds to a scale with 24 steps in an octave). Many Turkish modes are similar to the European ones, however, in Turkish theory they have special names: for example, natural major with supporting I and V steps and a step reduced to commune VI is called Mahhur, with the same supporting steps and a step reduced to commune III - rast

- Musical Encyclopedia [22]

Other indisputable evidence is the special signs of alteration, prescribing kommaticheskie increase / decrease of notes.

In Turkey, the use of a system of 53 artificial comms in an octave has spread, as a support for a theory compatible with the practice of playing music [23] .

In India, according to the ancient definition, the so-called shruti are perceived as sound-altitude intervals [24] . Three of their varieties are known: pramana, nyuna and purana sruti [25] . Varieties can be compared with numerical values: pramana shruti (70 cents), nyuna shruti (22 cents) and purana shruti (90 cents) [26] , which are obtained with good approximation from artificial commits of the 53rdo system [27] . This means that in ancient classical music, from ancient times, intervals comparable to comma have been known: they have their own names and are in demand on a par with all other intervals.

 
 
 
Courtesy of the Sciense Museum, London. Pam Fluke at the keyboard of the anharmonic harmonium R.H.M. Bozanketa (England, 1872; restored in 2006). 53RDO, 84 keys per octave
 

In Western music, several hundred years of the history of the appearance of numerous projects and even manufactured keyboard instruments of a fixed system of unusual temperament (or without it at all), where steps at a distance of the comma are specially provided, providing the possibility of practical study of their functional properties, can serve as confirmation of the constant desire to use coms [28 ] .

Didimova Komma plays in modern music science as important a role as Pythagorov in calculating uniform temperament, especially in works devoted to conducting, in contrast to all temperaments, a clean order (Hauptmann, Helmholtz, von Oettingen, Engel, Tanaka, etc. )

- G. Riemann [29]

One of those who showed this in practice was the Yugoslav composer I. Slavensky. The first part of his composition, “Music for the Natur-Tone System,” was written for the Bozanket's anharmonic harmonium (anharmonium) [30] , the first musical instrument in the world with octaves from 53 artificial comm lines .

Playing such instruments is inconceivable without the commotic notation first developed by Bozanket. Slavensky schematically stated it in the preamble of the score and explicitly applied it in the first part.

The Bozanket acoustic instrument built in 1871–72 was followed by an octave dividing system into 53 artificial comms of the harmonium of the American master J.P. White. One of his three built acoustic instruments has a nameplate:

Harmon No.3, Jess. Paul White, Inventor and Manufacturer, 1883

Original text
Harmon No.3, Jas. Paul White, Inventor & Maker, 1883

It is stored in the conservatory of Boston, USA [31] . The design of the keyboard and the device of White's harmonies are very different from the Bozanket prototype. However, the principle of maintaining the same fingering in the performance of the same play from different notes, implemented by Bozanket, is respected.

Like Bozanket's unique enharmonium and White's unique harmonies, acoustic instruments with full sets of artificial commos were made according to the designs mentioned by Riemann Ettingen in Germany (1914). The design of their keyboards claims to be an ergonomically advanced version of the Bozanket solution. It is significant that they were called orphotonophoniums, that is, sounding in true tones [32] . This emphasizes that the hearing perceives tonal music played in the 53 artificial comm system as sounding true. In the photo you can see one of the orthophonophoniums stored in Berlin. Several true chords of this instance can also be heard [33] . Another orthophonophonium is kept in Leipzig [34] .

Interesting Facts

  • In 1990, for the comma, which is the difference between the 665th fifths and 389th (in the source error: 359) octaves, the name Satanic was proposed [35] . Its value is less than 1 / 10th of a cent (more precisely, 0.076 cents), and the name mimics the name of the syntonic comma. The meaning of the name is also reflected in the fact that in the chain of pure fifths the 666th link turns out to be an indistinguishable fake of the true octave version of the initial link of this chain. Thus, the 666th fifth creates a satanically false closure of the fifth spiral. Another person, independently and much later [36] , noted that the first false closure of this spiral (with a difference in the Pythagorean commune) creates the 12th fifth, the 13th link of which in the spiral turns out to be the so-called hell dozen, and the last meaningful one is a satanic number .
  • Sometimes Mercator’s comma is called artificial commune , although Mercator himself gave her the artificial name.

Notes

  1. ↑ Great Russian Encyclopedia , vol. 14. M., 2009, p. 645.
  2. ↑ Dillon, Musenich 2009, p. 49: “C 53 = 1.002090314. C 53 is also known as Mercator’s Comma ( Eng. C 53 = 1.002090314. C 53 is also known as Mercator's comma) "
  3. ↑ Musical Dictionary 2008, Comma: “this is the name of the difference between the mathematical values ​​of two tones, approximately equal in height”
  4. ↑ For a clean system , for example, the difference between six minor thirds and one pure duodenum , the so-called brand ( en: Kleisma ), is about 8.1 cents and in the usual 12 RDO system is not destroyed, but degenerates there into a semitone (100 cents)
  5. ↑ Riemann 1898, p. 99: “According to the research of W. Preyer (" Ueber die Grenzen der Tonwahrnehmung ", 1876), experienced musicians can still distinguish the difference in height by 1/2 oscillations in the two-stroke octave; for g "with 792 oscillations, this would give a logarithmic value (at the base of 2) 0.00090, that is, barely 2/3 of the schism ”
  6. ↑ The interval of the natural pure fifth is equal to the interval of the natural scale between the 3rd and 2nd overtones.
  7. ↑ Fadeev, Allon 1973, p. 255-8
  8. ↑ If the frequency ratio of two sounds ( a ) and ( b ) is known, then the number of cents ( n ) in the interval between them:
    n=1200log2⁡(ab)≈3986logten⁡(ab){\ displaystyle n = 1200 \ log _ {2} \ left ({\ frac {a} {b}} \ right) \ approx 3986 \ log _ {10} \ left ({\ frac {a} {b}} \ right)}  
  9. ↑ The interval of the natural major third is equal to the interval of the natural scale between the 5th and 4th overtones.
  10. ↑ Barbieri 2008, p. 611 : “comma, definition:“ artificial ”(RTS 53), 350 ( eng. Comma, definition of:“ artifical ”(ETS 53), 350 )”
  11. ↑ Riemann 1898, p. 67
  12. ↑ The Ratio book: a documentation of The Ratio Symposium, Royal Conservatory, The Hague, December 14-16, 1992 .
  13. ↑ "Lux oriente": Begegnungen der Kulturen in der Musikforschung: Festschrift Robert Günther zum 65. Geburtstag. Kassel: G. Bosse Verlag, 1995. (= Kölner Beiträge zur Musikforschung, Bd. 188).
  14. ↑ Touma HH The Music of the Arabs, p.23. trans. Laurie Schwartz. - Portland, Oregon: Amadeus Press, 1996. ISBN 0-931340-88-8 .
  15. ↑ Kholopov 2003, p. 141: “we hear commatic difference”
  16. ↑ Riemann 1898, p. 63
  17. ↑ Kholopov 2003, p. 141: “Comma cannot be perceived as an interval itself (step)”
  18. ↑ Ogolevets 1941, ss. 61-2].
  19. ↑ Kogut 2005, p. 27
  20. ↑ Musical Encyclopedia 2008-11, Farabi
  21. ↑ Aliyeva 2011, p. ?
  22. ↑ Music Encyclopedia 2008-11, Turkish music
  23. ↑ Yarman 2007, p. 58: “Due to the excellent proximity of any 24-tone model to the corresponding tones of the octave when dividing it into 53 equal parts, the methodology of“ 9 comms per whole tone; 53 comms per octave ”is unanimously adopted in the Turkish vocabulary of music and teaching ( Eng. Because of the excellent proximity of either 24-tone model to the related tones of 53-equal divisions of the octave, the “9 commas per whole tone; 53 commas per octave” methodology is unanimously accepted in Turkish makam music parlance and education ) »
  24. ↑ Sarangadeva , Sangeet Ratnakar with comments of Kalinath, Anandasram edition, 1897.
  25. ↑ Lentz 1961, p. ?
  26. ↑ Datta, Sengupta, Dey and Nag 2006, p. 28: “Table 2.4 gives the length distribution of the predicted shruti. The smallest shruti are about 14 cents, and the largest 85 cents. These values ​​can be compared with the size of pramana shruti (70 cents), nyuna shruti (22 cents) and purana shruti (90 cents), as given in Western literature ( Table 2.4 gives a distribution of the length of the predicted shrutis. The smallest shruti is about 14 cents and the largest is 85 cents. These values ​​may be compared with the measure of pramana shruti (70 cents), nyuna shruti (22 cents) and purana shruti (90 cents) as given in western literatures ) »
  27. ↑ Khramov 2011, p. 32: “An ideal QI system is not closed, but can be approximated well in a closed 53RDO system. An interesting feature of this system is the proximity of its smallest microtone, or coma (22.642 ¢) to the size of the smallest microtone of the Indian scale, known as nyuna shruti (22 ¢). Pramana Shruti (70 ¢) and Purana Shruti (90 ¢) are close to the sums of three (67.925 ¢) and four (90.566 ¢) comm 53RDO systems respectively ( English The ideal JI system is nonclosed, but may be not bad approximated in the closed 53EDO system. As attractive feature of this system appears proximity its minimal microtone, or comma (22.642 ¢) to size of the minimal microtone of an Indian scale, which is known as nyuna shruti (22 ¢). Pramana shruti (70 ¢) and purana shruti (90 ¢) are accordingly close to sums of three (67.925 ¢) and four (90,566 ¢) commas of the 53EDO system ) »
  28. ↑ Barbieri 2008, 620 pp.
  29. ↑ Riemann 1898, p. 13
  30. ↑ The Harmonious Harmonium by R. H. M. Bozanket // London Science Museum.
  31. ↑ Barbieri 2008, p. 100-2
  32. ↑ Goldbach 2007, 29 pp.
  33. ↑ Orphotonophonium A. von Oettingen // Berlin Museum of Musical Instruments
  34. ↑ Orphotonophonium A. von Oettingen // Musical Instrument Museum of the University of Leipzig
  35. ↑ Jones 1990 according to Monzo 2005: << ... Satanic comm. The difference between the 665th quint and the 359th octave, less than 1/10 of the cent, about 1/15878 of the octave <...> [name] was invented in 1990 as a parody of the name of the synthonic coma . Satanic comma. The difference between 665 fifths and 359 octaves, less than 1/10 of a cent, around 1/15878 of an octave <...> coined 1990, as a parody on the name of the syntonic comma ) .. . >>
  36. ↑ Will 2005: commenting on his recently written work in a private conversation, G. Vol noted that the first and last closures of the theoretically infinite fifth spiral, which are conceivable for its physical embodiment in the form of keyboard instruments with fingering suitable for human hands, lead to the numbers 12 and 665, bordering on the wicked 13 and 666, respectively.

Links

  • Goldbach KT Arthur von Oettingen und sein Orthotonophonium im Kontext // Tartu ülikooli muusikadirektor 200, hrsg. v. Geiu Rohtla, Tartu 2007. — p. 29.
  • Datta AK, Sengupta R., Dey N., Nag D. Experimental Analysis of Shrutis from Performances in Hindustani Music . - Kolkata, India: SRD ITC SRA, 2006. - P. 103. - ISBN 81-903818-0-6 .
  • Dillon, Giorgio; Musenich R. The Huygens Comma: Some Mathematics Concerning The 31-Cycle // Thirty-One. The Journal of the Huygens-Fokker Foundation: journal / Gilmore, Bob. - Amsterdam: Stichting Huygens-Fokker Center for Microtonal Music, 2009. - Vol. 1 . - P. 49-56 .
  • Khramov, M. On Amount of Notes in Octave (Neopr.) // Ninãd, Journal of the ITC-SRA / Datta AK, Sengupta R. .. - Kolkata, India: ITC Sangeet Research Academy. Dept. of Academic Research, 2011 .-- December ( v. 25 ). - S. 31-37 . - ISSN 0973-3787 . Archived on October 18, 2012. Archived October 18, 2012 on Wayback Machine
  • Lentz DA Tones and Intervals of Hindu Classical Music // University of Nebraska Studies, New Series No.24, University of Lincoln, 1961.
  • Monzo, J. Tuning terms by Marc Jones (neopr.) . Tonalsoft Encyclopedia of Microtonal Music Theory (2005). Date of treatment October 22, 2012. Archived October 23, 2012.
  • Yarman, Ozan. A Comparative Evaluation of Pitch Notations in Turkish Makam Music: Abjad Scale & 24-Tone Pythagorean Tuning - 53 Equal Division of the Octave as a Common Grid // Journal of interdisciplinary music studies: journal /?. -?, 2007. - Fall ( vol. 1 , no. 2 ). - P. 43–61 . - ISSN ? . Archived on May 18, 2013.
  • Alieva, I. Microtonal notation through numerical refinement of the signs of alteration (on the example of the packaging scale) (Russian) // Musiqi Dünyası, publisistik musiki jurnali: journal / Mamedov T. A. .. - 2011.- T. 47 . - S.? . - ISSN 2219-8482 .
  • Barbieri P. Anharmonic instruments and music, 1470-1900 . - Latina: Il Levante libreria editrice, 2008 .-- 620 pp. ISBN 978-88-95203-14-0
  • Vol, Gennady Algebra of Tonal Functions (neopr.) (2005). Date of treatment October 23, 2012. Archived November 4, 2012.
  • Kogut G.A. Microtone music . - Kiev: Naukova Dumka, 2005 (in Russian). - 264 pp. ISBN 966-00-0604-7
  • Musical Encyclopedia. Source: Musical Encyclopedia in 6 vols., 1973-82 Turkish music (neopr.) . www.music-dic.ru (2008-11). Date of treatment October 24, 2012. Archived November 4, 2012.
  • Musical Encyclopedia. Source: Musical Encyclopedia in 6 vols., 1973-82 Farabi (neopr.) . www.music-dic.ru (2008-11). Date of treatment October 24, 2012. Archived November 4, 2012.
  • The musical dictionary. Based on the publication: Riemann G. Musical Dictionary [Transl. with him. B.P. Jurgenson, add. Russian Dep.], 1901 Comma (Neopr.) . Direct Media Publishing (2008). Date of appeal October 23, 2012. (unavailable link)
  • Ogolevets A. S. Fundamentals of harmonic language. M. - L., 1941.—? p .
  • Riemann, G. Acoustics from the point of view of music science . - Translation from German N. Kashkina. - Moscow: Tip-lit. K. Alexandrova, 1898 .-- P. 1 - 83.
  • Riemann, G. Acoustics from the point of view of music science . - Translation from German N. Kashkina. - Moscow: Tip-lit. K. Alexandrova, 1898 .-- P. 84 - 147.
  • Fadeev I. G., Allon S. M. Repair and tuning of pianos and pianos. - M.: Light Industry, 1973. - 304 p.
  • Kholopov Yu. N. Harmony: Theoretical course: Textbook. . - St. Petersburg, Moscow, Krasnodar: Publishing House "Lan", 2003. - P. 541. - ISBN 5-8114-0516-2 .

Literature

  • Zubov A. Yu. Comma // Big Russian Encyclopedia. T. 14. Moscow, 2009, p. 645.
Source - https://ru.wikipedia.org/w/index.php?title=Comm&oldid=101454335


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