19‑ET and the Hebrew calendar

19‑ET

Western music is based on three “consonant” intervals: the octave, the perfect fif‌th, and the major third. In the major scale – do, re, mi, fa, so, la, ti, do′ – the interval from do to do′ is an octave in size, the interval from do to so is a perfect fif‌th in size, and the interval from do to mi is a major third in size. There is a relation between these intervals: four perfect fif‌ths stacked on top of each other is the same size as two octaves plus a major third. If the terms perfect fif‌th and major third mean nothing to you, don’t worry: just regard them as arbitrary names and don’t chase the meanings of the individual words of which they are composed.

Notes an octave apart match so well to the ear that they are regarded as equivalent for some purposes, as for example the notes at the bottom and top of the major scale are both called do.

An interval between two notes sounds consonant, or sweet, if the ratio of the frequencies of the notes is the ratio of two small whole numbers, or is very close to such a ratio. If the ratio is exactly the ratio of two small whole numbers, the interval is said to be just, or justly intoned, or acoustically perfect. When an octave is tuned just, its frequency ratio is ²⁄₁. When a perfect fif‌th is tuned just, its ratio is ³⁄₂. When a major third is tuned just, its ratio is ⁵⁄₄. To clarify, the adjective perfect in the phrase perfect fif‌th has nothing to do with acoustic perfection: rather, it distinguishes the interval from the diminished fif‌th and the augmented fif‌th, which are different kinds of interval altogether and which will not concern us. With that clarification made, I will now drop the adjective perfect and refer to a perfect fif‌th simply as a fif‌th.

Tuning a keyboard instrument involves a compromise. We have seen that four stacked fif‌ths equals two octaves plus a major third. If the octaves and fif‌ths are tuned just, then each major third – four stacked fif‌ths minus two octaves – will be:

³⁄₂ × ³⁄₂ × ³⁄₂ × ³⁄₂ ÷ 2 ÷ 2
= ⁸¹⁄₆₄

But a justly intoned major third is ⁵⁄₄ = ⁸⁰⁄₆₄, not ⁸¹⁄₆₄: and a major third of ⁸¹⁄₆₄ sounds terribly wide and quite horrid. The solution is to temper all the fif‌ths: to narrow them slightly from just, until the major thirds sound reasonable. It is not necessary to temper the fif‌ths by much: since a major third is four fif‌ths minus two octaves, tempering all the fif‌ths by any given amount narrows the major thirds by four times as much.

Exactly how far to temper the fif‌ths is a musical judgement that has varied down the centuries. But a justly intoned fif‌th is a trifle larger than ⁷⁄₁₂ of an octave; and the modern solution, known as 12‑Equal Temperament, 12‑ET or simply Equal Temperament, is to temper each of the fif‌ths until its size is exactly ⁷⁄₁₂ of an octave. Then the octave consists of 12 small intervals of equal size known as semitones, each with a frequency ratio equal to the 12th root of 2, and the keyboard instrument can be played equally well in any key.

Figure 1 shows an octave in 12‑ET. Notice that so and mi are not quite in line with the right‑hand ends of the red lines. You can see:

• the 12 equal intervals that comprise the octave, known as semitones;
• the 13 notes that span the octave, including those at bottom and top;
• which of these notes the major scale do re mi fa so la ti do′ maps to;
• how the sizes of the fif‌ths and the major thirds compare with their justly intoned counterparts.

Note how the fif‌ths have been tempered only very slightly: you can see from Figure 1 that the fif‌th from do to so is only a whisker narrow of just. Consequently the major thirds (exemplified by the interval from do to mi) are still markedly wide of just. Indeed in 12‑ET the major thirds are wide of just by about 14 cents (14 hundredths of a 12‑ET semitone). They sound acceptable to our modern ears, but only because we are used to them. They would have sounded very peculiar to a sophisticated audience in the 18th century or earlier.

Figure 1 shows 13 notes, including the notes at bottom and top. But there are 12 notes per octave, not 13, because the notes at bottom and top are shared with the octaves below and above. Likewise the picture shows a major scale of 8 notes, including do and do′ (which is why it is called an octave), but the major scale has 7 notes per octave and not 8 when we eliminate the double counting.

If you have no training in musical theory, you may be puzzled as to why the major scale maps onto the particular notes shown. This is because the major scale is constructed from a chain of fif‌ths, beginning with fa: from fa to do′, from do to so, from so to re (up a fif‌th and down an octave), from re to la, from la to mi (again modulo an octave), and from mi to ti. No other 7‑note scale contains a greater number of consonant intervals, and no other 7‑note scale in 12‑ET has its notes distributed more evenly round the octave.

History favoured a temperament with 12 notes per octave, because existing keyboards had 12 notes per octave. If keyboards could have been redesigned from scratch, there would have been better alternatives. One of these better alternatives is 19 Equal Temperament or 19‑ET. A justly intoned fif‌th is a bit more than ¹¹⁄₁₉ of an octave, and in 19‑ET each of the fif‌ths is tempered so that its size is exactly ¹¹⁄₁₉ of an octave. The octave is thereby divided into 19 intervals of equal size.

Figure 2 shows an octave in 19‑ET. You can see:

• the 19 equal intervals (which are no longer called semitones);
• the 20 notes labelled 0 to 19;
• which of these notes the major scale do re mi fa so la ti do′ maps to;
• how the sizes of the fif‌ths and the major thirds compare with their justly intoned counterparts.

Note how the fif‌ths have been tempered more heavily than in 12‑ET: you can see from Figure 2 that the fif‌th from do to so is more narrow of just than it was in 12‑ET. Consequently the major thirds (exemplified by the interval from do to mi) are considerably narrower than in 12‑ET. Indeed now the major thirds are somewhat narrow of just. They are however much closer to just than they were in 12‑ET: they are about 7 cents narrow of just, whereas in 12‑ET they were about 14 cents wide of just. The main drawback of 12‑ET – its dubious major thirds – has been overcome.

In 19‑ET, as in 12‑ET, no other 7‑note scale has its notes distributed more evenly round the octave than the major scale has.

19‑ET is impractical for general keyboard use, because it requires a non‑standard keyboard with 19 notes per octave, or else an electronic keyboard with switching gear. Nevertheless it has its advocates, including the composer Joel Mandelbaum.

Finally, observe which notes the major scale maps onto: 0, 3, 6, 8, 11, 14, 17, 19. We shall meet these numbers again.

The Hebrew calendar

The average time from one March equinox to the next is known as the tropical year. The tropical year therefore comprises a complete cycle of the seasons. Because of the biological and socioeconomic importance of the seasons, the tropical year is the basis of most calendars. The tropical year is 365.2422 days to 7 significant figures.

Some calendars, such as the Islamic calendar, are based not on the year but on the phases of the moon. The average time from one new moon to the next is known as the synodic lunar month, and it is 29.53059 days to 7 significant figures. There are between 12 and 13 such lunar months in a year.

The Hebrew calendar is hybrid. It uses lunar months of 29 or 30 days, but it also uses “years” consisting of whole numbers of lunar months. In the Hebrew calendar, most years consist of 12 lunar months (about 354 days), but in every second or third year an “intercalary” month is inserted, to give a “leap year” consisting of 13 lunar months (about 384 days).

How often should an intercalary month be inserted? In the fif‌th century BC, the Greek astronomer Meton of Athens discovered that 19 tropical years is almost exactly equal to 235 synodic lunar months. Indeed they differ by less than 13 parts per million. The cycle of 19 years is known as the Metonic cycle after him. The cycle is believed to be a mere coincidence, and does not reflect any known physical resonance in the solar system.

The Hebrew calendar follows the Metonic cycle. In every cycle of 19 years, 7 of the years include an intercalary month. Thus every cycle of 19 years contains 19 × 12 + 7 = 235 lunar months, as the Metonic cycle demands.

The position of a given year in the Metonic cycle is calculated by dividing the Hebrew year number by 19 and taking the remainder. If the remainder is 0, it doesn’t matter whether we think of the position as 0 or as 19; traditionally it is labelled 19, but as a mathematician and a non‑Jew I find it more natural to think of it as 0.

And specifically which 7 of the 19 years have an intercalary month? You may have guessed. They are those labelled 0 (or 19), 3, 6, 8, 11, 14, and 17. Exactly the same numbers as the notes of 19‑ET that map to the major scale.

A series of coincidences

The above curiosity arises through a series of coincidences. 19‑ET and the Hebrew calendar are each based on a cycle of 19 nodes. In both cases, 7 of the 19 nodes are to be special. In both cases, the 7 special nodes fall to be distributed round the cycle of 19 as evenly as possible. And finally, the reference node of the Hebrew calendar, labelled 0 or 19, which could have been arbitrary, happens to be the one that corresponds to the bottom and top of the major scale.

Is this original?

I don’t know if I was the first person to spot this matter. I discovered it in 2009, and I made edits to the Wikipedia articles on 19‑ET and on the Hebrew calendar drawing attention to it. My edit on the 19‑ET page was soon reverted by a gentleman who found it inappropriate for an encyclopaedia article. At that time I could not find any mention of the matter on the Web. Now, 8 years later, there are mentions of the matter in a few places on the Web: but at least some of them are traceable to my work on Wikipedia.

Wikipedia stores permanently the entire history of every page, and you can find my lapsed contribution on the 11 August 2009 revision of its page on 19 equal temperament.