tuning library

Equal Temperament 12 notes

The most common Western tuning: twelve equal divisions of the octave, with steps of 100 cents.

5-Limit Just Intonation 12 notes

A twelve-note just-intonation tuning built from ratios whose prime factors are 2, 3, and 5. It includes a pure 5:4 major third and a pure 6:5 minor third.

Changing the Key rebuilds every ratio from the new tonic.

7-Limit Just Intonation 12 notes

A twelve-note just-intonation tuning built from ratios whose prime factors are 2, 3, 5, and 7.

This one includes 7:5 at 582.51 cents and the harmonic seventh, 7:4, at 968.83 cents.

Changing the Key rebuilds every ratio from the new tonic.

Slendro 5 notes

Slendro is one of the essential tuning systems used in gamelan instruments that have a pentatonic scale (5 notes per octave).

This tuning is a model of slendro, dividing the octave equally into 5 steps of 240 cents. Gamelans are usually tuned by ear, so the steps vary by region, ensemble and instrument. The next tuning, Javanese Slendro, is measured from real gamelans.

Javanese Slendro 5 notes

A slendro of Central Java, measured by Wasisto Surjodiningrat, P. J. Sudarjana and Adhi Susanto. It is an average of 30 gamelans of Yogyakarta and Surakarta. Its steps are uneven, from 231 to 253 cents, and its octave is stretched to 1208 cents.

Javanese Pelog 7 notes

The seven-note pelog of the gamelan Swastigitha in Yogyakarta, led by Suprapto Atmosutijo. William Sethares measured it. This tuning is the median across the ensemble in the middle register. The other registers stretch or shrink the octave by up to 11 cents.

Pieces use five of the seven notes at a time: 1 2 3 5 6 in pathet lima and nem, and 2 3 5 6 7 in pathet barang. Both are in the Scale menu.

Banyumas Calung 5 notes

The bamboo calung of Banyumas in Central Java. These are the notes of the gambang from one set, measured by Mukhlis Anton Nugroho. It starts on note 1, at 529 Hz, and its octave is stretched to about 1248 cents.

Balinese Gong Kebyar 5 notes

A five-note gong kebyar tuning of the begbeg type. A Balinese tuner gave Andrew Toth three example tunings, called begbeg, sedang and tirus, all with pure octaves. Toth says begbeg suits fast modern kebyar and accompanying legong.

Sundanese Salendro Sejati 5 notes

Kusumadinata's model of salendro, from his Sundanese laras theory. He called it sejati ("true"). His theory divides the octave into seventeen equal steps of about 70 cents. Salendro's steps of 282 and 212 cents are four and three of those.

Sundanese Pelog Degung Sejati 5 notes

Kusumadinata's model of pelog degung, the five-note laras of the Sundanese gamelan degung. Its steps are 424, 70, 212, 424 and 70 cents. So the scale splits into two matching halves a fifth apart. Each half is a 424-cent step followed by a 70-cent one.

Sundanese Sorog Madenda Sejati 5 notes

Kusumadinata's model of madenda, the third Sundanese laras. Madenda is his school term. Musicians call it sorog. It shares four notes with pelog degung and raises the middle one, the fifth at 706 cents, to 918.

Indian 12-Svara 12 notes

Indian classical music names the notes of the octave with seven svaras, Sa Re Ga Ma Pa Dha Ni (similar to do re mi). Sa and Pa are fixed. Re, Ga, Dha and Ni also have a flat form (komal), and Ma has a sharp form (tīvra). That makes twelve notes, as many as the keys of a piano.

Here they are tuned as five-limit just ratios. Each takes one of the twenty-four notes listed by Clements. Sa is the tonic. There is no fixed concert pitch, so performers set Sa wherever suits their voice or instrument. Choosing a Key moves Sa and rebuilds the ratios from it, as in the Just Intonation tunings.

The Scale menu holds Bhatkhande's ten thāṭs, the parent scales of North Indian music. It also holds twenty Hindustani rāgas and the 72 melakartas, the parent scales of South Indian (Carnatic) music. A rāga also has its own ornaments and its own ways of moving up and down. The scales here give only its notes.

Arabic Maqām (24 Quarter-Tones) 24 notes

Arabic music is built on maqāmāt (singular maqām), modes that describe which notes a melody uses and how it moves between them.

Many maqāmāt use notes that fall between the keys of a piano, so Arabic theory divides the octave into 24 equal quarter tones instead of 12 semitones. Mīkhāʼīl Mishāqa described this grid around 1840. The 1932 Cairo Congress of Arabic Music debated making it the standard but did not adopt it.

The tonic of Rast, the central maqām, sits on C. The Scale menu holds thirty maqāmāt: twelve principal ones, then the rest of each family as listed by Maqam World. Some maqāmāt are another maqām started on a different note, such as Muḥayyar, Farahfazā and Shadd ʿArabān. These appear as Key choices of that maqām. Sīkāh Baladī is left out because its intervals fit no quarter-tone grid. Musicians play the in-between notes somewhat above or below the exact quarter tones. How far depends on the region and the maqām. A maqām is also a melodic path, so the scales here give only its notes.

al-Fārābī 22-Note ʿŪd 22 notes

The twenty-two notes al-Fārābī found in each octave of the ʿūd, from his Great Book of Music (Baghdad, tenth century). He starts from an ʿūd tuned in fourths: its open strings and its index, Zalzal middle-finger, ring and little-finger frets. Then he adds frets until both octaves hold the same notes.

The result mixes Pythagorean ratios with Zalzal's neutral third, 27:22, and the 18:17 family of the Persian middle-finger fret. The ratios are Forster's reading of the treatise.

Ṣafī al-Dīn 17-Note 17 notes

The seventeen-note tuning from Ṣafī al-Dīn al-Urmawī's Book of Cycles (Baghdad, thirteenth century). Later Arabic, Persian and Turkish theory grew from this Pythagorean ʿūd fretting. It is a chain of sixteen pure fifths starting on C. Ellis lists it from E down to D double flat.

Ellis argues that its steps of 384 and 882 cents were meant to stand in for Zalzal's neutral third and sixth. They also sit within a schisma of the just major third and sixth.

Turkish Makam (Arel-Ezgi-Uzdilek) 24 notes

Turkish classical music is built on makams, modes that set which notes a melody uses and how it moves between them. Its theory calls the notes perdes, and many of them fall between the keys of a piano.

Arel-Ezgi-Uzdilek is named after three theorists: Hüseyin Sadettin Arel, Suphi Ezgi and Salih Murat Uzdilek. It has been the standard theory since the mid-20th century, taught in conservatories and used in notation. It defines 24 perdes per octave, taken from a chain of pure fifths. Intervals are counted in komas, small steps of which 53 make an octave.

It differs from Rauf Yekta's earlier 24-perde set at two pitches. Performers play several of its intervals differently, segâh among them. Yarman 31 is built to match how they play. Rast sits on C and hüseynî on A.

Turkish makams and Arabic maqāmāt share their roots and many names, but their theories differ. Arabic theory divides the octave into 24 equal quarter tones. The Turkish 24 perdes are unequal, picked from 53 komas. So makams with the same name, such as Hicaz and Hijaz, can use different notes.

Yarman 31 31 notes

Ozan Yarman proposed this tuning to bring makams closer to how Turkish musicians play them. The standard Arel-Ezgi-Uzdilek theory is further from their playing. Yarman's 24-perde system is built on a core of twelve tempered notes. This 31-perde version adds perdes for the middle seconds of Uşşak, Segâh, Hüzzam and Sabâ. A middle second is an interval between a semitone and a whole tone.

Rast sits on C and hüseynî on A, at the Sipürde concert pitch Yarman proposes. The Scale menu offers the 24-perde core and the notes of sixteen makams, each on its own karar (tonic).

Turkish 53 Koma (Holdrian) 53 notes

Turkish music theory measures intervals in komas, small steps of which 53 make an octave. This tuning divides the octave into these 53 equal steps. Each step is called a Holdrian comma, after the 17th-century English theorist William Holder.

In komas, a whole tone is 9, the Arel-Ezgi-Uzdilek semitones are 4 and 5, and the augmented second is 12 or 13. The 24 perdes of Arel-Ezgi-Uzdilek sit within a cent of 24 of these steps. The other 29 steps leave room for the inflections performers use between them. Rast sits on C, and hüseynî on A, 40 komas up. The Scale menu offers the makams as koma patterns that can start on any karar (tonic).

Persian Dastgāh (Vaziri) 24 notes

Persian classical music is organised into dastgāhs. A dastgāh is a family of traditional melodies (gushehs) that share a mode. Musicians learn them by heart from a repertoire called the radif.

Like Arabic and Turkish music, it uses notes that fall between the keys of a piano. In the 1920s Ali-Naqi Vaziri modelled them as 24 equal quarter tones. He also introduced the koron and sori accidentals, which lower and raise a note by about a quarter tone. Persian staff notation still uses them.

Musicians place many of these notes away from the exact quarter tones. Dastgāh Practice (Farhat), also in this list, is measured from real playing.

The Scale menu has seven dastgāhs and five āvāzes. Āvāzes are smaller modes that branch off Shūr and Homāyun. Each scale starts on its finalis (tonic).

Dastgāh Practice (Farhat) 17 notes

Seventeen notes per octave, averaged from the tunings Hormoz Farhat measured on the tār and setār in his 1966 research. These are the long-necked lutes of Persian music.

The Scala archive gives the notes in this form. They are a measured alternative to Vaziri's equal quarter tones. Players still adjust single notes from one dastgāh to the next.

Japanese Gagaku (Jūni-ritsu) 12 notes

Gagaku is the court music of Japan, played at the imperial court for more than a thousand years and adapted from music brought from China and Korea. Its theory has twelve notes, the jūni-ritsu, from Shinsen through Banshiki. Here they form a Pythagorean chain of fifths.

The Scale menu offers six classical chōshi (Ichikotsuchō, Hyōjō, Sōjō, Ōshikichō, Banshikichō, Taishikichō), plus five-note ritsu and ryō modes that start on any Key. The notes follow Takaoka's Columbia table for the shō, the mouth organ of the ensemble. It is a chain of pure fifths from D♭ to F♯, and Shinsen sits on C. Ōshiki sits on A, at 430 Hz. Gagaku ensembles still usually tune to A = 430 Hz. Many shō players keep a second instrument at 440 or 442 Hz for contemporary music.

Koto Hirajōshi 5 notes

The koto is a long Japanese zither with thirteen strings. Hirajōshi has been its standard tuning since the 17th century. Players tune it by ear, so it differs from player to player.

This tuning follows a teacher at Sohgakusha, a koto school in Sapporo and Yokohama. The teacher tuned hirajōshi by ear, starting on D, in pure octaves, fifths and fourths. Then they checked each string on a tuner set to A = 440 Hz. D was in tune. E♭ was 9 cents low, G 2 cents low, A 2 cents high and B♭ 7 cents low. That is close to a Pythagorean tuning, with narrow semitones.

Other players also narrow the semitones. How much depends on the piece, the school, the weather and the player. This tuning starts on the second string, G.

Chinese Twelve Lǜ (Shí-èr-lǜ) 12 notes

Traditional Chinese theory has twelve notes per octave, called the twelve lǜ. They were first fixed as the pitches of bronze bells and bamboo pipes. Melodies use five or seven of them.

The lǜ are built from a starting note, Huángzhōng, by a method called sanfen sunyi ("subtract or add a third"). Each new pipe is cut to 2/3 or 4/3 the length of the one before. Brought into one octave, this gives a chain of pure fifths.

The pitch of Huángzhōng changed from period to period. Here it sits on middle C. The fifths follow the order of the Lǚshì chūnqiū, which starts Huángzhōng, Línzhōng, Tàicù and ends on Zhònglǚ. Zhònglǚ lands at 521.5 cents. That is a Pythagorean comma above a pure fourth.

Guanzi Pentatonic 5 notes

The five-note scale at the heart of Chinese music. It is built from 3:2 fifths, as described in the 'Dì yuán' chapter of the Guǎnzǐ (3rd century BCE).

The Guǎnzǐ starts from the number 81 for gōng and adds a third of it to reach 108 for zhǐ. It then takes away or adds a third of the last number in turn: shāng 72, yǔ 96 and jué 64. This tuning starts on gōng, which sits on C: C D E G A.

Thai Seven-Equal Model 7 notes

Thai classical music is played by ensembles of xylophones, gong circles, fiddles and oboes. Its scale has no counterpart on the piano. Thai theory divides the octave into seven equal steps of about 171 cents, each between a Western semitone and whole tone. This tuning follows that model. Measured instruments often depart from it.

Melodies mostly use five of the seven notes, so the Scale menu offers that five-note scale starting on any note.

Maranao Kolintang (Taraka) 7 notes

The kolintang is a row of eight small bronze gongs in a wooden frame, played with two sticks. It is the lead melody instrument of the Maranao, Maguindanao and other peoples of the southern Philippines.

This one is from Taraka, in the Maranao area of Mindanao, measured by Usopay Cadar and Robert Garfias. Its eighth gong sits 1543 cents above the first, so this tuning repeats at that span.

Maranao Kolintang (Taraka, Octave) 6 notes

The Taraka kolintang, folded into a pure octave. Gongs 7 and 8 sit close to the octaves of gongs 2 and 3, so here they become those octaves.

Mandinka Balafon (Gambia) 7 notes

The balafon is the West African xylophone, with wooden keys over gourd resonators. It is central to the music of the Mande peoples.

These seven notes are from a Mandinka balafon in Gambia. They come from the Scala scale archive, which does not say who measured them. The last note is at 1141 cents, and this tuning repeats at that span.

Mandinka Balafon (Gambia, Octave) 8 notes

The Mandinka balafon from Gambia, with a pure octave added above its last note at 1141 cents.

Mandinka Kora (Knight) 14 notes

The kora is a West African harp with 21 strings over a large gourd. Mandinka jalis (griots) play it. They are hereditary musicians who keep history and family lines in song.

The Scale menu has four kora tunings documented by Roderic Knight: Silaba lowers the 3rd and 7th and raises the 2nd and 6th. Hardino raises the 3rd and 7th. Tomora lowers the 3rd and 7th by a semitone or more, and keeps Silaba's raised 2nd and 6th. Sauta is Hardino with a raised 4th.

Chopi Timbila 7 notes

The timbila is the xylophone of the Chopi people of southern Mozambique, played in large orchestras that accompany dance.

Sena Valimba (Malawi) 7 notes

The valimba is the xylophone of the Sena people in the Lower Shire Valley of Malawi. The Sena tune toward seven equal steps, with their octaves averaging 1213 cents.

Shona Mbira (John Kunaka) 7 notes

The mbira is the Shona instrument of Zimbabwe: metal keys on a wooden board, plucked with the thumbs and a finger. The mbira dzavadzimu ('mbira of the ancestors') is played at ceremonies that call ancestral spirits.

Ethiopian Qəñət (tempered reference) 12 notes

Qəñət is an old Amharic word that means “to start a tune, an air, a melody, to give the tuning, to sing”. In the secular music of the Amhara, in the northern highlands, it is now usually taught as four five-note modes: Tizita (meaning longing or memory), Bati, Ambassel and Anchihoye.

You can find the four modes as Scale options. The Yared School of Music teaches this system and many azmaris (singer-musicians) use it. It is written down only from the 1960s on, and Weisser and Falceto question how old it is and how well it fits what musicians play. Church music has its own modes and is not covered here.

This tuning places the modes on the 12-EDO grid for reference. Real intervals vary from player to player.

'Are'are Panpipes 7 notes

Melanesia is the island region of the southwest Pacific that includes Papua New Guinea, the Solomon Islands, Vanuatu and Fiji. The 'Are'are of Malaita in the Solomon Islands tune their panpipes to seven roughly equal steps per octave.

This tuning is the equal version of that scale, which is 7-EDO. In practice, tunings will vary. One kind of ensemble, the 'au paina, tunes only five tubes to the octave. The Scale menu offers that five-note scale as an approximation.

Great Highland Bagpipe 7 notes

The chanter scale of the Great Highland bagpipe, as Ewan Macpherson found pipers tuning it in the 1990s. It uses just intervals on low A, with high G at the harmonic seventh, 7:4.

Modern pipes put low A around 470 to 480 Hz. This tuning puts it at 440 Hz so the Key names match the pipers' note names.

The Scale menu adds the gapped scales from a 1915 letter to the Oban Times. These are five-note scales starting on G, A and D, and a six-note scale starting on D.

Byzantine (72 Moria) 72 notes

The 72-moria octave fixed by the Patriarchal Music Committee of Constantinople in 1881 to 1883. Greek Orthodox chant still teaches it. Ni sits on C.

Each genus is built from two tetrachords of 30 moria plus a 12-moria tone. The tetrachord steps are: diatonic 12-10-8, soft chromatic 8-14-8, hard chromatic 6-20-4, enharmonic 12-12-6.

Chanters bend notes depending on melodic direction and attraction, so sung pitches depart from the grid.

Harmonic Series 32 notes

A sequence of frequencies generated by whole-number ratios ( 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, …). When heard all at once, its harmony is called “chord of nature”.

This is a 32 note subset of the harmonic series, incorporating the first 64 possible frequencies (or octaves thereof).

Subharmonic Series 32 notes

A sequence of frequencies generated by the reciprocals of whole numbers (1/1, 1/2, 1/3, 1/4, 1/5, 1/6, …), the harmonic series mirrored downward. Hugo Riemann used it to explain the minor chord.

This is a 32 note subset of the subharmonic series, incorporating the first 64 possible frequencies (or octaves thereof).

Bohlen-Pierce 13 notes

A 13 tone macrotonal (each step larger than a semitone) scale with thirteen equal divisions of a 3:1 tritave instead of a 2:1 octave.

The result is an unusual sequence of tones providing many consonant intervals and hence the promise of extensive musical possibilities to those willing to explore non-traditional sounds.

Curtis Roads describes it as “both more sweet and more sour than 12-note equal temperament”.

Bohlen-Pierce (Just) 13 notes

Heinz Bohlen's original ratio form of the tritave scale: thirteen steps to 3:1 built from odd-number ratios such as 9:7, 7:5, 5:3 and 7:3. In contrast, Bohlen-Pierce uses equal steps of 146.31 cents.

Bohlen 833 Cents 7 notes

Heinz Bohlen's non-octave scale on the golden ratio (1.618, 833 cents). He stacked 833-cent intervals and kept the harmonics and subharmonics of each tone up to the fourth that fall between them, which gives seven unequal steps to a period of 833 cents.

Ten steps make an exact octave, since 2/φ above φ is 2, and most two-step intervals come close to 8:7 or 7:6.

Harry Partch 43 43 notes

Harry Partch's 43-note just-intonation scale, built from an 11-limit tonality diamond. Partch chose the 11 limit because the 11th harmonic is the first that is particularly foreign to Western ears.

Almost all of Partch's music is written in the 43-tone scale, and although most of his instruments can play only subsets of the full scale, he used it as an all-encompassing framework.

Harry Partch 29 29 notes

Harry Partch's first just-intonation system, drafted in 1928: 29 notes of an 11-limit scale, symmetric about the octave.

He composed for it on his Adapted Viola, a cello neck on a viola body that he built in New Orleans in 1930. It later grew into his 43-tone Genesis scale.

La Monte Young's The Well-Tuned Piano 12 notes

The 12-note tuning of La Monte Young's The Well-Tuned Piano as he has played it since 1981, the version on the 1987 Gramavision recording.

Michael Harrison Revelation 12 notes

Michael Harrison's tuning for Revelation, his 2001 work for harmonically tuned piano.

All twelve notes are overtones of a low F made from 3 and 7 alone: the white keys are a chain of pure fifths, F C G D A E B, and each black key sits a 7:4 above one of them.

Harp of New Albion (Terry Riley) 12 notes

Terry Riley's 12-note just-intonation tuning for The Harp of New Albion. It starts on C♯. The twelve ratios are the inversion of Malcolm's Monochord.

Lou Harrison 16 (Ptolemy Duple) 16 notes

Lou Harrison's 16-tone superparticular Ptolemy Duple tuning, built for an aluminum-bar instrument.

The Scale list adds the slendro Harrison and William Colvig tuned for the Mills College gamelan, Si Darius/Si Madeleine, in steps of 8/7, 7/6, 9/8, 8/7 and 7/6.

Lou Harrison Cinna 12 notes

Lou Harrison's 12-note tuning for his incidental music to Corneille's Cinna (1955-57), written for a tack piano, an upright with thumbtacks in its hammers. On G it is five-limit just intonation with two septimal notes, the 7/6 minor third (B♭) and the 7/4 seventh (F).

Every note but the 25/18 augmented fourth (C♯) is also in Partch's 43-tone scale.

Ben Johnston 22 22 notes

The 22 pitches Ben Johnston reaches at the climax of his String Quartet No. 4 "Amazing Grace" (1973): a seven-limit just-intonation scale on G, which is 1/1 here and sits on middle C.

The quartet is a theme and eight variations. The hymn opens on a Pythagorean pentatonic, and each variation adds pitches. Variation 4 builds a major set from seven-based ratios such as 7/6, 21/16 and 7/4. Variation 6 turns the tune upside down and uses the inverted ones, 8/7, 12/7 and 27/14. Variations 7 and 8 combine the two sets into these 22. The piece finally thins back out to the pentatonic.

Ben Johnston 81 81 notes

The 81 pitches of Ben Johnston's Sonata for Microtonal Piano, a just-intonation matrix of pure fifths and major thirds built out from the C major triad, with C always at 1/1. Johnston tuned the 88 keys to 81 different pitches, so almost no key has its octave elsewhere on the keyboard, but each note keeps its letter name in every register. Here the 81 are folded into one octave.

The key names follow Johnston's notation: a sharp raises a note by 25/24 and a plus by the syntonic comma, 81/80. The scales are pitch sets John Jeffrey Gibbens traces in the piece, all on C.

Carlos Alpha 15 notes

Wendy Carlos's Alpha scale, with equal 78-cent steps and no octave equivalence. Four steps closely approximate a minor third, five a major third, and nine a perfect fifth.

About Alpha, Carlos writes: "Try some harmonies and you'll find they're amazingly pure. The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before. This is a scale well worth exploring."

Carlos Beta 19 notes

Wendy Carlos's Beta scale, with equal 63.8-cent steps and no octave equivalence. Eleven steps closely approximate a perfect fifth.

About Beta, Carlos writes: "Beta is very like Alpha in its harmonies, but with 5 steps to the minor third, 6 to the major third, and 11 to the perfect fifth, melodic motions are different, rather more diatonic in effect than Alpha."

Carlos Gamma 34 notes

Wendy Carlos's Gamma scale, with equal 35.1-cent steps and no octave equivalence. Eleven steps closely approximate a major third and twenty a perfect fifth.

About Gamma, Carlos writes: "Gamma's scale is yet a "third flavor," sort of intermediate to 'alpha' and 'beta', although a melodic diatonic scale is easily available."

Wendy Carlos's Harmonic Scale 12 notes

Wendy Carlos's 12-note Harmonic Scale, whose notes are taken from the harmonic series and reduced to one octave. She used it on Beauty in the Beast (1986). There, a bank of 144 pitches per octave lets the music change key while staying in tune.

The Harmonic scale can be used as an alternative tuning for regular twelve-tone pianos to play spectral or otonal music, although it finds more regular use merely as a unique microtonal retuning of 12-EDO.

Stockhausen Stimmung 5 notes

The pitches of Karlheinz Stockhausen's Stimmung (1968) for six vocalists: harmonics 2, 3, 4, 5, 7 and 9 of a low B♭, which make the chord B♭, F, B♭, D, A♭, C. Here they are combined as one octave with 1/1, 9/8, 5/4, 3/2 and 7/4, on B♭ at 57 Hz.

Colundi 120 notes

121 frequencies taken from Aleksi Perälä's Colundi chart posted on Instagram (from 30 Hz to 30800 Hz).

In Colundi there are no octaves. Every frequency is its own note. Perälä says it is “not a tuning system; it's a sequence based on specific frequencies, often not related to each other in a traditional musical way”. Grant Wilson-Claridge, who devised the set of frequencies, gives each frequency a tolerance of ±0.455% and emphasizes to use subsets of the frequencies. Right-click the note selector in the control bar to open the detailed grid of frequencies and save custom scales you are particularly drawn to.

Hexany (1-3-5-7) 6 notes

Erv Wilson's six-note hexany. Its notes are the products of each pair from 1, 3, 5 and 7. Here they are divided by 1·3 so that 1/1 is one of them: 1/1, 7/6, 5/4, 35/24, 5/3, 7/4. This seven-limit set is symmetric and has no required tonic.

Eikosany (1-3-5-7-9-11) 20 notes

A 20-tone 1.3.5.7.9.11 eikosany combination-product set with 1.3.5 as tonic triad.

Centaur (7-limit) 12 notes

Kraig Grady's 1987 Centaur, a fixed 12-note seven-limit just-intonation tuning. Its intervals include 7:6, 7:5, 14:9, and 7:4 alongside familiar five-limit ratios. The Scale list adds the 10-tone constant structure Erv Wilson drew from it, the ten 7-tone scales inside that, and its three complete 7-limit tetrads.

5-Limit JI (Wolfed) 12 notes

The 5-limit just-intonation tuning, starting on C and staying there when the Key changes.

Only C, F and G are pure. On D and B♭ the fifth is 22 cents flat (the wolf). On E, A and B the major third is 41 cents sharp (32:25). On D♭, E♭ and A♭ the minor third is 41 cents narrow. F♯ is off on both its fifth and its third. aquirosa pulls a sound's harmonics onto these notes. So when a sound's fundamental is away from C, its own harmonics bend too. The sound goes metallic and beats. Honestly just a bit fucked but i like how it sounds.

7-Limit JI (Wolfed) 12 notes

The 7-limit just-intonation tuning, starting on C and staying there when the Key changes.

The 7:5 and 7:4 take the places of F♯ and B♭, so C carries a pure harmonic seventh. Away from C it gets rougher than the 5-limit version. E♭'s fifth is 49 cents flat. B♭ has a 9:7 major third and a fifth 27 cents sharp. F♯'s fifth is also 27 cents sharp. D keeps the flat wolf fifth, and E, A and B keep the 32:25 third. When a sound's fundamental is away from C, its own harmonics bend, so the sound goes metallic and beats. Honestly just a bit fucked but i like how it sounds.

88CET (Gary Morrison) 14 notes

Gary Morrison's 88-cent equal-step construction: 14 steps per 1232-cent stretched period.

Dreyblatt Overtone Lattice 20 notes

Arnold Dreyblatt's unequal just-intonation lattice, built from the 3rd, 5th, 7th, 9th, and 11th harmonics and their products. The source lists 20 notes within the octave and starts on F.

Pythagorean Twelve Tone 12 notes

A twelve-note Pythagorean tuning built from pure 3:2 fifths and reduced to one octave.

Its major thirds are wider than those of 12-EDO. The chain of fifths runs from E♭ to G♯ (E♭ B♭ F C G D A E B F♯ C♯ G♯), so the wolf fifth falls between G♯ and E♭.

Quarter-comma Meantone 12 notes

A sixteenth-century temperament with pure 5:4 major thirds and fifths narrowed by one quarter of the syntonic comma. With only twelve fixed notes, one fifth is left far out of tune (the wolf), so some keys sound much less consonant than others.

Half-comma Meantone 12 notes

A meantone temperament in which each fifth is narrowed by half a syntonic comma from a pure 3/2. As a result, minor sevenths are exactly 9/5 and major seconds exactly 10/9.

Werckmeister III 12 notes

A late seventeenth-century well temperament with four tempered fifths and eight pure fifths.

All keys remain usable, but the size of their thirds and fifths differs from key to key. The four fifths C-G, G-D, D-A and B-F♯ are each narrowed by a quarter of the Pythagorean comma. It starts on C, with the standard values. The name 'III' comes from counting just intonation and meantone as Werckmeister's first two tunings.

Kirnberger II 12 notes

Johann Philipp Kirnberger's tuning from the first part of Die Kunst des reinen Satzes (1771; second edition 1774). E, B and F♯ are pure 5:4 thirds above C, G and D. D♭, A♭, E♭, B♭ and F follow pure fifths down from C. A sits half a syntonic comma above 5:3 (270:161), so the fifths D-A and A-E split the comma between them. All other fifths are pure, except for a schisma at F♯-C♯.

Silbermann (after Sorge) 12 notes

The organ temperament that Georg Andreas Sorge attributed to Gottfried Silbermann in 1748. Eleven fifths are each a sixth of a Pythagorean comma narrow (698.0 cents), with a wolf from G♯ to E♭. Sorge, who argued for equal temperament, did not say which comma he meant. Later study of Silbermann's surviving organs disputes that he tuned this way. Barbour's version divides the syntonic comma and sits within 2.6 cents of this one.

Werckmeister Septenarius 12 notes

Andreas Werckmeister's monochord tuning from Musicalische Temperatur (1691). The string is divided into 196 parts, and every note is set by a whole-number string length. No fifth is tempered by a fraction of a comma. This tuning keeps D at the printed length of 176, which tempers G-D by more than half a comma. Werckmeister probably meant 175: on his plate, the label 176 sits one place too far to the right.

3-EDO 3 notes

Three equal divisions of the octave, with steps of 400 cents. Its three notes form an augmented triad, as in 12-EDO.

4-EDO 4 notes

Four equal divisions of the octave, with steps of 300 cents. Its four notes form a cycle of minor thirds, as in 12-EDO.

5-EDO 5 notes

Five equal divisions of the octave, with steps of 240 cents. It is the smallest EDO with intervals that 12-EDO lacks. Also called the equipentatonic scale, it is close to tunings in many musical traditions, such as the slendro scales of Javanese gamelan and the music of the Baganda in Uganda.

6-EDO 6 notes

Six equal divisions of the octave, with steps of 200 cents. Also known as the whole-tone scale.

7-EDO 7 notes

Seven equal divisions of the octave, with steps of 171.43 cents. Also called equiheptatonic. Thai fixed-pitch instruments are tuned close to it. Several other traditions use nearly equal seven-note scales, and these often fit the third and fifth harmonics better.

8-EDO 8 notes

Eight equal divisions of the octave, with steps of 150 cents.

9-EDO 9 notes

Nine equal divisions of the octave, with steps of 133.33 cents.

10-EDO 10 notes

Ten equal divisions of the octave, with steps of 120 cents.

11-EDO 11 notes

Eleven equal divisions of the octave, with steps of 109.09 cents. Orgone and Machine are 7- and 6-note scales built by stacking its 3- and 2-step intervals.

12-EDO 12 notes

Twelve equal divisions of the octave, with steps of 100 cents. The same as Equal Temperament.

13-EDO 13 notes

Thirteen equal divisions of the octave, with steps of 92.31 cents. Seven steps give a 646.15-cent fifth, far narrower than 3:2. Oneirotonic is an 8-note scale built by stacking its 5-step interval.

14-EDO 14 notes

Fourteen equal divisions of the octave, with steps of 85.71 cents. It contains two interleaved 7-EDO cycles.

15-EDO 15 notes

Fifteen equal divisions of the octave, with steps of 80 cents. It contains 5-EDO. Its fifth is 720 cents, its minor third 320 cents (close to 6:5), and its major third 400 cents, the same as in 12-EDO. Porcupine gives 7- and 8-note scales built by stacking its 2-step interval. Blackwood gives two 5-EDO rows a step apart.

16-EDO 16 notes

Sixteen equal divisions of the octave, with steps of 75 cents. Five steps give a 375-cent third and nine give a 675-cent fifth. The Diminished scale alternates steps of 3 and 1, and repeats every quarter octave.

17-EDO 17 notes

Seventeen equal divisions of the octave, with steps of 70.59 cents. Ten steps give a 705.88-cent fifth and six give a 423.53-cent third. The Western modes are built by stacking its 10-step fifth, which also gives the Key names their sharps and flats. The neutral-third scale is a 7-note scale built by stacking its 5-step interval.

18-EDO 18 notes

Eighteen equal divisions of the octave, with steps of 66.67 cents. It contains the complete 6-EDO cycle and divides every 12-EDO whole tone into three parts.

19-EDO 19 notes

Nineteen equal divisions of the octave, with steps of 63.16 cents. Its thirds support meantone harmony, and eleven steps give a 694.74-cent fifth. The Western modes are built by stacking its 11-step fifth, which also gives the Key names their sharps and flats.

20-EDO 20 notes

Twenty equal divisions of the octave, with steps of 60 cents. It contains two interleaved 10-EDO cycles and four interleaved 5-EDO cycles.

21-EDO 21 notes

Twenty-one equal divisions of the octave, with steps of 57.14 cents. It divides each 7-EDO step into three equal parts.

22-EDO 22 notes

Twenty-two equal divisions of the octave, with steps of 54.55 cents. The Western modes are built by stacking its 13-step fifth. Porcupine and Orwell are built by stacking its 3- and 5-step intervals.

23-EDO 23 notes

Twenty-three equal divisions of the octave, with steps of 52.17 cents.

24-EDO 24 notes

Twenty-four equal divisions of the octave, with quarter-tone steps of 50 cents. It contains 12-EDO as every second step. The Western modes are built by stacking its 14-step fifth. The neutral maqāmāt Rāst, Bayātī, Ṣabā, Sīkāh and Huzām use the same steps as on the Arabic Maqām (24 Quarter-Tones) tuning.

26-EDO 26 notes

Twenty-six equal divisions of the octave, with steps of 46.15 cents. It contains two interleaved 13-EDO cycles. The Western modes are built by stacking its 15-step fifth.

27-EDO 27 notes

Twenty-seven equal divisions of the octave, with steps of 44.44 cents. It contains 9-EDO as every third step. The Western modes are built by stacking its 16-step fifth.

29-EDO 29 notes

Twenty-nine equal divisions of the octave, with steps of 41.38 cents. Seventeen steps give a 703.45-cent fifth, a little wider than 3:2, and its diatonic scale has 5-step tones and 2-step semitones. The Western modes are built by stacking its 17-step fifth.

31-EDO 31 notes

Thirty-one equal divisions of the octave, with steps of 38.71 cents. Its 696.77-cent fifth and 387.10-cent major third come close to quarter-comma meantone. The Western modes are built by stacking its 18-step fifth. Mohajira and Orwell stack its 9- and 7-step intervals. Miracle decimal and Blackjack stack its 3-step interval of 116.13 cents, called the secor.

34-EDO 34 notes

Thirty-four equal divisions of the octave, with steps of 35.29 cents. It contains two interleaved 17-EDO cycles. The Western modes are built by stacking its 20-step fifth.

35-EDO 35 notes

Thirty-five equal divisions of the octave, with steps of 34.29 cents. It is the smallest division that holds both 5-EDO (every seventh step) and 7-EDO (every fifth step).

36-EDO 36 notes

Thirty-six equal divisions of the octave, with steps of 33.33 cents. It contains 12-EDO as every third step and 18-EDO as every second step. The Western modes are built by stacking its 21-step fifth.

41-EDO 41 notes

Forty-one equal divisions of the octave, with steps of 29.27 cents. Its 702.44-cent fifth closely approaches 3:2. The Western modes are built by stacking its 24-step fifth. Miracle decimal, Blackjack and Canasta stack its 4-step interval of 117.07 cents, called the secor. Magic stacks its 13-step major third.

43-EDO 43 notes

Forty-three equal divisions of the octave, with steps of 27.91 cents. Its 697.67-cent fifth is within a tenth of a cent of fifth-comma meantone. So it plays meantone harmony, and its chain of fifths closes after 43 steps. Joseph Sauveur described the division in 1701. The Western modes are built by stacking its 25-step fifth.

46-EDO 46 notes

Forty-six equal divisions of the octave, with steps of 26.09 cents. Twenty-seven steps give a 704.35-cent fifth and fifteen give a 391.30-cent major third. The Western modes are built by stacking its 27-step fifth.

48-EDO 48 notes

Forty-eight equal divisions of the octave, with steps of 25 cents. It contains 24-EDO as every second step and 12-EDO as every fourth step. The Western modes are built by stacking its 28-step fifth.

50-EDO 50 notes

Fifty equal divisions of the octave, with steps of 24 cents. It is a meantone. Twenty-nine steps give a 696-cent fifth, close to the 7/26-comma temperament that William Woolhouse recommended in 1835. The Western modes are built by stacking its 29-step fifth.

53-EDO 53 notes

Fifty-three equal divisions of the octave, with steps of 22.64 cents. Its 701.89-cent fifth is exceptionally close to 3:2. The Western modes are built by stacking its 31-step fifth.

55-EDO 55 notes

Fifty-five equal divisions of the octave, with steps of 21.82 cents. Its 698.18-cent fifth matches sixth-comma meantone. Eighteenth-century writers such as Tosi and Telemann described this tuning with a whole tone of nine parts and a diatonic semitone of five. The Western modes are built by stacking its 32-step fifth.

72-EDO 72 notes

Seventy-two equal divisions of the octave, with steps of 16.67 cents. It contains 12-EDO as every sixth step and comes close to several seven-limit and eleven-limit intervals. The Western modes sit on its 12-EDO subset. Miracle decimal, Blackjack and Canasta are 10-, 21- and 31-note scales built by stacking its 7-step interval of 116.67 cents, called the secor.

96-EDO 96 notes

Ninety-six equal divisions of the octave, with steps of 12.5 cents. These are Julián Carrillo's sixteenth tones. It contains 12-, 24- and 48-EDO, which Carrillo also wrote in.