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
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."Just intonation", The New Harvard Dictionary of Music. Edited by Don Randel. Belknap Press of Harvard University Press, 1986. 362f.
In the harmonic series on C, the 1st and 2nd harmonics 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.Lindley, Mark. "Just intonation." Grove Music Online. Oxford University Press, 2001. 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.
Just intonation also describes a tuning system that contains five or more pure intervals in an octave. There have been many attempts to construct scales composed completely of justly tuned intervals.
History
Musicians instinctively perform in just intonation when possible. Singers and string players gravitate towards pure intervals.Thompson, Thomas Perronet. Theory and Practice of Just Intonation With a View to the Abolition of Temperament. London: Effingham Wilson, 1850. 90. Brass players default to just tuning when possible.Carlos, Wendy. "Tuning: At the Crossroads", Computer Music Journal, vol. 11, no. 1. 1987. 29–43." Barbershop quartets naturally sing in just intonation.Heller, Eric J. Why You Hear What You Hear: An Experiential Approach to Sound, Music, and Psychoacoustics. Princeton University Press, 2013. 526.Averill, Gage. Four Parts, No Waiting: A Social History of American Barbershop Quartet. Oxford University Press, 2003. 167.
In Ancient Greece, intervals like the octave, fifth, and fourth were recognized as consonances, respectively referred to as diapason, diapente, and diatessaron. Using a monochord, Pythagoras discovered that simple fractions of the string length correspond to these consonant intervals.Johnson, Charles William Leverett. Musical Pitch and the Measurement of Intervals Among the Ancient Greeks. Baltimore: J. Murphy, 1896. 45.Sethares, William A. (2005). Tuning, Timbre, Spectrum, Scale. . 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. Clashing overtones will result in acoustic beats.Buck, Percy Carter. Acoustics for Musicians. Clarendon Press, 1918. 142f. When an interval is performed without audible beats, it was historically described as pure or just.
Constructing a scale out of just intervals requires compromise. A completely chromatic scale in just intonation was an impossible ideal.Jorgensen, Owen. Tuning the Historical Temperaments by Ear. Northern Michigan University Press, 1977. 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.Dolata, David. Meantone Temperaments on Lutes and Viols. Indiana University Press, 2016.
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.Bod, Rens. A New History of the Humanities: The Search for Principles and Patterns from Antiquity to the Present. Oxford University Press, 2013. 37.West, M. L. "The Babylonian Musical Notation and the Hurrian Melodic Texts", Music & Letters, vol. 75, no. 2. 1994. 164. Ptolemy and Didymus the Musician developed their own versions of the system.Barbour, James Murray. Tuning and Temperament: A Historical Survey. Dover Publications, 2004.
In China, the guqin draws on just intonation for its tuning system.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. Indian music has an extensive theoretical framework for tuning in just intonation.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.
Just intonation fettered music to a limited range of harmony and keys. Emulating its pure sound was impractical. 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.Purves, Dale. Music as Biology: The Tones We Like and Why. Harvard University Press, 2017. 45f. 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 practical system.Harlan, Brian T. One Voice: A Reconciliation of Harry Partch's Disparate Theories. University of Southern California, 2007.
In the 20th century, many composers returned to just intonation. Some developed their own scales or instruments in order to use the tuning.Haluska, Jan. The Mathematical Theory of Tone Systems. Slovakia, CRC Press, 2003. 283. 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.Partch, Harry, and Johnston, Ben. Barstow: Eight Hitchhiker Inscriptions from a Highway Railing at Barstow, California. (1968 Version). American Musicological Society, 2000. xxvi.The John Adams Reader. Bloomsbury Academic, 2006. 212.Lavezzoli, Peter. The Sawn of Indian Music in the West. Bloomsbury Academic, 2006. 247f.Gagné, Nicole V. Historical Dictionary of Modern and Contemporary Classical Music. Bloomsbury Publishing, 2019. 186f. Computers greatly aided the continuing quest for just intonation through software that could facilitate dynamic tuning.Stange, Karolin et al. "Playing Music in Just Intonation: A Dynamically Adaptive Tuning Scheme." Computer Music Journal 42 (2017): 47-62.
Scales
Pythagorean tuning generates a scale by relying on the just intonation of fifths and octaves in their natural ratios, 3:2 and 2:1. Just fifths are tuned in the same way violinists tune their open strings.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. By creating a series of fifths, a justly tuned pentatonic scale can easily be formed. Pythagorean tuning was used on early Renaissance keyboard instruments.Lindley, Mark. "Pythagorean intonation", Grove Music Online. Oxford University Press, 2001.
Using this technique to tune a chromatic scale creates a problem. A stack of 12 justly tuned fifths will terminate in a slightly different place than a stack of 7 perfect octaves.Greated, Clive. "Comma." Grove Music Online. Oxford University Press, 2001. The final note in the series of fifths is 23.5 cents wide of its destination, which should be seven octaves higher than the initial note. This gap is the Pythagorean comma. It is impossible to return to the unison.
Another major flaw of the Pythagorean tuning system are its badly mistuned major thirds. One solution is to build a scale around a major triad in just intonation with a 5:4 ratio for the major third and a 3:2 ratio for the perfect fifth.
In his second century AD book Harmonics, Ptolemy calculated an intense diatonic scale with ratios of string lengths 120, , 100, 90, 80, 75, , and 60.Barker, Andrew. Greek Musical Writings. Cambridge University Press, 1989. 350.Barker, Andrew. Scientific method in Ptolemy's Harmonics. Cambridge University Press, 2000. 153. This scale allows for the just major third in its natural 5:4 ratio.Schlesinger, Kathleen. "The Greek Foundations of the Theory of Music", The Musical Standard, vol. XXVII, no. 488. June 5, 1926. 177. Harry Partch described this scale as "one of the world's fundamentally beautiful tonal sequences".Partch, Harry. Genesis Of A Music: An Account Of A Creative Work, Its Roots, And Its Fulfillments. Second Edition. Hachette Books, 1974.
The ratios of just intonation can be governed by three prime numbers: 2, 3, 5. These primes can be factors which generate the ratios governing a scale. Harry Partch originated the idea that the limit of a scale was its highest prime factor.Wolf, Daniel James. 2003. "Alternative Tunings, Alternative Tonalities", Contemporary Music Review 22 (1–2). 2003. 3–14. By this classification, the Pythagorean scale is a 3-limit scale. A scale with the 5:4 major third is in 5-limit tuning.Wright, David. Mathematics and Music. Mathematical World. Vol. 28. Providence, RI: American Mathematical Society, 2009. Modern composers expanded the limit to 7, which creates far more complex tuning solutions.Tsuji, Kinko, and Müller, Stefan C.. Physics and Mathematics in Musical Composition: A Comparative Study. Germany, Springer Nature Switzerland, 2025. 32. Partch experimented with prime number limits as high as 17.
Notation
Justly tuned scales often yield multiple versions of the same interval, which can be managed through notation.Johnston, Ben. Maximum Clarity and Other Writings on Music. Chicago: University of Illinois Press, 2006. Moritz Hauptmann developed a system of notation to describe scales.Hauptmann, Moritz. Die Natur der Harmonik und der Metrik: Zur Theorie der Musik. Germany, Breitkopf & Härtel, 1873. 26. 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.Helmholtz, Hermann von. On the Sensations of Tone as a Physiological Basis for the Theory of Music. Longmans, Green, 1885.
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.Benson, David J. Music: a mathematical offering. Cambridge University Press, 2007. 173f.Eitz, Carl. Das mathematisch-reine Tonsystem. Leipzig, 1891
In the 1960s, Ben Johnston developed an extended just intonation. He also used + and − signs in his notation.Von Gunden, Heidi. The Music of Ben Johnston. Bloomsbury Academic, 1986. 148.
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.Wannamaker, Robert. The Music of James Tenney: Volume 2: a Handbook to the Pieces. University of Illinois Press, 2021. 415.Wannamaker, Robert. The Music of James Tenney, Volume 1: Contexts and Paradigms. University of Illinois Press, 2021. 288-89. Sagittal notation uses arrows as accidentals. The size of the symbol indicates the size of the alteration.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.
Audio examples
An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.