The Evenest Way

What the dial does

A perfect fifth is the interval a string makes when you stop it at two thirds of its length. Start on C, go up a fifth to G, up another to D, then A, E, B, and bring each one back down by octaves until it sits inside the same octave as the C. On a circle where one lap is one octave, every new note is the last one turned by the same angle: 701.955 cents out of 1,200, which is log₂(3/2) of a turn.

Now look at the gaps between neighbouring notes. With 7 fifths you get the major scale's two steps: five whole tones of 203.910 cents (the ratio 9:8) and two half steps of 90.225 (the ratio 256:243). With 8 fifths a third size appears. With 12 you are back to two: 113.685 and 90.225, the chromatic scale's two unequal semitones. The strip under the dial lights every count where there are exactly two sizes. For the pure fifth they are 2, 3, 5, 7, 12, 17, 29, 41, 53, and every count in between has three.

Never four. That is the three-gap theorem, conjectured by Hugo Steinhaus and proved in the late 1950s by Vera Sós, János Surányi and Stanisław Świerczkowski: however many marks you drop at a fixed step, the gaps take at most three lengths, and when there are three the longest is exactly the other two added together. Only Three Gaps lets you drop the marks yourself. This page asks the next question: at which counts is the third length missing?

The two-step moments of the pure fifth

Two columns are worth reading down. The short step at 17 is 23.460 cents, the Pythagorean comma, the amount by which twelve fifths overshoot seven octaves (The Comma lets you hear it). And the ratio of long step to short is nearest to one at 12 (1.260) and at 53 (1.182). Those are the two counts where the chain is closest to cutting the octave into equal pieces, which is why 12-tone and 53-tone equal temperaments are the famous ones. At 7 the ratio is 2.260 and at 17 it is 3.846: two sizes, but far from equal. Keep going past 53 and the next mark opens a third gap of exactly 3.615 cents, the amount by which 53 fifths overshoot 31 octaves. Jing Fang's 54th pitch is that mark. His table's rounded integers put it 3.63 cents from the start.

Fifty-three was reached another way as well. William Holder's Treatise of 1694 reports that Nicholas Mercator, working with logarithms, proposed 1/53 of an octave as an "artificial comma" to measure every interval by. That unit is now called the Holdrian comma, and Turkish makam theory still counts its 24 Pythagorean pitches in it.

Why those counts and not others

Write the fifth as a fraction of the octave and expand it as a continued fraction: log₂(3/2) = [0; 1, 1, 2, 2, 3, 1, 5, 2, 23, …]. Cut it off at each term and you get the best fractions for the fifth: 1/2, 3/5, 7/12, 24/41, 31/53, 179/306. The counts where the chain has two step sizes are exactly the bottoms of those fractions (2, 5, 12, 41, 53, 306) together with the in-between fractions a term larger than 1 allows (3, 7, 17, 29, 94, 147, 200, 253, 359). The verifier checks that match for every count up to 400, for each of the steps on this page, by exact arithmetic. The long pause after 53 is the partial quotient 23: it makes 31/53 an unusually good fraction, and it leaves no new two-step count until 94.

So a two-step moment is a place where the step has just been caught by a fraction as well as it will be caught for a while. The chain has used up one good approximation and not yet started on the next. Erv Wilson was calling these scales "moments of symmetry" by 1975, and Norman Carey and David Clampitt (1989) called them well-formed.

Narrow the fifth and the counts move

Many Renaissance and Baroque keyboards did not tune pure fifths. Quarter-comma meantone narrows each fifth to 696.578 cents so that four of them land on a pure major third. Press that preset and 17, 29, 41 and 53 go dark. The two-step counts become 5, 7, 12, 19, 31, then 50 and 81, and the one where the steps come nearest to equal is 31 (long ÷ short 1.173). Narrow it further, to the third-comma fifth of 694.786 cents, and the nearest to equal is 19, where the two steps differ by only a point and a half per cent (1.015). Francisco de Salinas (1577) described a keyboard of 19 notes to the octave in third-comma meantone, and its fifth is within 0.049 cents of the fifth of 19 equal steps. Christiaan Huygens (a letter of 1691; the Latin version was published in 1724) argued that 31 equal steps come very close to quarter-comma meantone, and they do: the fifths differ by 0.196 cents. A similar division is implicit in Nicola Vicentino's work of 1555.

The pentatonic, the diatonic and the chromatic survive all of this. Any fifth between 685.714 and 720 cents (4/7 and 3/5 of an octave) has two step sizes at 5, 7 and 12 notes, because every step in that window has a continued fraction that begins [0; 1, 1, 2, …], and those first terms are what fix the first few counts. Drag the dial just below the window, to 685 cents, and 12 drops out; drag it just above, to 721, and 7 does. Only the larger systems depend on exactly which fifth you tuned.

The same dial is a sunflower

Put a seed at the centre, turn by the step, move out a little (to radius √k for the kth seed) and put another. That is Helmut Vogel's 1979 model of a sunflower head, and with the golden angle, 137.508° of a turn, it packs the seeds with no visible spokes. Its continued fraction is all ones, so it has no good fractions at all, and its two-length moments are 2, 3, 5, 8, 13, 21, 34, 55, 89: the Fibonacci numbers.

Now press the pure-fifth preset and look at the flower again. It grows arms. The panel under the dial reports each seed's nearest neighbour, and around the thousandth seed it is 53 seeds away, every time: the flower has 53 arms, because 31/53 is the fifth's good fraction. Quarter-comma meantone gives 31 arms and third-comma gives 19. The 12-tone fifth gives 12 straight spokes, because 7/12 is not an approximation but the step itself. The golden flower's nearest neighbours are Fibonacci numbers: 89 seeds away around the thousandth seed, and by the two-thousandth split between 89 and 144 as the next spiral count takes over. The fifth's flower keeps its 53 over that whole range. A sunflower cannot use a musical fifth for the same reason a keyboard can: the fifth is easy to approximate by a fraction. The Most Irrational Number has the other half of that story.

Evenest, on twelve keys

On a piano the fifth is exactly 7 of 12 semitones, and the chain closes: the 12th fifth up lands back on the start. Here a stronger idea than "two step sizes" becomes available. John Clough and Jack Douthett called a set of k notes out of q maximally even when every span of i consecutive members covers either ⌊iq/k⌋ or ⌈iq/k⌉ steps and nothing else. Chains of piano fifths are maximally even for 1, 5, 7, 11 and 12 notes. Chains of 2, 3, 8, 9 and 10 have two step sizes but are not evenest (two fifths, C and G, leave gaps of 5 and 7 semitones where 6 and 6 would be evener). Seven fifths from F are the white keys and five fifths from F♯ are the black keys: the two maximally even sets everyone has touched.

Maximally even sets are also rhythms. Spread 7 beats over 12 pulses as evenly as possible and you get the same necklace as the white keys, rotated. Godfried Toussaint (2005) pointed out that this 7-in-12 rhythm is a common West African bell pattern (his example is the mpre of the Ashanti), and one of its rotations is the Afro-Cuban bembé. Press "play it as a rhythm" with 7 marks of the 12-tone fifth to hear the white keys as a bell pattern. The Algorithm That Drums builds these rhythms with Euclid's algorithm, and Each Interval, a Different Number of Times shows the other thing that makes the white keys rare.

What this does not show

That a count is a two-step moment does not mean a culture chose it for that reason, and many didn't. Bharata's Nāṭyaśāstra counts 22 śrutis to the octave and gives the steps between notes in śrutis (4, 3, 2, 4, 4, 3, 2), but no ratios and no chain of fifths. The 24 equal quarter-tones that Mikhāʾīl Mishāqa described around 1840 are an equal division, not a chain. Real sunflowers are untidy as well: in a citizen-science count of 657 heads (Swinton and others, 2016), roughly one spiral count in five had no Fibonacci structure at all. The theorem says where a chain of one interval comes out in two sizes. It only speaks to scales that were built as such a chain. Where a theorist did build one, the table shows where it stopped.

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