Just intonation

{{short description|Musical tuning based on pure intervals}} [[File:Harmonic series klang.png|thumb|Partials 1–5 of the harmonic series.<ref>Brown, Colin. ''[https://www.google.com/books/edition/Music_in_common_things_lects/gMGx5sBBR3MC?hl=en&gbpv=1&pg=PA10 Music in Common Things, Part I: Music in a Sound]''. London & Glasgow: William Collins, Sons, 1874. 10.</ref>]] '''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.<ref name=Harvard>"[https://archive.org/details/newharvarddictio00rand/page/422/mode/1up Just intonation]", ''The New Harvard Dictionary of Music''. Edited by Don Randel. Belknap Press of Harvard University Press, 1986. 362f.</ref>

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.<ref name=Grove>Lindley, Mark. "[https://www.oxfordmusiconline.com/grovemusic/view/10.1093/gmo/9781561592630.001.0001/omo-9781561592630-e-0000014564 Just intonation]." ''Grove Music Online''. Oxford University Press, 2001.</ref> 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.<ref name=Helmholtz1885/>{{rp|[https://www.google.com/books/edition/On_the_sensations_of_tone_as_a_physiolog/Xy4DAAAAQAAJ?hl=en&gbpv=1&pg=PA26 26]}}

Just intonation also describes a tuning system that contains five or more pure intervals in an octave.<ref name=Harvard/> There have been many attempts to construct scales composed completely of justly tuned intervals.<ref name=Dolata/>{{rp|18}}

==History== Musicians instinctively perform in just intonation when possible. Singers and string players gravitate towards pure intervals.<ref>Thompson, Thomas Perronet. ''[https://www.google.com/books/edition/Theory_and_Practice_of_Just_Intonation/IZDJ30DpqHoC?hl=en&gbpv=1&pg=PA90 Theory and Practice of Just Intonation With a View to the Abolition of Temperament]''. London: Effingham Wilson, 1850. 90.</ref> Brass players default to just tuning when possible.<ref name=CMJ>Carlos, Wendy. "[https://doi.org/10.2307/3680176 Tuning: At the Crossroads]", ''Computer Music Journal'', vol. 11, no. 1. 1987. [https://www.jstor.org/stable/3680176 29–43]."</ref>{{rp|42}}<ref name=Barbour/>{{rp|200}} Barbershop quartets naturally sing in just intonation.<ref>Heller, Eric J. ''Why You Hear What You Hear: An Experiential Approach to Sound, Music, and Psychoacoustics''. Princeton University Press, 2013. 526.</ref><ref>Averill, Gage. ''Four Parts, No Waiting: A Social History of American Barbershop Quartet''. Oxford University Press, 2003. 167.</ref>

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.<ref>Johnson, Charles William Leverett. ''[https://www.google.com/books/edition/Musical_Pitch_and_the_Measurement_of_Int/nu5V39L9P2QC?hl=en&gbpv=1&pg=PA45 Musical Pitch and the Measurement of Intervals Among the Ancient Greeks]''. Baltimore: J. Murphy, 1896. 45.</ref> 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.<ref name=Strange/> Clashing overtones will result in acoustic beats.<ref>Buck, Percy Carter. ''[https://www.google.com/books/edition/Acoustics_for_Musicians/FKcNAAAAIAAJ?hl=en&gbpv=1&pg=PA142&printsec=frontcover Acoustics for Musicians]''. Clarendon Press, 1918. 142f.</ref><ref name=CMJ/>{{rp|33f}} When an interval is performed without audible beats, it was historically described as pure or just.<ref name=Grove/>

Constructing a scale out of just intervals requires compromise.<ref name=Strange/>{{rp|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.<ref name=Dolata>Dolata, David. ''Meantone Temperaments on Lutes and Viols''. Indiana University Press, 2016.</ref>{{rp|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.<ref>Bod, Rens. ''A New History of the Humanities: The Search for Principles and Patterns from Antiquity to the Present''. Oxford University Press, 2013. 37.</ref><ref>West, M. L. "[http://www.jstor.org/stable/737674 The Babylonian Musical Notation and the Hurrian Melodic Texts]", ''Music & Letters'', vol. 75, no. 2. 1994. 164.</ref> Ptolemy and Didymus the Musician developed their own versions of the system.<ref name=Barbour>Barbour, James Murray. ''[https://archive.org/details/tuningtemperamen0000barb_e7s0 Tuning and Temperament: A Historical Survey]''. Dover Publications, 2004.</ref>{{rp|2}}

In China, the guqin draws on just intonation for its tuning system.<ref>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.</ref> Indian music has an extensive theoretical framework for tuning in just intonation.<ref>Vasudevan, D. V. K, et al. "[https://doi.org/10.1007/978-3-031-27199-1_12 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.</ref>

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.<ref name=Barbour/>{{rp|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.<ref>Purves, Dale. ''Music as Biology: The Tones We Like and Why''. Harvard University Press, 2017. 27.</ref> 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.<ref name=Harlan>Harlan, Brian T. ''[https://www.proquest.com/dissertations-theses/one-voice-reconciliation-harry-partchs-disparate/docview/304821097/se-2 One Voice: A Reconciliation of Harry Partch's Disparate Theories]''. University of Southern California, 2007.</ref>{{rp|4}}

In the 20th century, many composers returned to just intonation. Some developed their own scales or instruments in order to use the tuning.<ref>Haluska, Jan. ''The Mathematical Theory of Tone Systems''. Slovakia, CRC Press, 2003. 283.</ref> 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.<ref>Partch, Harry, and Johnston, Ben. ''Barstow: Eight Hitchhiker Inscriptions from a Highway Railing at Barstow, California''. (1968 Version). American Musicological Society, 2000. xxvi.</ref><ref>''The John Adams Reader''. Bloomsbury Academic, 2006. 212.</ref><ref>Lavezzoli, Peter. ''The Sawn of Indian Music in the West''. Bloomsbury Academic, 2006. 247f.</ref><ref>Gagné, Nicole V. ''Historical Dictionary of Modern and Contemporary Classical Music''. Bloomsbury Publishing, 2019. 186f.</ref> Computers greatly aided the continuing quest for just intonation.<ref name=Strange>Stange, Karolin et al. "[https://www.semanticscholar.org/reader/620b3a07a2d2f99b2286c41aaff076e6dd9074c1 Playing Music in Just Intonation: A Dynamically Adaptive Tuning Scheme]." ''Computer Music Journal'' 42 (2017): 47-62.</ref>

==Scales== {{See also|Five-limit tuning|Ptolemy's intense diatonic scale|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.<ref>Naylor, Edward Woodall. ''[https://www.google.com/books/edition/An_Elizabethan_Virginal_Book/j4ACAAAAYAAJ?hl=en&gbpv=1&pg=PA101 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.</ref> By creating a series of fifths, a justly tuned pentatonic scale can easily be formed. Pythagorean tuning was used on early Renaissance keyboard instruments.<ref>Lindley, Mark. "[https://www.oxfordmusiconline.com/grovemusic/view/10.1093/gmo/9781561592630.001.0001/omo-9781561592630-e-0000022604 Pythagorean intonation]", ''Grove Music Online.'' Oxford University Press, 2001.</ref>

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.<ref name="C&G"/> This gap is the Pythagorean comma.<ref>''Music of the past, instruments and imagination: proceedings of the harmoniques International Congress, Lausanne 2004''. Austria, Peter Lang, 2006. 111.</ref>

[[File:Just intonation diatonic scale derivation.png|thumb|right|Derivation of the 5-limit just diatonic scale from the major triad.<ref name="C&G">Campbell, Murray and Clive Greated. ''The Musician's Guide to Acoustics''. Oxford University Press, 2023. 172–3.</ref>]] 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.<ref name="C&G"/>

In his second century AD book ''Harmonics'', Ptolemy calculated an intense diatonic scale with ratios of string lengths 120, {{sfrac|112|1|2}}, 100, 90, 80, 75, {{sfrac|66|2|3}}, and 60.<ref name=Barker>Barker, Andrew. ''Greek Musical Writings''. Cambridge University Press, 1989. 350.</ref><ref>Barker, Andrew. ''Scientific method in Ptolemy's Harmonics''. Cambridge University Press, 2000. 153.</ref> This scale allows for the just major third in its natural 5:4 ratio.<ref>Schlesinger, Kathleen. "[https://www.google.com/books/edition/Musical_Standard/sCP39ahAeqYC?hl=en&gbpv=1&dq=ptolemy%20intense%20diatonic%205%3A4&pg=PA177 The Greek Foundations of the Theory of Music]", ''The Musical Standard'', vol. XXVII, no. 488. June 5, 1926. 177.</ref><ref name=Johnson2006/>{{rp|100}} Harry Partch described this scale as "one of the world's fundamentally beautiful tonal sequences".<ref name=Partch>Partch, Harry. ''Genesis Of A Music: An Account Of A Creative Work, Its Roots, And Its Fulfillments''. Second Edition. Hachette Books, 1974.</ref>{{rp|167}}

The ratios of just intonation can be governed by three prime numbers: 2, 3, 5.<ref name=Grove/> 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.<ref name=DW>Wolf, Daniel James. 2003. "[https://doi:10.1080/0749446032000134715 Alternative Tunings, Alternative Tonalities]", ''Contemporary Music Review'' 22 (1–2). 2003. 3–14.</ref>{{rp|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.<ref name=Wright>Wright, David. ''[https://archive.org/details/mathematicsmusic0000wrig/page/140/mode/1up Mathematics and Music]''. Mathematical World. Vol. 28. Providence, RI: American Mathematical Society, 2009.</ref>{{rp|p=137–39}} Modern composers expanded the limit to 7, which creates far more complex tuning solutions.<ref>Tsuji, Kinko, and Müller, Stefan C.. ''[https://www.google.com/books/edition/Physics_and_Mathematics_in_Musical_Compo/hI9qEQAAQBAJ?hl=en&gbpv=1&pg=PA32 Physics and Mathematics in Musical Composition: A Comparative Study]''. Germany, Springer Nature Switzerland, 2025. 32.</ref> Partch experimented with prime number limits as high as 17.<ref name=DW/>{{rp|7}}

==Notation== [[File:Notation of partials 1-19 for 1-1.png|thumb|Ben Johnston's notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C.<ref>Fonville, John. "[https://doi.org/10.2307/833435 Ben Johnston's Extended Just Intonation: A Guide for Interpreters]", ''Perspectives of New Music'', vol. 29, no. 2, 1991. 121.</ref>]] Justly tuned scales often yield multiple versions of the same interval, which can be managed through notation.<ref name=Johnson2006>Johnston, Ben. ''[https://archive.org/details/maximumclarityot0000john/mode/1up Maximum Clarity and Other Writings on Music]''. Chicago: University of Illinois Press, 2006.</ref>{{rp|77}} Moritz Hauptmann developed a system of notation to describe scales.<ref>Hauptmann, Moritz. ''[https://www.google.com/books/edition/Die_Natur_der_Harmonik_und_der_Metrik/j4sTO3Jn0u0C?hl=en&gbpv=1&pg=PA26 Die Natur der Harmonik und der Metrik: Zur Theorie der Musik]''. Germany, Breitkopf & Härtel, 1873. 26.</ref> 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.<ref name=Helmholtz1885>Helmholtz, Hermann von. ''On the Sensations of Tone as a Physiological Basis for the Theory of Music''. Longmans, Green, 1885.</ref>{{rp|[https://archive.org/details/onsensationston00unkngoog/page/276/mode/1up 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.<ref>Benson, David J. ''[https://logosfoundation.org/kursus/music_math.pdf Music: a mathematical offering]''. Cambridge University Press, 2007. 173f.</ref><ref>Eitz, Carl. ''[https://www.google.com/books/edition/Das_mathematisch_reine_Tonsystem/lMIPAAAAYAAJ?hl=en&gbpv=1&dq=Das%20mathematisch-reine%20Tonsystem&pg=PP3 Das mathematisch-reine Tonsystem]''. Leipzig, 1891</ref><ref name=Barbour/>{{rp|vi}}

In the 1960s, Ben Johnston developed an extended just intonation. He also used + and − signs in his notation.<ref>Von Gunden, Heidi. ''[https://archive.org/details/musicofbenjohnst0000vong/page/148/mode/1up The Music of Ben Johnston]''. Bloomsbury Academic, 1986. 148.</ref><ref name=Johnson2006/>{{rp|pages=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.<ref>Wannamaker, Robert. ''The Music of James Tenney: Volume 2: a Handbook to the Pieces''. University of Illinois Press, 2021. 415.</ref><ref>Wannamaker, Robert. ''The Music of James Tenney, Volume 1: Contexts and Paradigms''. University of Illinois Press, 2021. 288-89.</ref> Sagittal notation uses arrows as accidentals. The size of the symbol indicates the size of the alteration.<ref>Secor, George D. and David C. Keenan. "[http://sagittal.org/sagittal.pdf Sagittal: A Microtonal Notation System]", ''Xenharmonikôn: An Informal Journal of Experimental Music, Vol. 18''. Sagittal.org, 2006. 1–2.</ref>

==Audio examples== {{Unsourced section|date=April 2026}} *{{audio|Primary triads in C just.mid|C major primary triads in just intonation}} * {{audio|A_Major_Scale,_Triads,_and_Fifths_Just.ogg|Just intonation}} An A-major scale, followed by three major triads, and then a progression of fifths in just intonation. * {{audio|A_Major_Scale,_Triads,_and_Fifths_Equal.ogg|Equal temperament}} An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. The beating in this file may be more noticeable after listening to the above file. * {{audio|Just vs equal.ogg|Equal temperament and just intonation compared}} A pair of major thirds, followed by a pair of full major chords. The first in each pair is in equal temperament; the second is in just intonation. Piano sound. * {{audio|A-major-triad-equal-temperament-compared-to-just-intonation-6-2008C.ogg|Equal temperament and just intonation compared with square waveform}} A pair of major chords. The first is in equal temperament; the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just intonation between the two chords. In the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz. In the just intonation triad, this roughness is absent. The square waveform makes the difference between equal temperament and just intonation more obvious. *{{audio|Just major third on C.mid|Just major third on C}} *{{audio|Notation of partials 1-19 for 1-1.mid|Partials 1–19}}

==See also== ; Lists: {{div col begin |colwidth=16em}} * List of compositions in just intonation * List of intervals in 5-limit just intonation * List of meantone intervals * List of pitch intervals {{div col end}}

; Article topics: {{div col begin |colwidth=12em}} * Dynamic tonality * Electronic tuner * Hexany * Microtonal music * Microtuner * Music and mathematics * Musical interval * Pythagorean interval * Regular number * Superparticular ratio * Whole-tone scale {{div col end}}

==References== {{reflist|25em}}

==External links== {{sister project|project=Wikiversity |text=Wikiversity discusses a simplified just scale.}} ;Primers *[http://alum.mit.edu/www/nowitzky/justint/ Just Intonation] by Mark Nowitzky. *[https://en.xen.wiki/w/Just_intonation Just Intonation] at the Xenharmonic Wiki. *[http://www.kylegann.com/tuning.html Just Intonation Explained] by Kyle Gann *[http://www.chrysalis-foundation.org/just_intonation.htm Just Intonation: Two Definitions] by The Chrysalis Foundation. *[https://justintonation.fiu.edu/ Organ Music in Just Intonation] by Eric Zuurbier at Florida International University. *[http://www.patmissin.com/tunings/tun0.html Why does Just Intonation sound so good?] by Pat Missin.

;Repertoire *[http://www.ubu.com/sound/tellus_14.html Tellus #14 'Just Intonation' (1986)] by Tellus Audio Cassette Magazine, archived at UbuWeb. *[https://web.archive.org/web/20110514002750/http://artofthestates.org/cgi-bin/genresearch.pl?genre=microtonal%2Fjust%20intonation Microtonal/just intonation repertoire list] by Art of the States.

;Instruments *[http://www.mediafire.com/download.php?ljr44lwzoyj 22 Note Just Intonation Keyboard Software with 12 Indian Instrument Sounds] Libreria Editrice *Barbieri, Patrizio. [http://www.patriziobarbieri.it/1.htm Enharmonic instruments and music, 1470–1900]. (2008) Latina, Il Levante *[https://web.archive.org/web/20100630223629/http://users.rcn.com/dante.interport/justguitar.html Dante Rosati's 21 Tone Just Intonation guitar] *[http://www.anaphoria.com/wilson.html The Erv Wilson Archives]

;Notations *[http://www.plainsound.org Plainsound Music Edition] – JI music and research, information about the Helmholtz-Ellis JI Pitch Notation *[http://tonalsoft.com/enc/h/hewm.aspx Helmholtz/Ellis/Wolf/Monzo system] at ''Tonalsoft Encyclopaedia''.

;Videos *"[https://www.youtube.com/watch?v=DzUhjxNEyOs Marc Sabat: Tuning Bach (2024.10.19) Lecture-Performance with Sara Cubarsi and Xenia Gogu] by PLAINSOUND. *"[https://www.youtube.com/watch?v=d2I1zNw2w-c PACHELBEL's CANON, Tuning Comparison: Just intonation, Meantone and 12-equal]" by Claudi Meneghin.