Just intonation

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Harmonic series, partials 1-5 numbered Harmonic series klang.png
Harmonic series, partials 1–5 numbered
Primary triads in C in just intonation. Primary triads in C.png
Primary triads in C in just intonation.

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.

Contents

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. [3]

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. [4] : 18 

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. [5] 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. [6] Clashing overtones will result in acoustic beats. [7] 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. [6] : 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. [4] : 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. [8] [9] Ptolemy and Didymus the Musician developed their own versions of the system. [10]

In China, the guqin draws on just intonation for its tuning system. [11] Indian music has an extensive theoretical framework for tuning in just intonation. [12]

20th century composers often returned to just intonation for inspiration. Many developed their own scales or instruments in order to use the tuning. [13]

Scales

Derivation of a just diatonic scale from the major triad. Just intonation diatonic scale derivation.png
Derivation of a just diatonic scale from the major triad.

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. [15] 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. [16] 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. [14]

In his second century AD book Harmonics, Ptolemy calculated a "tense diatonic" scale with ratios of string lengths 120, ⁠112+1/2⁠, 100, 90, 80, 75, ⁠66+2/3⁠, and 60. [17] [18] Harry Partch described this scale as "one of the world's fundamentally beautiful tonal sequences". [19] : 167  Justly tuned scales often yield multiple versions of the same interval, which can be managed through notation. [20] : 77 

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. [21] Modern composers expanded the limit to 7, which creates far more complex tuning solutions. [22]

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. [23]

Notesaregamapadhanisa
Ratio1:19:85:44:33:25:315:82:1
Cents020438649870288410881200

Some accounts of Indian intonation system cite a given 12 swaras being divided into 22  shrutis. [24] [25] 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 [26] in Indian music theory):

NoteCD♭↓D♭D↓DE♭↓E♭EE↑FF↑F♯F♯↑GA♭↓A♭AA↑B♭↓B♭BB↑C
Ratio1:1256:24316:1510:99:832:276:55:481:644:327:2045:32729:5123:2128:818:55:327:1616:99:515:8243:1282:1
Cents0901121822042943163864084985205906127027928148849069961018108811101200

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 [27] or the praman [26] in Indian music theory. These notes are known as chala. [26] The distance between two letter names comes in to sizes, poorna (256:243) and nyuna (25:24). [26] One can see the symmetry, looking at it from the tonic, then the octave.

(This is just one example of explaining a 22 Śhruti scale of tones. There are many different explanations.)

Practical difficulties

Some fixed just intonation scales and systems, such as the diatonic scale above, produce wolf intervals when the approximately equivalent flat note is substituted for a sharp note not available in the scale, or vice versa. The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32:27 for D→F, and still worse, a minor tone next to a fourth giving 40:27 for D→A. Flattening D by a comma to 10:9 alleviates these difficulties but creates new ones: D→G becomes 27:20, and D→B becomes 27:16. This fundamental problem arises in any system of tuning using a limited number of notes.

One can have more frets on a guitar (or keys on a piano) to handle both As, 9:8 with respect to G and 10:9 with respect to G so that A→C can be played as 6:5 while A→D can still be played as 3:2. 9:8 and 10:9 are less than ⁠1/53⁠ of an octave apart, so mechanical and performance considerations have made this approach extremely rare. And the problem of how to tune complex chords such as C6 add 9 (C→E→G→A→D), in typical 5 limit just intonation, is left unresolved (for instance, A could be 4:3 below D (making it 9:8, if G is 1) or 4:3 above E (making it 10:9, if G is 1) but not both at the same time, so one of the fourths in the chord will have to be an out-of-tune wolf interval). Most complex (added-tone and extended) chords usually require intervals beyond common 5 limit ratios in order to sound harmonious (for instance, the previous chord could be tuned to 8:10:12:13:18, using the A note from the 13th harmonic), which implies even more keys or frets. However the frets may be removed entirely – unfortunately, this makes in-tune fingering of many chords exceedingly difficult, due to the construction and mechanics of the human hand – and the tuning of most complex chords in just intonation is generally ambiguous.

Some composers deliberately use these wolf intervals and other dissonant intervals as a way to expand the tone color palette of a piece of music. For example, the extended piano pieces The Well-Tuned Piano by La Monte Young and The Harp of New Albion by Terry Riley use a combination of very consonant and dissonant intervals for musical effect. In "Revelation", Michael Harrison goes even further, and uses the tempo of beat patterns produced by some dissonant intervals as an integral part of several movements.

When tuned in just intonation, many fixed-pitch instruments cannot be played in a new key without retuning the instrument. For instance, if a piano is tuned in just intonation intervals and a minimum of wolf intervals for the key of G, then only one other key (typically E♭) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes modulation within a piece, or playing a repertoire of pieces in different keys, impractical to impossible.

Synthesizers have proven a valuable tool for composers wanting to experiment with just intonation. They can be easily retuned with a microtuner. Many commercial synthesizers provide the ability to use built-in just intonation scales or to create them manually. Wendy Carlos used a system on her 1986 album Beauty in the Beast , where one electronic keyboard was used to play the notes, and another used to instantly set the root note to which all intervals were tuned, which allowed for modulation. On her 1987 lecture album Secrets of Synthesis there are audible examples of the difference in sound between equal temperament and just intonation.

Many singers (especially barbershop quartets) and fretless instrument players naturally aim for a more just intonation when playing:

"Don't be scared if your intonation differs from that of the piano. It is the piano that is out of tune. The piano with its tempered scale is a compromise in intonation." — Pablo Casals [ citation needed ]

In trying to get a more just system for instruments that is more adaptable like the human voice and fretless instruments, the tuning trade-offs between more consonant harmony versus easy transposability (between different keys) have traditionally been too complicated to solve mechanically, though there have been attempts throughout history with various drawbacks, including the archicembalo.

Since the advent of personal computing, there have been more attempts to solve the perceived problem by trying to algorithmically solve what many professional musicians have learned through practice and intuition. Four of the main problems are that consonance cannot be perfect for some complex chords, chords can have internal consistency but clash with the overall direction of the piece, and naively adjusting the tuning only taking into account chords in isolation can lead to a drift where the end of the piece is noticeably higher or lower in overall pitch rather than centered.

Software solutions like Hermode Tuning often analyze solutions chord by chord instead of taking in the global context of the whole piece like it's theorized human players do. Since 2017, there has been research to address these problems algorithmically through dynamically adapted just intonation and machine learning. [6]

Staff notation

Fig. 1: Legend of the Helmholtz-Ellis accidentals within the 23-limit Helmholtz-Ellis JI pitch notation microtonal accidents legend.gif
Fig. 1: Legend of the Helmholtz-Ellis accidentals within the 23-limit

Originally a system of notation to describe scales was devised by Hauptmann and modified by Helmholtz (1877); the starting note is presumed Pythagorean; a "+" is placed between if the next note is a just major third up, a "−" if it is a just minor third, among others; finally, subscript numbers are placed on the second note to indicate how many syntonic commas (81:80) to lower by. [3] :  276   For example, the Pythagorean major third on C is C+E ( Play ⓘ ) while the just major third is C+E1 ( Play ⓘ ). A similar system was devised by Carl Eitz and used in Barbour (1951) in which Pythagorean notes are started with and positive or negative superscript numbers are added indicating how many commas (81:80, syntonic comma) to adjust by. [28] For example, the Pythagorean major third on C is C−E0 while the just major third is C−E−1. An extension of this Pythagorean-based notation to higher primes is the Helmholtz / Ellis / Wolf / Monzo system [29] of ASCII symbols and prime-factor-power vectors described in Monzo's Tonalsoft Encyclopaedia. [29]

While these systems allow precise indication of intervals and pitches in print, more recently some composers have been developing notation methods for Just Intonation using the conventional five-line staff. James Tenney, amongst others, preferred to combine JI ratios with cents deviations from the equal tempered pitches, indicated in a legend or directly in the score, allowing performers to readily use electronic tuning devices if desired. [30] [31]

Beginning in the 1960s, Ben Johnston had proposed an alternative approach, redefining the understanding of conventional symbols (the seven "white" notes, the sharps and flats) and adding further accidentals, each designed to extend the notation into higher prime limits. His notation "begins with the 16th-century Italian definitions of intervals and continues from there." [32] Johnston notation is based on a diatonic C Major scale tuned in JI (Fig. 4), in which the interval between D (9:8 above C) and A (5:3 above C) is one syntonic comma less than a Pythagorean perfect fifth 3:2. To write a perfect fifth, Johnston introduces a pair of symbols, + and − again, to represent this comma. Thus, a series of perfect fifths beginning with F would proceed C G D A+ E+ B+. The three conventional white notes A E B are tuned as Ptolemaic major thirds (5:4) above F C G respectively. Johnston introduces new symbols for the septimal ( 7 rightside up.png & 7 upside down.png ), undecimal (↑ & ↓), tridecimal ( 13 rightside up.png & 13 upside down.png ), and further prime-number extensions to create an accidental based exact JI notation for what he has named "Extended Just Intonation" (Fig. 2 & Fig. 3). [20] : 77–88  For example, the Pythagorean major third on C is C-E+ while the just major third is C-E♮ (Fig. 4).

Fig. 2: Staff notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C using Johnston notation Notation of partials 1-19 for 1-1.png
Fig. 2: Staff notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C using Johnston notation
Fig. 3: Just harmonic seventh chord (4:5:6:7:8) on C in Johnston notation. The size of the 7th is 968.826 cents: 48.77 cents lower than B tuned 9:5 above C. Harmonic seventh chord just on C.png
Fig. 3: Just harmonic seventh chord (4:5:6:7:8) on C in Johnston notation. The size of the 7th is 968.826 cents: 48.77 cents lower than B♭ tuned 9:5 above C.

In 2000–2004, Marc Sabat and Wolfgang von Schweinitz worked in Berlin to develop a different accidental-based method, the Extended Helmholtz-Ellis JI Pitch Notation. [34] Following the method of notation suggested by Helmholtz in his classic On the Sensations of Tone as a Physiological Basis for the Theory of Music , incorporating Ellis' invention of cents, and continuing Johnston's step into "Extended JI", Sabat and Schweinitz propose unique symbols (accidentals) for each prime dimension of harmonic space. In particular, the conventional flats, naturals and sharps define a Pythagorean series of perfect fifths. The Pythagorean pitches are then paired with new symbols that commatically alter them to represent various other partials of the harmonic series (Fig. 1). To facilitate quick estimation of pitches, cents indications may be added (e.g. downward deviations below and upward deviations above the respective accidental). A typically used convention is that cent deviations refer to the tempered pitch implied by the flat, natural, or sharp. A complete legend and fonts for the notation (see samples) are open source and available from the Plainsound Music Edition website. [35] For example, the Pythagorean major third on C is C-E♮ while the just major third is C-E♮↓ (see Fig. 4 for "combined" symbol)

Fig. 4: Comparison of Helmholtz-Ellis JI Pitch Notation and Johnston Notation. Unaltered naturals in Helmholtz-Ellis may be omitted if desired. HEJI Johnston.png
Fig. 4: Comparison of Helmholtz-Ellis JI Pitch Notation and Johnston Notation. Unaltered naturals in Helmholtz-Ellis may be omitted if desired.
Fig. 5: Just harmonic thirteenth chord (4:5:6:7:9:11:13) on G in Sagittal notation (with mnemonics) Just harmonic thirteenth chord on G Sagittal notation.png
Fig. 5: Just harmonic thirteenth chord (4:5:6:7:9:11:13) on G in Sagittal notation (with mnemonics)

Sagittal notation (from Latin sagitta, "arrow") is a system of arrow-like accidentals that indicate prime-number comma alterations to tones in a Pythagorean series. It is used to notate both just intonation and equal temperaments. The size of the symbol indicates the size of the alteration. [36]

The great advantage of such notation systems is that they allow the natural harmonic series to be precisely notated. At the same time, they provide some degree of practicality through their extension of staff notation, as traditionally trained performers may draw on their intuition for roughly estimating pitch height. This may be contrasted with the more abstract use of ratios for representing pitches in which the amount by which two pitches differ and the "direction" of change may not be immediately obvious to most musicians. One caveat is the requirement for performers to learn and internalize a (large) number of new graphical symbols. However, the use of unique symbols reduces harmonic ambiguity and the potential confusion arising from indicating only cent deviations.

Audio examples

See also

Lists
Article topics

References

  1. 1 2 3 "Just intonation", The New Harvard Dictionary of Music . Edited by Don Randel. Belknap Press of Harvard University Press, 1986. 362f.
  2. 1 2 Lindley, Mark. "Just intonation." Grove Music Online. Oxford University Press, 2001.
  3. 1 2 Helmholtz, Hermann von (1885). On the Sensations of Tone as a Physiological Basis for the Theory of Music . Longmans, Green.
  4. 1 2 Dolata, David. Meantone Temperaments on Lutes and Viols. Indiana University Press, 2016.
  5. ↑ Johnson, Charles William Leverett.  Musical Pitch and the Measurement of Intervals Among the Ancient Greeks . Baltimore: J. Murphy, 1896. 45.
  6. 1 2 3 Stange, Karolin et al. "Playing Music in Just Intonation: A Dynamically Adaptive Tuning Scheme." Computer Music Journal 42 (2017): 47-62.
  7. ↑ Buck, Percy Carter.  Acoustics for Musicians . Clarendon Press, 1918. 142f.
  8. ↑ Bod, Rens. A New History of the Humanities: The Search for Principles and Patterns from Antiquity to the Present. Oxford University Press, 2013. 37.
  9. ↑ West, M. L. (May 1994). "The Babylonian musical notation and the Hurrian melodic texts". Music & Letters . 75 (2): 161–179. doi:10.1093/ml/75.2.161. JSTOR   737674.
  10. ↑ Barbour, James Murray.  Tuning and Temperament: A Historical Survey . Dover Publications, 2004. 2.
  11. ↑ 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.
  12. ↑ Vasudevan, D. V. K.; Molakatala, Nagamani; Priyatham, Nikil; Gautam, Ravikant; Rajender, M. (2023). "Equal Temperament and Just Intonation Feature Based Analysis of Indian Music". Intelligent Human Computer Interaction. Lecture Notes in Computer Science. Vol. 13741. Cham: Springer Nature Switzerland. pp. 109–120. doi:10.1007/978-3-031-27199-1_12. ISBN   978-3-031-27199-1.
  13. ↑ Haluska, Jan. The Mathematical Theory of Tone Systems. Slovakia, CRC Press, 2003. 283.
  14. 1 2 Campbell, Murray and Clive Greated. The Musician's Guide to Acoustics. Oxford University Press, 2023. 172–3.
  15. ↑ 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.
  16. ↑ Lindley, Mark. "Pythagorean intonation", Grove Music Online. Oxford University Press, 2001.
  17. ↑ Barker, Andrew (1989). Greek musical writings. Cambridge: Cambridge University Press. p. 350. ISBN   0-521-23593-6. OCLC   10022960.
  18. ↑ Barker, Andrew. Scientific method in Ptolemy's Harmonics. Cambridge University Press, 2000. 153.
  19. ↑ Partch, Harry. Genesis Of A Music: An Account Of A Creative Work, Its Roots, And Its Fulfillments. Second Edition. Hachette Books, 1974.
  20. 1 2 Johnston, Ben.  Maximum Clarity and Other Writings on Music . Chicago: University of Illinois Press, 2006.
  21. ↑ Wright, David. Mathematics and Music . Mathematical World. Vol. 28. Providence, RI: American Mathematical Society, 2009. 140–141.
  22. ↑ Tsuji, Kinko, and Müller, Stefan C..  Physics and Mathematics in Musical Composition: A Comparative Study . Germany, Springer Nature Switzerland, 2025. 32.
  23. ↑ Bagchee, Sandeep (1998). Nad: Understanding Raga music. India: BPI PVT Ltd. p. 23. ISBN   81-86982-07-8.
  24. ↑ Daniélou, Alain (1995). Music and the Power of Sound: The influence of tuning and interval on consciousness (Rep Sub ed.). Inner Traditions. ISBN   0-89281-336-9.
  25. ↑ Daniélou, Alain (1999). Introduction to the Study of Musical Scales. Oriental Book Reprint Corporation. ISBN   81-7069-098-6.
  26. 1 2 3 4 "22 shruti". 22shruti.com (main page). Retrieved 2023-06-28.
  27. ↑ Daniélou, Alain (1968). The Ragas of Northern Indian Music. London: Barrie & Rockliff. ISBN   0-214-15689-3.
  28. ↑ Benson, David J. (2007). Music: A Mathematical Offering . Cambridge University Press. p.  172. ISBN   978-0-521-85387-3.
    who cites Eitz, Carl A. (1891). Das mathematisch-reine Tonsystem. Leipzig.{{cite book}}: CS1 maint: location missing publisher (link)
  29. 1 2 Monzo. "Helmholtz / Ellis / Wolf / Monzo system". Tonalsoft Encyclopaedia. tonalsoft.com.
  30. ↑ Garland, Peter, ed. (1984). The Music of James Tenney. Soundings. Vol. 13. Santa Fe, New Mexico: Soundings Press. OCLC   11371167.
  31. ↑ Wannamaker, Robert, The Music of James Tenney, Volume 1: Contexts and Paradigms (University of Illinois Press, 2021), 288-89.
  32. ↑ Kyle Gann. "Just Intonation Explained". KyleGann.com. Retrieved 28 February 2016.
  33. ↑ Fonville, John (Summer 1991). "Ben Johnston's Extended Just Intonation: A guide for interpreters". Perspectives of New Music . 29 (2): 121, 106–137. doi:10.2307/833435. JSTOR   833435.
  34. ↑ Stahnke, Manfred, ed. (2005). "The Extended Helmholtz-Ellis JI Pitch Notation: eine Notationsmethode für die natürlichen Intervalle". Mikrotöne und Mehr – Auf György Ligetis Hamburger Pfaden. Hamburg: von Bockel Verlag. ISBN   3-932696-62-X.
  35. ↑ Sabat, Marc. "The Extended Helmholtz Ellis JI Pitch Notation" (PDF). Plainsound Music Edition. Retrieved 11 March 2014.
  36. ↑ Secor, George D.; Keenan, David C. (2006). "Sagittal: A Microtonal Notation System" (PDF). Xenharmonikôn: An Informal Journal of Experimental Music. Vol. 18. pp. 1–2 – via Sagittal.org.
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