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Just intonation

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Partials 1–5 of the harmonic series.[1]

Just intonation is the tuning of a musical interval without beats. The result is an acoustically pure sound that resonates within the harmonic series. The simplest relationship between pitches in this series can be expressed as small whole number ratios. Musicians around the world instinctively perform in just intonation.

Just intonation also describes any musical tuning system containing five or more pure intervals within an octave. Elaborate theories and instruments have been constructed in pursuit of a just intonation system that is fully chromatic.

Definition

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Any time an interval is sounded without acoustical beats it is in just intonation. The sound is also described as pure. The frequency of each note in a pure interval will correspond to the whole number ratios in the harmonic series.[2]

In the harmonic series on C, the 1st and 2nd notes form an octave in a 2:1 ratio. The fifth between the G and C is in a 3:2 ratio. The fourth is a 4:3 ratio.[3] When its frequency is doubled, A 440 Hertz sounds an octave higher at 880 Hz. The pitch sounds an octave lower when the frequency is halved to 220 Hz.[4]

Just intonation also describes a tuning system that contains five or more pure intervals in an octave.[2] There have been many attempts to construct scales composed completely of justly tuned intervals.[5]: 18 

History

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Musicians instinctively perform in just intonation when possible. Singers and string players gravitate towards pure intervals.[6] Brass players default to just tuning when possible.[7]: 42 [8]: 200  Barbershop quartets naturally sing in just intonation.[9][10]

In Ancient Greece, intervals like the octave, fourth, and fifth were recognized as consonances. Using a monochord, Pythagoras discovered that simple fractions of the string length correspond to these consonant intervals.[11] Pythagoras' ratios reflected a naturally sounding collection of overtones known as the harmonic series. When two notes are sounded together, the resulting interval is perceived as more consonant when their overtones are in accordance.[12] Clashing overtones will result in acoustic beats.[13][7]: 33f  When an interval is performed without audible beats, it was historically described as pure or just.[3]

Constructing a scale out of just intervals requires compromise.[12]: 2  Because of the difficulty of justly tuning fixed pitch instruments, the manifold attempts to do so have been likened to a quest for the Holy Grail in its simultaneous futility and worthiness.[5]: 18 

Pythagoras and Eratosthenes are credited with a solution that became known as Pythagorean tuning. However, the system is in evidence in much older Babylonian artifacts.[14][15] Ptolemy and Didymus the Musician developed their own versions of the system.[8]: 2 

In China, the guqin draws on just intonation for its tuning system.[16] Indian music has an extensive theoretical framework for tuning in just intonation.[17]

Just intonation fettered music to a limited range of harmony and keys. Emulating its pure sound was impractical. Johann Sebastian Bach was so adept at retuning his harpsichord, he could do it in fifteen minutes.[8]: 191  Several musical temperaments were developed that standardized intervals, stabilizing musicmaking and enabling wider tonal adventures for composers. The system that became standard was equal temperament.[18] With its division of the octave into twelve identical steps based on a ratio of the 12th root of 2 (≈1.0595), equal temperament uses irrational numbers to create a rational system. Just intonation generally relies on rational numbers to generate irrational systems.[19]: 4 

In the 20th century, many composers returned to just intonation. Some developed their own scales or instruments in order to use the tuning.[20] Harry Partch, Lou Harrison, La Monte Young, Terry Riley, John Adams, and Glenn Branca are just a few of the contemporary composers that used just intonation.[21][22][23][24] Computers greatly aided the continuing quest for just intonation.[12]

Scales

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See also: Five-limit tuning, Ptolemy's intense diatonic scale, and Pythagorean tuning

Pythagorean tuning relies on the just intonation of fifths to create a scale. The intervals are tuned in the same way violinists tune their open strings.[25] By creating a series of fifths, a justly tuned pentatonic scale can easily be formed. Pythagorean tuning was used on early Renaissance keyboard instruments.[26]

When justly tuned fifths are stacked to generate all twelve chromatic tones, the final note in the series is wide of its destination, which should be seven octaves higher than the initial note.[27] This gap is the Pythagorean comma.[28]

Derivation of the 5-limit just diatonic scale from the major triad.[27]

Additionally, any Pythagorean scale with more than five notes has inherent tuning problems, particularly with thirds. A solution is to begin with a major triad that uses the 5:4 just major third as a reference for the remaining notes.[27]

In his second century AD book Harmonics, Ptolemy calculated an intense diatonic scale with ratios of string lengths 120, ⁠112+1/2⁠, 100, 90, 80, 75, ⁠66+2/3⁠, and 60.[29][30] This scale allows for the just major third in its natural 5:4 ratio.[31][32]: 100  Harry Partch described this scale as "one of the world's fundamentally beautiful tonal sequences".[33]: 167 

The ratios of just intonation can be governed by three prime numbers: 2, 3, 5.[3] These primes can be factored to generate the ratios governing a scale. Harry Partch originated the idea that the limit of a scale was its highest prime factor.[34]: 13  By this classification, the Pythagorean scale is a 3-limit scale. A scale with the 5:4 major third is in 5-limit tuning.[35]: 137–39  Modern composers expanded the limit to 7, which creates far more complex tuning solutions.[36] Partch experimented with prime number limits as high as 17.[34]: 7 

Notation

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Ben Johnston's notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C.[37]

Justly tuned scales often yield multiple versions of the same interval, which can be managed through notation.[32]: 77  Moritz Hauptmann developed a system of notation to describe scales.[38] Hermann von Helmholtz adapted it in On the Sensations of Tone as a Physiological Basis for the Theory of Music (1877). The system used a combination of + and - signs in addition to subscript numbers.[4]: 276 

Carl Eitz developed a similar system which was adapted by J. Murray Barbour. Superscript numbers indicate the number of syntonic commas to apply to the tuning. The basic just intonation scale appears as C0 – D0 – E−1 – F0 – G0 – A−1 – B−1 – C0.[39][40][8]: vi 

In the 1960s, Ben Johnston developed an extended just intonation. He also used + and − signs in his notation.[41][32]: 77–88 

Composers like James Tenney employed just intonation by marking cents deviations from equal tempered pitches in his scores. Musicians often employ tuning devices during performances.[42][43] Sagittal notation uses arrows as accidentals. The size of the symbol indicates the size of the alteration.[44]

Audio examples

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See also

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Lists
Article topics

References

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  1. ^ Brown, Colin. Music in Common Things, Part I: Music in a Sound. London & Glasgow: William Collins, Sons, 1874. 10.
  2. ^ a b "Just intonation", The New Harvard Dictionary of Music. Edited by Don Randel. Belknap Press of Harvard University Press, 1986. 362f.
  3. ^ a b c Lindley, Mark. "Just intonation." Grove Music Online. Oxford University Press, 2001.
  4. ^ a b Helmholtz, Hermann von. On the Sensations of Tone as a Physiological Basis for the Theory of Music. Longmans, Green, 1885.
  5. ^ a b Dolata, David. Meantone Temperaments on Lutes and Viols. Indiana University Press, 2016.
  6. ^ Thompson, Thomas Perronet. Theory and Practice of Just Intonation With a View to the Abolition of Temperament. London: Effingham Wilson, 1850. 90.
  7. ^ a b Carlos, Wendy. "Tuning: At the Crossroads", Computer Music Journal, vol. 11, no. 1. 1987. 29–43."
  8. ^ a b c d Barbour, James Murray. Tuning and Temperament: A Historical Survey. Dover Publications, 2004.
  9. ^ Heller, Eric J. Why You Hear What You Hear: An Experiential Approach to Sound, Music, and Psychoacoustics. Princeton University Press, 2013. 526.
  10. ^ Averill, Gage. Four Parts, No Waiting: A Social History of American Barbershop Quartet. Oxford University Press, 2003. 167.
  11. ^ Johnson, Charles William Leverett. Musical Pitch and the Measurement of Intervals Among the Ancient Greeks. Baltimore: J. Murphy, 1896. 45.
  12. ^ a b c Stange, Karolin et al. "Playing Music in Just Intonation: A Dynamically Adaptive Tuning Scheme." Computer Music Journal 42 (2017): 47-62.
  13. ^ Buck, Percy Carter. Acoustics for Musicians. Clarendon Press, 1918. 142f.
  14. ^ Bod, Rens. A New History of the Humanities: The Search for Principles and Patterns from Antiquity to the Present. Oxford University Press, 2013. 37.
  15. ^ West, M. L. "The Babylonian Musical Notation and the Hurrian Melodic Texts", Music & Letters, vol. 75, no. 2. 1994. 164.
  16. ^ Hui, Yu and Chen Yingshi. "Theorizing 'Natural Sound': Ancient Chinese Music Theory and Its Contemporary Applications in the Study of Guqin Intonation". In The Oxford Handbook of Music in China and the Chinese Diaspora. Oxford University Press, 2023. 38–41.
  17. ^ Vasudevan, D. V. K, et al. "Equal Temperament and Just Intonation Feature Based Analysis of Indian Music", Intelligent Human Computer Interaction. Edited by Hakimjon Zaynidinov et al., vol. 13741. Springer International Publishing AG, 2023. 109–20.
  18. ^ Purves, Dale. Music as Biology: The Tones We Like and Why. Harvard University Press, 2017. 27.
  19. ^ Harlan, Brian T. One Voice: A Reconciliation of Harry Partch's Disparate Theories. University of Southern California, 2007.
  20. ^ Haluska, Jan. The Mathematical Theory of Tone Systems. Slovakia, CRC Press, 2003. 283.
  21. ^ Partch, Harry, and Johnston, Ben. Barstow: Eight Hitchhiker Inscriptions from a Highway Railing at Barstow, California. (1968 Version). American Musicological Society, 2000. xxvi.
  22. ^ The John Adams Reader. Bloomsbury Academic, 2006. 212.
  23. ^ Lavezzoli, Peter. The Sawn of Indian Music in the West. Bloomsbury Academic, 2006. 247f.
  24. ^ Gagné, Nicole V. Historical Dictionary of Modern and Contemporary Classical Music. Bloomsbury Publishing, 2019. 186f.
  25. ^ Naylor, Edward Woodall. An Elizabethan Virginal Book: Being a Critical Essay on the Contents of a Manuscript in the Fitzwilliam Museum at Cambridge. J. M. Dent, 1905. 101.
  26. ^ Lindley, Mark. "Pythagorean intonation", Grove Music Online. Oxford University Press, 2001.
  27. ^ a b c Campbell, Murray and Clive Greated. The Musician's Guide to Acoustics. Oxford University Press, 2023. 172–3.
  28. ^ Music of the past, instruments and imagination: proceedings of the harmoniques International Congress, Lausanne 2004. Austria, Peter Lang, 2006. 111.
  29. ^ Barker, Andrew. Greek Musical Writings. Cambridge University Press, 1989. 350.
  30. ^ Barker, Andrew. Scientific method in Ptolemy's Harmonics. Cambridge University Press, 2000. 153.
  31. ^ Schlesinger, Kathleen. "The Greek Foundations of the Theory of Music", The Musical Standard, vol. XXVII, no. 488. June 5, 1926. 177.
  32. ^ a b c Johnston, Ben. Maximum Clarity and Other Writings on Music. Chicago: University of Illinois Press, 2006.
  33. ^ Partch, Harry. Genesis Of A Music: An Account Of A Creative Work, Its Roots, And Its Fulfillments. Second Edition. Hachette Books, 1974.
  34. ^ a b Wolf, Daniel James. 2003. "Alternative Tunings, Alternative Tonalities", Contemporary Music Review 22 (1–2). 2003. 3–14.
  35. ^ Wright, David. Mathematics and Music. Mathematical World. Vol. 28. Providence, RI: American Mathematical Society, 2009.
  36. ^ Tsuji, Kinko, and Müller, Stefan C.. Physics and Mathematics in Musical Composition: A Comparative Study. Germany, Springer Nature Switzerland, 2025. 32.
  37. ^ Fonville, John. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", Perspectives of New Music, vol. 29, no. 2, 1991. 121.
  38. ^ Hauptmann, Moritz. Die Natur der Harmonik und der Metrik: Zur Theorie der Musik. Germany, Breitkopf & Härtel, 1873. 26.
  39. ^ Benson, David J. Music: a mathematical offering. Cambridge University Press, 2007. 173f.
  40. ^ Eitz, Carl. Das mathematisch-reine Tonsystem. Leipzig, 1891
  41. ^ Von Gunden, Heidi. The Music of Ben Johnston. Bloomsbury Academic, 1986. 148.
  42. ^ Wannamaker, Robert. The Music of James Tenney: Volume 2: a Handbook to the Pieces. University of Illinois Press, 2021. 415.
  43. ^ Wannamaker, Robert. The Music of James Tenney, Volume 1: Contexts and Paradigms. University of Illinois Press, 2021. 288-89.
  44. ^ Secor, George D. and David C. Keenan. "Sagittal: A Microtonal Notation System", Xenharmonikôn: An Informal Journal of Experimental Music, Vol. 18. Sagittal.org, 2006. 1–2.
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