Tuning systems, and why equal temperament cannot represent them

What equal temperament actually did

Twelve-tone equal temperament divides the octave into twelve identical steps of exactly 100 cents. It has one enormous virtue and one enormous cost, and both are consequences of the same decision.

The virtue: every key sounds the same. You can modulate anywhere, transpose anything, and build a keyboard that plays all of it. The cost: no interval except the octave is in tune. The fifth is two cents narrow, the major third is fourteen cents wide, and every chord you have ever heard on a piano is a compromise that a violin section quietly corrects.

That trade is worth making for music that modulates constantly. It is a bad trade for music that does not — and most of the music on this page stays in one mode for an entire performance, which means it has no use for the flexibility and no reason to pay for it.

The problem is not “extra notes between the keys”

The common framing is that these traditions use quarter tones — notes halfway between the piano keys — and that a fine enough grid would capture them. It would not, for three separate reasons, and each one defeats a different proposed fix.

1. The intervals are not halves of anything. A Persian neutral second is around 125–145 cents, not 150. A Turkish segâh is 384 cents, not 350.
2. They are ranges, not values. Egyptian and Syrian players place the same degree 14 cents apart, and both are correct. A grid has to pick one.
3. The same written note moves depending on what it is doing. An E half-flat as the third of Rast and an E half-flat as the second of Bayati are the same notehead and different pitches. No equal division of any size can express that, because an equal division has exactly one E half-flat.

1. Turkish makam — the comma system

Turkish art music is codified by the Arel-Ezgi-Uzdilek system, which counts the octave in 53 commas. Its intervals are Pythagorean: stacked perfect fifths, exactly as European music was before meantone.

Three makams built entirely from that vocabulary:

Ahenk: the pitch level, set by the ney

One more variable that has nothing to do with the scale. Turkish art music is written at a fixed nominal pitch — the degree called rast is written as G — and sounds wherever the ney in the room dictates, because every other instrument can be retuned or refretted and the ney cannot. The levels are named after ney lengths:

2. Arabic maqam — and the 24-quarter-tone simplification

The 1932 Cairo Congress of Arab Music adopted a division of the octave into 24 equal quarter tones. That convention is why an Arabic score can be typeset at all: it gives the half-flat a notehead and a fixed meaning of 50 cents.

It is a notation, and it was adopted knowing it was not a measurement. We offer it on this site as an explicit, labelled simplification — the quarter-tone marker set on the playable oud is exactly that — and the reason it needs a label is below.

Fourteen cents separates Cairo from Damascus, which is comfortably audible and identifies a player’s school the way an accent identifies a speaker. The Turkish degree of the same name is 35 cents above the theory and is not a quarter tone in any sense — it is a flattened major third.

Small whole-number ratios make a useful ruler for reading those numbers. These are arithmetic rather than measurements, and nobody claims a player aims at them:

Read against that, the Egyptian 342 and Syrian 356 both sit near the simple neutral thirds, and the Turkish 385 sits near none of them. The traditions are not disagreeing about where a quarter tone goes; they are using different intervals.

Per maqam family — and where we stop

Rast: Fourteen cents between Cairo and Damascus is comfortably audible and identifies a player the way an accent identifies a speaker. The Turkish degree is 35 cents above the theoretical value and is not a quarter tone in any sense.

Bayati: Reported consistently BELOW the 150 cents that 24-EDO notation implies, and below the corresponding degree of Rast, even though both are written as the same half-flat. The direction of the difference is well attested; the size of it varies by region and by performer. No cent value is printed because this repo has not reviewed one. Marcus (1993) is the source to measure against; until an entry here carries a figure and a page reference, the page states the direction and not a number.

Sikah / Huzam: The family whose tonic IS the neutral degree, so every question above applies to the tonic rather than to a note inside the scale — which is why a Sikah piece cannot be transposed onto a piano at all, not even badly. The absolute placement of the tonic follows whichever school the player belongs to; the Rast measurements above are the nearest reviewed figures this repo has.

Saba: Saba is the family that breaks the octave: its fourth degree is lowered, so the scale does not repeat at 1200 cents in the way the others do. That structural fact is not in dispute; the exact sizes are. No cent values shipped. A reviewed measurement of the Saba tetrachord would complete this entry.

3. Persian dastgah — koron and sori

Persian notation carries two accidentals Western notation does not: the koron, which lowers, and the sori, which raises. Ali-Naghi Vaziri’s early-20th-century reform defined both as exactly half a semitone — 50 cents — which makes the system 24-EDO and makes a koron the same object as an Arabic half-flat.

Hormoz Farhat rejects that outright. In The Dastgāh Concept in Persian Music (Cambridge, 1990) he argues Persian music has no fixed interval sizes at all, and describes practice in terms of a handful of sizes rather than a grid:

Work the arithmetic through and the consequence is concrete. A koron applied to a whole tone lowers it by 59 to 79 cents, not fifty — because 204 minus a neutral second of 125 to 145 is 59 to 79. So a Persian koron and an Arabic half-flat, written almost identically, are not interchangeable pitches.

Farhat’s wider point is not a footnote to his numbers, it ishis argument: the sizes move with the performer, with the teaching line, and with where in the dastgah the note falls. He treats that variability as the system’s defining property rather than as imprecision in it — so a page printing one number for a koron would be contradicting its own best source.

4. Indian shruti — a reference frame, not a tuning

The octave is traditionally described as 22 shrutis. The ratio table below is the one reproduced in modern Indian music theory and in most Western writing about it — but three things have to travel with it.

1. The Natyashastra gives no ratios. It describes 22 shrutis and assigns intervals of 4, 3 and 2 of them to the notes of the scale, and never states an absolute size. Every ratio table is a later reconstruction, and the common ones disagree — this list uses 45/32 and 64/45 for the tritone pair where others use 729/512 and 1024/729.
2. The shrutis are not a tuning. No instrument is fretted to 22 notes. A performer tunes a drone and places each note of the raga by ear against it, which is why the same swara sits differently in different ragas.
3. What is tuned precisely is the drone — and that is the part a dial genuinely helps with.

The tanpura, which really is tuned to targets

The fourth string is the tonic an octave down, the middle two are the tonic, and the first takes the degree the raga wants — at a pure ratio, not a tempered one:

Two cents sounds like nothing and is not. A tanpura sustains, and against a sustained drone a two-cent error is an audible beat rather than a rounding error — the drone pulses instead of sitting still. This is the one place on the site where following the needle to the centre makes the instrument sound worse. The tanpura tuner computes all four strings at pure ratios from whatever Sa you choose.

5. Gamelan — slendro and pelog

The rule is stronger here than anywhere else on this page: there is no such thing as “the” slendro or “the” pelog.A gamelan is tuned as a set, by its maker, and its scale belongs to that instrument. Two gamelans in the same city are not interchangeable and are not expected to be — players discuss a gamelan’s embat, its particular tuning character, as an identifying trait rather than as a deviation from anything.

Which means a tuner cannot tell you that a gamelan is out of tune. It can tell you what the gamelan in front of you is, which is a different and more useful job.

Slendro: Five steps per octave, near enough to equal that the equal-step model is a useful first approximation and far enough from it that no real gamelan matches. Measured instruments vary by tens of cents per degree, and that variation is the instrument’s identity rather than an error in it. The equal-step idealisation is 240 cents a step — 0, 240, 480, 720, 960, 1200 — and it is shipped here labelled as an idealisation, because it is what people mean when they say “slendro is five equal steps” and it is worth being able to hear how far a real instrument is from it.

Pelog: Seven unequal steps per octave, of which any one piece uses five, selected by pathet. The steps are genuinely uneven — some near a semitone, some wider than a whole tone — so no equal division of any size approximates it.

All five, at a glance

Sources

Every cent value on this page is either computed from a ratio in lib/tuning-systems.ts or carries a source there. Where this site does not have a reviewed figure, it says so rather than filling the gap. The reading behind each system:

• 12-tone equal temperament — ISO 16 (concert pitch); the equal division is arithmetic, not a measurement..
• Turkish makam — Arel-Ezgi-Uzdilek — H. S. Arel, Turk Musikisi Nazariyati Dersleri (1930s); Karl Signell, Makam: Modal Practice in Turkish Art Music (1977); Ozan Yarman, on pitch notation in Turkish makam music (2007-08).
• Arabic maqam — Cairo Congress of Arab Music, 1932, for the 24-tone convention; Scott Marcus, "The Interface between Theory and Practice: Intonation in Arab Music", Asian Music 24/2 (1993); Habib Hassan Touma, The Music of the Arabs (1996).
• Persian dastgah — Ali-Naghi Vaziri, for the koron/sori notation; Hormoz Farhat, The Dastgah Concept in Persian Music (Cambridge, 1990).
• Indian shruti — Bharata, Natyashastra, ch. 28; Alain Danielou, Introduction to the Study of Musical Scales (1943).
• Gamelan — slendro and pelog — Surjodiningrat, Sudarjana and Susanto, Tone Measurements of Outstanding Javanese Gamelans in Jogjakarta and Surakarta (1972).

Tune the instrument

Every one of these systems has a page here where it stops being theory.

• Oud — Arabic, Turkish and Iraqi tunings side by side, with fretless intonation targets for each degree of Rast.
• Tanbur — The Turkish comma system as fifty tied frets, with the millimetre positions worked out.
• Bağlama / saz — Why a düzen is a relationship rather than a set of notes.
• Ney — The instrument that sets everyone else’s pitch and cannot be tuned itself.
• Qanun — Diatonic pegs, and mandal levers that produce every microtone afterwards.
• Tar — Movable gut frets, and the Persian neutral second in millimetres.
• Setar — Koron and sori, and how far a koron really lowers a note.
• Santur — Retuning a 72-string instrument by a quarter tone to change dastgah.
• Kamancheh — Fretless, so every neutral interval is the hand.
• Tanpura — A pure 3/2 fifth, and why a twelve-tone tuner will argue with it.
• Sitar — Sa is yours, and eleven sympathetic strings follow the raga.
• Sarod — A fretless metal fingerboard with no markers at all.

And if you want to hear the difference between an Egyptian and a Syrian sikah rather than read about it, the playable oud draws both as a band on a fretless neck, because a marker can show a range where a fret has to pick one.

Frequently asked questions