Memory chips, log tables, Beethoven’s 3rd piano concerto and Who wants to be a millionaire?
– an interactive essay
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Who wants to be a millionaire?
“Here is your cheque for 64 thousand pounds,” grins Chris Tarrant, host of the TV game show Who wants to be a millionaire? “Will you take it home, or will you try to double it to 125 thousand pounds?”
But twice 64 is 128, not 125. On the face of it, Chris has got his sums wrong.
What mathematically oriented child has not lain in bed puzzling over the powers of 2, mentally computing 1, 2, 4, 8, 16, 32, 64… all the way up to 1024? And what such child has not found it fascinating that 1024 is so close to 1000? I was such a child. Or perhaps I mean I wasn’t: these double negatives are so confusing. Either way, I began halving those numbers again to discover the source of the approximation. I reasoned: 1024 is nearly 1000; because 512 is nearly 500; because 256 is nearly 250; because 128 is nearly 125.
But why is 128 nearly 125? We cannot delve in search of a deeper reason. As the mathematician Leopold Kronecker put it, God gave us the natural numbers. To delve deeper would be to delve into the mind of God. Yet 128 is 27, while 125 is 53: and it offends against my intuition for the harmony of mathematics to suppose that two such interesting numbers lie so close by pure chance. When two interesting numbers lie close, interesting things happen. And God knew it when he put them there.
If God was no fool, then neither was Chris Tarrant. He “doubled” 64 thousand to 125 thousand so that three further doublings would make a million. He could have doubled his money exactly, but “Who wants to be a One‐million‐and‐twenty‐four‐thousand‑aire?” doesn’t sound quite so snappy.
The Greek word for a thousand is chilioi, from which we get the prefix kilo‑. A kilometre is 1000 metres, and a kilogram is 1000 grams. A kilobar is 1000 bars, and a kilocycle is 1000 cycles (but probably not 500 bicycles). Amounts of data are measured in bytes, and a kilobyte is 1000 bytes. Except that perhaps it’s 1024 bytes, because physical storage devices like memory chips generally have a capacity, in bytes, of some power of 2. Who knows? Some sources, mainly the theoretical ones, say that a kilobyte is properly 1000 bytes, loosely 1024. Some others, mainly the practical ones, say that a kilobyte is properly 1024 bytes, loosely 1000. Some say that a kilobyte means 1000 bytes when it refers to data transfer, but 1024 bytes when it refers to data storage. Some say that a kilobyte with a lower‐case k is 1000 bytes, and a Kilobyte with an upper‐case K is 1024 bytes. Some say that a kilobyte (or Kilobyte) is 1000 bytes, and use the term kibibyte for 1024 bytes. What a mess! – and all because 128 and 125 are nearly the same but not quite. The confusion escalates for higher powers of 1000. A megabyte may be 1,000,000 bytes, or 220 = 1,048,576 bytes, and I have even seen the term used of the hybrid 1,024,000 bytes. A gigabyte may be 1,000,000,000 bytes, or 230 = 1,073,741,824 bytes, or conceivably either of two hybrids (“tribrids”?) in between.
Tables of logarithms are nearly obsolete. I have just used today’s most popular Internet search engine to search for “log tables”, and the most prominent search results are adverts for rustic wooden furniture. But when I was at high school in the 1960s, before the days of pocket calculators, we all routinely used 4‑figure log tables to the base 10 for what we should nowadays call floating‐point multiplication. (For the uninitiated modern reader, I should explain that logarithms are used to transform a messy multiplication into a straightforward addition.) Anyone who has used such log tables is certain to have noticed that log(2) is strikingly close to 0.3 (0.3010, I recall). And why is that, I wonder? It’s because:
log(2) ≈ 0.3
log(210)= 10 × log(2)
≈ 10 × 0.3
210 ≈ 1000
1024 ≈ 1000
128 ≈ 125
Beethoven’s 3rd piano concerto
If you are an experienced mathematician, then perhaps all these equivalences have been obvious to you. Kids’ stuff, you are thinking. If so, may this last equivalence rouse you from your slumber:
128 ≈ 125
27 ≈ 53
and dividing both sides by 26
2 ≈ (5⁄4)3
Now think musically. Two notes with a frequency ratio of 2 : 1 span an octave, and two notes with a frequency ratio of 5 : 4 span a major third. We have proved that three major thirds are approximately one octave.
Did I say “approximately”? But surely, since circular tuning systems such as equal temperament became standard in the 19th century, three major thirds have been exactly one octave? Yes indeed! We have just shown that circular tuning systems are founded on the assumption that 128 = 125.
To understand this, we need one small corner of musical theory.
The names of successive notes on a keyboard are shown in Figure 1.
If you are not familiar with these names, don’t worry: just think of them as arbitrary labels. The interval between successive notes is called a semitone. Do not assume, at this stage, that the semitones are all of the same size. Click on Figure 1 for a demonstration with animation and sound.
Twelve successive semitones make an octave. As already mentioned, two notes that span an octave have a frequency ratio of 2 : 1. Notes an octave apart “match” so well to the ear that for some purposes they can be regarded as equivalent. Indeed they have the same name, so that if one note is an A, for example, then the note an octave above or below it is an A also, as shown in Figure 1. Motivated by that equivalence, we can join the ends of the scale to make a circle, as in Figure 2. Click on Figure 2 for a demonstration.
After the unison and the octave, the greatest musical consonance is an interval known as the perfect fifth, which consists of 7 consecutive semitones. I’ll just call it a fifth. For completeness I would mention that an acoustically pure fifth has a frequency ratio of 3 : 2, though we will not be using that fact. Figure 3 shows the “chain of fifths”, linking each note with the note that is a fifth above it.
Notice that the chain is not a complete circle. The interval from G♯ to E♭ is not a fifth. It happens to be close to a fifth but, as we have described things so far, it is not in general close enough for making music. Click on Figure 3 for a demonstration.
The chain of fifths does not merely show relationships between notes. It also shows relationships between tonalities. The tonality of a piece of music can be thought of as the network of chords that provide its harmonic context. The tonality of a piece of music depends largely on its tonic or keynote (do, the note that feels like home) and on its mode (major or minor, to oversimplify somewhat). The keynote and the mode together are said to comprise the key. So for example G major is a key, and E minor is a key. Keys with closely related tonalities are shown close together in Figure 4. Major keys are shown in upper case letters, minor keys in lower case.
In tuning a keyboard instrument, making the octaves and the fifths sound good is not sufficient. It is necessary also to make the major thirds sound good. A major third is made of four successive fifths. (In a general context I should rather say that four successive fifths make two octaves plus a major third: but here we continue to factor the octaves out.) As already mentioned, an acoustically pure major third has a frequency ratio of 5 : 4. Two of the major thirds – from C to E and from E to G♯ – are shown in Figure 5. They cannot be completed to form a triangle: the interval from G♯ to C is not a major third, though it is close to a major third. Click on Figure 5 for a demonstration.
To make the fifths and the major thirds sound good entails compromising the acoustic purity of either the fifths or the major thirds or both. There are many tuning systems that are designed to do this: many temperaments, as they are called. Until the end of the 18th century, the most commonly used temperament was the “quarter‐comma mean‐tone” temperament, which compromised, or tempered, all the fifths by equal amounts in order to keep the major thirds acoustically pure.
Quarter‐comma mean‐tone temperament, in common with all temperaments in common use much before the end of the 18th century, is an open temperament: it makes no attempt to join up the two ends of the chain of fifths. This meant that a keyboard performer had to be ready to retune some of the notes, depending on the key of the piece he wanted to play. For example, to play a piece in C minor he would be unlikely to need G♯, but he would certainly need A♭, continuing the chain of fifths backwards one place beyond E♭. So he would retune all the G♯ on his instrument to A♭. The adjustment, known as a comma or a diesis, was a fraction of a semitone but, annoyingly, a big enough fraction to matter. (The audio demonstrations that accompany Figure 1, Figure 2, Figure 3 and Figure 5 are all in quarter‐comma mean‐tone temperament.)
The comma between G♯ and A♭ – the ratio between their frequencies – depends on the tuning system. In quarter‐comma mean‐tone temperament the ratio is 125 : 128, and the comma is commonly known as the Great Diesis. There’s our numbers 125 and 128 again. That comma has commanded a mystic fascination down the ages, and there is an organisation devoted to it, the Order of the Great Diesis. Figure 6 shows their logo:
As music became harmonically more complex, and as keyboard instruments became heavier and harder to tune, composers and performers came to demand a circular temperament: a temperament that, by a further series of compromises, identifies G♯ and A♭, completes the circle of fifths, and allows the instrument to be played in all keys without retuning. This is shown in Figure 7. Click on Figure 7 for a demonstration in the modern Equal Temperament, which is the best‐known circular temperament nowadays.
In a circular temperament three major thirds become an octave, or in other words, as we showed before, 128 = 125. But only roughly, you will be glad to hear, because the major thirds are no longer acoustically pure.
It so happens that the keyboard of the traditional upright piano is just long enough to illustrate this in whole‐number arithmetic. Its compass, from its bottom note to its top note, is exactly 7 octaves, and so the frequency of the top note is exactly 27 = 128 times the frequency of the bottom note. Divide the 7 octaves (84 semitones) into three equal intervals, each of 2 octaves and a major third (28 semitones). If these three intervals were tuned to be acoustically pure, instead of in a circular system, their frequency ratios would be exactly 5 : 1, and the frequency of the top note of the piano would be 53 = 125 times the frequency of the bottom note instead of 128.
J S Bach advocated a circular temperament in the first half of the 18th century, and wrote The Well‐Tempered Keyboard to show it to best advantage. However, circular temperaments took several decades after that to catch on. I might add in passing that modern scholarship has shown that the particular circular temperament that Bach advocated is not, as formerly believed, the modern equal temperament, in which all the semitones are the same size.
This is where Beethoven comes in. Beethoven wrote his 3rd piano concerto in 1800. The concerto has three movements, as was (and is) standard. The first movement is in the key of C minor, and the second movement is in the key of E major. Now, the tonalities of C minor and E major are very remote from each other. See how far apart they are in Figure 4! They are much more remote than a classical audience of the time would have expected. Indeed they are so remote that the concerto is only playable on a piano tuned in a circular temperament. And the work proceeds to show off the circularity in a stunning way. Keep your eye on Figure 7.
The 2nd movement ends on a loud chord of E major (Figure 8). The top note of the chord is G♯, a major third above the keynote of E, and it is accented, to highlight the major third, and to throw into relief what comes next. Click on Figure 8 to hear and see what is happening.
The 3rd movement is in the same key as the first movement, as would be expected: C minor. Its first strong melody note, again accented, is A♭, a major third below the keynote of C (Figure 9). Click on Figure 9.
But that is a huge musical pun, because, on a keyboard, one and the same ivory represents G♯ and A♭, and in a circular temperament it represents them both with equal conviction. Beethoven is saying, “You thought that C minor and E major were unrelated? Well, in these new‐fangled circular temperaments, here is the relation between them!” The effect is stunning, and must have struck his first audiences as quite shocking. Then throughout the 3rd movement Beethoven repeatedly flits between G♯ and A♭ like a child with a new toy.
And all of this is only possible because of circular tuning. And circular tuning is based on the assumption that 128 = 125.
The best‐known circular temperament nowadays is the modern equal temperament, in which all the major thirds are exactly the same size, and 3 major thirds still make an octave, so that each major third has a frequency ratio equal to the cube root of 2. The cube root of 2 is 1.25992… As you can see, that is suspiciously close to 1.26. Is there a reason? You bet there is! Observe that 1.25 × 1.25 × 1.28 = 2. (There’s our numbers 125 and 128 again!) It follows that the cube root of 2 is the geometric mean of 1.25, 1.25 and 1.28. But these three numbers are close together, and so their geometric mean is very close (second‐order) to their arithmetic mean. And their arithmetic mean is 1.26.
The lovely thing is that that is not merely an arithmetical curiosity: it has musical meaning. In the old quarter‐comma mean‐tone temperament which was standard until about 1800, the interval from C to E was a justly‐intoned major third of 5 : 4 or 1.25, the interval from E to G♯ was likewise a justly‐intoned major third of 5 : 4 or 1.25, and the rogue interval from G♯ to C was what was left of the octave, namely 1.28. We saw these three intervals in Figure 5. Equal Temperament arose by redistributing the octave equally among these three intervals, so that they all became usable major thirds. So an equally tempered major third does not just happen to be the geometric mean of 1.25, 1.25 and 1.28: rather, that is how it arose historically.
A sum of the form a + b = c, like 3 + 125 = 128, linking three whole numbers that have no common divisor and surprisingly few prime factors among them, is what number‐theorists call an abc‑triple. There is a natural measure of the “quality” of an abc‑triple, giving credit for large numbers with repeated small prime factors. Here is how you work out the quality:
Take the largest number:
c = 128
Take its log:
log(128) = 2.1072
Multiply the prime factors of a, b and c counting each prime factor
2 × 3 × 5 = 30
Take its log:
log(30) = 1.4771
Divide the first log by the second log to get the quality:
2.1072 ÷ 1.4771 = 1.4266
I have used base 10 for these logs, but any base gives the same result, as the base cancels out when you divide one log by the other.
To qualify as an abc‑triple, the quality must be greater than 1. To qualify as a “good” abc‑triple, by an arbitrary convention the quality must be at least 1.4. So far (as of 2 March 2019), 241 good abc‑triples have been found. The best one of those found so far, and almost certainly the best one overall, is the beautiful 2 + 310 × 109 = 235, discovered by Eric Reyssat, with a quality of 1.6299. There are only a finite number of them in total, if recent work by mathematician Shinichi Mochizuki proves to be accurate. Statistical considerations suggest that most of them have been found. But if God created the natural numbers starting at 0 and working his way upwards, then “our” abc‑triple 3 + 125 = 128 must be close to his heart, for it is his very first good one.
So: the computer buff who orders a gigabyte of memory, thinking of 1000 megabytes but expecting to receive 1024 megabytes; the game show host who moves from a thousand to a million by repeated doubling; the schoolboy who observes that log(2) = 0.3; and Beethoven with his pun on G♯ and A♭ in his 3rd piano concerto: each of these people, in his individual way, is asserting that 128 = 125. How much of that did God foresee, I wonder, when he planted two such interesting numbers so close together?
That 128 = 125 is not the only assumption that circular temperaments rely on. All (tonal) temperaments, and not just circular temperaments, also assume, as a quite separate matter, that 81 = 80. But that’s another story.
Come to think of it, that is another story! I think I’ll call it Tones in two sizes, the hole in the chessboard, and The Queen of Sheba. And unless and until I write it down, you have no conceivable way of knowing what the Queen of Sheba has to do with it.