Equal temperament
Equal temperament[a] is a musical temperament (tuning system) that divides the octave into 12 identical parts. It is a logarithmic scale with a ratio equal to the 12th root of 2 ( ≈ 1.05946). The scale unit is known as a semitone or half step.
![{\textstyle {\sqrt[{12}]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f276a910a28604e3b8afd8ba62a4678a37193480)
Equal temperament has been the predominant tuning system of Western music since the 18th century. It is also used in other cultures.
Equal temperament is usually tuned relative to a standard pitch of 440 Hz.[1]
History
[edit]The two figures frequently credited with the exact calculation of equal temperament are Chinese mathematician and theorist Zhu Zaiyu (also romanized as Chu-Tsaiyu, Chinese: 朱載堉) and Flemish scholar Simon Stevin.[2]
Zaiyu is sometimes credited with the discovery of equal temperament.[3] In 1584, Zaiyu wrote, "I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations."[4]
Simon Stevin's independent solution came the following year and was slightly less precise. Both men primarily approached the problem as a computing procedure rather than a practical musical solution.[5]
China
[edit]Early history
[edit]A complete set of bronze chime bells, among many musical instruments found in the tomb of the Marquis Yi of Zeng (early Warring States, c. 5th century BCE in the Chinese Bronze Age), covers five full 7-note octaves in the key of C Major, including 12 note semi-tones in the middle of the range.[6]
The earliest numerical approximation for equal temperament was described by He Chengtian, a mathematician of the Southern and Northern Dynasties who lived from 370 to 447.[7][8]
Zhu Zaiyu
[edit]
Zhu Zaiyu (朱載堉), a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar 律暦融通 published in 1580. This was followed by the publication of a precise numerical specification for equal temperament in his 5,000-page work Complete Compendium of Music and Pitch (Yuelü quan shu 樂律全書) in 1584.[9] An extended account is also given by Joseph Needham.[10] Zhu obtained his result mathematically by dividing the length of string and pipe successively by 12√2 ≈ 1.059463, and for pipe length by 24√2,[11] such that after twelve divisions (an octave) the length was divided by a factor of 2:
![{\displaystyle \left({\sqrt[{12}]{2}}\right)^{12}=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0e94dc870dfc2af0eb35a7b362eb3776054719)
Similarly, after 84 divisions (7 octaves) the length was divided by a factor of 128:
![{\displaystyle \left({\sqrt[{12}]{2}}\right)^{84}=2^{7}=128}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efc85da07cabc4f33cc3fc965e9ef6e74361e7c7)
Zhu Zaiyu has been credited as the first person to solve the equal temperament problem mathematically.[12] At least one researcher has proposed that Matteo Ricci, a Jesuit in China recorded this work in his personal journal[12][13] and may have transmitted the work back to Europe. (Standard resources on the topic make no mention of any such transfer.[14]) In 1620, Zhu's work was referenced by a European mathematician.[who?][13] Murray Barbour said, "The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyü's brilliant solution remains an enigma."[2] The 19th-century German physicist Hermann von Helmholtz wrote in On the Sensations of Tone that a Chinese prince (see below) introduced a scale of seven notes, and that the division of the octave into twelve semitones was discovered in China.[15]
Zhu Zaiyu illustrated his equal temperament theory by the construction of a set of 36 bamboo tuning pipes ranging in 3 octaves, with instructions of the type of bamboo, color of paint, and detailed specification on their length and inner and outer diameters. He also constructed a 12-string tuning instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890, Victor-Charles Mahillon, curator of the Conservatoire museum in Brussels, duplicated a set of pitch pipes according to Zhu Zaiyu's specification. He said that the Chinese theory of tones knew more about the length of pitch pipes than its Western counterpart, and that the set of pipes duplicated according to the Zaiyu data proved the accuracy of this theory.
Europe
[edit]
Early history
[edit]One of the earliest discussions of equal temperament occurs in the writing of Aristoxenus in the 4th century BC.[16]
Vincenzo Galilei was one of the first practical advocates of equal temperament. He composed a set of dance suites on each of the 12 notes of the chromatic scale in all the "transposition keys", and published also, in his 1584 "Fronimo", 24 + 1 ricercars.[17] He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).[18]
Galilei's countryman and fellow lutenist Giacomo Gorzanis had written music based on equal temperament by 1567.[19] Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino wrote a "Recercare de tutti li Toni" (Ricercar in all the Tones) as early as 1507.[20] In the 17th century lutenist-composer John Wilson wrote a set of 30 preludes including 24 in all the major/minor keys.[21][22] Henricus Grammateus drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by Giovani Maria Lanfranco in his "Scintille de musica".[23] Zarlino in his polemic with Galilei initially opposed equal temperament but eventually conceded to it in relation to the lute in his Sopplimenti musicali in 1588.
Simon Stevin
[edit]The first mention of equal temperament related to the twelfth root of two in the West appeared in Simon Stevin's manuscript Van De Spiegheling der singconst (c. 1605), published posthumously nearly three centuries later in 1884.[24] However, due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values.[14] As a result, the frequency ratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is claimed by Gene Cho as incorrect.[25]
The following were Simon Stevin's chord length from Van de Spiegheling der singconst:[26]
| Tone | Chord 10000 from Simon Stevin | Ratio | Corrected chord |
|---|---|---|---|
| semitone | 9438 | 1.0595465 | 9438.7 |
| whole tone | 8909 | 1.0593781 | |
| tone and a half | 8404 | 1.0600904 | 8409 |
| ditone | 7936 | 1.0594758 | 7937 |
| ditone and a half | 7491 | 1.0594046 | 7491.5 |
| tritone | 7071 | 1.0593975 | 7071.1 |
| tritone and a half | 6674 | 1.0594845 | 6674.2 |
| four-tone | 6298 | 1.0597014 | 6299 |
| four-tone and a half | 5944 | 1.0595558 | 5946 |
| five-tone | 5611 | 1.0593477 | 5612.3 |
| five-tone and a half | 5296 | 1.0594788 | 5297.2 |
| full tone | 1.0592000 |
A generation later, French mathematician Marin Mersenne presented several equal tempered chord lengths obtained by Jean Beaugrand, Ismael Bouillaud, and Jean Galle.[27]
In 1630 Johann Faulhaber published a 100-cent monochord table, which contained several errors due to his use of logarithmic tables. He did not explain how he obtained his results.[28]
Baroque era
[edit] | This article needs attention from an expert in music history. The specific problem is: The "Baroque era" section appears to conflate well and equal temperaments; for keyboard instruments, at least, I have heard numerous times that equal temperament was rare until the 20th century, and specifically that Bach's Well-Tempered Clavier was not promoting equal temperament specifically. This conflicts with the article's statement (without a citation) that "[12-TET] was a convenient fit for the existing keyboard design" (my understanding, reinforced by subsequent sentences in this section of the article, is that 12-TET was difficult to achieve on keyboard instruments prior to widespread use of beat rates for tuning). The rest of this sentence, and the following one, seems to apply equally to any form of "well-tempered" keyboard tuning: "[12-TET] permitted total harmonic freedom with the burden of moderate impurity in every interval, particularly imperfect consonances. This allowed greater expression through enharmonic modulation, which became extremely important in the 18th century in music of such composers as [list of composers]." An expert, or at least a reliable source, is needed to establish exactly how common 12-TET was prior to the 20th century, and whether it's true that composers expected or relied on it.. (September 2026) |
From 1450 to about 1800, plucked instrument players (lutenists and guitarists) generally favored equal temperament,[29] and the Brossard lute manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to Bocquet written in all keys, including the last prelude, entitled Prélude sur tous les tons, which enharmonically modulates through all keys.[30][clarification needed] Angelo Michele Bartolotti published a series of passacaglias in all keys, with connecting enharmonically modulating passages. Among the 17th-century keyboard composers Girolamo Frescobaldi advocated equal temperament. Some theorists, such as Giuseppe Tartini, were opposed to the adoption of equal temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music, although Andreas Werckmeister emphatically advocated equal temperament in his 1707 treatise published posthumously.[31]
Equal temperament took hold for a variety of reasons. It was a convenient fit for the existing keyboard design, and permitted total harmonic freedom with the burden of moderate impurity in every interval, particularly imperfect consonances. This allowed greater expression through enharmonic modulation, which became extremely important in the 18th century in music of such composers as Francesco Geminiani, Wilhelm Friedemann Bach, Carl Philipp Emmanuel Bach, and Johann Gottfried Müthel.[citation needed] Equal temperament did have some disadvantages, such as imperfect thirds, but as Europe switched to equal temperament, it changed the music that it wrote in order to accommodate the system and minimize dissonance.[citation needed]
A precise equal temperament is possible using the 17th century Sabbatini method of splitting the octave first into three tempered major thirds.[32] This was also proposed by several writers during the Classical era. Tuning without beat rates but employing several checks, achieving virtually modern accuracy, was already done in the first decades of the 19th century.[33] Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century.[34] The ultimate precision was available with 2 decimal tables published by White in 1917.[35] The intervals of equal temperament closely approximate some intervals in just intonation.[36]
Varieties of equal temperament
[edit]Other equal temperaments divide the octave differently. The Arab tone system uses 24-tone equal temperament. Music has also been written in 19 equal temperament and 31 TET.
Instead of dividing an octave, an equal temperament can also divide a different interval. The equal-tempered version of the Bohlen–Pierce scale divides the just interval of an octave and a fifth into 13 equal parts.
Similar tuning systems
[edit]At least a century before He Chengtian, Chinese musicians developed a series of 12 fundamental tones known as Shí-èr-lǜ.[37] Developed by the ancient Greeks, Pythagorean tuning was the predominant system in Europe until the Renaissance. A series of unequal temperaments were developed that made more chromatic notes available in acceptable tunings.[38]
Comparison with just intonation
[edit]The intervals of 12 TET closely approximate some intervals in just intonation.[36] The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.
In the following table, the sizes of various just intervals[39][additional citation(s) needed] are compared to their equal-tempered counterparts, given as a ratio as well as cents.[citation needed]
| Interval Name | Equal temperament interval | Cents in equal temperament | Just intonation interval | Cents in just intonation | Tuning error |
|---|---|---|---|---|---|
| Unison | 20⁄12 = 1 | 0 | 1/1 = 1 | 0.00 | 0.00 |
| Minor second | 21⁄12 | 100 | 16/15 | 111.73 | −11.73 |
| Major second | 22⁄12 | 200 | 9/8 | 203.91 | −3.91 |
| Minor third | 23⁄12 | 300 | 6/5 | 315.64 | −15.64 |
| Major third | 24⁄12 | 400 | 5/4 | 386.31 | +13.69 |
| Perfect fourth | 25⁄12 | 500 | 4/3 | 498.04 | +1.96 |
| Tritone | 26⁄12 | 600 | 45/32 | 590.22 | +9.78 |
| Perfect fifth | 27⁄12 | 700 | 3/2 | 701.96 | −1.96 |
| Minor sixth | 28⁄12 | 800 | 8/5 | 813.69 | −13.69 |
| Major sixth | 29⁄12 | 900 | 5/3 | 884.36 | +15.64 |
| Minor seventh | 210⁄12 | 1000 | 9/5 | 1017.60 | −17.60 |
| Major seventh | 211⁄12 | 1100 | 15/8 | 1088.27 | +11.73 |
| Octave | 212⁄12 = 2 | 1200 | 2/1 = 2 | 1200.00 | 0.00 |
See also
[edit]References
[edit]Footnotes
[edit]- ↑ Also known as 'twelve-tone equal temperament' ('12-TET'), '12-tone equal division of the octave' ('12-TEDO'), '12 equal division of 2/1' ('12-ED2'), '12 equal division of the octave' ('12-EDO'); informally abbreviated to 'twelve equal' or referred to as 'equal temperament' without qualification in Western countries.[citation needed]
Citations
[edit]- ↑ von Helmholtz & Ellis 1885, pp. 493–511.
- 1 2 Barbour 2004, p. 7.
- ↑ Robinson 1980, p. vii: Chu-Tsaiyu the first formulator of the mathematics of "equal temperament" anywhere in the world
- ↑ Needham, Ling & Robinson 1962, p. 223.
- ↑ Kuttner 1975, p. 163, 200.
- ↑ Kwang-chih Chang, Pingfang Xu & Liancheng Lu 2005, p. 140.
- ↑ Goodman, Howard L.; Lien, Y. Edmund (April 2009). "A Third Century AD Chinese System of Di-Flute Temperament: Matching Ancient Pitch-Standards and Confronting Modal Practice". The Galpin Society Journal. 62. Galpin Society: 7. JSTOR 20753625.
- ↑ Barbour 2004, pp. 55–56.
- ↑ Hart 1998.
- ↑ Needham, Ling & Robinson 1962, p. 221.
- ↑ Needham & Ronan 1978, p. 385.
- 1 2 Cho 2010.
- 1 2 Lienhard 1997.
- 1 2 Christensen 2002, p. 205.
- ↑ von Helmholtz & Ellis 1885, p. 258.
- ↑ True 2018, pp. 61–74.
- ↑ Galilei 1584, pp. 80–89.
- ↑ Barbour 2004, p. 8.
- ↑ de Gorzanis 1981.
- ↑ "Spinacino 1507a: Thematic Index". Appalachian State University. Archived from the original on 25 July 2011. Retrieved 14 June 2012.
- ↑ Wilson 1997.
- ↑ Jorgens 1986.
- ↑ "Scintille de musica", (Brescia, 1533), p. 132
- ↑ Cohen 1987, pp. 471–488.
- ↑ Cho 2003, p. 223.
- ↑ Cho 2003, p. 222.
- ↑ Christensen 2002, p. 207.
- ↑ Christensen 2002, p. 78.
- ↑ Lindley, Mark. Lutes, Viols, Temperaments. ISBN 978-0-521-28883-5
- ↑ Vm7 6214
- ↑ Andreas Werckmeister (1707), Musicalische Paradoxal-Discourse
- ↑ Di Veroli 2009, pp. 140, 142 and 256.
- ↑ Moody 2003.
- ↑ von Helmholtz & Ellis 1885, p. 548.
- ↑ White, William Braid (1946) [1917]. Piano Tuning and Allied Arts (5th enlarged ed.). Boston, Massachusetts: Tuners Supply Co. p. 68.
- 1 2 Partch 1979, p. 134
- ↑ Needham, Ling & Robinson 1962, pp. 169-171.
- ↑ Benward & Saker 2003, p. 56f.
- ↑ Benson, Dave (14 December 2008). "Music: a Mathematical Offering" (PDF). Professor David J. Benson (homepage). University of Aberdeen. p. 160 (chapter 5). Archived from the original on 24 April 2015. Retrieved 23 September 2026.
[...] Thus we obtain the following table of ratios for a just major scale: [... table ...] The complementary intervals (mi–do) of 8:5 and (la–do) of 6:5 are called the just minor sixth and the just minor third.
Book version: Cambridge University Press, Nov 2006. ISBN 0521853877
Sources
[edit]- Barbour, James Murray (2004). Tuning and Temperament: A Historical Survey. Courier Corporation. ISBN 978-0-486-43406-3.
- Benward, Bruce; Saker, Marilyn (2003). Music in Theory and Practice. Vol. 1. McGraw-Hill. ISBN 978-0-07-294261-3.
- Cho, Gene J. (2003). The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. E. Mellen Press. ISBN 978-0-7734-6941-9.
- Cho, Gene J. (2010). "The Significance of the Discovery of the Musical Equal Temperament in the Cultural History". Journal of Xinghai Conservatory of Music. Archived from the original on 15 March 2012. Retrieved 6 April 2020.
- Christensen, Thomas (2002). The Cambridge History of Western Music Theory. Cambridge University Press. ISBN 978-0-521-62371-1.
- Cohen, H. Floris (1987). "Simon Stevin's equal division of the octave". Annals of Science. 44 (5). Informa UK Limited: 471–488. doi:10.1080/00033798700200311. ISSN 0003-3790.
- de Gorzanis, G. (1981). Intabolatura di liuto: I-III. Intabolatura di liuto: I-III (in Italian). Minkoff. ISBN 978-2-8266-0721-2.
- Di Veroli, Claudio (2009). Unequal Temperaments: Theory, History and Practice (2nd ed.). Bray, Ireland: Bray Baroque.
- Galilei, Vincenzo (1584). Il Fronimo. Venice: Girolamo Scotto.
- Hart, Roger (1998), Quantifying Ritual: Political Cosmology, Courtly Music, and Precision Mathematics in Seventeenth-Century China, Departments of History and Asian Studies, University of Texas, Austin, archived from the original on 5 March 2012, retrieved 20 March 2012
- Jorgens, Elise Bickford (1986). English Song, 1600-1675: Facsimiles of Twenty-six Manuscripts and an Edition of the Texts. Garland. ISBN 9780824082314.
- Kuttner, Fritz A. (May 1975). "Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory" (PDF). Ethnomusicology. 19 (2): 163–206. doi:10.2307/850355. JSTOR 850355.
- Kwang-chih Chang; Pingfang Xu; Liancheng Lu (2005). "The eastern Zhou and the growth of regionalism". The Formation of Chinese Civilization: An Archaeological Perspective. Xu Pingfang, Shao Wangping, Zhang Zhongpei, Wang Renxiang. Yale University Press. ISBN 978-0-300-09382-7.
- Lienhard, John H. (1997). "Equal Temperament". The Engines of Our Ingenuity. University of Houston. Retrieved 5 October 2014.
- Moody, Richard (February 2003). "Early Equal Temperament, An Aural Perspective: Claude Montal 1836". Piano Technicians Journal. Kansas City.
- Needham, Joseph; Ling, Wang; Robinson, Kenneth G. (1962). Science and Civilisation in China. Volume 4 – Part 1. Cambridge University Press.
- Needham, Joseph; Ronan, Colin A. (1978). The Shorter Science and Civilisation in China. Volume 4 – Part 1. Cambridge University Press.
- Partch, Harry (1979). Genesis of a Music (2nd ed.). Da Capo Press. ISBN 0-306-80106-X.
- Robinson, Kenneth (1980). A Critical Study of Chu Tsai-yü's Contribution to the Theory of Equal Temperament in Chinese Music. Volume 9 of Sinologica Coloniensia. Wiesbaden: Steiner. ISBN 978-3-515-02732-8.
- True, Timothy (2018). "The Battle Between Impeccable Intonation and Maximized Modulation". Musical Offerings. 9 (2): 61–74. doi:10.15385/jmo.2018.9.2.2.
- von Helmholtz, Hermann; Ellis, Alexander J. (1885). On the Sensations of Tone as a Physiological Basis for the Theory of Music (2nd ed.). London: Longmans, Green.
- Wilson, John (1997). "Thirty preludes in all (24) keys for lute [DP 49]". The Diapason Press. Retrieved 27 October 2020.
Further reading
[edit]- Duffin, Ross W. How Equal Temperament Ruined Harmony (and Why You Should Care). W.W. Norton & Company, 2007.
- Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0-87013-290-3
- Khramov, Mykhaylo. "Approximation of 5-limit just intonation. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave", Proceedings of the International Conference SIGMAP-2008[permanent dead link], 26–29 July 2008, Porto, pp. 181–184, ISBN 978-989-8111-60-9
- Sethares, William A. (2005). Tuning, Timbre, Spectrum, Scale (2nd ed.). London: Springer-Verlag. ISBN 1-85233-797-4.
- Surjodiningrat, W., Sudarjana, P.J., and Susanto, A. (1972) Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta, Gadjah Mada University Press, Jogjakarta 1972. Cited on https://web.archive.org/web/20050127000731/http://web.telia.com/~u57011259/pelog_main.htm. Retrieved May 19, 2006.
- Stewart, P. J. (2006) "From Galaxy to Galaxy: Music of the Spheres"
- Sensations of Tone a foundational work on acoustics and the perception of sound by Hermann von Helmholtz. Especially Appendix XX: Additions by the translator, pp. 430–556, (pdf pp. 451–577)]
External links
[edit]- Xenharmonic wiki on EDOs vs. Equal Temperaments
- Huygens-Fokker Foundation Centre for Microtonal Music
- A.Orlandini: Music Acoustics
- "Temperament" from A supplement to Mr. Chambers's cyclopædia (1753)
- Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900 Archived 2009-02-15 at the Wayback Machine. (2008) Latina, Il Levante Libreria Editrice
- Fractal Microtonal Music, Jim Kukula.
- All existing 18th century quotes on J.S. Bach and temperament
- Dominic Eckersley: "Rosetta Revisited: Bach's Very Ordinary Temperament"
- Well Temperaments, based on the Werckmeister Definition
- "Favored Cardinalities of Scales" by Peter Buch