Equal temperament
Equal temperament is a musical tuning system in which all intervals are approximated by multiples of a single interval. The most common equal temperament represents the octave as 12 exactly equal semitones.[a][1]
The 12-note chromatic scale employs a semitone ratio equal to the 12th root of 2 ( ≈ 1.05946). Equalizing the semitones in this way causes other intervals to deviate slightly from their "pure" sounds; the fourths and fifths produced by this method are closer to their just intonation than the thirds and sixths are.[2]
![{\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 20th century.[1] It is also used in other cultures. The system is usually tuned relative to a standard pitch of 440 Hz.[3]
History
[edit]The two figures frequently credited with determining how to calculate equal temperament are Chinese mathematician and theorist Zhu Zaiyu (also romanized as Chu-Tsaiyu, Chinese: 朱載堉) and Flemish scholar Simon Stevin.[4]: 7
Zhu is sometimes credited with the discovery of equal temperament.[5] In 1584, he 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."[6]
Simon Stevin's independent solution came the following year and was slightly less precise. Both men approached the problem primarily as a computing procedure rather than as a practical musical solution.[7]
China
[edit]Early history
[edit]In 433 BCE, a massive set of bronze chime bells were manufactured that could play five octaves of a C major scale. The middle range of the set included bells for all twelve semitones. The instruments were discovered intact in the tomb of Marquis Yi of Zeng.[8]
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.[9][4]: 55f
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 1581.[10] 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.[11] An extended account is also given by Joseph Needham.[12] 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,[13] 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. Zaiyu's work may have been transmitted to Europe by Jesuit missionary Matteo Ricci. [14][15] There is no evidence to confirm such transmission.[citation needed] James Murray Barbour wrote, "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."[4]: 7 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.[16]
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]Writing in 4th century BC Greece, Aristoxenus criticized the irrational numbers that arose in the Pythagorean system and proposed to simply divide the octave into equal parts.[17][18] During the Renaissance, Aristoxenus was often cited as the originator of equal temperament.[4]: 2
Francesco Spinacino's Recercare de tutti li Toni (Ricercar of all the Tones, 1507) is emblematic of a genre of preludes that would establish the tonality of a work by modulating through all of them until the correct key is reached.[19][20]
In 1567, lutenist Giacomo Gorzanis published a set of pieces for every major and minor mode. Such projects were more feasible in equal temperament. Vincenzo Galilei followed this pattern in his 1584 Fronimo Dialogo.[21][22][23]: 31f Galilei used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).[4]: 8
In the 17th century lutenist-composer John Wilson wrote a set of 30 preludes including 24 in all the major/minor keys.[24][25] Henricus Grammateus drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by Giovanni Maria Lanfranco in his "Scintille de musica".[26][1] 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.[27] Stevin's chord lengths are slightly incorrect due to inaccurate calculation.[28]: 205 Instead of a unified ratio, Simon Stevin's chords have one ratio per tone.[29]: 222f
| 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.[30]
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.[31]
Baroque era
[edit] | This article needs attention from an expert in music theory. 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 did not promote 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 before 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 before the 20th century, and whether composers expected or relied on it.. (September 2026) |
From 1550 to about 1650, lutes and viols were equal temperament instruments. Equally spaced frets are also depicted in 15th-century paintings.[32] The Brossard lute manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to Mlle Bocquet written in all keys, including the last prelude, Prélude sur tous les tons, which enharmonically modulates through all keys.[33][clarification needed] Angelo Michele Bartolotti published a series of passacaglias in all keys, with modulating passages connecting enharmonically. The 17th-century keyboard composer Girolamo Frescobaldi advocated equal temperament. Some theorists, such as Giuseppe Tartini, opposed equal temperament; they felt that degrading the purity of each chord diminished the music's aesthetic appeal, although Andreas Werckmeister emphatically advocated equal temperament in his 1707 treatise published posthumously.[34]
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.[35] 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.[36] Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century.[37] In the 20th century, one piano tuning manual included a table of beats per second for important intervals.[38] The intervals of equal temperament closely approximate some intervals in just intonation.[39]
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 equal temperament.
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.[40] 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.[41]
Comparison with just intonation
[edit]The intervals of 12-tone equal temperament closely approximate some intervals in just intonation (JI).[39] The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.
| Interval Name | Interval (ET) | Cents (ET) | Interval (JI) | Cents (JI)
[42]: 160 |
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]- ↑ The system is sometimes called 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);[43][unreliable source][additional citation(s) needed] informally abbreviated to twelve equal[citation needed] or referred to as equal temperament without qualification in Western countries.[additional citation(s) needed]
Citations
[edit]- 1 2 3 "Temperament", The New Harvard Dictionary of Music. Edited by Don Randel. Belknap Press of Harvard University Press, 1986. 837f.
- ↑ Lindley, Mark. "Equal temperament", Grove Music Online. Oxford University Press, 2001.
- ↑ von Helmholtz & Ellis 1885, pp. 493–511.
- 1 2 3 4 5 Barbour, James Murray. Tuning and Temperament: A Historical Survey. Dover Publications, 2004.
- ↑ 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.
- ↑ Liancheng, Lu. "The Eastern Zhou and the Growth of Regionalism". In The Formation of Chinese Civilization: An Archaeological Perspective. Yale University Press, 2005. 241–5.
- ↑ Goodman, Howard and Y. Edmund Lien. "A Third Century AD Chinese System of Di-Flute Temperament: Matching Ancient Pitch-Standards and Confronting Modal Practice", The Galpin Society Journal, vol. 62, 2009. 7.
- ↑ Li, Xifan. A General History of Chinese Art: Ming Dynasty, Volume 5. Berlin, Boston: De Gruyter, 2022. 162–65.
- ↑ Hart 1998.
- ↑ Needham, Ling & Robinson 1962, p. 221.
- ↑ Needham & Ronan 1978, p. 385.
- ↑ 卓仁祥. "从文化史角度看十二平均律的发现", Xinghai yin yue xue yuan xue bao, no. 2, June 2010. 1–4.
- Cho, Gene J. "The Significance of the Discovery of the Musical Equal Temperament in the Cultural History", Journal of Xinghai Conservatory of Music. Archived March 15, 2012.
- ↑ Temple, Robert K. G. The Genius of China: 3,000 Years of Science, Discovery, and Invention. Prion, 1991. 209–13.
- ↑ von Helmholtz & Ellis 1885, p. 258.
- ↑ Staufer, Amanda N. "The Unifying Strands: Formalism and Gestalt Theory in the Musical Philosophies of Aristoxenus, Descartes, and Meyer", Musical Offerings, vol 9, no. 1. 2018. 34.
- ↑ Bowman, Wayne D. Philosophical Perspectives on Music. New York: Oxford University Press, 1998. 137.
- ↑ Ness, Arthur J. "Prelude", The New Harvard Dictionary of Music. Edited by Don Randel. Belknap Press of Harvard University Press, 1986. 653.
- ↑ Spinacino, Francesco. Intabulatura de Lauto, Libro Primo. Tree Edition, 2011. 41f.
- ↑ Gorzanis, Giacomo de. Intabolatura di Liuto, Libro I. Venice: Antonio Gardano, 1561.
- Libro II. Venice: Girolamo Scotto, 1563.
- Libro III. Venice: Antonio Gardano, 1564.
- ↑ Galilei, Vincenzo. Il Fronimo. Venice: Girolamo Scotto, 1584.
- ↑ Dolata, David. Meantone Temperaments on Lutes and Viols. Indiana University Press, 2016.
- ↑ Wilson 1997.
- ↑ Jorgens 1986.
- ↑ Lanfranco, Giovanni Maria. Scintille de musica. Brescia: Lodovico Britannico, 1533. 132f.
- ↑ Cohen 1987, pp. 471–488.
- ↑ Rasch, Rudolf. "Tuning and Temperament", The Cambridge History of Western Music Theory. Edited by Thomas Christensen. The Cambridge History of Music. Cambridge University Press, 2002.
- ↑ Cho, Gene J. The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. E. Mellen Press, 2003. ISBN 978-0-7734-6941-9.
- ↑ Christensen 2002, p. 207.
- ↑ Christensen 2002, p. 78.
- ↑ Lindley, Mark. Lutes, Viols, Temperaments. Cambridge University Press, 1984. 19f. 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. Piano Tuning and Allied Arts. Fifth edition. Boston: Tuners Supply Co, 1964. 67f.
- 1 2 Partch 1979, p. 134
- ↑ Needham, Ling & Robinson 1962, pp. 169-171.
- ↑ Benward, Bruce; Saker, Marilyn (2003). Music in Theory and Practice, Volume 1. Seventh Edition. McGraw-Hill, 2003. ISBN 978-0-07-294261-3. 56f, 342.
- 1 2 Benson, David J. Music: a mathematical offering. Cambridge University Press, 2007.
- Sevish (25 June 2016). "What's the difference between ET and EDO?". sevish.com. Archived from the original on 13 June 2026. Retrieved 7 October 2026.
12-ET and 12-EDO both refer to the exact same tuning which has 12 equal notes per octave. [...] Some people use ED2 or ED2/1 synonymously with EDO.
Sources
[edit]- 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.
- Di Veroli, Claudio (2009). Unequal Temperaments: Theory, History and Practice (2nd ed.). Bray, Ireland: Bray Baroque.
- 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.
- 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.
- 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.
- "The Grail of Harmony: Just Intonation vs. Equal Temperament. An Annotated Bibliography". University of Denver, 2022.
- 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. "From Galaxy to Galaxy: Music of the Spheres". ResearchGate. January 2016.
- 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)]