Tuning systems

What is a tuning system?

A tone system, also referred to as a “scale system,” refers to the organisation and arrangement of pitches within a musical framework. It forms the basic structure from which different scales and modes are derived. In Western music, the most common tone system is based on equal temperament, in which the octave is divided into twelve equal semitones. This system allows modulation between different keys without noticeable detuning. However, the purity of certain intervals in this tuning leaves something to be desired. In addition, the absence of tones that can approximate the seventh and eleventh overtones can be experienced as problematic. Over time, various composers and theorists have therefore experimented with alternative tone systems. For example, the Dutch scientist Christiaan Huygens developed a 31-tone system, which was later applied by Adriaan Fokker in the 31-tone organ he built, and in the previous century the Salzburg composers Franz Richter Herf and Rolf Maedel advocated the 72-tone system with their Ekmelic Music. These systems strive for greater purity in intonation and offer new possibilities for microtonal music.
The concept of “tone system” is closely connected to terms such as “scale.” A scale is a succession of ascending or descending tones within an octave, such as the major or minor scale. A scale system, on the other hand, refers to the broader collection of all possible tones and the relationships between them. These systems determine the characteristic sound and harmony of the music and vary according to musical style and culture. Alongside the Western tone system, there are, after all, many other systems worldwide. In Indian music, for example, the concept of “that” is used, a system of scales that serves as the basis for ragas. Owing to the global diversity of tone systems, music possesses a rich palette of expressive possibilities that reflect various cultures.

The meantone tuning

The meantone tuning (or meantone temperament) is a system for tuning keyboard instruments that was particularly popular from the early 16th to the 18th century. It focuses on optimising major thirds, such as the interval C–E, which spans four semitones. In this system, keyboards are tuned so that the major third is evenly divided between the outer notes, for example the root and the fifth. This is achieved by slightly reducing the fifth, by about 5.38 cents, making it a little smaller than a natural fifth. When a sequence of four meantone fifths is tuned, such as C–G, G–d, d–a and a–e′, and the redundant octaves are removed, this results in a pure major third, such as c–e′.

Meantone tuning offered an alternative to just intonation, in which the correct tuning of all the intervals in the scale was obtained by various additions and subtractions of perfect natural fifths and thirds, in accordance with the fifths and thirds found in the natural harmonic series. This process resulted in whole tones of two different sizes. When an instrument tuned in C was played in G, the large and small whole tones appeared in the wrong order, making the instrument sound out of tune. Meantone tuning replaced this with a single, average whole tone.

Different combinations of meantone fifths were used to determine the correct tuning of each of the 12 notes per octave on the keyboard. This produced a strikingly pleasant sound for triads, the prevailing chord type consisting of a root, a third and a fifth, such as c–e–g. When tuning the black keys, however, notes such as F♯ and G♭, which share the same key, did not have the same pitch. A given black key could therefore only be used for one of the two possible notes, usually C♯, E♭, F♯, G♯ and B♭ (three sharps and two flats). If an instrument was played in a key that required an alternative note, for example A♭ instead of G♯, a strong dissonance arose, known as the “wolf,” because the combination of sounds was so out of tune that it howled like a wolf. This disadvantage led in the 18th century to the replacement of meantone temperament by equal temperament. Nevertheless, it remained in use in England until the mid-19th century and, for expressive reasons, experienced a modest revival among some composers in the late 20th and early 21st centuries. Below, the intervals between the notes in standard meantone tuning are shown schematically.

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The division of the larger diatonic and smaller chromatic semitone in standard meantone temperament, in which the semitones in a scale are not at equal distances from one another.

The calculation of meantone tuning can be illustrated by means of quarter-comma meantone tuning, a specific variant within this system. In this tuning, the perfect fifth is reduced by a quarter of a syntonic comma, resulting in a fifth of about 696.578 cents. By stacking a series of such diminished fifths, the scale can be constructed. The aim is to obtain pure major thirds with a frequency ratio of 5:4. Meantone tunings are built according to the same principle as Pythagorean tuning, by the succession of fifths, but temper the fifths in order to improve the purity of the thirds. The term “meantone” indicates that all whole-tone intervals are equal to one another and form a kind of average.

In practice, different variants of meantone tuning were used, depending on the extent to which the fifths were tempered in order to optimise the thirds. A well-known example is quarter-comma meantone tuning, but other forms were also developed to meet specific musical needs. Although meantone tuning had its limitations, such as the presence of the so-called “wolf fifth” and its unsuitability for certain keys, it played a crucial role in the development of Western music and harmony. It provided a system in which harmonies and cadences in different keys had a unique tonal colour, contributing to the expressiveness of the music of that period.

Pietro Aaron, an Italian music theorist of the early 16th century, is often mentioned as the first to describe meantone tuning, specifically aimed at musicians and instrument builders. In his work Toscanello in musica from 1523, he gave practical instructions for tuning instruments which indicated a system in which fifths were reduced in order to obtain pure thirds. Although Aaron did not explicitly use the term “meantone tuning,” he nevertheless laid the foundation for this system of tuning by describing techniques that later became characteristic of meantone tuning. His work marks a transitional period in music history, in which the focus shifted from purely melodic structures to harmonic principles, and his contributions significantly influenced the development of tuning systems in Western music during the Renaissance. Earlier indications of concepts resembling meantone tuning can be found in the works of music theorists such as Marchetto da Padova in the 14th century. Marchetto introduced ideas about small adjustments in intervals, which may be regarded as precursors to meantone tuning. Also in the late Middle Ages and early Renaissance there were other theorists who experimented with alternative tuning systems in order to make thirds more pure.

The 19-tone system

The 19-tone system is an equal tuning that divides the octave into 19 equal intervals of about 63.16 cents per step and offers an interesting alternative to the traditional 12-tone tuning. The 19-tone tuning, like the 31-tone tuning, is also a solution from the Renaissance to the problems of meantone tuning, in which the major thirds tune better and the fifths are slightly narrowed, and within which it is possible to modulate to all keys. As early as the 16th century, the Spanish music theorist and organist Francisco de Salinas (1513–1590) was one of the first theorists to lay the foundations for this type of tuning. In his work De Musica Libri Septem (1577), he described systems such as 1/3- and 1/4-comma meantone, which are closely related to the 19-tone system. Salinas also advocated in this work a tuning in which both chromatic and enharmonic notes could easily be played. He designed a keyboard with 19 keys per octave, enabling him to realise more complex melodies and harmonies without the “wolf fifth,” a frequently disturbing element in traditional meantone tuning. The system was also discussed by composers and theorists such as Quirinus van Blankenburg, Guillaume Costeley, Ferruccio Busoni, Joseph Yasser and Joel Mandelbaum. The composer Guillaume Costeley (1530/1531–1606) even used such a system as early as 1558 for a song. And organist and musicologist Joseph Yasser (1893–1981) pointed out that composers such as Alexander Scriabin may already have applied ideas related to the 19-tone system, and he suggested that a 19-tone instrument could be a natural evolution of musical instruments. Joel Mandelbaum particularly emphasised the unique sound and melodic power of the system, which makes it especially attractive for experimental music.

The tuning offers some remarkable advantages. Minor thirds sound extremely pure (with a negligible deviation of +0.15 cents in relation to the overtones), and major thirds are more harmonious than in 12-tone tuning (–7.36 cents as opposed to +13.69 cents). At the same time, the fifths have a less pure and waning character (–7.22 cents), which some composers actually regard as expressive. The system, however, eliminates the limitations of traditional meantone tuning and opens new possibilities for melodic expression. It also fits well with existing notational practices, which makes its application simpler. In this way, 19-tone music can easily be realised with standard accidentals, since in this system (as in many other tone systems with more than 12 tones) a distinction is made between, for example, C♯ and D♭. Below is an example of the notation: 


Nevertheless, the 19-tone system has some limitations for use in contemporary music. It cannot, for example, approximate certain overtones, such as the 7th and 11th (–21.46 cents and +17.10 cents, respectively). Other tunings, such as the 22- and 31-tone systems, are often experienced as more complete. Today, the 19-tone system is mainly applied for specific musical purposes, more as an idiosyncratic mode than as a replacement for existing tunings. And for those who are not looking for an accurate 7th and/or 11th overtone, the advantage of the 19-tone system is that with relatively few added tones it is possible to modulate to all keys within meantone tuning, which has the advantage of being physically easier to realise nowadays. Instruments such as specially adapted keyboards or retuned harpsichords make its use practically possible. Moreover, it enables composers to create rich, expressive harmonies that are often more difficult to achieve with 12 tones. The system can thus offer a balance between technical feasibility and artistic freedom.

 

The 24-tone system

The 24-tone system, also known as the quarter-tone system, is an equal tuning in which the octave is divided into 24 equal intervals, each interval being exactly 50 cents. This system therefore doubles the traditional 12-tone scale and introduces quarter-tones, which are found exactly between the usual semitones of 100 cents. It opens new musical possibilities with finer intervals and richer harmonies. Quarter-tones are often defined as half of a semitone or a quarter of a whole tone, and form an integral part of this system. The quarter-tone between, for example, A4 (440 Hz) and B♭4 (ca. 466 Hz) is A♯4 (440 × 1.02930 ≈ 453 Hz), lying at a distance of 50 cents between the first two notes. In this way, the 11th overtone can be approached almost perfectly, with a difference of only 1.32 cents. The equal quarter-tone tuning therefore has much in common with intervals based on the eleventh overtone of the harmonic series: 11:8 = 551 cents, 11:9 = 347 cents, 11:6 = 1049 cents and 12:11 = 151 cents. But intervals such as 7:6 and 11:10 also fall within the range. These intervals can make new chords possible. The third and the fifth, however, are exactly the same as in the 12-tone system.

The history of the quarter-tone began in the early nineteenth century, when the French composer Antonin Reicha (1770–1836) speculated about its use. The quarter-tone system only really took shape in the early 20th century. The German composer Richard H. Stein (1882–1942) published the first composition with quarter-tones in 1906, Zwei Konzertstücke für Violoncello und Klavier, Op. 26. He was followed by Arthur Lourie (1892–1966), who in 1915 composed his Prélude, Opus 12 for quarter-tone piano. Not much later, other pioneers such as Charles Ives (1874–1954) with Three Quarter-Tone Pieces, and Jörg Mager (1880–1939), Willi von Möllendorf (1872–1934), Julián Carrillo (1875–1965), Mildred Couper (1887–1974), Alois Hába (1893–1973) and Ivan Wyschnegradsky (1893–1979) further explored the 24-tone system. Hába was an important figure in Prague and even designed special instruments, such as quarter-tone pianos and harmoniums, to make the practical performance of quarter-tone music possible. He saw microtonal music as a natural evolution of Western harmonic theory by expanding the chromatic scale and broadening its expressive possibilities. Wyschnegradsky, who introduced the term “ultrachromatic” for systems such as the 24-tone system, worked from Paris and emphasised in his works the expressive possibilities of quarter-tones, particularly in dramatic passages. 
While composers such as Hába (in his string quartets and operas, such as Matka) and Wyschnegradsky (in, among others, his piano compositions) fully embraced the system, others used quarter-tones sporadically as a colour effect. Béla Bartók (1881–1945), for example, experimented with quarter-tones in his Sonata for Solo Violin, although in the final version these were largely removed. 
In Russia, Georgy Rimsky-Korsakov (1901–1965) played a prominent role in the development of quarter-tone music, with the founding of the Petrograd Society for Quarter-Tones in 1923. He introduced, among other things, a quarter-tone harmonium, and organised concerts to promote quarter-tone music. The Italian composer Giacinto Scelsi (1905–1988) also used the quarter-tones of the 24-tone system in almost all of his works from 1956 onwards. Other composers of the post-war period, such as the Hungarian-Austrian György Ligeti (1923–2006) and Polish Krzysztof Penderecki (1933–2020), usually employed quarter-tones as textural or harmonic additions. Ligeti’s Ramifications is a well-known example in which quarter-tones play a key role as clusters.

Quarter-tones also play a role in improvised music, such as jazz. Here, quarter-tones are often used as expressive effects, such as the “blue notes” of the blues, which connect with the quarter-tone concept. In Arabic music, Mikha’il Mishaqah in the 19th century formalised a system with 24 equal intervals per octave, in which quarter-tones played a central role in maqamat (modal systems), for example neutral intervals such as the three-quarter-tone. In world music, such as the Indian 22 Śrutis and sometimes the Indonesian gamelan, the use of quarter-tones and other microtonal approaches can also be found.

Quarter-tones are notated with specific symbols to distinguish them from traditional semitones. The simplest notation is as follows:


When music is less abstract and provided with concrete pitch relations, more accidentals are needed. In addition to the half-flat and half-sharp to indicate quarter-tone-lowered and raised notes, a three-quarter-flat/sharp (plus double flat/sharp) is also desirable. Below is the extended Tartini-Couper quarter-tone notation:

Traditional instruments were not designed for quarter-tones, which sometimes made special instruments necessary. Examples include the quarter-tone piano with double keyboards, guitars with modified frets, and the double horn, in which the two integrated horns can be tuned a quarter-tone apart. The 96-tone piano (Carrillo piano) is also capable of playing quarter-tones perfectly, since these tones form part of the much more extensive 96-tone system. 

The 24-tone system remains the most practical microtonal system for musicians and composers. With advances in digital technologies, such as synthesisers and music software, it has become easier than ever in the 21st century to explore and use quarter-tones. This technological development contributes to the accessibility and popularity of the 24-tone system. Although the quarter-tone system provides a solid basis for microtonal exploration, it is ultimately one of many systems within the broader context of microtonality. Critics in the microtonal music field sometimes regard the quarter-tone system as simply a doubled 12-tone system, and as a tuning with twice the same “bad” intervals. Nevertheless, the 24-tone system remains one of the most practical methods for composing and performing microtonal music, since on traditional instruments quarter-tones can often be executed with alternative fingerings.

 

The 31-tone system

The 31-tone system, an equal tuning based on meantone tuning, divides the octave (of 1200 cents) into 31 equal intervals of approximately 38.71 cents per step. These 31 intervals are one-fifth of a whole tone and are called ‘dieses’. The system is perhaps the best solution from the Renaissance and early Baroque for the problems of meantone tuning, because it allows modulation to all keys. This equal 31-tone tuning offers a refined approach to the 1/4-comma meantone tuning, because the pure fifth is tempered to 696.77 cents and is therefore only 0.19 cents wider than in 1/4-comma meantone tuning. This makes the 31-tone system excellently suited for the approximation of both meantone and septimal harmony (harmonies based on the 7-limit, such as the harmonic seventh chord), including intervals such as 7:6, 8:7 and 7:5, which in other tunings, such as the 12-tone system, are hardly or inaccurately represented. The system is also suitable for unconventional chords and intervals, such as the neutral third, which (like, for example, the harmonic seventh chord) cannot be performed in 12-tone equal tuning. Just as in meantone tuning, the major thirds are much better tuned than in 12-tone equal tuning. For example, this interval (with a deviation of only +0.79 cents) is 12.9 cents closer to the pure third in the 31-tone system than in the 12-tone system. The minor third is also, with a deviation of -5.96 cents, a substantial 9.68 cents better. The fifths, however, are slightly narrowed, so that these, with a deviation of -5.19 cents compared to the pure fifth in the overtones, are -3.23 cents smaller. As a result, in the 31-tone system one gains a great deal of third compared to the 12-tone system and loses a small amount of fifth. The harmonic seventh and eleventh overtone can also be approached almost perfectly and reasonably well respectively, which is entirely impossible in the 12-tone system. The harmonic seventh has a deviation of no more than -1.09 cents from the 7th overtone. The eleventh overtone can be approximated with an acceptable deviation of -9.38 cents.
Below is an overview of important intervals in the 31-tone system and their deviations compared to the pure overtones:

• Minor third (6:5): 309.68 cents (deviation: -5.96 cents) versus 315.64 cents
• Major third (5:4): 387.10 cents (deviation: +0.79 cents) versus 386.31 cents
• Pure fifth (3:2): 696.77 cents (deviation: -5.19 cents) versus 701.96 cents
• Harmonic seventh (7:4): 967.74 cents (deviation: -1.09 cents) versus 968.83 cents
• Harmonic eleventh (11:8): 541.94 cents (deviation: -9.38 cents) versus 551.32 cents

In the 31-tone system, uneven semitones arise because the whole tones are divided into five dieses. This means that one semitone has a size of 3/5 of a whole tone and the other semitone 2/5 of a whole tone. In this tuning system, the first-mentioned larger interval is called the diatonic semitone and the last-mentioned smaller interval the chromatic semitone. Due to the division of the whole tone into five equal parts in the 31-tone system, one can effectively speak of a five-tone system, instead of the quarter-tone system as a synonym for the 24-tone system. These five-tone notes, if divided over six conventional whole tones in the octave, would produce only 30 notes. However, in the 31-tone system six whole tones do not make an octave; an extra five-tone note is needed. In the division of the five-tone notes in this system, one starts from the diatonic notes (white keys on the keyboard), where the five whole tones are divided into five five-tone notes and the two semitones into three five-tone notes. Added together (5×5 and 2×3), this makes 31 notes in the octave. An important fact is that, due to the uneven division of the semitones, there is a five-tone difference between, for example, the notes C♯ and D♭, which are therefore not enharmonically equivalent.

The simplest notation of the 31-tone system can be realised with quarter-tone accidentals from the Tartini-Couper notation system, because the extra five-tone note within the whole tone arises as a result of the fact that, for example, the note G♯ and the note A♭ are different notes and are specifically notated as such. In the example below with 31 notes per octave, this is clearly visible:

A more advanced notation system combines part of the Tartini-Couper quarter-tone accidentals with other accidentals from Giuseppe Tartini (1692–1770) and Quirinus van Blankenburg (c. 1654–1739), who gave names to the 31 notes and devised some accidentals, such as for the half chromatic raising/lowering.
Below is an overview of the extended 31-tone notation with either only sharps or only flats, which can be combined in scores:

The 31-tone system was theoretically developed by the physicist Christiaan Huygens in the 17th century and practically implemented by Adriaan Fokker in the 20th century, notably via the renowned Fokker organ, built in 1950 and now managed by the Huygens-Fokker Foundation. However, the history of the 31-tone system begins already in the Renaissance of the 16th century, with roots that even go back to the ancient Greeks, such as Pythagoras (ca. 570 BCE – ca. 500 BCE) and Aristoxenus (ca. 360 BCE – ca. 300 BCE). They studied pitch and its ratios, which led to systems going beyond the diatonic scale. These ideas later inspired the Italian composer and music theorist Nicola Vicentino (1511–1576), who in Rome in 1555 proposed a more extensive meantone tuning with 31 notes per octave, inspired by his desire to reconstruct the enharmonic and chromatic tuning systems (such as the tetrachord) of the ancient Greeks. He argued this in his most famous work, L’antica musica ridotta alla moderna prattica (‘ancient music adapted to modern practice’), in which he fully set out his ideas to connect contemporary Renaissance works with ancient Greek music theory. This work is both a theoretical treatise on music and a collection of four musical examples with 31 notes in the octave illustrating Vicentino’s vision, of which the best-known part, Musica prisca caput, has been preserved in full. He also designed the archicembalo, a harpsichord with a series of extra keys (with the five black keys divided into four and two extra keys divided into two) to play five-tone notes. He also had an archiorgano built, a small organ with one stop based on a 31-tone tuning. Vicentino’s work and 31-tone music was revolutionary, but his instruments and theories found little practical application due to the technical limitations of his time. Nevertheless, the 31-tone tuning proposed by Nicola Vicentino was not entirely equal.

After the Italian naturalist and botanist Fabio Colonna (1567–1640) had also worked on 31 notes in the octave, it was the relatively unknown Italian mathematician, philosopher, astronomer and Hellenist Lemme Rossi (1601–1673) from Perugia who in 1666 was the first to publish a discussion of the 31-tone equal tuning. Independently, the famous Dutch mathematician, physicist, engineer, astronomer and inventor Christiaan Huygens (1629–1695), considered a key figure in the scientific revolution of the 17th century, presented a more practical approach to the 31-tone system 25 years later. He also proposed a tuning in which an octave is divided into 31 equal steps and discovered through the use of logarithms that the 31-tone equal tuning could well approximate the 1/4-comma meantone tuning. Christiaan Huygens described these ideas in his Lettre touchant le cycle harmonique (Rotterdam, 1691) and in Novus cyclus harmonicus (Leiden, 1724). Already in 1661 he had made notes in which he, among other things, demonstrated the close connection between meantone tuning and 31-tone tuning, and recognised the possible consonant nature of septimal intervals such as 4:7 and 5:7, which according to Huygens could be approximated in meantone tuning and in 31-tone tuning. Huygens also designed a (probably never realised) keyed instrument with 31 strings per octave, where a keyboard with twelve normal keys per octave could be moved above 31 strings by sliding the keyboard. Against the background of the emerging 12-tone equal tuning, Huygens considered the system with 31 equal intervals a full-fledged alternative to be able to modulate to all keys and yet retain the qualities of meantone tuning.

In the twentieth century, the Dutch physicist Professor Adriaan Fokker (1887–1972) brought the 31-tone theories of Christiaan Huygens to life by designing and realising a 31-tone organ (the Fokker organ) and, as a result,initiating the completion of the translation for the remaining portion of your text to maintain continuity.bringing about a true 31-tone movement in the Netherlands. He conducted research into musical tuning systems and published various articles on the subject. He also liaised with numerous scientists and inspired many composers at home and abroad, such as Ivan Wyschnegradsky, Henk Badings, Alois Hába, Hans Kox, Joel Mandelbaum, among others. He regularly gave lectures on topics including the 31-tone system, during which he, whenever possible, demonstrated on the Fokker organ.
This 31-tone organ, which resided for fifty years in the Teylers Museum in Haarlem and has been housed since 2008 in the Muziekgebouw aan ’t IJ in Amsterdam, is a technological masterpiece. It was specifically built to exploit the possibilities of the 31-tone tuning and enables the performance of both traditional and experimental compositions. The instrument, built in 1950, became one of the most important instruments for microtonal music in general. The organ features a 31-tone keyboard with unique keys in black, white, and baby blue, providing access to all 31 tones. Additionally, the 31-tone organ can also be played with 12-tone keyboards and, following a large-scale renovation in 2008/2009, includes advanced MIDI functionality. Consequently, the Fokker organ was also one of the first hyperorgans in the world, as it can be fully controlled by computers. This instrument is regularly played during concerts, presentations, demonstrations, and educational projects organised by the Huygens-Fokker Foundation.
Besides the Fokker organ, other 31-tone instruments have been designed, such as the Archifoon, an analogue electronic instrument from 1970 with a keyboard somewhat similar to that of the 31-tone organ. Additionally, the 31-tone guitar, with 19 extra frets in the octave, is a microtonal instrument that can be relatively easily realised by modifying a standard guitar. Furthermore, digital synthesizers and music software nowadays provide an important aid for composers of contemporary microtonal music and can be easily adapted to the 31-tone system.

In all these ways, the 31-tone system can be used for both historical and innovative musical applications, because it combines the pure thirds of meantone tuning with the freedom to modulate. Thus, the system allows music from the Renaissance and (early) Baroque, originally written in meantone tuning, to be performed authentically. Moreover, twentieth-century composers such as Henk Badings and Joel Mandelbaum embraced the system for new compositions, creating unique sonic worlds beyond the possibilities of the 12-tone system. In the twenty-first century, the 31-tone system has, due to the advent of software and new technological possibilities, experienced an enormous surge even more than other microtonal systems, as many composers have found in it a means to provide their own musical language with greater expressivity.

The 41-tone system

The 41-tone system, also known as 41-tone equal tuning, is a Pythagorean tuning that divides the octave into 41 equal steps of approximately 29.27 cents each, meaning that each step corresponds to the 41st root of 2 in frequency ratio. This provides a very fine subdivision of the octave and more consonant sounds, as the system approximates certain pure intervals more accurately than the 12-tone system. For example, the perfect fifth is almost perfectly pure, with a deviation of +0.48 cents (thus the perfect fourth −0.48 cents). The pure major third can be found in the 41-tone system with an acceptable deviation of −5.83 cents. The best minor third has a deviation of +6.31 cents compared to the pure minor third in the overtones. Furthermore, the system contains a very satisfactory harmonic seventh (−2.97 cents deviation) and a functional eleventh overtone (+4.78 cents deviation). The distribution of the 41 steps across the seven diatonic notes in the octave is as follows: the five whole tones are divided into seven steps and the two semitones into three steps.

Notably, the 41-tone system also contains the Bohlen-Pierce scale, since five steps of 29.26 (one step being 29.268 cents) from this system fill one “macrotonal” step of 146.3 cents in the Bohlen-Pierce scale, with a deviation of +0.04 cents. In the 41-tone system, 40 steps effectively complete the slightly smaller and impure octave (−30 cents) of the non-octave BP scale.

Regarding the notation of the 41-tone system, it is better to avoid a problematic extended Pythagorean notation and instead use a system with conventional accidentals and arrows pointing up and down, placed before the notes and any accidentals. One arrow (or half-arrow) raises or lowers the note by one of the 41 steps, and two arrows by two steps. Unlike in the 31-tone system, for example, A♭ is considered one step lower than G♯, whereas in meantone tuning it is the opposite. It is also important to note that, for example, the major third C–E is a Pythagorean major third (with a +1.94 cents deviation), meaning a lowering accidental must be placed on the note E to hear the pure third. Below is an overview of the simplest notation for the 41-tone system:

Since the 41-tone equal tuning is not a meantone tuning, it distinguishes, unlike the 31-tone system, between classical and Pythagorean major thirds, together with the intervals 10:9 and 9:8. It is more accurate in the 13-limit than the 31-tone equal system. Some intervals (commas and diesis) are not tempered in the 41-tone system, such as the syntonic comma (81:80), which can make modulation in this system more difficult than in the 31-tone system.
Adriaan Fokker once wrote that if the appreciation of the eleventh overtone were to increase in the future, the 41-tone system could be an interesting alternative to the 31-tone system (with an improvement of 3.55 cents). The Dutchman Dirk de Klerk from Leiden wrote in 1979, based on the book Musikalische Tonsysteme (1927) by E.M. von Hornbostel, that the Hungarian pianist Paul von Jankó (1856–1919), also inventor of the Jankó keyboard (1882), had already developed a system with 41 intervals in the octave in 1906.
With the constant development of new instruments and (computer) technologies in the 21st century, it has become increasingly simple to explore the 41-tone system and integrate it into contemporary musical practice.

The 48-tone system

The 48-tone system is an equal tuning that divides an octave into 48 equal steps of 25 cents each. In this case, one speaks of eighth-tones, and sometimes of the eighth-tone system. Because 48 is a multiple of 12, the 48-tone system contains all the intervals of the traditional 12-tone system, but provides additional intermediate notes, such as quarter-tones and eighth-tones. This allows more precise approximations of certain intervals. For the fifth (3:2), this is not necessary, as it is equally well represented in the 48-tone system as in the 12-tone system, with a deviation of −1.96 cents. The major third (5:4), which in this system can measure 375 or 400 cents, is respectively −11.31 cents too low or +13.69 cents too high compared to the pure major third (386.31 cents). The minor third (6:5), which can measure 300 or 325 cents, is respectively −15.64 cents too low or +9.36 cents too high compared to the pure minor third. For both thirds, it is not entirely clear which variant is better. The harmonic seventh (7:4), on the other hand, can be approximated very well in the 48-tone system, with a deviation of +6.17 cents, and represents an improvement over the 24-tone system, in which the quarter-tone with the best approximation is −18.83 cents too low. The eleventh overtone is equally remarkable in both systems, being only −1.32 cents too low. Therefore, they share many similarities with intervals based on the eleventh overtone (11:8) of the overtone series. Nevertheless, for the new generation of composers and music theorists, in their search for purer intervals, these approximations are often no longer sufficiently accurate. As a result, the eighth-tone system currently attracts less interest than it did in the 20th century.

Despite this, important composers in the previous century made use of eighth-tones. Gérard Grisey (1946–1998), for example, employed microtonal systems with eighth-tones in works such as Partiels (1975, from Les Espaces Acoustiques) to model the sound spectrum and overtones of instruments in detail. Tristan Murail (1947) composed works such as Ethers (1978), in which quarter- and eighth-tones are used to create complex layers of sound. The spectral composers often worked with electronic analysis of sounds and used microtones to imitate physical acoustic phenomena in acoustic music. Earlier, Alois Hába and Julián Carrillo had already explored the use of eighth-tones. In the 20th century, it was a logical step for avant-garde music to subdivide the twelve conventional tempered tones into halves and quarters. With the 21st-century reassessment of consonant music, however, many composers have focused on systems with more pure tones, such as the 31-, 53-, and 72-tone tunings. The 48-tone system remains relevant in the context of MIDI, where a step in 48edo is known as a ‘doamu’ (second MIDI-resolution unit, 2mu), corresponding to four equal divisions of a semitone in the 12edo system.

In practical music-making, the eighth-tone is often regarded as a subdivision of the quarter-tone, from which performing musicians usually derive the eighth-tone. The 96-tone piano (Carrillo piano) is among the specific microtonal instruments suitable for the precise execution of eighth-tones, since half of its 96 tones belong to the eighth-tone system. On this piano, the eighth-tones can be played chromatically using a whole-tone scale. There are two variants of this. In the notation of the 48-tone system, special accidentals can be used to indicate the extra microtones. Because all quarter-tones occur in the 48-tone system, and this system is also part of the 96-tone system, a notation extending quarter-tone accidentals is logical. This allows the 96-tone notation to be used for writing eighth-tones. This corresponds to the standard quarter-tone notation plus the addition of double arrows ‘up and down’ from the Sagittal Evo notation, combined with the quarter-tone signs of the Tartini-Couper notation. Below is an overview of the notation with sharps and flats between the semitone and the whole tone:

A practical (Tartini-Couper-Sagittal) 48-tone notation, based on the 96-tone notation, showing the division of the semitone and the whole tone

In addition to this method of notation, a notation using Tartini-Couper quarter-tone signs in combination with associated arrows is also a proven option for the 48-tone system, should a composition not be intended for the 96-tone piano. This does not correspond to the proposed notation of the 96-tone system, but it is a notation that is reasonably self-explanatory and easy to learn if one is already familiar with reading the quarter-tone signs in use. Below is an overview of the notation with sharps and flats between the semitone and the whole tone:

A suitable alternative for notating music in the 48-tone system, showing the division of the semitone and the whole tone

  • Read more about the 96-toonssysteem, of which the 48-tone system is a part.
  • For more information on the specifications of the 48-tone system, see xen.wiki.

 

The 53-tone system

The 53-tone system is an equal tuning in which an octave is divided into 53 equal steps of approximately 22.64 cents. This system has the characteristic that 53 perfect fifths are almost equal to 31 octaves, with a small discrepancy known as Mercator’s comma. This makes the system particularly suitable for very accurately approximating traditional ratios from just intonation, such as the 5-limit ratios. Because a distance of 31 steps in this scale is almost exactly equal to a pure fifth, it can in theory be regarded as a lightly tempered form of the Pythagorean tuning extended to 53 tones. The diatonic notes in the 53-tone system are distributed as follows: the whole tones are divided into 9 steps and the diatonic semitones (e-f and b-c) into 4 steps. The chromatic semitone (for example c-c♯) then consists of 5 steps.

The special feature of this tuning system is that intervals can be approximated very precisely. For example, the fifth, measuring 701.89 cents, is perfectly pure (with a negligible deviation of −0.07 cents). The major third, at 384.91 cents, is also nearly pure (with a deviation of −1.40 cents). Likewise, the minor third, at 316.98 cents, has a deviation of +1.34 cents. Less conventional but prominent intervals, such as the harmonic seventh (973.59 cents with a deviation of +4.76 cents) and the harmonic eleventh (543.40 cents with a deviation of −7.92 cents), can also be approximated reasonably well. This makes the system, on average, very accurate. However, this requires 53 tones per octave, which renders it somewhat less practical.
To notate the 53-tone system, the 53-EDO Evo Sagittal notation can be used, which also incorporates traditional accidentals:

Historically, this system was already proposed in antiquity by the Chinese mathematician Jing Fang (78–37 BCE) and later independently rediscovered by the German-English mathematician Nicholas Mercator (c.1620–1687) in the 17th century. After Mercator, William Holder (1616–1698) published a treatise in 1694 pointing out that 53-tone equal tuning is also very close to the pure major third (within 1.4 cents), and that consequently the 53-tone equal tuning can accommodate the intervals of 5-limit just intonation very well. Furthermore, unpublished manuscripts of Isaac Newton (1643–1727) from 1664–1665 indicate that he was already aware of the favourable properties of this system. He appears to have realised that the 53-tone tuning not only perfectly aligns with the physical intervals of fifths but also provides an excellent approximation for the major third. Later, the British scientist Robert Holford Macdowall Bosanquet (1841–1912) developed a special keyboard, the ‘generalized keyboard’, suitable for playing in the 53-tone system. This system allows complex harmonies to be realised that are not possible on traditional 12-tone systems. In modern times, it was further analysed by the Japanese musicologist Shōhei Tanaka (1862–1945), who discovered that the system eliminates both the ‘schisma’ and the ‘kleisma’ (subtle interval differences), making it perfectly aligned with the principles of 5-limit tuning. The Russian composer Leonid Sabaneyev (1881–1968) proposed the 53-tone equal tuning because it approximates just intonation most closely while also allowing modulation to all other keys.

Today, the Dutch-Canadian music theorist and keyboard designer Siemen Terpstra (1948) is often associated with the 53-tone tuning. His special focus on this system, the 31-tone system, and other tuning systems led, after in-depth research into the mathematical structures of various tuning systems, to the concepts of innovative musical instruments and notation systems that simplify the playing and understanding of microtonal systems. One of his most remarkable creations is the ‘Terpstra Keyboard’, an ergonomically designed musical instrument optimised for performing microtonal music and in particular 53-tone music. This keyboard integrates important historical breakthroughs in its design and offers musicians the ability to perform within tuning systems with greater flexibility and precision. It was through his ideas that, in the 2010s in Canada, the production of the ‘Lumatone Keyboard’ began, based on Terpstra’s important design. With this instrument, a master keyboard for microtonal music has, for the first time, become a standard, allowing musicians to perform in the 53-tone system on a global scale. 

 

The 72-tone system

The 72-tone system, also known as the 72-tone equal tuning, divides the octave into 72 equal intervals. Each interval in this system measures approximately 16.67 cents, allowing the natural overtones to be approximated many times more accurately than with the 12-tone division used in standard Western music. From a scientific perspective, the 72-tone system is therefore remarkable because of its mathematical structure. Embedded within the system are the 36- and 18-tone systems, corresponding to sixth- and third-tones, as well as the quarter tones of the 24-tone system, each further divided into three small steps of 1/72 tone. This microtonal system is often used in experimental music because of its versatility and its ability to precisely render both the septimal minor third (7:6 ratio with a deviation of less than 1 cent) and quarter-tone intervals, while also improving the accuracy of traditional harmonies and other tuning theories developed by composers such as Alois Hába and Harry Partch. The fifth, as in the 12-tone system, is fairly pure at 700 cents (with a deviation of −1.96 cents). The major third, at 383.33 cents, is also nearly pure (with a deviation of −2.98 cents). Even more so, the minor third at 316.67 cents has a deviation of +1.03 cents. Less conventional intervals, such as the harmonic seventh (966.67 cents, deviation −2.16 cents) and the harmonic eleventh (550 cents, deviation −1.32 cents), can also be approached very accurately, making this tuning system, on average, very precise. However, this requires 72 tones per octave, which renders the system somewhat less practical. The 72-tone tuning is sometimes compared with the 31-tone system due to its versatility in realising both traditional and non-traditional harmonies.

In practical terms, using the 72-tone system often requires customised instruments or electronic synthesizers. Some quarter-tone instruments and microtonal keyboards can execute intervals within this system, but often not all of them. Thanks to modern technology, however, it has become easier to apply the complex harmonic structures of the 72-tone system in compositions and performances, enabling new musical expressions.

Regarding the notation of the 72-tone system, various initiatives have been undertaken. Two standard notations were developed independently in Austria and the United States, where notable 72-tone movements took place. The Austrian movement for Ekmelische Musik originally used a notation with arrows above the notes, while standard accidentals were still placed in front of the notes. The arrows indicate deviations from the 12-tone equal tuning. In this ekmelic notation, a single arrow and a double arrow have the opposite function compared with similar Sagittal symbols in the notation of the 96-tone system. The double arrow indicates the smallest raising, comparable to a flag of a sixteenth note, whereas the Sagittal double arrow represents a double raising. This can lead to confusion. In Sims’ notation, no double arrow is used, but a regular arrow and its derivatives are placed in front of the note. The differences between the ekmelic notation of the 72-tone system and Ezra Sims’ notation are shown in the overview below.

The 72-tone system has historical roots in both theory and practice. It is an extension of ideas about microtonality, first explored in the 16th century. In the 20th century, the system was further developed and promoted by composers such as Franz Richter Herf, Joe Maneri, and Ezra Sims, who were interested in expanding the possibilities of the conventional tonal system. In Boston, United States, jazz saxophonist, clarinettist, pianist, and composer Joe Maneri (1927–2009) was known for his pioneering work in microtonal music, including as founder of the Boston Microtonal Society in 1988. He developed a system based on the 72-tone equal tuning and was influenced by his study of Turkish and Albanian music, which incorporated quarter tones and other microtonal intervals. He designed a customised keyboard with 72 keys per octave to play these intervals precisely. In addition, he taught microtonal music at the New England Conservatory in Boston, training students to compose and improvise within the 72-tone system. He also presented his ideas at the Mozarteum in Salzburg and at Harvard University near Boston. The American composer Ezra Sims (1928–2015) was also a pioneer, beginning in 1960 to compose microtonal pieces. Later, he developed a system of asymmetrical modes with 18 tones per octave, based on a division of the octave into 72 equal parts, with which he composed exclusively from 1971 onwards. This system allowed him to use precise approximations of overtone intervals and explore new expressive possibilities. Sims also designed a notation system for this microtonal music, later adopted by other composers, including Joseph Maneri. Like Maneri, Sims lived and worked in the Boston area.

In Salzburg, Austria, the Ekmelische Musik movement was founded by composers Franz Richter Herf (1920–1989) and Rolf Maedel (1917–2000), who together developed an electronic fine-tuning organ in 1974 at the Mozarteum in Salzburg, where Richter Herf served as rector from 1979 to 1983. The term ‘ekmelic’ derives from the Greek ekmelēs, meaning ‘impure’ or ‘non-harmonic’, and refers to music that uses microtonal systems beyond the traditional 12-tone tuning. The movement focused on exploring and composing music in the 72-tone system to discover new expressive possibilities. Richter Herf and Maedel published works and organised events to promote the theory and practice of Ekmelische Musik, including the international symposium Mikrotöne in 1985. Later, the composer and musician Johannes Kotschy (1949) played a prominent role within the ekmelic movement in Salzburg. Under his leadership, a new Ekmelic Organ was built in 1995 by organ builder Walter Senn, commissioned by the International Ekmelic Music Society. This organ represented a further development and refinement of the original concept and was presented at the 1st International Symposium Mikrotöne in Salzburg in 1995. To this day, the ekmelic movement continues to organise concerts and symposia.

Both Maneri and Sims, as well as the Ekmelische Musik movement, contributed to the development of the 72-tone system, albeit in different musical contexts and geographical locations. Their work paved the way for further exploration of microtonality in both jazz and contemporary classical music.

 

The 96-tone system

The 96-tone system, a microtonal tuning that divides the octave into 96 equal steps, originates from experiments with quarter tones and their further subdivisions. This system divides each semitone into eight equal steps and each whole tone into sixteen steps, resulting in intervals of 1/16 tone, making it possible to create extremely subtle intervals. Dividing an octave into 96 equal parts means that each step measures 12.5 cents (a cent being 1/100 of a semitone in the 12-tone scale). Compared with the standard 12-tone system, in which each step is 100 cents, the 96-tone system offers new possibilities for musical expression. In this system, various intervals are considered particularly interesting because of their closeness to pure harmonic ratios. The minor third (6:5) and major third (5:4) are approximated very accurately, with deviations of −3.14 and +1.19 cents respectively. The fifth (3:2) is as precise as in the 12-tone system, with a deviation of −1.96 cents. The harmonic seventh (7:4) can also be approximated reasonably well (+6.17 cents deviation). Ratios such as the septimal major third (9:7) or the harmonic eleventh (11:8), which cannot be rendered in 12-tone equal tuning, are extremely precise in the 96-tone system, with deviations of +2.42 and −1.32 cents respectively. This opens the door to the use of intervals rarely audible in standard music. The system also enables other microtonal intervals, such as the quarter tone (50 cents) and eighth tone (25 cents), as the 24- and 48-tone systems are embedded within the 96-tone system. Of course, the smallest interval, the sixteenth tone (12.5 cents), can also be sounded, producing an ultrachromatic scale with a highly mysterious timbre.

The system is often associated with the Mexican composer, violinist, and theorist Julián Carrillo (1875–1965), a pioneer in microtonal music. He introduced the concept of microtonal music in his work El Sonido 13 (The Thirteenth Tone) and developed specialised instruments, such as modified string instruments (Carrillo harp) and pianos, to realise his ideas. In the 1990s, the German piano builder Sauter once again constructed a limited series of these 1/16-tone pianos, specifically for the performance of Carrillo’s music. The Huygens-Fokker Foundation has been the proud owner of one of these pianos since 2011. This has led to a modest 96-tone movement centred around the instrument. Carrillo’s work inspired later composers and theorists, although practical use of the 96-tone system often remained limited due to the complexity of the instrumentation.

Regarding notation for the 96-tone system, several initiatives have been undertaken. Julián Carrillo himself developed a unique numeric notation for his 96-tone system to record microtonal intervals precisely. In this notation, traditional notes are replaced by numbers indicating the pitch, with each number corresponding to a specific frequency in the 96-tone system. This approach was intended to allow musicians to perform Carrillo’s compositions exactly as written. In practice, however, these scores proved not to be easy to read. The parts for the 96-tone piano (Carrillo piano) with 97 keys are typically notated in the same way as conventional piano parts. This means that the full range of one octave (from c′ to c″) on this piano is divided and notated across eight octaves, so in principle any sufficiently trained pianist can perform the parts. Notating the sounding pitches in the score, however, requires a different approach. Since the 96-tone system is a further subdivision of the quarter tones of the 24-tone system, it is logical to use the corresponding accidentals as a basis; in this case, the accidentals of the Tartini-Couper notation system. These pitches and accidentals can then be further subdivided using half-arrows and double-arrows from Sagittal notation systems. An example of the most practical 96-tone notation is shown below: 

The most practical (Tartini-Couper-Sagittal) 96-tone notation proposed by the Huygens-Fokker Foundation, with the division of semitone and whole tone

The Bohlen-Pierce scale

The Bohlen-Pierce scale (or BP scale) is an alternative musical tuning that diverges from the traditional octave system by using the ‘tritave’, a frequency ratio of 3:1, as the fundamental interval instead of the octave ratio of 2:1. This means that a note with a frequency three times higher than another is considered equivalent, rather than twice as high, as in the octave system. The scale divides this tritave into 13 equal steps, resulting in a 13-tone system that evenly distributes the tritave (13-EDT). Each step in this scale represents a frequency ratio of 3^(1/13), equivalent to approximately 146.3 cents per step. This contrasts with the traditional 12-tone equal tuning system, where each step (semitone) measures roughly 100 cents.
A distinctive feature of the Bohlen-Pierce scale is its emphasis on odd harmonics, such as the 3rd, 5th, and 7th overtones, while even overtones are avoided. This produces a timbre that differs from that of the traditional octave system, where both even and odd harmonics are present. The absence of octaves and the focus on odd harmonics give the Bohlen-Pierce scale its unique sonic character.
There are two main variants of the Bohlen-Pierce scale: the equal-tempered version (13-EDT) and the just intonation version. In the equal-tempered version, the 13 steps are evenly distributed across the tritave, whereas in the just intonation version, specific frequency ratios based on simple fractions without factors of 2 are used, resulting in a scale without octaves and without even overtones. Remarkably, the 41-tone system also contains the Bohlen-Pierce scale, because five steps of 29.26 cents each (one step being 29.268 cents) fill one ‘macrotonal’ step of 146.3 cents in the BP scale, with a deviation of only +0.04 cents. These two tuning systems are therefore highly compatible.

Regarding notation for the BP scale, several proposals have been made over time. Manuel Op de Coul and Georg Hajdu independently developed notations for the system. An overview of the sounding pitches (in cents) of the BP scale (13-EDT) is shown below:

The Bohlen-Pierce scale was discovered and developed by several researchers, namely Heinz Bohlen, John Pierce, and Kees van Prooijen, each of whom independently contributed to the formulation and popularisation of this system, leading to renewed interest in alternative tuning systems. The development of the Bohlen-Pierce scale reflects a collective search for alternative musical expression, free from the constraints of the traditional octave system.
Heinz Bohlen (1935–2016) was a German engineer specialised in microwave electronics and communication technology. In the early 1970s, he became interested in alternative tuning systems after being involved in recording concerts at the Hochschule für Musik und Theater in Hamburg. His curiosity about why music exclusively used the 12-tone equal tuning led him to explore other possibilities. In 1972, he introduced the Bohlen-Pierce scale, based on the tritave instead of the octave. Since 2008, the Huygens-Fokker Foundation has managed Bohlen’s BP website and holds his physical archive of original articles and writings.
John R. Pierce (1910–2002) was an American engineer and author, best known for his work in electronics and communications. In 1984, he independently discovered the Bohlen-Pierce scale. When publishing his findings, together with colleagues such as Max Mathews, he acknowledged Bohlen’s earlier work and named the scale the ‘Bohlen-Pierce scale’. Kees van Prooijen (1952), a Dutch software engineer, artist, and microtonal theorist, independently discovered the same scale in 1978. He is also known for his contributions to microtonal theory, including the development of the ‘Kees height’, a measure of the complexity of pitch classes in just intonation.

In practice, the Bohlen-Pierce scale has been applied to various musical instruments, such as adapted guitars, clarinets, and electronic instruments, designed or modified to accommodate the specific requirements of this tuning. For instance, the ‘Stredici’ was developed, a string instrument designed by David Lieberman. There is also the Bohlen-Pierce clarinet, played by pioneering musicians such as Nora-Louise Müller and Ákos Hoffman. Additionally, Elaine Walker developed adapted MIDI keyboards for the Bohlen-Pierce scale and, as an electronic musician and microtonal composer, has created various works in this scale, following in the footsteps of her inspiration, Richard Boulanger. Composer Georg Hajdu has also applied the possibilities of the Bohlen-Pierce scale in his music. Other composers include Curtis Roads, Juan Reyes, Ami Radunskaya, Manfred Stahnke, and Charles Carpenter.

Understanding and applying the Bohlen-Pierce scale requires a shift from traditional thinking about music and harmony, as the system fundamentally differs from the conventional octave-based system. For professional musicians, however, this non-octave-based musical system provides a challenging and enriching alternative for developing innovative musical ideas.

  • Read more about the Bohlen-Pierce scale on the BP-website of Heinz Bohlen, facilitated by the Huygens-Fokker Foundation.
  • For more detailed information on Bohlen-Pierce scale, see wiki and xen.wiki.

Other interesting tuning systems

• The 5-tone system
The 5-tone system is a 5-tone equal tuning that divides the octave into five equal parts of 240 cents each. This means that each step corresponds to a frequency ratio of the fifth root of 2. The fifth measures 720 cents, which deviates by +18.04 cents from the pure just fifth. A major third is not present in the system. However, the first step comes close to the septimal minor third (−26.87 cents deviation) or the septimal whole tone (+8.83 cents deviation). Furthermore, there is a diminished fourth in the system with a deviation of −18.04 cents. The fifth tone in the system approximates the harmonic seventh reasonably well, with a deviation of −8.83 cents. The 5-tone system is remarkable because it is the smallest equal tuning that contains microtonal intervals (2-, 3- and 4-tone equal tunings are in fact contained within the 12-tone system). According to the renowned Dutch ethnomusicologist Jaap Kunst (1891–1960), Indonesian gamelans are tuned in five equal parts within the octave. However, their tuning often varies considerably and may even contain stretched octaves. Only the Indonesian ‘slendro’ somewhat resembles a 5-tone equal tuning. It is in fact true that the builders of gamelan instruments distinguish themselves with their own timbre. By this is often not meant a different sound, but rather a slightly different tuning, which therefore cannot be neatly categorised. The floating sound of, among others, gongs, xylophone- and bell-like instruments makes the precise tuning of the notes in the gamelan ensemble of secondary importance.
At the World Exhibition of 1889 in Paris, Claude Debussy was introduced to gamelan music from Java, which contained non-Western pentatonic scales. Inspired by these sounds, he developed the whole-tone scale, a 6-tone system within the 12-tone equal tuning, in which all notes are equally spaced (200 cents). This gave rise to a similar directionless sound that is characteristic of his impressionist style. (More specifications on xen.wiki)

• The 17 and 34-tone system
The 17-tone system, also known as 17-tone equal tuning, divides the octave into 17 equal steps, with each step having a frequency ratio of the 17th root of 2, amounting to approximately 70.6 cents. This system was theoretically described in the 13th century by the Middle Eastern musician Safi al-Din Urmawi, who developed a system of seventeen tones to describe Arabic and Persian music, although these tones were not distributed equally across the octave. In modern times, the American composer Easley Blackwood Jr. (1933–2023) proposed a simple notation system for the 17-tone system, in which the enharmonic equivalents differ from those in 12-tone tuning, since, for example, the note F sharp and the note G flat are not identical. The same is true of these notes in other systems such as the 31-tone system. What is different, however, is that in the 17-tone system G flat is lower than F sharp. This results visually in a divergent chromatic scale. An important aspect of the 17-tone system is how the various just intervals are approximated. Thus, the minor third (6:5), with a deviation of −33.29 cents, is far too small, the major third (5:4), with a deviation of −33.37 cents, is also much too small, the perfect fifth (3:2), with a deviation of only +3.93 cents, is strikingly good, the harmonic seventh (7:4), with a deviation of +19.41 cents, is unusably large, and the eleventh harmonic (11:8), with a deviation of +13.39 cents, lies at the boundary of usability.
The 17-tone system is also part of the 34-tone system, which divides the octave into 34 equal steps, each step equal to the 34th root of 2, or about 35.29 cents. In contrast to other tunings such as 19, 31 or 53 tones per octave, which stem from older musical theories, the 34-tone division did not arise ‘naturally’ from earlier music theory. The first recognition of the potential of the 34-tone system appeared in a 1979 article entitled Equal Temperament by the Dutch music theorist Dirk de Klerk. The approximations of the harmonics in the 34-tone system are much more precise than in the 17-tone system, although 17 additional tones are required. Thus, the minor third (6:5) has a deviation of only +2.01 cents, the major third (5:4) a deviation of no more than +1.92 cents, the perfect fifth (3:2) a deviation of 3.93 cents, the harmonic seventh (7:4) a deviation of −15.89 cents, and the eleventh harmonic (11:8) a deviation of +13.39 cents. Taken together, these are fine specifications for consonant music in the 5-limit. (More specifications on wiki/xen.wiki)

• The 22-tone system
The 22-tone system, also known as 22-tone equal tuning (22-TET), divides the octave into 22 equal steps of approximately 54.55 cents each. This system was introduced in the 19th century by the English music theorist R.H.M. Bosanquet (1841–1912), who was inspired by the 22-tone division in Indian music theory and observed that an equal division into 22 steps per octave could represent music within the 5-limit with reasonable accuracy. In the 22-tone system, the minor third (6:5) and the major third (5:4) deviate respectively by +11.63 and −4.49 cents from their just counterparts, while the perfect fifth (3:2) is about +7.14 cents sharper. The harmonic seventh (7:4) is approximated fairly well with +12.99 cents, and the eleventh harmonic (11:8) is quite precise with −5.87 cents. A remarkable feature of this system is that it does not temper the syntonic comma (81:80), but rather enlarges it to one step, thereby preserving the distinction between major and minor whole tones. There are several notations for this tone system. The 22-tone system offers the possibility of exploring new musical territories while preserving good approximations of consonances from common practice. (More specifications on wiki/xen.wiki

• The 26-tone system
The 26-tone system is an equal tuning that divides the octave into 26 equal steps of about 46.15 cents each. An important property of the system is the accurate approximation of the harmonic seventh (7:4), with a negligible deviation of +0.4 cents. The eleventh harmonic is also good, with a deviation of +2.53 cents. Although it is a meantone tuning, it is otherwise a very flat tuning, because the syntonic comma (81/80) is tempered in the 5-limit. As a result, four stacked perfect fifths, each with a deviation of −9.65 cents, form a major third that is about 17 cents too low (equivalent to the neutral third of 11:9). The system can also be seen as two parallel 13-tone systems and offers two types of minor thirds and two minor sixths. The Dutch violinist, music theorist and microtonal thinker Leo de Vries (1924–2018), who wrote many treatises, came to the unexpected conclusion, after a lifetime of microtonal research and shortly before his death at the age of 94, that the 26-tone system was of unique quality. (More specifications on xen.wiki)

• The 36- and 18-tone system
The 36-tone system is an equal tuning in which each whole tone is divided into six steps of 33.33 cents, giving rise to sixth-tones and even to the sixth-tone system. The 36-tone system is a doubling of the 18-tone system with third-tones. The need for 36 tones arose in the 20th century partly from dissatisfaction with this equal 18-tone tuning, because this system lacked the possibilities to play classical semitones, as each whole tone is divided into three. The result was that the sound of third-tones was experienced as thin and meagre, and only partially conventional sounds based on 12-tone equal tuning could be performed with it. Thus, the fourth and the fifth in the 18-tone system, with a deviation of 31.38 cents, are in fact unusable within consonant harmonies. Ferruccio Busoni (1866–1924) was already experimenting in Berlin in the first decades of the 20th century with third-tones and sixth-tones, as evidenced clearly by his treatise Sketch of a New Aesthetic of Music. Around the same time, the Czech Alois Hába (1893–1973), encouraged by Busoni, also began to concern himself with the division of the tones into three. Both Busoni and Hába had a keyboard instrument built in the 1920s that could be played in this tuning. Hába, however, was able to put his ideas truly into practice. In addition to three string quartets, a violin duo and two solo pieces for violin and cello respectively in sixth-tones, he also composed the opera Přijď královtví Tvé (Zukomme uns den Reich) in seven acts from 1942 in sixth-tones. He also wrote in his book Neue Harmonielehre about third- and sixth-tones, and even about the twelfth-tones of the 72-tone system, of which the 36-tone system is in fact a part. Besides Hába, Ivan Wyschnegradsky (1893–1979), with Music for three pianos in sixths of tones among other works in the same period, also made a significant contribution to the repertoire of the equal 36-tone tuning, in which his pansonoric sound continuum could contain not just 36 tones but as many as 72. Sixth-tones are also used by the French spectralists, such as Gérard Grisey (1946–1998) and Tristan Murail (1947), in an attempt to reconstruct the natural harmonics. (More specifications on xen.wiki)

• The 43-tone system
The 43-tone system is known both as equal tuning and as the unequal tuning of Harry Partch. The equal variant divides the octave into 43 equal parts, with a step size of about 27.9 cents. This system tempers the syntonic comma (81/80) in the 5-limit and is related to the 1/5-comma meantone tuning, often associated with both 7- and 11-limit temperings. The 43-tone system provides reasonably good approximations of the pure harmonics on average. The major third (5:4) in the system has a deviation of +4.39 cents, the minor third (6:5) a deviation of −8.66 cents, the perfect fifth (3:2) a deviation of −4.29 cents, the harmonic seventh (7:4) of +7.91 cents, and the 11th harmonic (11:8) of +6.82 cents. Because the 43-tone system is a meantone system, it is easier to adapt traditional Western notation to it than to other tunings. For instance, A♯ and B♭ are different (respectively low and high), and the distance between them is one méride (1/43 of an octave). Of the seven diatonic tones in the octave, the five whole tones are divided into seven mérides and the two semitones into four mérides, which together make up 43 tones.
The system of 43 equal tones in the octave had already been advocated in the 17th century by the French mathematician, physicist and acoustician Joseph Sauveur (1653–1716). He introduced the idea of 43 proportional steps to represent smaller intervals more accurately, naming one step a “méride”. He also devised the “eptaméride”, which was one-seventh of a méride. He was furthermore the originator of the term “acoustique” (acoustics). Strangely enough, Sauveur had both a hearing and speech impairment, but despite this he is remembered for his detailed studies of sound. (More specifications on xen.wiki)

• The 43-tone scale of Harry Partch
The 43-tone scale of Harry Partch is an unequal tuning with 43 tones in the octave, developed by the American music theorist and composer Harry Partch (1901–1974). It is a microtonal scale that divides the octave into 43 unequal steps. In contrast to the traditional 12-tone equal tuning, where the octave is divided into twelve equal parts, Partch aimed for a system that came closer to the natural harmonic series. His microtonal scale is derived from the 11-limit just intonation, in which the intervals are defined by simple fractional ratios, such as 3:2 for the perfect fifth, 5:4 for the major third and 7:4 for the harmonic seventh. An intervallic leap such as 11:8 (the 11th harmonic) is specifically 551.32 cents in Partch’s system. This approach leads to intervals that sound more harmonious and natural, but also results in a more complex system with unequal pitch steps. Partch’s 43-tone system is based on the “eleven-limit tonality diamond”, a collection of pitch ratios with odd factors up to and including the highest prime factor 11, which arranges these ratios into a matrix that contains both upward and downward intervals. This tonal diamond is comparable to the “seven-limit diamond” developed earlier by Max Friedrich Meyer (1873–1967) and was refined by Partch. This principle results in 29 basic ratios within the octave, which are then supplemented by 14 additional ratios to complete the scale. These supplementary intervals fill the gaps and provide a more cohesive and versatile scale. Partch chose the 11-limit (that is, all rational numbers in which the odd factors of numerator and denominator are no higher than 11) as the basis for his music, reasoning that the 11th harmonic was the first to be completely alien to Western ears.
Harry Partch was dissatisfied with the limitations of the traditional 12-tone system, and he strove for a musical style that was not confined to traditional concert halls or academic circles, and stood apart from formal classical music. Partch wanted his music to be more rooted in the human experience. In 1935 he therefore abandoned his academic and conventional musical life. He burned all of his earlier compositions, which had been written in the traditional 12-tone system, and began his search for a new musical language. He travelled across the United States, often as a “hobo”, hitching rides on freight trains, as many unemployed people did during the Great Depression. During his travels Partch collected stories, experiences and texts from other wanderers and migrant workers. In this way he sought a direct, emotional and cultural connection with the voices of the “forgotten people”. His goal was to create an artistic expression that was deeper and more sincere than what he found in traditional musicology. This influence is clearly visible in his later work, such as the cycle The Wayward, which is based on the experiences of wanderers. Around 1943 Partch settled again and began to work more intensively on his instruments and compositions. His nomadic years had clearly shaped his artistic vision, and he used the experiences he had gained as inspiration for his idiosyncratic musical works, which often had a theatrical approach in which visual and physical elements played a major role.
His experiences as an outsider strengthened his desire to design new instruments that could escape the limitations of traditional Western musical instruments. To realise his musical vision, from 1945 onwards Partch built a series of unique instruments, such as the Chromelodeon (adapted 43-tone harmonium, c. 1945), the Harmonic Canon (elongated string instrument developed from 1945), the Diamond Marimba (marimba with a diamond-shaped keyboard pattern, 1946), the Cloud Chamber Bowls (cut Pyrex crystal bowls, 1948), the Kithara (large string instrument with multiple sets of strings, developed from 1950), and the Spoils of War (percussion and string instruments for unconventional sounds, 1950), which were specifically designed to bring out the precise pitches of his scale and the subtleties of human vocal intonation. Earlier, in 1930, he had already had his Adapted Viola built. Almost all of Partch’s music is written in the 43-tone scale, and although most of his instruments can only play subsets of the full scale, he used them as elements within his all-encompassing musical framework. (More specs on wiki/xen.wiki)

• Non-western tuning systems
There are various non-Western tone systems used in different musical traditions around the world. One of the best-known is the Indian music system (Śruti system), which is based on śrutis, the smallest intervals of the system. Traditionally, there are 22 śrutis within an octave, but in practice music is usually performed within ragas, which are based on specific scales of seven basic svaras (tones). The śruti system consists of seven basic tones in the octave (Sa, Ri, Ga, Ma, Pa, Da, Ni), similar to the Western diatonic tones, with five intermediate tones, comparable to the black keys on the piano, resulting in 12 tones. Apart from the tonic and perfect fifth, the other ten tones each have two variants, resulting in 22 tones (śrutis). The tones are tuned purely, comparable to Just Intonation. It is therefore in fact an unequal 22-tone tuning. Hindustani music from North India uses the That system, consisting of ten scales that serve as the basis for ragas. Carnatic music from South India employs the Melakarta system, which includes 72 different types of heptatonic scales. (specs: wiki)
In Arabic, Turkish and Persian music, the maqam system is used, a modal musical system that is complex and expressive. A maqam is therefore more than just a scale. Maqams are based on intervals that can be smaller than the semitones in the standard Western system, such as quarter tones, which means there are more than the usual twelve tones per octave. This allows musicians to play subtle nuances and microtonal variations, which are essential to the expressive nature of this music. Each maqam has fixed rules for ascending and descending, and can be combined with other maqams by means of modulation. Because it includes specific melodic movements, pitch bends and characteristic phrases, each maqam has a unique atmosphere and is used to convey particular emotions or states of mind. For instance, Maqam Bayati is often considered melancholic and intimate, while Maqam Rast conveys strength and stability. Although tones from the 24-tone tuning (quarter tones) are often used, the exact tuning of the tones in a maqam can also vary and is not always standardised, which contributes to the rich expressiveness of the music. Traditionally, these scales and their nuances are passed on orally and learned through intensive listening. (specs: wiki)
Javanese-Balinese gamelan music from Indonesia makes use of two unique tunings: Slendro, a five-tone system (see 5-tone tuning) with approximately equal tone distances, and Pelog, a seven-tone system, from which in practice a subset of five (or sometimes six) tones is always used. The pentatonic Slendro has the note names S (Singgul), G (Galimer), P (Panelu), L (Loloran) and B (Barang/Tugu), while the heptatonic Pelog mainly uses the note names S (Singgul), G (Galimer), P (Panelu), U (Bungur), L (Loloran), B (Barang) and O (Sorog). Although a gamelan ensemble contains instruments tuned in both Pelog and Slendro, the two tunings are never used simultaneously in the traditional repertoire, except in cases of modulation when they are applied alternately. Unlike the Western tone system, the intervals are not standardised, which means gamelan ensembles can have unique tunings. The Gong Ageng is the largest gong in the gamelan ensemble and functions as the principal tone on which the music rests. In both the Pelog and Slendro tunings, the entire gamelan is tuned to this gong. (specs: wiki)
Traditional Chinese music is largely based on pentatonic scales, consisting of five tones per octave. These scales are related to the black keys on the piano and are associated with harmony and balance. In some styles, a heptatonic system is used, which corresponds to Western major and minor scales, but with unique intonation and ornamentation. Many African musical styles are also pentatonic, such as the music of the Mande culture in West Africa. However, the rich African musical traditions are diverse and make use of different tone systems, often in combination with percussive rhythms. Some tone systems use microtones, with tones that lie between the Western semitones, especially in sung melodies and in the use of string instruments. (More specifications on wiki/wiki)

• Just Intonation
Just Intonation, also called pure intonation or pure tuning, is a tuning system in which the frequencies of tones relate to each other as simple whole-number ratios. This leads to intervals that correspond to the natural harmonics of a fundamental tone. For example, a perfect fifth has a frequency ratio of 3:2, which means that one tone has exactly 1.5 times the frequency of the other. This creates highly consonant and harmonious sounds. Intervals such as the fifth, the third and the harmonic seventh are therefore completely pure in this system. In Just Intonation the precise size of all intervals in the scale is calculated through various additions and subtractions of pure thirds and fifths from the natural harmonic series (overtone series, see below). This pure tuning was used until the Middle Ages and early Renaissance. However, the system proved impractical for polyphonic music and was gradually replaced by meantone tuning around the year 1500.
The origin of pure tuning goes back to the ancient Greeks, particularly Pythagoras (c. 570 BC – c. 500 BC), who studied intervals based on simple ratios such as 2:1 (octave), 3:2 (fifth) and 4:3 (fourth). This system, known as Pythagorean tuning, laid the foundations for later developments in tunings. In the Renaissance, Just Intonation was further developed and applied, with theorists such as Gioseffo Zarlino (1517–1590) advocating the use of pure thirds with a ratio of 5:4. This led to the so-called 5-limit Just Intonation, in which intervals were built up from the first prime numbers 2, 3 and 5. Applying Just Intonation in practice presents both possibilities and challenges. One of the advantages is the complete purity of the consonances. Instruments without fixed pitch, such as string instruments and voices, can relatively easily adapt to Just Intonation. The major disadvantage of this tuning, however, is the limited flexibility in modulations to other keys. Because the intervals are based on specific frequency ratios, modulations quickly lead to tones that fall outside the original system, resulting in dissonant intervals or the need to add new pure tones. This problem is known as the “comma problem”, where small tuning discrepancies accumulate and lead to undesirable deviations in pitch. To circumvent these limitations, various tuning systems were developed, such as meantone tuning and the 31-tone system. The most well-known and most common tuning system for several centuries has been the 12-tone equal tuning, in which the octave is divided into twelve equal semitones. Although this system makes it possible to modulate freely between keys, it comes at the expense of the purity of many intervals, since most do not correspond to whole-number ratios. Over time, composers, music theorists and scientists have sought ways to combine the advantages of Just Intonation with the flexibility of other tuning systems, which has led to experiments with microtonality. (More specifications on wiki/xen.wiki)