A scale is not a set of pitches

Raga Bhupali uses the notes Sa, Re, Ga, Pa, Dha. Raga Deshkar uses the notes Sa, Re, Ga, Pa, Dha. The two sets are identical — the common pentatonic, in Western terms — and the two ragas are not the same raga. They are performed at different times of day, they have different characteristic phrases, they weight different notes, and a listener familiar with the tradition will not confuse them.

If a mode were a set of pitches, that would be impossible.

What the two ragas differ in #

The distinctions are not subtle to anyone inside the tradition and they are all invisible to a pitch set.

Which notes are structural. Hindustani theory names a vadi, the note a raga leans on most, and a samvadi, the second. Bhupali’s are Ga and Dha; Deshkar’s are Dha and Ga. That is the same pair with the emphasis reversed, and the effect is that Bhupali sits in the lower half of its octave and Deshkar in the upper.

Which notes appear where. Deshkar’s ascent commonly omits Re — Sa, Ga, Pa, Dha — while its descent uses all five. Bhupali uses all five in both directions.

Characteristic phrases. Each raga has a pakad, a short figure that identifies it. Two ragas on one pitch set are distinguished in performance largely by these, and a phrase is an ordering, which a set does not have any more than a chord has a voicing.

Intonation. Performers of the two are widely described as shading Ga and Dha differently — Deshkar’s slightly higher. The published measurements of this are thin and inconsistent, so it is stated here as what the tradition says rather than as a number, and it belongs on the list of things a twelve-position ruler cannot show.

Which computation produced the number #

Everything drawn in these figures comes from the tradition’s own units, converted into cents.

Hindustani theory locates degrees within a framework of twenty-two śruti; the five notes of Bhupali fall on positions that correspond, in the standard modern reckoning, to 0, 200, 400, 700 and 900 cents. Those are the values plotted.

The Turkish system works in a different unit. Its degrees are counted in Holdrian commas — one fifty-third of an octave, $1200/53 = 22.6415$ cents. Maqam Rast in the Arel–Ezgi–Uzdilek codification is the sequence 9, 8, 5, 9, 9, 8, 5 commas, which sums to 53 and therefore closes the octave exactly. Accumulating them gives 0, 203.8, 384.9, 498.1, 702.0, 905.7 and 1086.8 cents, and those are computed from the comma counts rather than transcribed.

The Arabic codification of the same maqam works in quarter-tones — twenty-four equal steps of 50 cents — and gives Rast as 0, 200, 350, 500, 700, 900, 1050.

Both of those are drawn from the theory that produced them, and the interesting thing is what happens when they are drawn on the same axis.

Thirty-five cents is not a rounding error #

The two systems disagree about the third degree of Rast by 34.9 cents. That is more than a third of a semitone, larger than any comma in this site’s tuning essays, and unmistakable in a sustained note.

The disagreement is not a mistake by either party. It is what happens when two committees are asked to write down an oral practice in a fixed notation, and choose different fixed notations.

The Arabic convention comes out of the Cairo Congress of Arab Music in 1932, which adopted twenty-four equal quarter-tones — a decision that had the enormous practical advantage of being writable, teachable and printable. It has the disadvantage that the intervals of Arabic practice are not quarter-tones, and everyone involved knew it. The 350-cent third is a notational position, not a measurement.

The Turkish system, codified in the 1920s and 30s, chose 53 divisions instead, and arrived at a Rast third of 384.9 cents, which is within 1.4 cents of a pure 5:4. That is a much more plausible number acoustically and it is not obviously more accurate as a description of performance.

Two traditions naming one maqam disagree about where its degrees are, and neither disagreement is visible on a keyboard. A set of pitches cannot record which of them is meant, because the difference between the two readings is smaller than the grid the set would have to be written on.

Measurements of actual performance land between the two and vary by region, by performer and by context. That is the point rather than an embarrassment: the third of Rast is a region that is approached and inflected, and both codifications flatten it to a point in order to be able to write it down.

The disagreement is two decisions, not seven #

The claim that the Turkish third is “more plausible acoustically” was made above from one degree. Made from all seven it survives, and it also changes shape, because the two codifications turn out not to disagree about the scale at all — they agree about five of its seven degrees and disagree about two, twice, in the same direction.

The Turkish Rast is a Pythagorean scale with its third and seventh lowered by exactly one Holdrian comma. Degrees two, four, five and six sit within 0.2 cents of their Pythagorean values — that is a chain of pure fifths, which is what a 53-fold division is built to hold — and degrees three and seven sit one comma below theirs, which lands them within 1.5 cents of a pure 5:4 and a pure 15:8. It is not a just scale: its sixth is Pythagorean and therefore twenty-one cents above 5:3. It is a Pythagorean skeleton with two degrees deliberately moved.

The Arabic Rast makes the same two moves and makes them larger. Its third is fifty cents below the equal-tempered third and its seventh fifty cents below the equal-tempered seventh, and its other five degrees are the equal-tempered ones. So both academies singled out the same two degrees and lowered both; they disagreed about how far, by twenty-seven cents.

That reframes the thirty-five. It is not two rulers disagreeing about where a scale falls. It is one agreed structural decision — the third and the seventh of Rast are lower than the diatonic ones — implemented at two sizes.

And the size is not forced by either notation. Averaged over all seven degrees the Turkish values sit 3.5 cents from the just ones and the Arabic values 14.0 — but the closest a quarter-tone system could come, choosing the nearest available step at every degree, is 7.0 cents. Four hundred cents was in the Cairo system and was not chosen for the third. The Arabic codification is twice as far from the simple ratios as its own notation permitted, which is as clear a statement as one could want that its 350 is not a failed approximation to 386 but a different note being named.

The same failure inside European practice #

Before this becomes a story about distant traditions, it is worth noticing that the set-of-pitches model fails in Europe too, and for the same reason.

The church modes are conventionally presented exactly as this figure presents them: one collection, seven starting points. That presentation is a Renaissance theoretical tidy-up of a much older practice, and the practice distinguished modes by things the tidy-up discards — the finalis a chant ends on, the repercussa it recites on, the ambitus it moves within, and the formulae by which it opens and closes. Two chants on the same seven notes with different reciting tones were different modes, exactly as Bhupali and Deshkar are different ragas.

The authentic and plagal pairs make the point sharpest. Dorian and Hypodorian have the same final and the same collection; they differ in the range the melody occupies and in the note it recites on. As pitch sets they are indistinguishable. As modes they were never confused.

What happened next is the interesting part. As European music moved toward key rather than mode, the behavioural apparatus was gradually replaced by a harmonic one — a mode’s identity moved into its chords and cadences — and the surviving residue of the old system is the eight-note collection with a name attached. The set-of-pitches model is what is left after the grammar was relocated, not what was there first.

The tuning question underneath #

There is a second thing a pitch set cannot record, and it is one this site is otherwise well equipped to talk about.

The seven note names of a European scale do not fix seven frequencies. Which frequencies they name depends on the temperament, and the spread between historically ordinary answers reaches sixteen cents — half the disagreement between the two codifications of Rast, arrived at without leaving one continent or one instrument.

So the difference between the European case and the others is a difference of degree in one respect and of kind in another. In degree: everybody’s notation under-determines the pitches. In kind: European practice eventually built instruments that fixed them and a notation that assumed they were fixed, and then mistook that arrangement for what a scale is.

What a mode is instead #

Both traditions have a word for the thing a pitch set leaves out.

In Turkish practice it is seyir: the characteristic path a maqam takes — where it starts, which degrees it dwells on, how it ascends, where it turns, how it closes. Two maqamat can share a scale and differ entirely in seyir, and when they do, the seyir is what the name refers to.

In Hindustani practice the equivalent apparatus is aroha and avaroha, the vadi and samvadi, the pakad, and the chalan — the characteristic movement. A raga is often described as having a personality, and the word is doing real work: it names a set of behaviours, not a set of positions.

A useful way to put it is that a European scale is a domain and a maqam or raga is closer to a grammar. The scale says which notes are available and stops. The grammar says what is done with them, and the availability is one clause of it.

Drawn as a cycle of twelve filled positions the two ragas are the same object, and every property the census measures says so. Drawn as a path they are not. That is the difference between a scale and a set of pitches in one picture, and it is the whole of this essay’s title.

A word that gets in the way #

The intervals in these figures are routinely called microtonal, and the word does more harm than any of the numbers.

It means “smaller than a semitone”, which defines these traditions by reference to a semitone they do not use. Maqam Rast’s third is not a sharpened minor third or a flattened major one; it is a degree in its own right, with a name — sīkāh — that is older than any European theory of thirds. Calling it microtonal is like calling a metre a micromile.

The framing has practical consequences. It encourages the idea that these systems are twelve-tone systems with corrections applied, which leads directly to the assumption that a keyboard can play them with a bit of retuning. It cannot, and not because the pitches are hard to hit: the pitches are the least of what is missing, as the two ragas on one pitch set already showed.

A cleaner way to say what is going on is that all of these traditions — including the European one — have more pitch material available than their notation records, and differ in how much of the difference matters. European practice minimised it deliberately, by fixing twelve pitches and making them transposable, and gained an enormous amount: an instrument that plays in every key, a notation anybody can read, and an orchestra that can be assembled from strangers. What it gave up is the material this essay is about, and the trade was a good one for the music it was made for.

Where the model stops #

This essay is still drawing cents rulers. Every figure above places a tradition’s degrees on a horizontal axis calibrated in cents and compares them to a keyboard. That framing is itself the European one, and using it to criticise European framing is a limitation worth stating. It is used because it is the framing this whole site is built in, and because the alternative — describing a raga in its own terms — is a different kind of writing.

Maqam and raga are not the same kind of object. They are grouped here because both defeat the set-of-pitches model, and that is all they have in common for present purposes. The systems differ in almost every other respect, including how modulation works, what improvisation consists of, and whether the tonic is fixed.

The Pythagorean reading of the Turkish values is a description, not a claim about intent. Four degrees landing within 0.2 cents of a chain of fifths is not a coincidence — a 53-fold division exists because its fifth is 0.07 cents from pure, so any scale built in it will tend to fall on that chain — but whether Arel and his colleagues chose those commas because they are Pythagorean, or arrived at them from the earlier comma theory that already held them, is a question about documents rather than about arithmetic.

The codified values are not performance values. The numbers in these figures come from theoretical writings. What performers do differs from them, is not uniform, and has been measured only patchily. Any claim that a raga’s Ga “is” 400 cents is a claim about a document.

Two ragas on one set is not the general case. Bhupali and Deshkar are the standard textbook example precisely because the coincidence is striking. Most pairs of ragas differ in their pitch sets as well, and the argument here does not need them to.

Nothing here is about fixed-pitch instruments. All of the above concerns traditions in which pitch is continuous, or in which fretting and tuning are adjusted to the piece. The moment a fixed-pitch keyboard enters, the tradition acquires all the problems European tuning has, and the twentieth-century history of the Arabic and Turkish systems is substantially a history of that arrival.

Whose music, and when #

The codifications are recent, and their datedness is part of what they are.

The Cairo Congress of 1932 was convened by the Egyptian government and attended by, among others, Béla Bartók, Paul Hindemith and Erich von Hornbostel alongside musicians from across the Arab world. It made recordings, argued about scales, and adopted the twenty-four-tone framework partly because it was compatible with Western staff notation and Western instruments. The delegates who objected — including several who argued that the quarter-tone was not what anybody played — lost.

The Arel–Ezgi–Uzdilek system in Turkey was consolidated over the 1920s and 30s from earlier comma-based theory. Its 53-fold division has a long ancestry in the region and connects to a much older tradition of describing intervals as counts of a small unit.

Hindustani and Carnatic theory both trace to the Natya Shastra, conventionally placed between 200 BC and 200 AD, whose twenty-two śruti remain the framework. What the śruti actually were is contested; the modern reckoning is a reconstruction, and rival reconstructions differ.

Gamelan is the case that goes furthest. There is no standard pitch, no standard tuning, and no expectation of one; instruments are made as a set and tuned to each other, and the resulting character of a particular gamelan is valued. Asking which pitches slendro consists of is like asking what colour a forest is — answerable only for a particular one, and the variation is not noise.

The general lesson is not that European theory is parochial, which is unsurprising, but that the specific abstraction it settled on — a mode as an unordered set of pitch classes — was available to it only because twelve equal semitones and free transposition had already made ordering and inflection unrepresentable. The abstraction is a consequence of the notation, and it travels badly because the notation does.

The ladder from here #

Later rungs on this anchor: the maqam families and how modulation between them works, which has no European counterpart. The 22 śruti and the reconstructions that disagree about them. Measured intonation in performance — what has actually been recorded and analysed, and how much of it supports the theory. Gamelan tuning as a design practice, and the embat variation between instrument sets. And the question this essay raises and does not answer: what a notation would look like that could carry seyir, and why nobody has built one that the traditions themselves adopted.

Two ragas, one pitch set, and everything that matters is in the part that was not written down.