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In music, just intonation or pure intonation is the
tuning Tuning can refer to: Common uses * Tuning, the process of tuning a tuned amplifier or other electronic component * Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice ** Guitar tunings ** Piano tuning, adjusti ...
of musical
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...
as whole number
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s (such as 3:2 or 4:3) of
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and chords created by combining them) consist of tones from a single harmonic series of an implied
fundamental Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
. For example, in the diagram, if the notes G3 and C4 (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth. In Western musical practice, instruments are rarely tuned using only pure intervals—the desire for different keys to have identical intervals in Western music makes this impractical. Some instruments of fixed pitch, such as electric pianos, are commonly tuned using
equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
, in which all intervals other than octaves consist of irrational-number frequency ratios. Acoustic pianos are usually tuned with the octaves slightly widened, and thus with no pure intervals at all.


Terminology

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
, or 3-limit tuning, allows ratios including the numbers 2 and 3 and their powers, such as 3:2, a
perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...
, and 9:4, a
major ninth In music, a ninth is a compound interval consisting of an octave plus a second. Like the second, the interval of a ninth is classified as a dissonance in common practice tonality. Since a ninth is an octave larger than a second, its ...
. Although the interval from C to G is called a perfect fifth for purposes of
music analysis Musical analysis is the study of musical structure in either compositions or performances. According to music theorist Ian Bent, music analysis "is the means of answering directly the question 'How does it work?'". The method employed to answe ...
regardless of its tuning method, for purposes of discussing tuning systems musicologists may distinguish between a ''perfect fifth'' created using the 3:2 ratio and a ''tempered fifth'' using some other system, such as
meantone Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...
or
equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
. 5-limit tuning encompasses ratios additionally using the number 5 and its powers, such as 5:4, a
major third In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
, and 15:8, a
major seventh In music from Western culture, a seventh is a musical interval encompassing seven staff positions (see Interval number for more details), and the major seventh is one of two commonly occurring sevenths. It is qualified as ''major'' because it i ...
. The specialized term ''perfect third'' is occasionally used to distinguish the 5:4 ratio from major thirds created using other tuning methods. 7-limit and higher systems use higher partials in the overtone series. Commas are very small intervals that result from minute differences between pairs of just intervals. For example, the 5:4 ratio is different from the Pythagorean (3-limit) major third (81:64) by a difference of 81:80, called the
syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
. A Cent is a measure of interval size. It is logarithmic in the musical frequency ratios. The octave is divided into 1200 steps, 100 cents for each semitone.


History

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
has been attributed to both
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
and
Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ;  – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...
by later writers, but may have been analyzed by other early Greeks or other early cultures as well. The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts. During the second century AD,
Claudius Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
described a 5-limit diatonic scale in his influential text on music theory ''Harmonics'', which he called "intense diatonic". Given ratios of string lengths 120, , 100, 90, 80, 75, , and 60, Ptolemy quantified the tuning of what would later be called the
Phrygian scale The Phrygian mode (pronounced ) can refer to three different musical modes: the ancient Greek ''tonos'' or ''harmonia,'' sometimes called Phrygian, formed on a particular set of octave species or scales; the Medieval Phrygian mode, and the modern ...
(equivalent to the
major scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double i ...
beginning and ending on the third note) – 16:15, 9:8, 10:9, 9:8, 16:15, 9:8, and 10:9. Non-Western music, particularly that built on pentatonic scales, is largely tuned using just intonation. In China, the
guqin The ''guqin'' (; ) is a plucked seven-string Chinese musical instrument. It has been played since ancient times, and has traditionally been favoured by scholars and literati as an instrument of great subtlety and refinement, as highlighted b ...
has a musical scale based on
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
positions. The dots on its soundboard indicate the harmonic positions: , , , , , , , , , , , , . Indian music has an extensive theoretical framework for tuning in just intonation.


Diatonic scale

The prominent notes of a given scale may be tuned so that their frequencies form (relatively) small whole number ratios. The 5-limit diatonic major scale is tuned in such a way that major triads on the tonic,
subdominant In music, the subdominant is the fourth tonal degree () of the diatonic scale. It is so called because it is the same distance ''below'' the tonic as the dominant is ''above'' the tonicin other words, the tonic is the dominant of the subdomina ...
, and dominant are tuned in the proportion 4:5:6, and minor triads on the
mediant In music, the mediant (''Latin'': to be in the middle) is the third scale degree () of a diatonic scale, being the note halfway between the tonic and the dominant.Benward & Saker (2003), p.32. In the movable do solfège system, the mediant note i ...
and
submediant In music, the submediant is the sixth degree () of a diatonic scale. The submediant ("lower mediant") is named thus because it is halfway between tonic and subdominant ("lower dominant") or because its position below the tonic is symmetrical to t ...
are tuned in the proportion 10:12:15. Because of the two sizes of wholetone – 9:8 (major wholetone) and 10:9 (minor wholetone) – the
supertonic In music, the supertonic is the second degree () of a diatonic scale, one whole step above the tonic. In the movable do solfège system, the supertonic note is sung as ''re''. The triad built on the supertonic note is called the supertonic chor ...
must be microtonally lowered by a
syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
to form a pure minor triad. The 5-limit diatonic major scale (
Ptolemy's intense diatonic scale Ptolemy's intense diatonic scale, also known as the Ptolemaic sequence, justly tuned major scale, Ptolemy's tense diatonic scale, or the syntonous (or syntonic) diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy, and corresp ...
) on C is shown in the table below: In this example the interval from D up to A would be a
wolf fifth In music theory, the wolf fifth (sometimes also called Procrustean fifth, or imperfect fifth) Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction', p.165. Theodore Baker, trans. G. Schirmer. ...
with the ratio , about 680 cents, noticeably smaller than the 702 cents of the pure ratio. For a justly tuned diatonic minor scale, the mediant is tuned 6:5 and the submediant is tuned 8:5. It would include a tuning of 9:5 for the
subtonic In music, the subtonic is the degree of a musical scale which is a whole step below the tonic note. In a major key, it is a lowered, or flattened, seventh scale degree (). It appears as the seventh scale degree in the natural minor and descendin ...
. For example on A:


Twelve-tone scale

There are several ways to create a just tuning of the twelve-tone scale.


Pythagorean tuning

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
can produce a twelve-tone scale, but it does so by involving ratios of very large numbers, corresponding to natural harmonics very high in the harmonic series that do not occur widely in physical phenomena. This tuning uses ratios involving only powers of 3 and 2, creating a sequence of just fifths or fourths, as follows: The ratios are computed with respect to C (the ''base note''). Starting from C, they are obtained by moving six steps (around the
circle of fifths In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...
) to the left and six to the right. Each step consists of a multiplication of the previous pitch by (descending fifth), (ascending fifth), or their inversions ( or ). Between the
enharmonic In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written n ...
notes at both ends of this sequence is a pitch ratio of , or about 23 cents, known as the
Pythagorean comma In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C ...
. To produce a twelve-tone scale, one of them is arbitrarily discarded. The twelve remaining notes are repeated by increasing or decreasing their frequencies by a power of 2 (the size of one or more
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
s) to build scales with multiple octaves (such as the keyboard of a piano). A drawback of Pythagorean tuning is that one of the twelve fifths in this scale is badly tuned and hence unusable (the
wolf fifth In music theory, the wolf fifth (sometimes also called Procrustean fifth, or imperfect fifth) Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction', p.165. Theodore Baker, trans. G. Schirmer. ...
, either F–D if G is discarded, or B–G if F is discarded). This twelve-tone scale is fairly close to
equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
, but it does not offer much advantage for tonal harmony because only the perfect intervals (fourth, fifth, and octave) are simple enough to sound pure. Major thirds, for instance, receive the rather unstable interval of 81:64, sharp of the preferred 5:4 by an 81:80 ratio. The primary reason for its use is that it is extremely easy to tune, as its building block, the perfect fifth, is the simplest and consequently the most
consonant In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wit ...
interval after the octave and unison. Pythagorean tuning may be regarded as a "three-limit" tuning system, because the ratios can be expressed as a product of integer powers of only whole numbers less than or equal to 3.


Five-limit tuning

A twelve-tone scale can also be created by compounding harmonics up to the fifth: namely, by multiplying the frequency of a given reference note (the base note) by powers of 2, 3, or 5, or a combination of them. This method is called five-limit tuning. To build such a twelve-tone scale (using C as the base note), we may start by constructing a table containing fifteen pitches: The factors listed in the first row and column are powers of 3 and 5, respectively (e.g., = 3). Colors indicate couples of
enharmonic In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written n ...
notes with almost identical pitch. The ratios are all expressed relative to C in the centre of this diagram (the base note for this scale). They are computed in two steps: # For each cell of the table, a ''base ratio'' is obtained by multiplying the corresponding factors. For instance, the base ratio for the lower-left cell is # The base ratio is then multiplied by a negative or positive power of 2, as large as needed to bring it within the range of the octave starting from C (from 1:1 to 2:1). For instance, the base ratio for the lower left cell () is multiplied by 2, and the resulting ratio is 64:45, which is a number between 1:1 and 2:1. Note that the powers of 2 used in the second step may be interpreted as ascending or descending
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
s. For instance, multiplying the frequency of a note by 2 means increasing it by 6 octaves. Moreover, each row of the table may be considered to be a sequence of fifths (ascending to the right), and each column a sequence of major thirds (ascending upward). For instance, in the first row of the table, there is an ascending fifth from D and A, and another one (followed by a descending octave) from A to E. This suggests an alternative but equivalent method for computing the same ratios. For instance, one can obtain A, starting from C, by moving one cell to the left and one upward in the table, which means descending by a fifth and ascending by a major third: Since this is below C, one needs to move up by an octave to end up within the desired range of ratios (from 1:1 to 2:1): A 12-tone scale is obtained by removing one note for each couple of enharmonic notes. This can be done in at least three ways, which have in common the removal of G, according to a convention which was valid even for C-based Pythagorean and quarter-comma meantone scales. Note that it is a
diminished fifth Diminished may refer to: *Diminution In Western music and music theory, diminution (from Medieval Latin ''diminutio'', alteration of Latin ''deminutio'', decrease) has four distinct meanings. Diminution may be a form of embellishment in which ...
, close to half an octave, above the tonic C, which is a disharmonic interval; also its ratio has the largest values in its numerator and denominator of all tones in the scale, which make it least harmonious: all reasons to avoid it. This is only one possible strategy of five-limit tuning. It consists of discarding the first column of the table (labeled ""). The resulting 12-tone scale is shown below:


Extension of the twelve-tone scale

The table above uses only low powers of 3 and 5 to build the base ratios. However, it can be easily extended by using higher positive and negative powers of the same numbers, such as 5 = 25, 5 = , 3 = 27, or 3 = . A scale with 25, 35 or even more pitches can be obtained by combining these base ratios.


Indian scales

In
Indian music Owing to India's vastness and diversity, Indian music encompasses numerous genres in multiple varieties and forms which include classical music, folk (Bollywood), rock, and pop. It has a history spanning several millennia and developed ove ...
, 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''. Some accounts of Indian intonation system cite a given 22 Shrutis. 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): Where we have ''two'' ratios for a given letter name, we have a difference of 81:80 (or 22 cents), which is known as the
syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
. One can see the symmetry, looking at it from the tonic, then the octave. (This is just one example of explaining a 22-Śruti 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 interval In music theory, the wolf fifth (sometimes also called Procrustean fifth, or imperfect fifth) Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction', p.165. Theodore Baker, trans. G. Schirmer. ...
s 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. Moving D down 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
fret A fret is any of the thin strips of material, usually metal wire, inserted laterally at specific positions along the neck or fretboard of a stringed instrument. Frets usually extend across the full width of the neck. On some historical instrume ...
s on a
guitar The guitar is a fretted musical instrument that typically has six strings. It is usually held flat against the player's body and played by strumming or plucking the strings with the dominant hand, while simultaneously pressing selected stri ...
(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 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 C6add9 (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—this, unfortunately, 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 ''The Well-Tuned Piano'' is an ongoing, improvisatory, solo piano work by composer La Monte Young. Begun in 1964, Young has never considered the composition or performance "finished", and he has performed incarnations of it several times since i ...
'' by LaMonte 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. For many fixed-pitch instruments tuned in just intonation, one cannot change keys 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-flat) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes
modulation In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...
within a piece, or playing a repertoire of pieces in different keys, impractical to impossible.
Synthesizer A synthesizer (also spelled synthesiser) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis and ...
s have proven a valuable tool for composers wanting to experiment with just intonation. They can be easily retuned with a
microtuner A microtuner or microtonal tuner is an electronic device or software program designed to modify and test the tuning of musical instruments (in particular synthesizers) with microtonal precision, allowing for the design and construction of microton ...
. Many commercial synthesizers provide the ability to use built-in just intonation scales or to create them manually.
Wendy Carlos Wendy Carlos (born Walter Carlos, November 14, 1939) is an American musician and composer best known for her electronic music and film scores. Born and raised in Rhode Island, Carlos studied physics and music at Brown University before moving ...
used a system on her 1986 album ''
Beauty in the Beast ''Beauty in the Beast'' is a studio album from the American keyboardist and composer Wendy Carlos, released in 1986, on Audion Records, her first for a label other than Columbia Records since 1968. The album uses alternate musical tunings and s ...
'', 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.


Singing and scale-free instruments

The human voice is among the most pitch-flexible instruments in common use. Pitch can be varied with no restraints and adjusted in the midst of performance, without needing to retune. Although the explicit use of just intonation fell out of favour concurrently with the increasing use of instrumental accompaniment (with its attendant constraints on pitch), most
a cappella ''A cappella'' (, also , ; ) music is a performance by a singer or a singing group without instrumental accompaniment, or a piece intended to be performed in this way. The term ''a cappella'' was originally intended to differentiate between Ren ...
ensembles naturally tend toward just intonation because of the comfort of its stability.
Barbershop quartets A barbershop quartet is a group of four singers who sing music in the barbershop style, characterized by four-part harmony without instrumental accompaniment, or a cappella. The four voices are: the lead, the vocal part which typically carries t ...
are a good example of this. The unfretted stringed instruments such as those from the violin family (the violin, the viola, and the cello), and the double bass are quite flexible in the way pitches can be adjusted. Stringed instruments that are not playing with fixed pitch instruments tend to adjust the pitch of key notes such as thirds and
leading tone In music theory, a leading-tone (also called a subsemitone, and a leading-note in the UK) is a note or pitch which resolves or "leads" to a note one semitone higher or lower, being a lower and upper leading-tone, respectively. Typically, ''the ...
s so that the pitches differ from equal temperament. Trombones have a slide that allows arbitrary tuning during performance. French horns can be tuned by shortening or lengthening the main tuning slide on the back of the instrument, with each individual rotary or piston slide for each rotary or piston valve, and by using the right hand inside the bell to adjust the pitch by pushing the hand in deeper to sharpen the note, or pulling it out to flatten the note while playing. Some natural horns also may adjust the tuning with the hand in the bell, and valved cornets, trumpets, Flugelhorns, Saxhorns, Wagner tubas, and tubas have overall and valve-by-valve tuning slides, like valved horns. Wind instruments with valves are biased towards natural tuning and must be micro-tuned if equal temperament is required. Other wind instruments, although built to a certain scale, can be micro-tuned to a certain extent by using the
embouchure Embouchure () or lipping is the use of the lips, facial muscles, tongue, and teeth in playing a wind instrument. This includes shaping the lips to the mouthpiece of a woodwind instrument or the mouthpiece of a brass instrument. The word is of ...
or adjustments to fingering.


Western composers

Composers often impose a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
on how complex the ratios may become. For example, a composer who chooses to write in 7-limit just intonation will not employ ratios that use powers of prime numbers larger than 7. Under this scheme, ratios like 11:7 and 13:6 would not be permitted, because 11 and 13 cannot be expressed as powers of those prime numbers ≤ 7 (''i.e.'' 2, 3, 5, and 7).


Staff notation

Originally a system of notation to describe scales was devised by
Hauptmann is a German word usually translated as captain when it is used as an officer's rank in the German, Austrian, and Swiss armies. While in contemporary German means 'main', it also has and originally had the meaning of 'head', i.e. ' literally ...
and modified by
Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
(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. Note the use of the “+” between just major thirds, “−” between just minor thirds, “, ” between Pythagorean minor thirds, and “±” between perfect fifths. For example, the Pythagorean major third on C is C+E () while the just major third is C+E1 (). 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. 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'' of
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
symbols and prime-factor-power vectors described in Monzo's ''Tonalsoft Encyclopaedia''. 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 James Tenney (August 10, 1934 – August 24, 2006) was an American composer and music theorist. He made significant early musical contributions to plunderphonics, sound synthesis, algorithmic composition, process music, spectral music, microtonal ...
, 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. 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 limit In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term ''limit'' was introduced by Harry Partch, who used it to give ...
s. His notation "begins with the 16th-century Italian definitions of intervals and continues from there." 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 In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
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 ( & ), undecimal ( & ), tridecimal ( & ), 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). For example, the Pythagorean major third on C is C-E+ while the just major third is C-E (Fig. 4). In 2000–2004,
Marc Sabat Marc Sabat (born 22 September 1965) is a Canadian composer based in Berlin, Germany, since 1999. Works He has made concert music pieces, works with video, and installations with acoustic instruments and, in some recent pieces, computer-generate ...
and
Wolfgang von Schweinitz Wolfgang von Schweinitz (born 7 February 1953 in Hamburg) is a German composer of classical music and an academic teacher. Career Schweinitz studied composition at the Hochschule für Musik und Theater Hamburg, from 1971 to 1973 with Gernot Klu ...
worked in Berlin to develop a different accidental-based method, the Extended Helmholtz-Ellis JI Pitch Notation. Following the method of notation suggested by
Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
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. 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) 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. 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

* 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 Beat, beats or beating may refer to: Common uses * Patrol, or beat, a group of personnel assigned to monitor a specific area ** Beat (police), the territory that a police officer patrols ** Gay beat, an area frequented by gay men * Battery ...
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 Beat, beats or beating may refer to: Common uses * Patrol, or beat, a group of personnel assigned to monitor a specific area ** Beat (police), the territory that a police officer patrols ** Gay beat, an area frequented by gay men * Battery ...
can be heard at about 4 Hz and about 0.8 Hz. In the just intonation triad, this roughness is absent. The
square wave A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions b ...
form makes the difference between equal temperament and just intonation more obvious.


See also

* List of compositions in just intonation * List of intervals in 5-limit just intonation *
List of meantone intervals The following is a list of intervals of extended meantone temperament. These intervals constitute the standard vocabulary of intervals for the Western common practice era. Here 12-EDO refers to the size of the interval in 12 equal divisions of th ...
*
List of pitch intervals Below is a list of intervals expressible in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals. For commonly encountered harmonic or melodic intervals ...
*
Electronic tuner In music, an electronic tuner is a device that detects and displays the pitch of musical notes played on a musical instrument. "Pitch" is the perceived fundamental frequency of a musical note, which is typically measured in Hertz. Simple tune ...
*
Hexany In musical tuning systems, the hexany, invented by Erv Wilson, represents one of the simplest structures found in his combination product sets. It is referred to as an uncentered structure, meaning that it implies no tonic. It achieves this b ...
*
Microtonal music Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, also called "microintervals". It may also be extended to include any music using intervals not found in the customary Western tuning of tw ...
*
Microtuner A microtuner or microtonal tuner is an electronic device or software program designed to modify and test the tuning of musical instruments (in particular synthesizers) with microtonal precision, allowing for the design and construction of microton ...
*
Music and mathematics Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing mus ...
* Musical interval *
Pythagorean interval In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa.Benson, Donald C. (2003). ''A Smoother Pebble: Mathematical Explorations'', p.56. . ...
*
Regular number Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 602 = 3600 = 48 ×&nb ...
*
Superparticular ratio In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers. More particularly, the ratio takes the form: :\frac = 1 + \frac where is a positive integer. Thu ...
*
Whole-tone scale In music, a whole-tone scale is a scale in which each note is separated from its neighbors by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole-tone scales, both six-note or ''hexatonic'' s ...


References


External links


Art of the States: microtonal/just intonation
works using just intonation by American composers




Just Intonation
b
Mark Nowitzky

Just intonation
compared with
meantone Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...
and 12-equal temperaments; a video featuring Pachelbel's canon.
Just Intonation Explained
by
Kyle Gann Kyle Eugene Gann (born November 21, 1955, in Dallas, Texas) is an American professor of music, critic, analyst, and composer who has worked primarily in the New York City area. As a music critic for ''The Village Voice'' (from 1986 to 2005) and ...

A selection of Just Intonation works edited by the Just Intonation Network
web published on the
Tellus Audio Cassette Magazine Launched from the Lower East Side, Manhattan in 1983 as a subscription only bimonthly publication, the ''Tellus Audio Cassette Magazine'' utilized the audio cassette medium to distribute no wave downtown music and audio art and was in activity ...
project archive at
UbuWeb UbuWeb is a web-based educational resource for avant-garde material available on the internet, founded in 1996 by poet Kenneth Goldsmith. It offers visual, concrete and sound poetry, expanding to include film and sound art mp3 archives. Philosop ...

Medieval Music and Arts Foundation






* Barbieri, Patrizio

(2008) Latina, Il Levante
22 Note Just Intonation Keyboard Software with 12 Indian Instrument Sounds
Libreria Editrice
Plainsound Music Edition
– JI music and research, information about the Helmholtz-Ellis JI Pitch Notation {{Authority control *