Just intonation explained

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.

Just intonation also describes any musical tuning system containing five or more pure intervals within an octave.^[1]

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.^[1]

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.^[2] 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.^[1] There have been many attempts to construct scales composed completely of justly tuned intervals.

History

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.^[3] 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.^[4] When an interval is performed without audible beats, it was historically described as pure or just.^[2]

Constructing a scale out of just intervals requires compromise. 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]

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.^{[6] [7]} Ptolemy and Didymus the Musician developed their own versions of the system.^[8]

In China, the guqin draws on just intonation for its tuning system.^[9] Indian music has an extensive theoretical framework for tuning in just intonation.^[10]

20th century composers often returned to just intonation for inspiration. Many developed their own scales or instruments in order to use the tuning.^[11] Computers greatly aided the continuing quest for just intonation.^[12]

Scales

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.^[13] By creating a series of fifths in the ratio 3:2, a justly tuned pentatonic scale can easily be formed. Pythagorean tuning was used on early Renaissance keyboard instruments.^[14] When extended to scales with more notes, several tuning problems arise particularly with thirds. An alternate solution is to begin with a tuned major triad as a reference for the remaining notes.

In his second century AD book Harmonics, Ptolemy calculated a "tense diatonic" scale with ratios of string lengths 120,, 100, 90, 80, 75,, and 60.^{[15] [16]} Harry Partch described this scale as "one of the world's fundamentally beautiful tonal sequences".^[17] Justly tuned scales often yield multiple versions of the same interval, which can be managed through notation.^[18]

All of the ratios of just intonation are governed by four prime numbers: 1, 2, 3, and 5. Tuning solutions that rely on just this set of primes is sometimes called five-limit tuning.^[19] Modern composers expanded the limit to 7, which creates far more complex tuning solutions.^[20]

Indian scales

In Indian music, the just diatonic scale described above is used, though there are different possibilities, for instance for the sixth pitch (dha), and further modifications may be made to all pitches excepting sa and pa.^[21]

Note	sa	re	ga	ma	pa	dha	ni	sa
Ratio	1:1	9:8	5:4	4:3	3:2	5:3	15:8	2:1
Cents	0	204	386	498	702	884	1088	1200

Some accounts of Indian intonation system cite a given 12 swaras being divided into 22 shrutis.^{[22] [23]} According to some musicians, one has a scale of a given 12 pitches and ten in addition (the tonic, shadja (sa), and the pure fifth, pancham (pa), are inviolate (known as achala^[24] in Indian music theory):

Note	C	D	D	D	D	E	E	E	E	F	F	F	F	G	A	A	A	A	B	B	B	B	C
Ratio	1:1	256:243	16:15	10:9	9:8	32:27	6:5	5:4	81:64	4:3	27:20	45:32	729:512	3:2	128:81	8:5	5:3	27:16	16:9	9:5	15:8	243:128	2:1
Cents	0	90	112	182	204	294	316	386	408	498	520	590	612	702	792	814	884	906	996	1018	1088	1110	1200

Where we have two ratios for a given letter name or swara, we have a difference of 81:80 (22 cents), which is the syntonic comma^[25] or the praman^[24] in Indian music theory. These notes are known as chala.^[24] The distance between two letter names comes in to sizes, poorna (256:243) and nyuna (25:24).^[24] One can see the symmetry, looking at it from the tonic, then the octave.

Staff notation

Moritz Hauptmann developed a system of notation to describe scales.^[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.^[27]

Carl Eitz developed a similar system which was adapted by J. Murray Barbour (1951). 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.^{[28] [29]}

In the 1960s, Ben Johnston developed "Extended Just Intonation". He also used + and − signs in his notation.^{[30] [18]}

Beginning in 2000, Marc Sabat and Wolfgang von Schweinitz developed an accidental system suggested by Helmholtz's work that incorporated Alexander John Ellis' invention of cents. Their system, Extended Helmholtz Ellis JI Pitch Notation, adapts traditional accidentals combined with arrows to indicate alterations in tuning.^{[31] [32]}

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.^{[33] [34]} Sagittal notation uses arrows as accidentals. The size of the symbol indicates the size of the alteration.^[35]

Audio examples

• An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.
• 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.
• 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.
• 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.

See also

Lists:

• List of compositions in just intonation
• List of intervals in 5-limit just intonation
• List of meantone intervals
• List of pitch intervals

Article topics:

• Dynamic tonality
• Electronic tuner
• Hexany
• Microtonal music
• Microtuner
• Music and mathematics
• Musical interval
• Pythagorean interval
• Regular number
• Superparticular ratio
• Whole-tone scale

External links

• works using just intonation by American composers
• The Chrysalis Foundation – Just Intonation: Two Definitions
• Just Intonation by Mark Nowitzky
• Just intonation compared with meantone and 12-equal temperaments; a video featuring Pachelbel's canon.
• Just Intonation at the Xenharmonic Wiki.
• Just Intonation Explained by Kyle Gann
• A selection of Just Intonation works edited by the Just Intonation Network web published on the Tellus Audio Cassette Magazine project archive at UbuWeb
• Medieval Music and Arts Foundation
• Music Novatory – Just Intonation
• Why does Just Intonation sound so good?
• The Wilson Archives

Instruments

• 22 Note Just Intonation Keyboard Software with 12 Indian Instrument Sounds Libreria Editrice
• Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900. (2008) Latina, Il Levante
• Dante Rosati's 21 Tone Just Intonation guitar

Notations

• Plainsound Music Edition – JI music and research, information about the Helmholtz-Ellis JI Pitch Notation
• Helmholtz/Ellis/Wolf/Monzo system at Tonalsoft Encyclopaedia.

Notes and References

1. "Just intonation", The New Harvard Dictionary of Music. Edited by Don Randel. Belknap Press of Harvard University Press, 1986. 362f.
2. Lindley, Mark. "Just intonation." Grove Music Online. Oxford University Press, 2001.
3. Johnson, Charles William Leverett. Musical Pitch and the Measurement of Intervals Among the Ancient Greeks. Baltimore: J. Murphy, 1896. 45.
4. [Percy Buck|Buck, Percy Carter]
5. Dolata, David. Meantone Temperaments on Lutes and Viols. Indiana University Press, 2016.
6. Bod, Rens. A New History of the Humanities: The Search for Principles and Patterns from Antiquity to the Present. Oxford University Press, 2013. 37.
7. West. M. L.. Martin Litchfield West. The Babylonian musical notation and the Hurrian melodic texts . Music & Letters. 75. 2 . May 1994. 161–179. 737674. 10.1093/ml/75.2.161.
8. Barbour, James Murray. Tuning and Temperament: A Historical Survey. Dover Publications, 2004. 2.
9. 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.
10. Book: Vasudevan . D. V. K. . Molakatala . Nagamani . Priyatham . Nikil . Gautam . Ravikant . Rajender . M. . Equal Temperament and Just Intonation Feature Based Analysis of Indian Music . Lecture Notes in Computer Science . 2023 . 13741. Intelligent Human Computer Interaction . https://link.springer.com/chapter/10.1007/978-3-031-27199-1_12 . en . Cham . Springer Nature Switzerland . 109–120 . 10.1007/978-3-031-27199-1_12 . 978-3-031-27199-1.
11. Haluska, Jan. The Mathematical Theory of Tone Systems. Slovakia, CRC Press, 2003. 283.
12. Stange, Karolin et al. "Playing Music in Just Intonation: A Dynamically Adaptive Tuning Scheme." Computer Music Journal 42 (2017): 47-62.
13. [Edward Naylor|Naylor, Edward Woodall]
14. Lindley, Mark. "Pythagorean intonation", Grove Music Online. Oxford University Press, 2001.
15. Book: Barker, Andrew. Greek musical writings . 1989 . Cambridge University Press. Andrew Barker (classicist). 0-521-23593-6 . Cambridge . 350 . 10022960.
16. Barker, Andrew. Scientific method in Ptolemy's Harmonics. Cambridge University Press, 2000. 153.
17. Partch, Harry. Genesis Of A Music: An Account Of A Creative Work, Its Roots, And Its Fulfillments. Second Edition. Hachette Books, 1974.
18. Johnston, Ben. Maximum Clarity and Other Writings on Music. Chicago: University of Illinois Press, 2006.
19. Wright, David. Mathematics and Music. Mathematical World. Vol. 28. Providence, RI: American Mathematical Society, 2009. 140–141.
20. Tsuji, Kinko, and Müller, Stefan C.. Physics and Mathematics in Musical Composition: A Comparative Study. Germany, Springer Nature Switzerland, 2025. 32.
21. Book: Bagchee, Sandeep. 1998. Nad: Understanding Raga music. India. BPI PVT Ltd. 81-86982-07-8. 23.
22. Book: Daniélou, Alain . Alain Daniélou . 1995 . Music and the Power of Sound: The influence of tuning and interval on consciousness . Inner Traditions . Rep Sub . 0-89281-336-9 . registration .
23. Book: Daniélou, Alain . Alain Daniélou . 1999 . Introduction to the Study of Musical Scales . Oriental Book Reprint Corporation . 81-7069-098-6.
24. Web site: 22 shruti . 22shruti.com . main page . en . 2023-06-28.
25. Book: Daniélou, Alain. Alain Daniélou. The Ragas of Northern Indian Music . 1968 . Barrie & Rockliff . London . 0-214-15689-3.
26. Hauptmann, Moritz. Die Natur der Harmonik und der Metrik: Zur Theorie der Musik. Germany, Breitkopf & Härtel, 1873. 26.
27. Book: Helmholtz, Hermann von. Hermann von Helmholtz. 1885. On the Sensations of Tone as a Physiological Basis for the Theory of Music. Longmans, Green.
28. Benson, David J. Music: a mathematical offering. Cambridge University Press, 2007. 173f.
29. [Carl Eitz|Eitz, Carl]
30. Web site: Just Intonation Explained. Kyle Gann. KyleGann.com . 28 February 2016 . dmy-all.
31. Web site: Sabat . Marc . The Extended Helmholtz Ellis JI Pitch Notation . Plainsound Music Edition . March 11, 2014 . dmy-all.
32. Book: The Extended Helmholtz-Ellis JI Pitch Notation: eine Notationsmethode für die natürlichen Intervalle . Mikrotöne und Mehr – Auf György Ligetis Hamburger Pfaden . Manfred Stahnke . Stahnke, Manfred . von Bockel Verlag . Hamburg . 2005 . 3-932696-62-X.
33. Book: Garland. Peter. Peter Garland (composer). 1984. Soundings . 13 . The Music of James Tenney . Soundings Press . Santa Fe, New Mexico . 11371167.
34. Wannamaker, Robert, The Music of James Tenney, Volume 1: Contexts and Paradigms (University of Illinois Press, 2021), 288-89.
35. George Secor . Secor, George D.. Keenan, David C. . 2006 . Sagittal: A Microtonal Notation System . Xenharmonikôn: An Informal Journal of Experimental Music . 18 . 1–2 . Sagittal.org.