It warms my heart to see so many lighthouses beaming out into the chilly night in the far north of the British Isles, near Orkney and the Shetlands. But what are those two lighthouses due west of Orkney? They're well north of the Hebrides.
The more easterly one is on the small isle of Sule Skerry. Wikipedia writes:
There is a lighthouse at the centre high point of the island and a number of small cairns around the periphery. According to the Guinness Book of Records, the Sule Skerry lighthouse was the most remote manned lighthouse in Great Britain from its opening in 1895 to its automation in 1982. Its remote location meant that construction could only take place during the summer, thus it took from 1892–94 to complete.The more westerly one is on North Rona. This island was inhabited by Irish monks until 1680, when they all died due to an infestation of rats. It is the most remote island in the British Isles ever to have been inhabited on a long-term basis. It is also the closest neighbour to the Faroe Islands, far to the north.
It's also fun to see there's a lighthouse in the middle of Scotland! That's Fort Augustus, at the southern end of Loch Ness.
Much of Ireland is dark because it's not part of the British Isles.
This map, and other similar maps, were made by Terence on Mastodon.
This was in 1665, when he was 22: the year he fled Trinity College to avoid the Great Plague, went to the countryside, invented calculus, and discovered that a prism can recombine colors of light to make white light.
Check out this great new video — it explains everything very simply.
Newton's work is amazing because while an equal tempered scale — one with notes equally spaced — is standard now, it was very unusual in Newton's time. Much more common was just intonation, where the frequency ratios are simple fractions.
Another reason Newton's work is amazing: he compared just intonation not only to a 12-tone equal tempered scale, but also to equal tempered scales with 15, 19, 20, 24, 25, 29, 36, 41, 51, 53, 59, 100, 120 and 612 tones! He discovered that the 12, 53, 120 and 612 tone scales work especially well.
The 53-note equal tempered scale actually goes back to the Chinese music theorist Jing Fang (78–37 BC), who discovered that 53 just fifths is very nearly equal to 31 octaves: $$ \frac{(3/2)^{53}}{2^{31}} \approx 1.00209031404... $$ Much later the same observation was made by Nicholas Mercator. I don't know if Newton could have known of Mercator's work — but he was influenced by Descartes's Compendium Musicae. For more, read this:
Lots of people say they don't like math because they don't like numbers. In reaction, many mathematicians say that math is not really about numbers. Indeed, I don't spend most of my days messing with numbers: I spend a lot of time thinking about shapes, abstract structures, ideas from physics, and so on.
But some mathematicians do love numbers and spend a lot of time on them. I love them as a kind of hobby. The properties of the number 24, for example, are utterly mind-blowing, connecting higher-dimensional spheres to lattices and string theory.
The study of tuning systems offers humbler fun with numbers. If you go up a fifth you multiply the frequency of your sound by 3/2. Do this twelve times and you almost go up 7 octaves. But you're off by a factor of $$ \displaystyle{ \frac{(3/2)^{12}}{2^7} = \frac{531441}{524288} \approx 1.01346 } $$ This is called the Pythagorean comma — a glitch in the Pythagorean tuning system.
There's also a tuning system called 'just intonation', based on simple fractions as shown here:
In this setup if you play the sequence C G D A E C you don't get back where you started: you wind up higher by a factor of $$ \displaystyle{ \frac{3}{2} \cdot \frac{3}{4} \cdot \frac{3}{4} \cdot \frac{4}{5} = \frac{81}{80} \approx 1.0125 } $$ This is called the syntonic comma — a glitch in just intonation.
In the 6th century, Boethius noticed that these two commas are close but not quite the same — a kind of meta-glitch between glitches! He called their ratio the schisma. It's $$ \displaystyle{ \frac{531441/524288}{81/80} = \frac{32805}{32768} \approx 1.00113 } $$ It's also the ratio between 8 justly tuned perfect fifths plus a justly tuned major third and 5 octaves: $$ \displaystyle{ \frac{(3/2)^8 \cdot 5/4}{2^5} = \frac{3^8 \cdot 5}{2^{15}} = \frac{32805}{32768} \approx 1.00113 } $$ I find this fun!
An important early tuning system is Pythagorean tuning, where we force all frequency ratios to involve only powers of 2 and 3. In music, 3/2 is the 'fifth’: the most consonant of intervals except for the octave.
If we start with some frequency and go up and down by powers of 3/2, we create the 'circle of fifths’ shown above. It's almost a 12-pointed star, with one point for each note in the 12-tone equal-tempered scale.
Almost — but not quite! When we go up 12 fifths, we get a tone that's almost but not quite 2 times the frequency we started with. In other words, it's almost but not quite 7 octaves higher. So there's a glitch.
Here I've stuck that glitch at the opposite from the pitch labeled 1. That's a good place for it, because the spot directly opposite 1 is called the 'tritone’, or sometimes diabolus in musica: the 'devil in music’.
Let me explain the chart a bit more carefully. I started with any pitch and arbitrarily called its frequency 1. Then I climbed up 6 fifths, multiplying the frequency by 3/2 each time, getting pitches with frequencies
$$ 1, 3/2, 9/4, 27/8, 81/16, 243/32, 729/64 $$
Going all the way around the circle clockwise means going up an octave: that is, multiplying the frequency by 2. So each time I multiplied the frequency by 3/2, I went log_{2}(3/2) of the way clockwise around the circle: that is, about 0.585 of the way around, a bit more than half-way.
Then I climbed down 6 fifths, going counterclockwise and dividing the frequency by 3/2 each time, getting pitches with frequencies
$$ 1, 2/3, 4/9, 8/27, 16/81, 32/243, 64/729 $$
These are the reciprocals of the numbers we saw going up.
Why did I stop when I did? 729/64 is nowhere close to 64/729, but their ratio is almost a power of 2:
$$ \displaystyle{ \frac{729/64}{64/729} = \left(\frac{3}{2}\right)^{12} \! \approx 129.7 } $$
while
$$ \displaystyle{ 2^7 = 128 } $$
So our star is close to a 12-pointed star. But there's a glitch. And the size of the glitch is called the Pythagorean comma:
$$ \displaystyle{ \frac{(3/2)^{12}}{2^7} = \frac{3^{12}}{2^{19}} \approx 1.01364326477 } $$
This is one of the problems afflicting Pythagorean tuning, and later I'll say a bit about how people deal with it.
Now let's multiply the frequencies we've seen by powers of 2 to make them lie between 1 and 2. That gives us a scale that lies within a single octave:
These different pitches have names, by the way! Not just in Pythagorean tuning but in other related tuning systems, like the equal-tempered scale widely used in music today. Here they are: with their frequencies in the Pythagorean system: $$ \def\arraystretch{1.4} \begin{array}{lc} \textbf{tonic} & 1 \\ \textbf{minor 2nd} & \frac{256}{243} \\ \textbf{major 2nd} & \frac{9}{8} \\ \textbf{minor 3rd} & \frac{32}{27} \\ \textbf{major 3rd} & \frac{81}{64} \\ \textbf{major 4th} & \frac{4}{3} \\ \textbf{diminished 5th} & \frac{1024}{729} \\ \textbf{augmented 4th} & \frac{729}{512} \\ \textbf{perfect 5th} & \frac{3}{2} \\ \textbf{minor 6th} & \frac{128}{81} \\ \textbf{major 6th} & \frac{27}{16} \\ \textbf{minor 7th} & \frac{16}{9} \\ \textbf{major 7th} & \frac{243}{128} \\ \textbf{octave} & 2 \\ \end{array} $$
In the equal-tempered scale there's no difference between the augmented 4th and the diminished 5th: they're both the tritone. But in Pythagorean tuning they're different. And surprisingly, the augmented 4th is higher in pitch than the diminished 5th.
It may or may not help you to see the abbreviations for these different pitch names:
I could talk about this chart all day, but most of what I'd say would apply just as well to the equal-tempered scale. The big difference is that in Pythagorean tuning, unlike the equal-tempered scale, the augmented 4th (A4) and diminished fifth (d5) are not the same note. They are very close: the chart is not to scale, and if it were these two pitches would be almost on top of each other. But they're not the same!
It also may or may not help you to see the names for these pitches when the frequency we arbitrarily called 1 is the note called C:
We get a funny version of the scale with 13 notes, because F sharp (the augmented 4th in the key of C) is different from G flat (the diminished 5th).
Okay, but what if we want a scale with just 12 notes? We usually remove the diminished 5th, and make the augmented 4th do whatever jobs the diminished 5th would have done! So, we change our chart to this:
Or, in terms of frequencies, this:
This looks terrible, but more importantly it creates a badly out-of-tune pair of notes, namely those connected by the new red edge. These pitches have an ugly frequency ratio of
$$ \displaystyle{ \frac{729/512}{256/243} = \frac{3^{11}}{2^{17}} \approx 1.351524} $$
If we hadn't used the augmented 4th for a job the diminished 5th should be doing, we'd have gotten the much nicer-sounding ratio
$$\displaystyle{ \frac{1024/729}{256/243} = \frac{4}{3} \approx 1.333333} $$
The difference is audible and unpleasant: we've created what's called a 'wolf interval’, called that because it howls like a wolf. Unsurprisingly, the ugly ratio divided by the nicer ratio is our old nemesis, the Pythagorean comma:
$$\displaystyle{ \frac{3^{11}/2^{17}}{4/3} = \frac{3^{12}}{2^{19}} \approx 1.01364326477 } $$
It's interesting to look at the frequency ratios of neighboring notes in the Pythagorean scale: $$ \def\arraystretch{1.4} \begin{array}{lcccc} \textbf{minor 2nd / tonic} \phantom{ABC} & \frac{256}{243}\big/1 &=& 256/243 \\ \textbf{major 2nd / minor 2nd} & \frac{9}{8}\big/\frac{256}{243} &=& 2187/2048 \\ \textbf{minor 3rd / major 2nd} & \frac{32}{27}\big/\frac{9}{8} &=& 256/243 \\ \textbf{major 3rd / minor 3rd} & \frac{81}{64}\big/\frac{32}{27} &=& 2187/2048\\ \textbf{major 4th /major 3rd} & \frac{4}{3}\big/\frac{81}{64} &=& 256/243 \\ \textbf{augmented 4th / major 4th} & \frac{729}{512}\big/\frac{4}{3} &=& 2187/2048 \\ \textbf{perfect 5th / augmented 4th} & \frac{3}{2}\big/\frac{729}{512} &=& 256/243 \\ \textbf{minor 6th / perfect 5th} & \frac{128}{81}\big/\frac{3}{2} &=& 256/243 \\ \textbf{major 6th / minor 6th} & \frac{27}{16}\big/\frac{128}{81} &=& 2187/2048 \\ \textbf{minor 7th / major 6th} & \frac{16}{9}\big/\frac{27}{16} &=& 256/243 \\ \textbf{major 7th / minor 7th} & \frac{243}{128}\big/\frac{16}{9} &=& 2187/2043 \\ \textbf{octave / major 7th} & 2\big/\frac{243}{128} &=& 256/243 \\ \end{array} $$
In the equal-tempered scale the frequency ratio of neighboring notes is always 2^{1/12}, and it's called a semitone. But as you can see, in the Pythagorean scale some neighboring notes have a frequency ratio of 256/243, while others have a ratio of 2187/2048. So there are two kinds of semitones in the Pythagorean scale:
The Pythagorean chromatic semitone is bigger than the Pythagorean diatonic semitone. How much bigger? What's their ratio?
$$\displaystyle{ \frac{2187/2048}{256/243} = \frac{3^{12}}{2^{19}} \approx 1.01364326477 } $$
Yes, it's the Pythagorean comma! Like a bad penny, it keeps coming back to haunt us.
By the way, the word 'limma' is from a Greek word meaning 'remnant', and it's used for several small intervals in music. The word 'apotome' is from a Greek word meaning 'cutting off', and it's apparently used only for this particular interval, as well as other things in mathematics and optics.
The somewhat irregular pattern of semitones in my chart above would become symmetrical if we had kept the diminished 5th, but then there would be a Pythagorean comma between the augmented 4th and diminished 5th, like this: $$ \def\arraystretch{1.4} \begin{array}{lcccc} \textbf{minor 2nd / tonic} \phantom{ABC} & \frac{256}{243}\big/1 &=& 256/243 \\ \textbf{major 2nd / minor 2nd} & \frac{9}{8}\big/\frac{256}{243} &=& 2187/2048 \\ \textbf{minor 3rd / major 2nd} & \frac{32}{27}\big/\frac{9}{8} &=& 256/243 \\ \textbf{major 3rd / minor 3rd} & \frac{81}{64}\big/\frac{32}{27} &=& 2187/2048\\ \textbf{major 4th / major 3rd} & \frac{4}{3}\big/\frac{81}{64} &=& 256/243 \\ \textbf{diminished 5th / major 4th} & \frac{1024}{729}\big/\frac{4}{3} &=& 256/243 \\ \textbf{augmented 4th / diminished 5th} & \frac{729}{512}\big/\frac{1024}{729} &=& 531441/524288 \\ \textbf{perfect 5th / augmented 4th} & \frac{3}{2}\big/\frac{729}{512} &=& 256/243 \\ \textbf{minor 6th / perfect 5th} & \frac{128}{81}\big/\frac{3}{2} &=& 256/243 \\ \textbf{major 6th / minor 6th} & \frac{27}{16}\big/\frac{128}{81} &=& 2187/2048 \\ \textbf{minor 7th / major 6th} & \frac{16}{9}\big/\frac{27}{16} &=& 256/243 \\ \textbf{major 7th/minor 7th} & \frac{243}{128}\big/\frac{16}{9} &=& 2187/2043 \\ \textbf{octave / major 7th} & 2\big/\frac{243}{128} &=& 256/243 \\ \end{array} $$
Now the chart is symmetrical from top to bottom. The big nasty fraction in the middle is the Pythagorean comma:
$$\displaystyle{ \frac{531441}{524288} = \frac{3^{12}}{2^{19}} \approx 1.01364326477 }$$
Besides the Pythagorean comma — and the wolf interval we get if we try to avoid it — another big problem with Pythagorean tuning is that some very important intervals are represented by fairly complicated fractions. The ear seems to enjoy simple fractions! There are other tuning systems that do better at this, like 'just intonation’. For example, in Pythagorean tuning the minor third is 32/27 above the tonic, while in just intonation it's 6/5. In Pythagorean tuning the major third is a ridiculous 81/64 above the tonic, while in just intonation it's 5/4. Arguably just intonation gets these things right, while Pythagorean tuning gets them wrong.
I hope to write more about just intonation later! To finish off, here's a comparison between Pythagorean tuning and equal temperament:
I'm not trying to imply that equal temperament is 'correct', but at right I'm showing the Pythagorean frequency divided by the corresponding equal-tempered frequency. It's a bit remarkable how close they are! The biggest deviations occur for the augmented 4th and diminished 5th, which are the same in equal temperament but separated by a comma in the Pythagorean scale. The second biggest deviations occur for the minor 2nd and major 7th. All these are fairly dissonant intervals even in the best of worlds. But the third biggest occur for the major 3rd and minor 6th.
Here's a picture comparing Pythagorean tuning and equal temperament:
Equal-tempered is black and Pythagorean is green. You can see the diminished fifth and augmented fourth straddling the tritone.
So much more to say! And I want to get back to talking about modes, too. The appearance of the circle of fifths in both is far from a coincidence! But I'll quit here for now.
In my October 8th entry I explained how Pythagorean tuning, one of the older tuning systems, arises from the fact that twelve fifths is almost the same as seven octaves. In other words, multiplying by 3/2 twelve times is almost the same as multiplying by 2 seven times: \[ \displaystyle{ \left(\frac{3}{2}\right)^{12} \! \approx 129.7 > 128 = 2^7 } \]
But not quite! That's why the star above does not quite close.
In the most widely used modern scale, we deal with this discrepancy by using a fifth that does not have a frequency ratio of 3/2, but rather
\[ \displaystyle{ 2^{7/12} \approx 1.49830707688} \]
It's a bit off, but not much. So it sounds pretty good, and most of us have decided to accept it (though I love those of you who haven't). We then equally divide this fifth into 7 steps, each with a frequency ratio of 2^{1/12}. We thus divide the octave into 12 steps, each with a frequency ratio of 2^{1/12}. The result is called the equal-tempered 12-tone scale, or 12-TET for short.
The only question I want to discuss today is: why 12 tones? What if we tried an equal-tempered scale with some other number of tones?
The historical question of how Western music arrived at a 12-tone scale is complicated. It seems much of the evidence is lost in the mists of time. I'm not qualified to tackle this question. Indeed, nothing I've read so far has convinced me that anyone is qualified, though I want to read more. So instead I'll just talk about the math.
Furthermore, I'll take a very narrow view of the question! When choosing a tuning system there are many factors at work. At the very least, for each interval you might like to play, you should ask how well a given tuning system accommodates that interval. In common practice Western music this includes asking how all the major thirds and fifths sound in a given tuning system — since major triads, consisting of a tone, the major third above that tone, and the perfect fifth above that tone, are so fundamental to this music. It's generally thought that a really nice major triad has frequency ratios
\[ 1 : \frac{5}{4} : \frac{3}{2} \]
And so, mathematically, we can take any tuning system and ask how closely its major triads come to having these frequency ratios. For example, in just intonation we make some major triads have exactly these frequency ratios... while others are quite bad.
But because I've recently been thinking about Pythagorean tuning, which is about fifths, in this entry I won't talk about major thirds at all — much less the myriads of other issues — and focus with laser-like single-mindedness on the question of fifths in equal-tempered scales. I hope to talk about other things later.
Clearly \[ \displaystyle{ 2^{7/12} \approx 1.49830707688} \] is remarkably close to 3/2. How well could we approximate the frequency ratio 3/2 if we used an equal-tempered scale with some other number of tones?
For a scale with n tones, this amounts to finding the power of 2^{1/n} that comes closest to 3/2. Here's how it works:
\[ \begin{array}{llllcl} \textbf{1-TET} & 2^{1/1} & = & 2.00000 &\; \; & 33.3333\% \\ \textbf{2-TET} & 2^{1/2} & \approx & 1.41421 &\; \; & -5.7191\% \\ & 2^{2/3} & \approx & 1.58740 && +5.8267\% \\ & 2^{2/4} & \approx & 1.41421 && -5.7191\% \\ \textbf{5-TET} & 2^{3/5} & \approx & 1.51572 && +1.1048\% \\ & 2^{4/6} & \approx & 1.58740 && +5.8267\% \\ \textbf{7-TET} & 2^{4/7} & \approx & 1.48599 && -0.9337\% \\ & 2^{5/8} & \approx & 1.54221 && +2.8141\% \\ & 2^{5/9} & \approx & 1.46973 && -2.0177\% \\ & 2^{6/10} & \approx & 1.51572 && +1.1048\% \\ & 2^{6/11} & \approx & 1.45948 && -2.7013\% \\ \textbf{12-TET} & 2^{7/12} & \approx & 1.49831 && -0.1129\% \\ \\ \textbf{29-TET} & 2^{17/29} & \approx & 1.50129 && +0.08629\% \\ \textbf{41-TET} & 2^{24/41} & \approx & 1.50042 && +0.02796\% \\ \textbf{53-TET} & 2^{31/53} & \approx & 1.49994 && +0.00394\% \\ \end{array} \]
At right I show the percentage error of the approximate fifth: for example, 2^{1/2} is 5.7191% less than 3/2. The rows with names have a better perfect fifth than any row above. After 12-TET I got tired of showing you every row, and just showed the scales whose perfect fifth beats all previous scales.
◆ 1-TET is a ridiculous scale that only lets you play octaves, so its best approximate fifth is the octave.
◆ 2-TET also has a terrible approximate fifth: it's 2^{1/2}, the tritone, which is extremely dissonant.
◆ 5-TET is surprisingly good for a scale with so few tones, with an approximate fifth just 1.1048% too high. Many different pentatonic scales are used worldwide, but the pentatonic scale called slendro in Javanese and Balinese gamelan music is pretty close to 5-TET.
◆ 7-TET has an approximate fifth that's 0.9337% too low, just a bit better than 5-TET. Oddly I'm having trouble finding examples of music in 7-TET. Apparently it was used in some traditional Chinese music. I'd love to know more details.
◆ Then comes 12-TET, a marked improvement with a fifth that's just 0.1129% too low.
It's impossible not to notice that 5 + 7 = 12, and that our standard use of 12-TET on a piano divides the 12 tones into 5 black keys and 7 white keys.
The black keys form a pentatonic scale, while the white keys form the so-called diatonic scale. Both are widely used in Western music — and in the case of the diatonic scale that's a massive understatement: see my article on modes of the major scale for some ways it's used.
These pentatonic and diatonic scales are significantly different from 5-TET and 7-TET, since their notes are not close to evenly spaced. And yet I can't help but wonder if some faint shadow of 5-TET and 7-TET hangs over 12-TET, and the way we subdivide it into pentatonic and diatonic scales.
Mathematically, it seems to be a sheer coincidence that 5-TET, 7-TET and 12-TET give particularly good fifths and 5 + 7 = 12. But later I'll mention a few more facts that make this fact even more tantalizing!
◆ 29-TET is the next winner: its fifth is just 0.08629% too high. It has been argued that in 1318 the medieval Italian music theorist Marchetto da Padova proposed a system that is approximately 29-TET:
I'll admit I'm not convinced.
◆ 41-TET has a fifth that's just 0.02796% too high. This has been used or at least studied enough to have its own Wikipedia page:
The pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in the Hague. I recommend looking at the Wikipedia page to see how much more detailed it is than anything I'm saying here. Then look at this:
and see what true devotion looks like! Xenharmonic music is music that uses tuning systems other than 12-TET.
◆ 53-TET is a massive improvement, with a fifth that's only 0.00394% too high, so this scale has been on people's radar for a long time. Wikipedia again has an article on it:
and this time the historical section is interesting enough to quote in detail:
Theoretical interest in this division goes back to antiquity. Jing Fang (78–37 BCE), a Chinese music theorist, observed that a series of 53 just fifths (3⁄2^{53}) is very nearly equal to 31 octaves (2^{31}). He calculated this difference with six-digit accuracy to be 177147⁄176776. Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c. 1620–1687), who calculated this value precisely as
\[ \displaystyle{ \frac{3^{53}}{2^{84}} = \frac{19383245667680019896796723}{19342813113834066795298816} }\]which is known as Mercator's comma. Mercator's comma has a small value to begin with (≈ 3.615 cents), but 53 equal temperament flattens each fifth by only 1⁄53 of that comma (≈ 0.0682 cent ≈ 1⁄315 syntonic comma ≈ 1⁄344 Pythagorean comma). Thus, 53 tone equal temperament is for all practical purposes equivalent to an extended Pythagorean tuning.
After Mercator, William Holder published a treatise in 1694 which pointed out that 53 equal temperament also very closely approximates the just major third (to within 1.4 cents), and consequently 53 equal temperament accommodates the intervals of 5 limit just intonation very well. This property of 53-TET may have been known earlier; Isaac Newton's unpublished manuscripts suggest that he had been aware of it as early as 1664–1665.
Mercator's observation that 53-TET has a good approximate 'just major third' boils down to the fact that
\[ \displaystyle{ 2^{17/53} \approx 1.24898 }\]
is close to 5/4 = 1.25.
I find all the history here fascinating. Nicholas Mercator is not the guy with the Mercator projection — that was Gerardus Mercator. He's the guy who invented the term 'natural logarithm' and discovered that \[ \displaystyle{ \ln(1 + x) = 1 + x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots }\]
By the way, 'Mercator' just means 'merchant', and Nicholas Mercator's German name was 'Kauffman'. I hope sometime to say more about the work of Mercator and Newton on tuning systems.
By the way, remember my observation that three good scales 5-TET, 7-TET and 12-TET are related by 5 + 7 = 12? After I posted this article, Sylvain pointed out on Mastodon that 29-TET, 41-TET and 53-TET are related by similar formulas! \[ \begin{array}{ccl} 29 &=& 12+12+5 \\ 41 &=& 12+12+12+5 \\ 53 &=& 12+12+12+12+5 \end{array} \]
More coincidences? Or is something deeper at work here? I have no idea!
What's next? Is there a systematic way to get ahold of these equal-tempered scales with good approximate fifths?
Yes: the key is to study the number
since this number x has
so if we find a good rational approximation to x, say
then we get
so in an equal-tempered scale with q tones, the fifth will be close to p tones above the octave.
To find good rational approximations to x we can take its continued fraction expansion... I'll do it using Wolfram Alpha... and get
If we truncate this at some point we get a good rational approximation to x and thus an equal-tempered scale with a good approximate fifth. For example \[ \displaystyle{ \frac{1}{1 + \frac{1}{1 + \frac{1}{2 + \frac{1}{2}}}} = \frac{7}{12} } \]
This says that 12-TET has a pretty good approximate fifth, and we get it by going up 7 steps on this 12-tone scale.
If we do this systematically we get these rational approximations to x:
1
1/2
3/5
7/12
24/41
31/53
179/306
389/665
9126/15601
18641/31867
46408/79335
65049/111202
111457/190537
6195184/10590737
6306641/10781274
31421748/53715833
100571885/171928773
131993633/225644606
and so on. I thank Chris Grossack for showing me how to compute these using Sage.
This list includes all the 'best so far' equal-tempered scales on my previous list except for 7-TET. It skips straight from 5-TET to 12-TET. Why is that — what's the underlying math here? I imagine there's quite a bit to say, not about this one particular case but about the general theory of approximating numbers by rationals. I know continued fractions give good results, but what about all the other 'best so far' approximations: that is, rational approximations that are better than any with a smaller denominator? Are they common, rare, etc.?
We can also get new scales, using \[ \begin{array}{ccl} 2^{179/306} & \approx & 1.50000501098 \\ 2^{389/665} & \approx & 1.49999990153 \\ 2^{9126/15601} & \approx & 1.50000000175 \\ 2^{18641/31867} & \approx & 1.49999999966 \end{array} \]
and so on.
Of course, this is basically an argument for why you shouldn't let mathematicians get involved with tuning systems. A scale with 306 or more tones is not very practical — and even though we can compose and play music on such scales using computers, the ear will not greatly prefer the fifths in this scale to those in 53-TET.
To be frank, most people are perfectly happy with 12-TET! And I should lay my cards on the table: my overall goal is not to find better tuning systems, but to better understand the math behind the historically most important 12-tone tuning systems. I would like to write about these:
These dates are very rough, and I'd love to find some books that investigate the history more carefully... but anyway, I'm not sure I'll even get around to discussing all these systems. But I'd like to! There is some mildly fancy math that could be brought in, which I haven't seen people using.
Western music has long used chords with fractions built from the primes 2, 3, and 5. So far I've mainly talked about 2 and 3. Next comes 5, and that will be quite a story. But first: what about 7?
This book tells the story quite entrancingly:
Four hundred years ago, the possibility of using seven as a tuning ratio was still up in the air in Europe. Gioseffo Zarlino, in his Le institutioni harmoniche of 1558, had enshrined the numero senario, the number six, as the "sonorous number" beyond which no consonances were possible (although he had to create an exception for the minor sixth 8/5 as an acceptable inversion of the 5/4 major third). Over the next couple of centuries, though, various theorists and musicians made an argument for septimal (seven-based) intervals, only to retreat in the face of common practice.The mathematician Marin Mersenne (1588–1648) claimed in 1636, "I have not the slightest doubt that the dissonant intervals of which I have spoken [...] i.e., the ratios 7:6 and 8:7 that subdivide the fourth — may become pleasing if one accustoms oneself to hearing and bearing them [...] for diverse effects that ordinary music lacks." Later, however, he changed his tune and his tuning, writing that because 7/6 "is neither a consonance nor a difference of consonances, nature — which is harmonic — rejects it and prefers to interrupt its series of intervals and melodies than to move through an interval that serves no purpose except to wound the ear and the spirit."
The astronomer and scientist Christiaan Huygens (1629–1695) likewise wrote that the number seven "is not unable to produce a consonance" but then dismissed 7/6 and 8/7 as being "incompatible with the consonances already established." The Jesuit Honoré Fabri (1607–1688) wrote in 1670 that "if 8:1 is a consonance, I do not see 7:1 should not also be [...] nor will this be a disagreeable consonance, but, quite the contrary, more pleasing to the ear than 8:1: let anyone try for himself." Then he dismisses septimal intervals on the same incompatability argument.
And so it went, writer after writer admitting that septimal intervals could please the ear before deciding that employing them was inconvenient. It was as though if you came out publicly for the number seven, a couple of thugs from the Six-Based Mafia showed up at your office for a little persuasion. Ah, well.
Nothing is miserable unless you think it is so; and on the other hand, nothing brings happiness unless you are content with it.But how did he get there? In fact his story is quite dramatic. He was born in Rome a few years after the collapse of the Western Roman Empire. After mastering Latin and Greek in his youth, he rose to prominence as a statesman during the Ostrogothic Kingdom, becoming a senator by age 25, and later a personal advisor to the king, Theodoric the Great.
He tried to translate all the Greek classics into Latin. Though his project was unfinished, it helped the works of Aristotle survive in the West. This is probably the most important thing he did.
Alas, he became very unpopular among members of the Ostrogothic court after he denounced their corruption. He was imprisoned by Theodoric in 523, and tortured and executed a year later.
But here's what I hadn't known: earlier he wrote about math and music, including the math of tuning systems!
Perhaps most importantly, he invented the system of using letters for notes. He started with the lowest note he cared about and called it A. Then came B, C, D, E, F, G, H, I, K... yes, this was before the letter J was invented! Much later, this system got changed to the one we're familiar with. But if you look at a standard 88-key grand piano you'll see the lowest note is still A.
He also discovered some sophisticated concepts in tuning theory. On October 3rd I mentioned the Pythagorean comma, a glitch that shows up in the Pythagorean tuning system: $$ \displaystyle{ \frac{(3/2)^{12}}{2^7} = \frac{3^{12}}{2^{19}} = \frac{531441}{524288} \approx 1.01364326477\dots } $$ I also mentioned the syntonic comma, a glitch that shows up in just intonation: $$ \displaystyle{\frac{(3/2)^4}{2^2 \cdot 5/4} = \frac{81}{80} = 1.0125 } $$ These glitches are close but not equal. Some tuning systems try to exploit this fact, using one of these commas where you should really use the other. But this gives rise to a kind of meta-glitch: a glitch between glitches! It's almost undetectable, since the ratio of the Pythagorean and syntonic commas is $$ \displaystyle{\frac{3^{12}/2^{19}}{81/80} = \frac{32805}{32768} \approx 1.00112915039\dots } $$ But it's there nonetheless.
It seems that Boethius was the one who first thought about this meta-glitch. Apparently he discussed it in the third book of his De Institutione Musica, and even gave it a name: the schisma. It's so cool to imagine an advisor to a Gothic king doing this fancy math.
(Wikipedia claims that Boethius also discovered another musical fraction, the diaschisma, but that this was named much later by the German physicist and mathematician Helmholtz. Part of the fun of music theory is that it brings together famous figures from very different eras.)
I also just learned that most of On the Consolation of Philosophy was set to music in the Middle Ages! The melodies were considered lost because their notation relied on now-forgotten oral traditions. But Sam Barrett at Cambridge has tried to reconstruct them — and the ensemble Sequentia, who put out an amazing complete works of Hildegard von Bingen, performed his versions in 2016. You can learn more about this here:
https://www.youtube.com/watch?v=Ih2p_NiiwsU
Later, in 2018, they released an album of this music called Boethius: Songs of Consolation, which you can listen to here.
Three days ago I posed a puzzle on Mastodon: can you figure out what's going on in this picture from his De Arithmetica?
This version of the picture was modernized by Martin Kullman. You can see the whole book by Boethius here.
The only really good answer to my puzzle came from David Egolf who wrote:
I don't have a Mastodon account, but I wanted to respond to what you posted about there. You mentioned a somewhat mysterious figure from a book by Boethius.I was able to read the version of the book here:
- Gottfried Friedlein, Anicii Manlii Torquati Severini Boetii De institutione arithmetica libri duo, De institutione musica libri quinque, 1828.
To do this, I took screenshots of the text, converted them to "copy-pastable" text using https://www.imagetotext.info/ and then translated the text to English using chatGPT.
I believe the square is illustrating some simple properties of certain arithmetic and geometric sequences. Namely, if you take two terms \(a(n)\) and \(a(m)\) of an arithmetic sequence, then their sum \(a(n)+a(m)\) is equal to \(a(n-k) + a(m+k)\). Similarly, if you take two terms \(g(n)\) and \(g(m)\) of a geometric sequence, then their product \(g(n)g(m)\) is equal to \(g(n+k)g(n-k)\).
Each column of this square is an arithmetic sequence, and each row of the square is a geometric sequence. By the way, the reason Boethius is talking about this is because he is interested in studying "even-ness" and "odd-ness". Here, he is interested in numbers that are "somewhat even" but not "minimally" or "maximally" even. These are the numbers that have at least two factors of 2, and have at least one prime factor besides 2. The first column is the "least even" numbers satisfying these criteria — the odd multiples of 4 (skipping 4×1). The entries in the second column from the left have three factors of 2, and so its entries are the odd multiples of 8 (skipping 8×1). The "even-ness" continues to increase as one moves to the right.
The "arcs" on the outside of the square illustrate the property of arithmetic and geometric sequences I mentioned above. I first give some examples relating to geometric sequences. For example, 12 · 96 = 1152 = 24 · 48. We also have that 24 · 96 = 48 · 48 = 2304. Similarly, moving to the second row (which corresponds to the "inner arcs" on the top of the drawing), we have 20 · 160 = 3200 = 40 · 80. We also have 40 · 160 = 6400 = 80 · 80.
On the left of the square, we have some examples of the above mentioned property of arithmetic sequences. On the leftmost arcs, we have some examples illustrating properties of the arithmetic sequence in the leftmost column. For example, 20 + 36 = 56 = 28 + 28 and 12 + 36 = 20 + 28 = 48. Moving to the "right-most arcs" on the left side, these now correspond to properties of the arithmetic sequence in the second column from the left. For example, 112 = 40 + 72 = 56 + 56 and 96 = 40 + 56 = 24 + 72.
After having had all the fun of working from the Latin, I now see that there is a nice presentation of the key ideas in English here:
- Dorothy V. Schrader, DE ARITHMETICA, Book I, of Boethius, The Mathematics Teacher 61 (1968), 615–628.