### Month: June 2018

The Colour of Music : Notes of Equal Temperament

## The Colour of Music : Notes of Equal Temperament

Equal temperament and just intonation are two of the methods used to divide an octave in western classical music. While just intonation is based on ratios of simple integers, the equal temperament system uses logarithmic intervals. In part II of this series, the Nerd Druid talks about equal temperament.

## Music, sound, and frequency

Music is basically a set of well-ordered nice sounds. Sound is pressure waves in a medium–air, water, metal, the Earth. Not all pressure waves cause sound that we humans can hear. Earthquakes give rise to sounds of very long wavelength, too low for us to hear. Bats give off high frequency sounds, too sharp for us to hear. Our hearing range is $20Hz – 20kHz$; that is, we can[1] detect those pressure waves that oscillate at a rate of $20$ to $20000$ cycles per second.

Some animals are far better than us at this game. Cats and dogs can hear upto $40 kHz$, while bats and dolphins can hear sounds as sharp as $160 kHz$.

#### Notes and tunes

Since sound is music, it too is based on frequencies. Play a certain frequency and you get the fundamental musical species, a note. Play a series of frequencies, a set of notes, and you get a tune. Whether or not that is a nice tune depends very much on the notes you have chosen. Notes that have certain well-defined frequency relations among each other tend to produce nicer tunes.

In most well-developed music systems, such as western classical or Hindustani and Carnatic classical, the notes are labelled for ease of manipulation. Given a set of labelled notes, the problem is to assign frequencies to them so that they sound nice. In Part I of this series, The Colour of Music : Understanding Just Intonation, we discussed just intonation, a method of doing exactly this.

### Just intonation : A quick recap

In just intonation, the frequencies you assign to various notes are in simple numeric ratios to the first note, the unison. Consider the standard western C-major scale, comprising notes

$$C-D-E-F-G-A-B-C^{\ast} \tag{1}$$

where $C$ (frequency $f$, say) is the unison and $C^{\ast}$ is an octave above it, with frequency $2f$. $G$ is a perfect fifth above $C$, and has a frequency of $\frac{3}{2} f$. $E$ is a major third above $C$, and has a frequency of $\frac{5}{4} f$. Using these three ratios $\displaystyle {}^2\!/\!_1, {}^3\!/\!_2, {}^5\!/\!_4$ you can generate every other note in the C-major scale in five-limit tuning. I like to call these three ratios the three generators of the just intonation series. You can either ascend by multiplying by the ratio, or descend by dividing with it, which is the same as multiplying with the inverse ratio. Thus, for instance, if you wish to go to $D$ from $C$, you need to ascend by two fifths and descend by an octave ($\uparrow$ fifth $\uparrow$ fifth $\downarrow$ octave), meaning you’d have to multiply by ${}^3\!/\!_2 \times {}^3\!/\!_2 \times {}^1\!/\!_2$.

The table below is a slightly modified form of the one from the previous article, and will hopefully explains what’s going on.

Note Up/Down Frequency
$C$ (unison) = $f$
$D$ (major second) $\uparrow$ fifth $\uparrow$ fifth $\downarrow$ octave $\frac{9}{8}f$
$E$ (major third) $\uparrow$ third $\frac{5}{4}f$
$F$ (perfect fourth) $\uparrow$ octave $\downarrow$ fifth $\frac{4}{3}f$
$G$ (perfect fifth) $\uparrow$ fifth $\frac{3}{2}f$
$A$ (major sixth) $\uparrow$ octave $\uparrow$ third $\downarrow$ fifth $\frac{5}{3}f$
$B$ (major seventh) $\uparrow$ fifth $\uparrow$ third $\frac{15}{8}f$
$C^{\ast}$ (octave) $\uparrow$ octave $2 f$

For further explanation, read Part I.

#### Pure intervals

Intervals that involve the ratios of simple integers are called pure or just intervals, since they correspond to sounds created by vibrations in physical objects.

Just intonation is a good way to assign frequencies, and since the intervals are pure, the sound produced by justly tuned scales tend to be quite pleasing to the ear. However, the ratios in the rightmost column are given with respect to the unison, and thus, if we want to move from a non-unison note to another such note, we’d be in a bit of a pickle.

How much of a pickle? Here’s a problem for you, see if you can solve it. I am at $A$, major sixth, ${}^5\!/\!_3 f$. I want to go to $D$, major second, ${}^9\!/\!_8 f$. What is the simplest way to get there using the three generators? Answer in the footnote[2]. Don’t peek till you’ve tried it yourself.

As you’ll find, simply going from $A$ to $D$ in just intonation involves quite a ride! Enough of it, then. Time for equal temperament.

## Equal Temperament

Unlike just intonation, which has three generators, equal temperament has a single generator, and is thus mathematically simpler.

Well, sort of.

#### Black keys

So far I have discussed breaking up the octave into seven portions, seven notes in the C-major scale with frequencies assigned so that they sound nice. This corresponds to the seven white keys in a piano, as seen in the image above. Clearly, though, there is something incomplete about this picture. Where are the black keys? What frequencies do they represent? What are they called? Which scales include them?

Take a look at the table above. In the first column, I’ve written down the symbols designating each note. The names of the notes are in brackets. The adjective major appears quite a lot. In the C-major scale, where $C$ is the tonic or the root note, the white keys of the piano correspond to these major notes. The black keys sit in between the white keys, suggesting that their frequencies should be in-between those of the white keys.

#### Sharps $\sharp$ and flats $\flat$

Counting from $C$ on the left, you have 7 white keys, corresponding to $D-E-F-G-A-B$, and 5 black keys in between them, making up a total of 12 notes in an octave. The black notes are

$$C♯ – D♯ – F♯ – G♯ – A♯ \tag{2a}$$

where the note $C♯$ is pronounced C-sharp and corresponds to a frequency slightly higher than $C$. However, as you’ve no doubt noticed, the same note might well have been slightly lower than D, slightly flatter, if you will, and can thus be called $D♭$, D-flat. Thus the series might as well be called

$$D♭ – E♭ – G♭ – A♭ – B♭ \tag{2b}$$

or, if you are fastidious and have no eye for aesthetics, like this horribleness

$$C\sharp/D\flat – D\sharp/E\flat – F\sharp/G\flat – G\sharp/A\flat – A\sharp/B\flat \tag{2c}$$

The notes that can be both a $\sharp$ and a $\flat$ are called enharmonic equivalents.

Appending a $♯$ (sharp) to a note elevates the pitch (increases the frequency), while appending a $♭$ lowers the pitch (decreases the frequency).

But by how much?

Well, by a semitone.

### Semitone

A semitone (s) is the unit interval, the smallest frequency increment. In a piano, the interval between successive keys is a semitone. For instance, the interval between the white $C$ key and the black $C♯/D♭$ key, or that between the black $F\sharp/G\flat$ and the white $G$ key, are both semitones. Thus, if you want to get to the fifth, $G$, from the root, $C$, you would have to cross 7 semitones. Similarly, to get to the third, $E$, you’d need to cross 4 semitones.

This massively simplifies our five-limit tuning table. Instead of dealing with three generators and going up and down, as you found out earlier, going from $A$ to $D$ in equal temperament simply involves knowing how many semitones separate the two notes. You don’t even need to involve the unison, you can simply count off 7 semitones to the left of $A$ and reach $D$. Voila!

### Chopping up the octave

Time to get mathematical. Equal temperament tuning is based on logarithms, with the semitone as the unit. Clearly, if an octave has 12 semitone intervals, then we need to divide the range $f$ to $2f$ into twelve portions. A simple arithmetic method is to chop the interval $f$ up equally and assign $s = {}^f\!/\!_{12}$. If $f = 100 Hz$, then $s = 8.333…Hz$, and you are liable to be beaten up by people around you if use those notes.

Ok, not as simple as that, then. Remember that assigning frequencies to notes depends on whether those notes sound nice, not whether they are mathematically nice. Of course, in the case of just tuning, the ratios are mathematically nice. Clearly, we should attempt a method where the notes sound somewhat close to those in just intonation, but are also mathematically nicer.

#### The perfect(?) fifth

A good reference is the fifth, $G$. It is seven semitones away from the root, and its frequency ratio is $3:2$ with respect to unison. Therefore, while $G$‘s frequency is halfway between unison and the octave, $1.5 f$, there are 7 semitones on one side and 5 on the other. Clearly, then, we need a division method which gives smaller intervals early on and larger ones later on. Logarithms fit the bill perfectly.

#### Back to the semitone

I did get to logarithms in a roundabout way. Another simpler way to invoke logarithms is to realise that frequencies of the notes multiply , not add. Also, in equal temperament tuning, all semitones are equal. This immediately suggests that the best method of chopping up an octave into 12 intervals is to use, as the unit, a semitone interval of

$$s = \sqrt[ 12 ]{ 2 } = 2^{{}^1\!/\!_{12}} \tag{3}$$

Therefore, any note that is, say, seven notes away from another note, has a frequency that is higher by a factor of $7s = \sqrt[ 12 ]{ 2^7 } = 2^{{}^7\!/\!_{12}}$. This is the factor by which the frequency of a fifth $G$ in equal temperament is higher than the pitch[3] of the unison $C$. Similarly, the major third, $E$, is higher than the unison by a factor of $5s = \sqrt[ 12 ]{ 2^5 } = 2^{{}^5\!/\!_{12}}$, whereas the major sixth is higher by a factor of $10s = \sqrt[ 12 ]{ 2^{10} } = 2^{{}^5\!/\!_{6}}$.

#### Any note as root

Although I have used the unison as the root here, the same method can be used taking any key as root. For instance, I can proclaim that $A$ is root, and then descending to $D$ would simply involve lowering the pitch by a factor of $7s = \sqrt[ 12 ]{ 2^7 } = 2^{{}^7\!/\!_{12}}$. This also means that if I invert the system, and take $D$ as root, then $A$ is the fifth, a factor of $2^{{}^7\!/\!_{12}}$ higher.

Well, that’s all fine and dandy, but how does this fare with respect to that supreme condition of music, nicety? Do these logarithmic semitone intervals produce nice music? Is there any way to find out without tuning a piano and playing the notes?

### Is Equal Temperament nice?

Well, yes, compare it to the just intervals. If the notes of equal temperament are close enough to just, then our job is done.

For that, first, how much is the twelfth root of two, exactly? How does one find its value? And why do I keep bandying the word logarithm about, when we haven’t really calculated the log of anything yet?

Well, it’s time for that. Problem #2 : calculate the value of $2^{{}^1\!/\!_{12}}$ using logarithms. The steps are in the footnote[4].

Have you tried it yourself? If not, do try, it’s fun.

Since we’re dealing with logarithms and exponentials here, the value of $s$ is not a rational number. However, for our purposes, we’ll take the value of $s = 1.059463$ to be sufficiently correct.

#### 12-TET vs Just

Ok. Time to construct the twelve-tone equal temperament (12-TET) table. The frequency in equal temperament have been rounded off to the third place in decimal.

Name 12-TET Just
Unison (C) $2^{{}^0\!/\!_{12}} = 1.000$ $\frac{1}{1} = 1$
Minor Second (C♯ / D♭) $2^{{}^1\!/\!_{12}} = 1.059$ $\frac{ 16 }{ 15 } = 1.0666…$
Major Second (D) $2^{{}^2\!/\!_{12}} = 1.122$ $\frac{9}{8} = 1.125$
Minor Third (D♯ / E♭) $2^{{}^3\!/\!_{12}} = 1.189$ $\frac{6}{5} = 1.2$
Major Third (E) $2^{{}^4\!/\!_{12}} = 1.259$ $\frac{5}{4} = 1.25$
Perfect[5] Fourth (F) $2^{{}^5\!/\!_{12}} = 1.335$ $\frac{4}{3} = 1.333…$
Tritone (F♯ / G♭) $2^{{}^6\!/\!_{12}} = 1.414$ $\frac{7}{5} = 1.4$
Perfect[5:1] Fifth (G) $2^{{}^7\!/\!_{12}} = 1.498$ $\frac{3}{2} = 1.5$
Minor Sixth (G♯ / A♭) $2^{{}^8\!/\!_{12}} = 1.587$ $\frac{ 8 }{ 5 } = 1.6$
Major Sixth (A) $2^{{}^9\!/\!_{12}} = 1.682$ $\frac{5}{3} = 1.666…$
Minor Seventh (A♯ / B♭) $2^{{}^{10}\!/\!_{12}} = 1.782$ $\frac{ 16 }{ 9 } = 1.777…$
Major Seventh (B) $2^{{}^{11}\!/\!_{12}} = 1.888$ $\frac{15}{8} = 1.875$
Octave (C) $2^{{}^{12}\!/\!_{12}} = 2.000$ $\frac{2}{1} = 2$

#### The minor notes

This table also introduces additional five notes; the minor second, the minor third, the tritone, the minor sixth, and the minor seventh. Each of these notes is a semitone raised or lowered from one of the seven major notes. So, if you start playing at $C$ on a piano, these five notes correspond to the five black keys.

Equal temperament looks mathematically simple enough. All you need is to increment the power of the twelfth root of $2$ and you have a new note. However, as you can see, the frequencies of 12 tone equal temperament aren’t exactly equal to those of just intonation. This could potentially cause trouble, for as we know, the just intervals are pure, and intervals away from that might not sound nice.

#### Back to the perfect fifth

For instance, take a look at the fifth. In just, $G = 1.5f$, while in 12 tone equal temperament, $G = 1.498f$. That’s not equal, but that’s not massively far off too. However, it isn’t exactly $3:2$, and that is why I haven’t been calling it the perfect fifth anymore. However, the question still stands. Does an equal temperament $G$ sound more-or-less the same as the perfect fifth, or does it sound off?

Well, one way to verify this is to input the frequencies in some digital music software, have it play it back, and judge if the notes sound nice. However, clearly, not everyone is a good judge of good music. Besides, this is a very subjective (though perhaps more proper) way to analysing music.

#### The percentage difference

Instead, we science it out. In order for equal temperament to be nice enough, its frequencies needs to be within a certain percentage of the just frequencies. Agreed? All right, let’s take the normalised difference, or rather the percentage difference $\Delta$, given by

$$\Delta = 100 \times \frac{ E – J }{ J } \tag{4}$$

where $E$ and $J$ are the equal temperament and justly tuned frequencies, respectively.

I could list it out as another boring table, but diagrams, especially colourful diagrams, are far cooler. So, here’s a plot of the equal temperament and justly tuned frequencies, along with $\Delta$.

Blue dots (and line) are 12 tone equal temperament frequencies; red dots (and line) are justly tuned frequencies; yellow dots (and dashed line) are the percentage difference. $\Delta$ almost never goes beyond 1%. The highest difference is at $F\sharp$; the lowest are at $F$ and $G$. This means that, folks, equal temperament…or rather, 12-tone equal temperament should be quite good indeed.

And it is! You don’t really play it back to know that it is. Pianos are (almost always) tuned to 12-tone equal temperament. Just imagine a Chopin or a Mendelssohn or perhaps the Moonlight Sonata being played on an equal temperament piano.

Heaven!

### The Reference Frequency

But how do you tune a piano? I mean, in the table above, we have normalised the frequencies; $C$ (unison) is $1$, while $C^{\ast}$ (octave) is 2. We need the frequencies in actual units, in hertz, to be able to actually tune a piano and have a good player play the Moonlight. Clearly, we need a reference frequency.

Well, more on that, and a bit of the history of equal temperament, in part III. In the meantime, here’s Captain A. F. Haddock, “playing the piano”.

## Footnotes

1. Well, most of us. As we grow older, the upper limit starts diminishing. People in their thirties can only detect sound of upto 17-18 kHz in frequency. ↩︎
2. Beginning at unison $C$, I can get to $A$ using the following steps : $\uparrow$ octave $\downarrow$ fifth $\uparrow$ third. Writing this in the much more intuitive $+O-F+T$, we do the same for $D$, the major second: $-O+2F$. With a little bit of mental (or paper, if you wish), we see that we would need to add $-2O+3F-T$ to $A$ to get to $D$, since $(+O-F+T) + (-2O+3F-T)$ equals $(-O + 2F)$. This is the solution, then. Beginning at $A$, descend by two octaves, ascend by three fifths, and then descend by a third to get to $D$. Quite convoluted, as is obvious. Also, do note that we’d need to first know where the notes stand with respect to unison. There is no direct way to move from $A$ to $D$ in just, unlike in equal temperament, where you basically have to just…count. ↩︎
3. Frequency is the technical, objective term, and refers the rate of oscillations of a wave. Pitch is a human subjective term, and is the sensation of frequency. For instance, one would generally use pitch in a comparative context, such as stating that the wail of a police siren is at a higher pitch than the notes of a cello. In contrast, frequency is mostly used in an absolute sense. ↩︎
4. Let $s$ be the twelfth root of $2$, $s = 2^{{}^1\!/\!_{12}}$. Therefore, we can write $\log(s) = \frac{ \log(2) }{ 12 }$, which leads to
$$s = e^{ \left( {}^{\log(2)}/_{12} \right) }$$

which gives an approximate value of $s \approx 1.059463$. For our purposes we shall take this to be sufficiently equal. ↩︎

5. Clearly, the values of the fourth and the fifth in 12 tone equal temperament are not exactly equal to $4:3$ and $3:2$, are they? Which is why you no longer call them perfect. ↩︎ ↩︎
The Colour of Music : Understanding Just Intonation

## The Colour of Music : Understanding Just Intonation

Just intonation and equal temperament are two of the methods used to divide an octave in western classical music. While just intonation is based on ratios of simple integers, the modern equal temperament system uses logarithmic intervals. In part I of this series, the Nerd Druid talks about just intonation.

## Music, art or science?

What is your favourite piece of music?

Darned difficult question, I admit. In the modern world, where most urban and townspeople own a mobile phone, listening to music is pretty easy. If you have a smartphone, you can fill it up with all the music you want. If you have an ordinary mobile, you can still listen to radio[1]. And there are no dearth of musicians, and thus absolutely no lack of variety and choice when it comes to the type of music you’d want to hear. We live in a very musical world. Indeed, a most colourful world.

### Sound is a wave

But what is music? After all, it is but a series of sounds that are picked up by our ears and analysed by our brains. Pressure waves in air, like most sound, that can be transformed into electromagnetic waves for faster and longer transmission or for storage. And like all waves, sound waves have wavelengths and frequencies. A wavelength ($\lambda$) is the distance between two crests or two troughs, whereas the frequency ($\nu$) of a wave is the number of crests (or troughs) in a second. Multiply these and you have the velocity of the wave, $v = \lambda \nu$.

### Indian and western classical music

Consider Indian classical music. There are two primary branches : Hindustani classical, prevalent in the northern half, and Carnatic classical, popular in the southern half of the country. If you have had the fortune (or misfortune) of listening to a young music student practice his scales early in the morning[2], you’d probably have heard the following vocalisations repeated ad nauseam

Sa – Re – Ga – Ma – Pa – Dha – Ni – Sa*

If you’ve listened closely, you’ll know that each successive syllable here is at a higher pitch than the last; that is, it sounds more treble, sharper, while the ones previous sound more bass deeper.

This is true in Western music too. In the Italian genre, massively popularised by Julie Andrews in The Sound of Music, you have

Do – Re – Mi – Fa – Sol – La – Ti – Do*

while the German version, widely used nowadays, is

C – D – E – F – G – A – B – C*

You notice those asterisks attached to the notes at the end? We’ll come to that soon.

### Creating music

There are, of course, two ways for humans to produce music; either use one’s own body or use an external device. The first is usually called singing, while the second involves the use of musical instruments. Using either method effectively requires long periods of training. Untrained people, or those trained insufficiently, produce sound that is most unpleasant. A good example would be your neighbourhood musically enthusiastic but thoroughly ungifted early riser.

What does it mean to use either method effectively? Are there some criteria for good (or bad) music? Is it like modern art, entirely subjective, or can it be analysed, at least partially, in a scientific manner?

Well, of course it can! Most things in this world can be analysed scientifically. Even human stupidity can…but I digress.

## Music and Frequency

It has, of course, to do with frequencies. When your voice skips from a Sa to a Re, or down from an A to a D, your vocal chords attempt to, with some help from your brain, find the correct frequency. That is, the frequency at which the lower note sounds right. If it is off, even by a bit, even non-practitioners realise that something is not quite how it should be.

If it sounds a tad subjective, it isn’t, really. Our brains are wired to respond favourably to music where the frequency relation between various notes are, in want of a better term, nice.

### What is nice?

What is nice? Scores of people throughout history and geography have tried to find out. One of the first was Pythagoras, him of the right-angled triangle. The Greek polymath’s attempts culminated in the sixteenth century efforts of Zhu Zaiyu, a Chinese prince, who calculated the exact relation between the twelve notes in an octave that would make transposition simplest.

I might have gotten a little ahead of myself. Backing up now, slowly.

### The keys of a piano

Consider a piano. There are white keys, and there are black keys. However, these are not arranged randomly. The black keys are always in between two white keys, and two successive black keys have, alternately, two or three white keys in between them. This is the patterns in most keyboard instruments nowadays.

### Finding the C

If you have a piano at home, great! You’re rich! If you don’t, and instead have a keyboard, awesome. If not, don’t fret, use your imagination and trace your eyes to about the middle of the keyboard. The black keys are grouped in twos and threes, each black key always between two white keys . Pick a group of two black keys; any pair would do. Play the white key to the immediate right of the first of the two black keys[3]; that would be the cyan key in the second image below. The note you hear is a C. Find another such pair, play the first key. Again, the note you hear is a C. However, this C is the not the same as the first C. Depending on whether the second key you pressed is to the left or to the right of the first key you pressed, the note is lower or higher in pitch. For instance, in the first image below, if you play the C on the right, it will sound higher in pitch than the C on the left. However, somehow, these two still sound very similar.

The same is true for any of the other keys. If you shift your focus to the second key in the pair, the one to the immediate right of the C, then you have a D. All such white keys are D. Their frequencies are also closely related. Hit one of the C keys in the middle, then find the next occurrence of the C, and play it. The second frequency will be, provided the piano is in tune, exactly double the first.

### Frequency relations between the Cs

Which brings me back to the curious asterisks I had a while back, when I was writing out the DoReMi or the SaReGaMa or the CDE notations. Here’s the CDE again, for reference

C – D – E – F – G – A – B – C*

If you play this sequence on a piano[4] or a keyboard, you’ll find that the C* is twice the pitch of the C. So, if the first C has a frequency of 100 Hz, then the second C will have a frequency of 200 Hz. You could say

$$f_{C^{\star}} = 2 f_{C} \tag{1}$$

#### Heinrich Hertz

A quick detour about the units. Hertz, abbreviated as Hz, is the unit of frequency. In plain English, it means cycles per second. In plainer English, it refers to the number of troughs (or crests) that the wave has per second. The unit is named after Heinrich Hertz, a German physicist who was the first to conclusively demonstrate the existence of electromagnetic waves, first postulated by James Clerk Maxwell in his electromagnetic theory of light. In the twentieth century, most common people would associate the word hertz with the radio; in the twenty-first, they associate it with clock and bus speeds in computers and smartphones.

### Octave

Anyway, back to music. If C* has twice the frequency of C, 200 Hz to its 100 Hz, then C* is said to be an octave higher than C. Similarly, if you go down (left on a piano) to the lower C, of frequency 50 Hz, then this C will be said to be one octave below C.

### Scientific notation

All these C’s are a tad confusing. Which is why scientists use subscript indices to designate the various C notes. For our purposes, let out 100 Hz note be $C_1$, the 200 Hz be $C_2$, an octave higher at 400 Hz is $C_3$ etc. The 50 Hz note can be $C_0$, but hopefully we won’t need it.

On a piano, the keys begin from A, not C. There are a total of 88 keys, of which 52 are white keys, spanning a little more than seven octaves. The middle C is in the fourth octave, and is designated $C_4$.

This octave structure in music makes labeling simple. You only need to think about one note, its double, and the notes in between. Once you have figured that out, it is a simple thing to reapply it to the other octaves.

### Frequencies of intermediates

Right. We now know the frequencies of two notes, $C_1 = 100 Hz$ and $C_2 = 200 Hz$. We know $C_n$ too, where $n$ are integers, but they won’t be necessary here.

Our next job is to figure out what the frequency of the intermediate notes (D, E, F, G, A, B) could be. The problem is to chop up an interval of 100 Hz ($C_2 – C_1$) into seven pieces so that the notes don’t sound awful.

Do keep in mind that in actual music, the C notes are rarely at 100 Hz. What I am offering is just a simple pedagogical example. I will talk about actual musical frequencies later on.

## Just intonation

One simple way is to use ratios of simple integers. Pythagoras used a version of this method way back in ancient Greece. Over the centuries, this method has been modified and improved, and by the time of renaissance and post-renaissance Europe, it had evolved into what is known as just intonation.

Do keep mind that a piano is not tuned in just intonation. Instead, a method called equal temperament is used. I’ll get to that in part II of this series, The Colour of Music : Notes of Equal Temperament.

#### Octave (C*) & the perfect fifth (G)

The octave is a perfect example of simple ratios–$C_2$ is $2:1$ times $C_1$. $C_1$ itself is a trivial example, for it has a ratio of $1:1$. The next ratio is, naturally, $3:2$. Multiply $C_1$ with $3/2$ and you get the note G or Ga or Sol, with a frequency value of $150 Hz$. This note is called the perfect fifth. This is one of the most important intermediate notes, and playing this together with the unison, C, makes for pleasing hearing. Or, as musicologists like to call it, consonance.

#### Perfect fourth (F), major third (E), major second (D)

The next simplest ratio is $4:3$. This is the perfect fourth, designated F, with a frequency value of $133.333… Hz$. This is followed by $5:4$ at $125 Hz$; this is E, also called the major third. We skip the next three ratios and move straight on to $9:8$, at $112.5 Hz$; this is D, the major second.

Do remember that all these ratios are with respect to the base note, C ($= 100Hz$), and not with respect to the previous note[5].

You do realise why these are called perfect fifths and fourths and major thirds and seconds, don’t you? That is the order in which they appear after C in the series CDEFGAB. That way, we can expect A and B to be the major sixth and seventh notes. Why major? Wait till Part II.

#### C-D-E-F-G

Here’s a quick table to organise matters (rounding off to one place after the decimal):

Name Symbol Ratio Value
Perfect Unison C 1:1 100.0 Hz
Major Second D 9:8 112.5 Hz
Major Third E 5:4 125.0 Hz
Perfect Fourth F 4:3 133.3 Hz
Perfect Fifth G 3:2 150.0 Hz

That takes care of the lower half of the octave in just intonation. Clearly, we can’t do much more with ratios of consecutive numbers anymore. Going higher than $9:8$ will only decrease the frequency, whereas we want to go beyond 150 Hz. Also, we don’t want to beyond $2:1$, so keep that in mind.

#### Major sixth (A)

Thus the next viable candidate is $5:3$, with a frequency value of $166.66… Hz$; this is A, the major sixth.

I’m certain that you’ve noticed by now that I’m not really talking about the ratios that have 5 in the denominator. Had I done so, $6:5 = 120 Hz$, $7:5 = 140 Hz$, and $8:5 = 160 Hz$ should have featured by now.

Fear not, they shall, in the next part of this series. They just don’t quite fall into the white major-ity.

That was a dreadful pun and I apologise.

#### Major seventh (B)

Moving on. We need one more note, the major seventh, note B. The next viable candidate is, naturally, $7:4$, giving a frequency value of $175Hz$. Which completes our octave comprising seven notes in ra…

Err, no. That’s not what happens.

For some strange reason, detectable only to people who actually understand music, $7:4$ is not used as ratio. Neither is $9:5$. In fact, the next note comes a lot further on, at $15:8$! So B has a frequency of $187.5Hz$, and we can update our table:

Name Symbol Ratio Value
Perfect Unison C 1:1 100.0 Hz
Major Second D 9:8 112.5 Hz
Major Third E 5:4 125.0 Hz
Perfect Fourth F 4:3 133.3 Hz
Perfect Fifth G 3:2 150.0 Hz
Major Sixth A 5:3 166.7 Hz
Major Seventh B 15:8 187.5 Hz
Octave C* 2:1 200.0 Hz

### C major scale

In musical parlance, this comprises the C Major Scale in just intonation. This is the most basic scale[6] one uses, and has obvious pedagogic benefits. Musically, it’s pretty boring.

The following is an audio sample of C-major scale tuned using just intonation. Sound credit : Kyle Gann.

It is rather simple to generate these tunes using a computer nowadays. However, if you want to go old school, and get your hands dirty, here’s a nice way to find CDEFGAB while, hopefully, gaining better understanding of the system.

#### Finding the notes : octave, fifth, third

The tools at our disposal are ratios. Not just simple ratios, ratios of the first three prime numbers; $3:2$, $2:1$, and $5:2$. Since $5:2$ is greater than 2, we divide it by 2 and take it down to $5:4$. These are our principal ratios, the octave, the fifth, and the third, and it is possible to generate the C major scale from the C key by repeated combinations of these three ratios. So, for instance, to get to the perfect fourth, F, you need to start from C ($1:1$), go up an octave to C* ($2:1$) by multiplying with 2, then down a fifth to F ($4:3$) by dividing by 3/2.

The table will make things clearer:

Note Up/Down Multiply by… Final ratio
C Stay $\times \frac{1}{1}$ $1:1$
C* Up octave $\times \frac{2}{1}$ $2:1$
G Up fifth $\times \frac{3}{2}$ $3:2$
E Up third $\times \frac{5}{4}$ $5:4$
F Up octave, down fifth $\times \frac{2}{1} \times \frac{2}{3}$ $4:3$
B Up fifth, up third $\times \frac{3}{2} \times \frac{5}{4}$ $15:8$
D Up two fifths, down octave $\times \frac{3}{2} \times \frac{3}{2} \times \frac{1}{2}$ $9:8$
A Up octave, up third, down fifth $\times \frac{2}{1} \times \frac{5}{4} \times \frac{2}{3}$ $5:3$

Hmm. So that is why that weird 15:8 ratio. Makes sense now, doesn’t it?

#### Five limit tuning

I included the fourth column as an afterthought. This is why this system is called 5-limit tuning, because you only have powers of prime numbers below 5 in your toolbox. As you can see, it is enough for the C major scale.

This system of simple numerical ratios is called just intonation. In the next part in this series, The Colour of Music : Notes of Equal Temperament, I’ll take up equal temperament, a system involving logarithms. I’ll also look into scales more interesting than C major, and will discuss the minor notes.

Until then, here’s Signora Bianca Castafiore practising music. Poor Archibald. Poor poor Archibald.

## Footnotes

1. Although, truth be told, most Kolkata FM channels are more ad than music these days. ↩︎
2. At a time when you’d rather he shut up and let you sleep. ↩︎
3. First means the one on the left ↩︎
4. Easy enough. Find a C, then play all the white keys. ↩︎
5. A mistake I had made while trying to wrap my head around this stuff. Things had gotten pretty interesting after a while. ↩︎
6. The way I understand it, a scale is basically the set of notes you choose to play with during composition. As will be clear in Part II, the most natural way to divide up an octave is in twelve parts, not seven. You then choose 7 of these 12 notes available to you and form a sort of a scrabble rack. Then, you form musical words and phrases from this stack. Music is Scrabble. Hmm, never thought of it that way. ↩︎
Why Indigo? The Mystery behind Newton’s VIBGYOR

## Why Indigo? The Mystery behind Newton’s VIBGYOR

Four and a half centuries ago, Isaac Newton used a prism and a dark room to split white sunlight into its coloured components. He labelled seven primary colours, later known by the mnemonic VIBGYOR. Curiously, he chose indigo to be one of the seven, a colour that most people fail to pick out from the spectrum. Why did Newton choose this elusive colour to be one of his chosen seven? Why indigo? The Nerd Druid Investigates!

## VIBGYOR and the rainbow colours

In my previous article, I spoke at length about the history of the colours in the rainbow, what VIBGYOR and its inverse ROYGBIV mean, and how non-anglophone cultures perceive the seven rainbow colours. Here’s a quick recap.

### Rainbow in world cultures

Most cultures associate the rainbow with a god holding a bow. For the Estonians, it is the thunder god Erruk. For the Indians, it is either the thunder god Indra or the prince of Ayodhya, Rama. The Norse do not have a bow-wielding divine being. Instead, for them, the rainbow is the Bifröst bridge in divine Asgard.

### Aristotle and the Greeks

The Greeks ascribed three primary colours to the rainbow : porphyra (dark purple), khloros (green), and erythros (red). Aristotle allowed for a fourth, yellow, but made it quite clear that this was a composite, being darker than white but lighter than red.

### Newton and the Spectrum

In his 1671-72 paper New Theory about Light and Colours and then, in more detail, in the 1704 masterpiece Opticks, Newton describes how he placed a glass prism in front of a hole in his window, and, having darkened his room, observed on the opposite wall

the Spectrum … appear tinged with this Series of Colours, violet, indico, blue, green, yellow, orange, red, together with all their intermediate degrees in a continual succession perpetually varying…

### VIBGYOR and ROYGBIV in anglophone cultures

Post-Newton, anglophone cultures adopted ROYGBIV as a mnemonic for schoolchildren to learn and remember this seried of seven colours. Schoolchildren in the US were told that this refers to a man named Roy G. Biv, while those in the UK learnt a bit of history too, with Richard Of York Gave Battle In Vain.

The inverse, VIBGYOR, which agrees with Newton’s own colour order, is probably an Indian English thing.

### BayNeeAaShoHoKawLa, the Bangla rainbow

Bengali people too had a nice mnemonic for the rainbow colours. BayNeeAaShoHoKawLa (বেনীআসহকলা) was, like VIBGYOR, made out of the first letters of the seven colours. However, there is a crucial difference. While VIBGYOR is

Violet > Indigo > Blue > Green > Yellow > Orange > Red

BayNeeAaShoHoKawLa, when translated to its equivalent English colour names, reads

Violet > Blue > Cyan > Green > Yellow > Orange > Red

A quick comparison with a good modern spectrum shows that the Bangla BayNeeAaShoHoKawLa is probably more accurate than VIBGYOR. So, the question is…

## Why Indigo?

Most people were perfectly happy with indigo being the second (or the sixth, if you take ROYGBIV) for almost two and a half centuries after Newton’s spectrum. However, in the early part of the 20th century, some curious sould began to ponder the significance of seven primary colours. Some of these theories have long been discredited, others are perhaps more plausible. Analysis of Newton’s choice got a fresh impetus in the 1970s, and one would very much like to believe it was due a letter from a reader in a 1973 edition of the British popular science magazine New Scientist. The letter, by a Mrs H Davoll, of 10 Broadlands Avenue, Shepperton, Middlesex, UK, titled Why Indigo?, was published on 13 Dec 1973. In it, Mrs Davoll says that

Ever since I first learned the sequence of colours in the rainbow, I have been puzzled and annoyed by the inclusion of indigo in this sequence…I feel that there is an element of the Emperor’s clothes situation in this matter, most people not daring to admit that they cannot distinguish another colour between blue and violet.

Let’s find out why, shall we?

## Colour nomenclature : Sociopolitical Effects

Compare VIBGYOR with BayNeeAaShoHoKawLa again. I’ll lay them out one after the other, along with a modern spectrum :

Violet > Indigo > Blue > Green > Yellow > Orange > Red

Violet > Blue > Cyan > Green > Yellow > Orange > Red

Let’s imagine, for a moment, that both versions are accurate, and actually refer to the same thing. If that is so, then would we be amiss in thinking that, for some reason, what Newton thought was blue is what we know as cyan nowadays? There are precedents for this sort of linguisting colour labelling phenomenon. Homer, the blind Greek poet of the Epics, referred to

…the wine-dark sea…

while we, social media savvy modern humans, have great difficulties in figuring out the correct colour of an evening gown. Take the colour of the sky, for instance. Some would say it is cyan, some would prefer azure, while people with simpler vocabularies, such as me, would probably have to make do with sky-blue.

### Newton’s indigo is today’s blue?

Take a look at the RGB colour star. Would you say that the colour of the sky is what is labelled as azure in the diagram? Or is it perhaps closer to cyan? If it were up to me, I would probably opt to lighten azure a little and call it sky-blue.

Therefore, isn’t it at all possible that, just as we posited that Newton’s blue might be today’s cyan, Newton’s indigo might also be today’s blue? After all, indigo had been adopted in Europe as a natural dye since about a hundred years before Newton’s experiment. It is entirely possible Newton himself owned clothing that had been coloured blue using the indigo dye. Would it be impossible for him to possess and wear a blue bowtie, dyed with indigo, and state, “I wear an indigo bowtie now. Indigo bowties are cool!”

### Orange and Indigo : Fruit and Dye

Consider oranges. They are a fruit, but they are also a colour. Which came first?

Well, the fruit did. Portuguese merchants brought sweet Indian oranges to Europe in the late 15th century, displacing the bitter Persian oranges grown in southern Europe. The word orange itself has a circuitous route,

narangas (Sanskrit) > narang (Persian) > naranj (Arabic) > naranza > narancia > arancia (Italian) > orange (French, English)

After arriving on English shores, the fruit lent its name to the colour; the first English usage was in 1512. By Newton’s time, a century and a half later, the colour was not just connected to delicious sweet oranges, but had entered daily use, supplanting yellow-red, saffron, and citrine as the most popular word describing that particular colour.

Similarly, indigo, though originally a dye, must have been closely connected with anything dark blue, prompting Newton to use it in his spectrum.

All this was, of course, quite unnecessary when it came to the other five colours, Newton’s Originals. Red, Yellow, Green, Blue, and Violet were all in use for centuries by the time Newton landed in the scene, and had probably lost their charm as colour names. Perhaps orange and indigo were still exotic enough to be used fashionably.

### Why Indigo?

If so, why wasn’t Mrs H Davoll equally bothered about orange? That is because of how current affairs and recent history shaped our perception off colour and usage of certain colour names. Unlike the fruit orange, a daily item in our lives, the dye indigo isn’t as ubiquitous.

#### The various names for dark blue

For instance, British naval uniforms were first introduced in 1748. They were coloured blue using indigo dye. Soon, though, that particular dark shade of blue came to be known as navy blue. Over the years, indigo has lost out to navy as a label for a dark blue in the fashion world. Among painters, the preferred term for a dark blue pigment is ultramarine.

Ever since the discovery of the first artificial dye in the mid-19th century, the use of the dye indigo has dropped off steadily, and so has the use of the colour term. While navy blue and ultramarine have survived into the 21st century, indigo has fallen by the wayside. So much so that nowadays, the word is used only when referring to the rainbow, or recollecting Newton’s work.

## Colour perception : Psychological Effects

In 1920, Edridge-Green developed a theory of colour perception in which he separated colour vision into seven classes or psycho-physical units. Naturally, these units were based on Newton’s seven colours. According to Edridge-Green, a normal person is hexachromic; she should be able to see six colours–red, orange, yellow, green, blue, and violet. People with extraordinary vision, those who can all seven of Newton’s colours, are classified as heptachromic. These people have eyes and vision sensitive enough to distinguish and detect an indigo in between blue and violet.

It is difficult to believe Newton was a normal hexachromic like most other people; surely one of the greatest human geniuses must have extraordinary vision too, no?

Well, no. In fact, Newton’s vision was actually somewhat poor. So much so that, during his prism spectrum experiment, he had to ask

…an Assistant whose Eyes for distinguishing Colours were more critical than mine…

to draw the boundaries of the spectral colours. In his own words, this was

…because my owne eyes are not very criticall in distinguishing Colours…

### Newton’s assistant

Newton’s assistant in the prism experiment, the heptachromic, was possibly John Wilkins, his roommate at Cambridge. Wilkins and Newton were roommates for 20 years, and possibly enjoyed a sort of Holmes-Watson dynamic. Wilkins left Cambridge in the early 1680s and moved 240 km away to Stoke Edith. There he joined the parish, got married, and had a son. The departure of his Watson hit Newton hard; he buried himself even deeper into his research, and channeled his loneliness into work. In this period, a meeting with the astronomer Edmund Halley led him to channel his energies into what would ultimately become the Principia Mathematica, and with it the birth of calculus.

No, not the Tintin character.

## Colour perception : Physical Effects

### Raman and bright sunlight

Edridge-Green’s theory in no longer in favour, and has been discredited. Instead, C.V. Raman, in his book The Physiology of Vision (1968) had suggested that perhaps it was Newton’s use of sunlight that led to indigo. Raman’s hypothesis was based on the following fact; use a bright enough source of light for your spectrum experiment and you might well be able to make out a band of colour in between blue and violet, a colour that might as well be called indigo.

Sounds plausible. However, if you do indeed perform the experiment with sunlight, then, under ideal viewing conditions, you should be able to detect as many as 200 separate hues of colour. Whether you choose to name all of them, or merely a certain subset, would quite likely depend on your personal preferences, shaped by your culture and background.

Thus, I’m afraid we’re still on Why indigo?.

### The Glass Prism

Modern optical lenses and prisms are made out of two materials, crown glass and flint glass. The latter contains lead, is denser, and has a higher refractive index, which means light rays bend more while passing through it1 Newton probably used crown glass, although it seems his prism had a slightly higher refractive index than is usual. This was possibly because his prism had some lead in it too. Nevertheless, the spectrum it creates seems heavier on the blue-end.

This bias towards the blue end might have prompted Newton and Wilkins to identify an extra colour in between violet and blue.

## Colour number : Mystical Effects

In addition to mathematics and physics, Newton had considerable interest in alchemy, astrology, and theology. He wrote almost two million words on these subjects, a number that is almost twice the million words he wrote on science. His mystical interests have led him to be accused of being

…misled by a predilection for the number seven which during many ages has been regarded with a sort of mystical veneration…

It is entirely possible Newton was as fascinated with the number seven as the ancients were. There were enough references to that number in antiquity for Newton to have been influenced enough to have expanded his original five colours to seven, by inserting orange and indico.

#### Seven celestial bodies

The number of known celestial bodies in antiquity was seven : the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn. Uranus and Neptune were discovered after Newton’s time, and Pluto is now a dwarf planet.

#### Seven days in a week

There are seven days in a week. Be it the Gregorian or Julian calendars of Europe, or the Bangla and neighbouring calendars of India, we always seem to find seven days. Also, these calendars attribute the days of the week to the seven celestial bodies–Tuesday to Mars, Wednesday to Mercury, Thursday to Jupiter, Friday to Venus, and Saturday to Saturn, the first two being rather obvious.

#### Seven metals of antiquity

The number of known metals in antiquity was seven–Gold, Silver, Iron, Mercury, Tin, Copper, and Lead. These were associated with the days of the week as well as the planets.

#### Planets, Days, Metals, Gods

The table below shows the ancient associations.

Metal Day Planet Greek/Roman god Norse god
Gold Sun Sunday Helios/Sol Sunna/Sól
Silver Moon Monday Selene/Luna Máni
Iron Mars Tuesday Ares/Mars Tyr
Mercury Mercury Wednesday Hermes/Mercury Odin
Tin Jupiter Thursday Zeus/Jupiter Thor
Copper Venus Friday Aphrodite/Venus Freya

There are seven deadly sins in Christian theology

1. Envy
2. Greed
3. Pride
4. Lust
5. Gluttony
6. Sloth
7. Wrath

Of course, one could counteract these evil evilnesses by embodying the seven virtues.

#### Seven heavens and worlds

Different ancient cultures and religions believed in seven heavens, or worlds divided in parts of seven. For instance, in Hinduism, there are fourteen worlds, and they are divided thus (I’m quoting directly from Wikipedia)

According to some Puranas, the Brahmanda is divided into fourteen worlds. Among these worlds, seven are upper worlds which constitute of Bhuloka (the Earth), Bhuvarloka, Svarloka, Maharloka, Janarloka, Tapoloka and Satyaloka, and seven are lower worlds which constitute of Atala, Vitala, Sutala, Talatala, Mahatala, Rasatala and Patala.

#### Seven day Creation myth

Creation myths in the Abrahamic religions, particularly Christianity, propound the belief that god built Creation in seven days.

#### Seven Wonders of the Ancient World

There were Seven Wonders of the Ancient World

1. Great Pyramid of Giza
2. Hanging Gardens of Babylon
3. Temple of Artemis at Ephesus
4. Statue of Zeus at Olympia
5. Mausoleum at Halicarnassus
6. Colossus of Rhodes
7. Lighthouse of Alexandria.

#### Seven Liberal Arts

In ancient Greece, knowledge of the Seven Liberal Arts were considered essential for a free person. These were

1. Grammar
2. Logic
3. Rhetoric
4. Arithmetic
5. Geometry
6. Theory of music
7. Astronomy.

Correspondingly, or perhaps not, the number of core subjects one has to write one’s tenth level exam in Bengal, the dreaded Madhyamik, is also seven. Bengali and English are the languages, History and Geography make up the humanities, while Mathematics, Physical Sciences (Phys+Chem) and Life Sciences (Biology) make up the sciences.

However inspired Newton might have been from these many examples to expand his colour roster to seven, one should remember that he was, after all, a consummate logical scientist. Would he really have no scientific reason for inserting indigo and orange? Shouldn’t one justifiably expect scientific answers to why indigo, as also why orange? Answers that are not only scientific, but also reveal a crucial insight about how the world works?

## Colour pitch : Light and Music

DO – RE – MI – FA – SOL – LA – TI

If you have seen The Sound of Music, you will be familiar with these seven notes. Sung joyously by Julie Andrews, these seven are the names of the notes in the diatonic scale. These can also be represented as

C – D – E – F – G – A – B

or as

Sa – Re – Ga – Ma – Pa -Dha – Ni

in Hindustani classical music.

### Musical notes are audio frequencies

Musical notes are, essentially, audio frequencies. In Western music, the A key in the fourth CDEFGAB series of the piano, denoted as A4, is set to exactly 440 Hz, and other notes are derived from it by changing frequency in units called semitones. Each octave has 12 semitone intervals. For instance, C4 is 9 semitones lower than A4. Under the popular equal temperament scheme, this makes C4‘s frequency 261.63 Hz.

A larger unit in music theory is the whole tone. It is double the semitone, and together, these two are instrumental in setting scales. For instance, in a diatonic scale of seven notes, there are 5 whole tones (T) and 2 semitones (S), always in groups of TTS and TTTS. Where you start from determines which mode you are following.

#### Ionian Mode

The CDEFGAB series that I began this section with is in the oft-used Ionian mode, with intervals TTSTTTS. Thus, in order to go from C to D, or from D to E, you need to increase by a whole tone T. E to F is a semitone S. F to G, G to A, and A to B are three whole tone (T) intervals TTT. Finally, to get to the higher C, you need to increase by a semitone (S).

#### Dorian Mode

In Newton’s time, however, the Dorian mode was much more in vogue, with notes DEFGABCD and intervals TSTTTST. The table below might help

Mode Ionian Dorian
Notes C–D–E–F–G–A–B–C D–E–F–G–A–B–C–D
Intervals T–T–S–T–T–T–S T–S–T–T–T–S–T

In the Dorian mode, the semitones occur at positions 2 and 6, exactly at the positions orange and indigo appear. Adding Newton to the table above (and removing Ionian mode), we get the answer to why indigo.

Mode Dorian Newtonian
Notes D–E–F–G–A–B–C–D R-O-Y-G-B-I-V
Intervals T–S–T–T–T–S–T P-S-P-P-P-S-P

where P are primary colours and S secondary.

### Newton’s Insight

As Newton himself writes (Opticks, pg 92)

…I delineated therefore in a Paper the perimeter of the Spectrum FAPGMT, and … I found that the … rectilinear sides MG and FA were by the said cross lines divided after the manner of a musical Chord…to be in proportion to one another, as the numbers, 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2, and so to represent the Chords of the Key, and of a Tone, a third Minor, a fourth, a fifth, a sixth Major, a seventh, and an eighth above that Key: And the intervals Mα, αγ, γε, εη, ηι, ιλ, and λG, will be the spaces which the several Colours (red, orange, yellow, green, blue, indico, violet) take up.

Appreciate Newton’s insight and genius. What are colours? They are simply frequencies of light; their pitch are seen and not heard, their instruments of detection are the eyes and not the ears. They are both, ultimately, waves. Very different types of waves, granted, but waves nevertheless, with frequencies and wavelengths.

Why indigo? Because Music!

## References

### Books

1. Aristotle : Meterology, Greece (350 BCE).
2. Newton, Isaac : Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light . Also two treatises of the species and magnitude of curvilinear figures, Sam Smith & Benj. Walford, for the Royal Society (MDCCIV, 1704).

### Papers

1. Newton, Isaac : New Theory about Light and Colours, Philosophical Transactions (1672).
2. McLaren, K. : Newton’s Indigo and references therein, Color Research and Application (1985).

### Articles

1. Fisher, Len : Perceptual thresholds: Music inspired Newton’s rainbow, Nature (2015).
2. Morr, Kelly : Why are there 7 colors in the rainbow?, 99designs, (2016).
VIBGYOR : Newton’s Rainbow and Indigo

## VIBGYOR : Newton’s Rainbow and Indigo

VIBGYOR is a popular mnemonic for the seven rainbow colours. 450 years ago, Newton split white light into its coloured components and labelled seven of them. Curiously, most people see only six. The Nerd Druid Investigates!

## VIBGYOR

One of the very first English words I had learnt was VIBGYOR. Of course, it wasn’t really a word, but it was associated deeply with that thing that all childhood craves, colour.

Violet. Indigo. Blue. Green. Yellow. Orange. Red.

Red is the colour of passion, of love, of anger. Orange tastes sweet and sour, and reminds one of winter, and the Dutch; it is also the colour of greed. Yellow is the brightest of all colours, but is also associated with cowardice, and fear. Green is Life itself, of its great willpower to “…always find a way…“, as Ian Malcolm loves repeating. Blue, the colour of the skies and the seas, is calm, and instills hope. Indigo has a colourful history as a natural dye, and a confused one as regards its place in all this. And when a nor’wester, an April thunderstorm gathers clouds so deep that they look violet, you know the evening’s going to get more interesting.

### ROYGBIV

Richard Of York Gave Battle In Vain

Up until a few days ago, I used to think that VIBGYOR was the most common colour mnemonic in the English-speaking world. Turns out this is only true for India. For the US and the UK, the mnemonic of choice is ROYGBIV.

Now, that is of course simply VIBGYOR reversed; the Brits and the Yanks seem to prefer starting off with red. While ROYGBIV doesn’t quite pronounce as sweetly as VIBGYOR (vibjeeyohr), it does make up a (sort of) a name, Roy G. Biv. This is how pre-schoolers in the US learn their colours. The British are far more dextrous; they also have thicker history books. Thus,

Richard Of York Gave Battle In Vain

Richard of York was, of course, Richard III, the last king of England to die in battle. In 1485, he was defeated and killed at the Battle of Bosworth Field, an event that ended the War of the Roses1. The victor, Henry of the House of Tudor, ascended the throne as Henry VII.

## Aristotle

### Summer in Kolkata

Summers in India are hot. Summers in the Eastern metropolis of Kolkata are hot and very humid. Sweltering and suffocating are two English adjectives that attempt capture a Kolkata summer. They fall well short of the mark.

Curiously, the Kolkata summer this year, 2018, has been uncharacteristically…pleasant. Yes, it has been very hot and yes, it has been very humid. But, interspersed within these short bouts of I-want-to-run-away-to-the-hills, there has been rain and high wind and storms. Big, violent storms. And rain at times it usually does not do so in Kolkata. And the almost constant presence of clouds has made the sunsets absolutely gorgeous.

### Rainbows

…the rainbow necessarily has three colours, and these three and no others.

It is not difficult to imagine that, during these spells of rain, there might come instances, short periods of time when, looking up towards the heavens, one would see that slate-grey rain clouds covering half the firmament, delivering their watery loads to the thirsty patches below, while on the other end, the sun, having peaked out tentatively from its nebular veil, would shine gloriously for an instant, showering its silver rays through the curtain of rain water.

Essentially, one expects rainbows.

Rainbows are a staple of cultures throughout the geography and history of the world. In Estonian, for instance, the rainbow is the bow wielded by the thunder-god Erruk. In the Nordic Eddas, and now in the Marvel films, the rainbow comes from the Asgardian Bifrost, the bridge of many colours. In Bangla, the word is Ramdhonu, or Ram’s Bow. In Hindi, this transforms to Indradhanu, Indra’s Bow. Perhaps unsurprisingly, Indra is the thunder-god in the Hindu pantheon.

### The Greeks

Clearly, the ancients realised the correlation, if not the causation, behind rainbows and unsettled weather. Aristotle, the great Greek natural philosopher, was perhaps the first to peer closer to the rainbow in an attempt to classify the colours within, perhaps in a hope to divining its nature and purpose. Aristotle attempted to reconcile the colours of the rainbow with his theory that all colours came from white and black. In his book Meterologica, he says

…the rainbow necessarily has three colours, and these three and no others.

Aristotle’s triad of rainbow colours is the same as that suggested by his predecessor, Xenophanes of Colophon. These are porphyra (dark purple), khloros (green), and erythros (red). Aristotle allows for a fourth colour, yellow, a non-primary bright colour that is darker than white but lighter than red, and lives in between red and green.

### RGB, rods and cones

From a modern perspective, Aristotle and his predecessor is surprisingly correct. Greek colour names could be a bit…confusing, and porphyra could well be blue. Which means, to them, the three primary colours are blue, green, and red. RGB. All the other colours stem from them. Mix R and G and you have Y, yellow. G + B = C(yan); R + B = M(agenta). White and black, light and dark, add extra dimensions to these colours.

We now know why that is so. Our eyes have three classes of colour detecting cells. Some of these cones are sensitive to red light, some to green light, and others to blue light. The retina also has rod cells; these detect brightness (or darkness). Together, the three cone types and the rods recreate a gamut of colours for human stimulation.

Why, then, do we talk about the seven primary colours? How exactly did VIBGYOR come about?

## Newton

Blame Isaac.

Before Albert, these two words would be oft-heard in the Halls of Physics. Newton was single-handedly responsible for kick-starting and rejuvenating several prime physics disciplines. While he is most well-known for the incident with the apple, his prism comes a close second.

The Original or primary colours are, Red, Yellow, Green, Blew, and a Violet-purple, together with Orange, Indico, and an indefinite variety of Intermediate gradations.

After he was done with gravity in the 1660’s, Newton turned his attention to light. He had read Aristotle and the other Greeks, and was rather motivated by them. Like Aristotle, he too wanted to figure rainbows out, to find out what light is. And he had an analytical tool Aristotle did not; the glass prism.

### Newton’s Spectrum

On a bright and sunny day, Newton darkened his room, made a small hole in the window, and placed his prism in front of the hole. Sure enough, on the wall opposite, he saw a beautiful technicolour spectrum. In his own words;

…in order thereto having darkened my chamber, and made a small hole in my window-shuts, to let in a convenient quantity of the Suns light, I placed my Prisme at his entrance, that it might be thereby refracted to the opposite wall. It was at first a very pleasing divertisement, to view the vivid and intense colours produced thereby…

Once the prism was set, and the spectrum ready, all Newton had to do was to walk over to the wall and mark out the colours of the rainbow. Instead, he asked his assistant (possibly his Cambridge roommate, John Wickins) to do it, remarking later that

because my owne eyes are not very criticall in distinguishing Colours

Not only does this reveal an interesting facet of Newton—that the intensely immensely competitive man did not consider himself perfect–it also shows how English spelling has evolved and changed over the years.

### Newton’s Opticks

In 1671-72, Newton published “New Theory about Light and Colours” in the Philosophical Transactions of the Royal Society, in which he reported and explained his results. He wrote (emphasis mine)

There are therefore two sorts of Colours. The one original and simple, the other compounded of these. The Original or primary colours are, Red, Yellow, Green, Blew, and a Violet-purple, together with Orange, Indico, and an indefinite variety of Intermediate gradations.

More than 30 years later, in his Opticks, he modified his earlier statement slightly

the Spectrum … appear tinged with this Series of Colours, violet, indico, blue, green, yellow, orange, red, together with all their intermediate degrees in a continual succession perpetually varying

including updating the spelling of blue.

Of course, he did more than just label the colours. Careful observations and accurate sketching mixed with a healthy dose of Newtonian analytical genius told him that, as white sunlight passes through a prism, red refracts, that is, changes direction the least, while violet refracts the most. Nowadays we know that light is composed of photons, and that refraction is merely photons decelerating as they pass through an optically denser medium such as glass2. Photons of lower energies, such as those that appear red to us, decelerate the least, while the higher energy violet photons decelerate the most.

Which is all fine and dandy, and proves once again what an absolute magician Isaac Newton was.

But why seven colours?

And we thus come to the rub of the matter. Or the hub of the matter, where lies the rub…

Never mind. This is the central question. Why seven? Why indico, or indigo?

### Indigo

Well, first of all, a little bit about indigo. Indigo, as a dye, has ancient origins. According to Pliny the Elder, the Harappans extracted the dye from a certain plant (Indigofera tinctoria) that grew in the Indus valley. The Ancient Greek term for the dye was Ἰνδικὸν φάρμακον (Indikon farmakon). This later became indicum in Latin and later indigo in Portuguese. The Silk Route brought indigo to Europe, when Marco Polo reported about it in 1289. However, a further three centuries went by before the European textile landscape realised the potential of the dye, and started large-scale manufacture. A further three centuries went by before the first artificial dye was invented. Things have rather moved up nowadays, with quantum dots being the latest and best in the colour business.

Around Newton’s time, clothing dyed in indigo was quite in vogue. The dye itself had an air of exotic orientalism about it, and it might have been quite fashionable to say, “Ah, is that there an indigo doublet that I spy?” instead of saying “I want that blue vest.” Newton was human, though perhaps less so than most, and he must not have been entirely out of step with the times. He must have felt, “I’ll use indico a lot. Indico is cool.”

### Cyan

Which perhaps makes sense, except that to some people, it doesn’t. Most of the human race, when asked to identify the colours in a sunlight-prism spectrum, manage to name only six colours. A few sharp-eyed ones might manage seven, but there is a crucial difference. There is no indigo. The colour series is, for most,

Violet > Blue > Cyan > Green > Yellow > Orange > Red

What is cyan, then?

In the RGB colour star above, red, green, and blue are considered primary colours, in tune with human physiology. By mixing two of these primary colours in equal amounts, you get the secondary colours cyan (blue + green), yellow (green + red), and magenta (red + blue). Mix a primary with its adjoining secondary, and you have the tertiary colours. My favourite is obviously chartreuse green, the best colour there is.

Cyan (B+G) is basically what most people refer to as sky-blue. Although, truth be told, looking at the colour star, it kinda looks like azure might be closer to sky-blue. Either way, there is a pretty prominent colour between blue and green on this colour star, and it has non-English analogues as well.

### VIBGYOR in Bengal : BayNeeAaShoHoKawLa

I am from Bengal, India. Although we do a lot of English stuff, Bangla is our language. Our first words are in Bangla, and our first connections with this wondrous world is via words in Bangla. And does Bangla have an analogue to VIBGYOR?

Well, of course it does. Although, sadly, urban Bengali kids born after the nineties might not quite have heard of

BayNeeAaShoHoKawLa

In Bangla, বেনিআসহকলা.

I’ll break it down. BayNeeAaShoHoKawLaa3. The colours, and their English equivalents, are

1. Beguni (baygoonee, বেগুনী): Violet
2. Neel (kneel, নীল) : Blue
3. Aakashi (aakaashee, আকাশী)4 : Cyan/Sky-blue/Azure
4. Shobuj (showbooj, সবুজ) : Green
5. Holud (howlooð, হলুদ) : Yellow
6. Komola (kawmohlaa, কমলা) : Orange
7. Laal (laal, লাল) : Red

Notice any colour missing?

### The Modern Spectrum

Red photons have the least energy (among visible photons), and so have the longest wavelength. In the image, red appears at around 700 nm. Violet, at the other end of the visible spectrum, has the most energetic photons, and therefore the shortest wavelengths. In the image, violet appears around 400 nm. The human visual range is approximately that, roughly between 400nm to 700 nm. Any lower and we get high energy ultraviolet (UV) rays (< 380 nm); any higher and you have the low energy infrared rays (> 740 nm). Too much of UV is bad and can give you skin cancer; too much of infrared is also bad and can cook you.

Newton’s indico falls between 430 and 450 nm. Looking at the image, to me at least that looks rather blue, or at least blue-violet. I don’t know if you’ll do better. On the contrary, there is a clear cyan band between 475 nm and 500 nm.

Here we clearly find a violet, a blue, a cyan, a green, a yellow, and a red. No clear indigo, and, surprisingly, a somewhat muddled orange. Seems BayNeeAaShoHoKawLa works better than VIBGYOR.

Ever since I first learned the sequence of colours in the rainbow, I have been puzzled and annoyed by the inclusion of indigo in this sequence.

Newton wasn’t aware of BayNeeAaShoHoKawLa though, and generations of people after him have sworn by VIBGYOR (or ROYGBIV). No questions were asked till the early part of the 20th century, when people began to analyse indigo and why Newton’s colour spectrum has seven colours. However, the greatest boost probably came in 1973, from a very unlikely source.

### Helen Davoll and New Scientist

New Scientist is a popular science magazine5 based in London. Like other magazines, New Scientist too has a Letters section. In the issue dated 13 Dec 1973, among letters from readers providing valuable comments and input on issues as diverse as Jupiter’s Red Spot, methane power, and whether Uri Geller was a charlatan, there was one letter that was, perhaps, a little different. Titled Why Indigo?, the letter begins with these words :

Ever since I first learned the sequence of colours in the rainbow, I have been puzzled and annoyed by the inclusion of indigo in this sequence.

A sentiment, I’m sure, most of us share.

The letter was sent by a Mrs H Davoll, of 10 Broadlands Avenue, Shepperton, Middlesex, UK. She goes on to state that

I feel that there is an element of the Emperor’s clothes situation in this matter, most people not daring to admit that they cannot distinguish another colour between blue and violet.

Once again, I for one am completely in agreement with her.

This letter was followed by no less than five replies from various readers, and was preceded and succeeded by a number of printed peer-reviewed publications analysing the same question Mrs Davoll had asked, Why Indigo?

### Next on VIBGYOR

In the next installment of this series on VIBGYOR, The Nerd Druid shall investigate these explanations. The Nerd Druid shall also attempt to uncover who Mrs Davoll was, and will trace the sequence of comments and letters in the New Scientist. Until then, we should remember that this June…

## Pride

…is Pride month in the US. For the LGBTQ community, it is a time to come out in droves and celebrate life as normal human beings, to stand out from the stigma and oppression that accompanies them. It is a time for them to appreciate the full spectrum of which conspicuously omits indigo life. Which is why, appropriately, their symbols is the rainbow flag (six colours, no indigo):

## References

### Books

1. Aristotle : Meterology, Greece (350 BCE).
2. Newton, Isaac : Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light . Also two treatises of the species and magnitude of curvilinear figures, Sam Smith & Benj. Walford, for the Royal Society (MDCCIV, 1704).

### Papers

1. Newton, Isaac : New Theory about Light and Colours, Philosophical Transactions (1672).
2. McLaren, K. : Newton’s Indigo, Color Research and Application (1985).
Quantum Dots of All Hues : A Modern Marvel

## Quantum Dots of All Hues : A Modern Marvel

Quantum dots are fluorescent nanocrystalline semiconductors that seem to magically change colour with size. The Nerd Druid investigates this 21st century marvel.

Alexei Ekimov discovered quantum dots in a glass matrix in 1981. Four years later, in 1985, Louis Brus discovered quantum dots in colloidal solutions. Their discovery is now at the heart of nanotechnology.

### What are quantum dots?

Quantum dots are extremely small semiconducting crystallites. You could fit anywhere between 10-50 atoms along the diameter of a typical quantum dot. The entire structure contains hundred to thousands of atoms.

### How big are quantum dots?

The image above is another excellent way of comparing the size of quantum dots with everyday objects. The one on the very left is a standard football, while the image at the center is that of a human hair. The football is about 20 cm in diameter; the hair is about 80 μm (= 0.08 mm) in diameter. The inset of the rightmost image shows two quantum dots cirlced in yellow; each of them is about 2.5 nm in diameter1. That is about the width of DNA. You could fit about 30000 quantum dots side by side along the width of a human hair, and 80 million quantum dots side by side along the diameter of a football.

### What do quantum dots do?

Quantum dots fluoresce; shine light on them and they will glow. In addition to being photoluminescent, quantum dots are also electroluminescent. Pass electric current through them and they glow; put them in a high electric field and they glow.

#### Photoluminescence & electroluminescence in everyday life

Why, though, is that so awesome? Plenty of everyday items are photoluminescent and electroluminescent. For instance, your home fluorescent lamp is photoluminescent; electricity passing through the mercury vapour inside the tube causes it to fluoresce and emit ultraviolet radiation. Thankfully, the white layer of phosphor material coating the inner surface of the tube absorbs this UV light and fluoresces, emitting visible light2. Similarly, your car dashboard dials and your calculator backlight are electroluminescent.

### Quantum dots produce colour

No, the reason why quantum dots are awesome is their colour-producing capability. Take a look at the image of those coloured bottles above; each contains colloidal quantum dots and have been irradiated with UV light. You can see the entire VIBGYOR there; from left, Violet, Indigo/Blue, Blue/Cyan, Green, Yellow, Orange, Red, and, er, Red again! VIBGYORR, very pirate-y, savvy3?

#### But LCDs and LEDs also produce colour

It is not impossible to have coloured fluorescent lamps. With the invention of the blue LED (light-emitting diode) in the 1990s, we now have LED lamps of all colours. LCD (liquid crystal display) screens on televisions, computers, and mobile phones are nowadays capable of creating millions of colours. Organic LED screens and their evolved cousins, the AMOLED screens, have made it possible to have ultraHD TV, computer and mobiles.

Why quantum dots, then?

### Quantum dots can produce any colour

Take a magnifying glass and peer into the screen you are reading this on. If you don’t have one, use a drop of water. You’ll see that the pixels on the screens have three colours : red, blue, and green. Combinations of this RGB trio make up most of the colours we perceive. This is because our eyes also only have three types of colour-detecting cone cells, and these are sensitive in the red, green, and blue channels.

#### Red, Green, and Blue LEDs

LEDs are bulk semiconductors. The problem with them is that you can’t quite use the same material to make different colours. For instance, you can use AlGaAs (aluminium gallium arsenide) or GaAsP (gallium arsenide phosphide)4 to fabricate red LEDs. However, if you want a blue LED, you’d need to use InGaN (indium gallium nitride). You’ll not be able to make red LEDs out of InGaN, or green ones out of AlGaAs. That is how light-emitting semiconductors work.

#### Quantum dot colour depends on their size

Quantum dots are very different in that respect. You can tune their colour at will by simply changing their sizes! Larger quantum dots (5-6 nm) emit orange and red light. Medium sized ones (3 nm) emit green light, while smaller ones (2 nm) emit blue light. The ones in Swansea experiment, those circled in yellow in the earlier image, are about 2.5 nm in diameter and emit blue-green light. These technicolour quantum dots can all be fabricated from the same material.

#### An Amazing Technicolour Dreamcoat

If you think about it, this is quite remarkable. For LEDs, the only true colours you get are red, green, and blue5; the colour step-size for them are the differences in frequency between red and green, and that between green and blue. With quantum dots, however, the colours they emit is dependent on their sizes, which in turn is dependent on how many atoms they have. Thus, the colour step size here is devastatingly smaller than LEDs–take out one atom, or put one in, and you get a different frequency. Thus, a different colour. Voila! Human eyes are quite incapable of distinguishing that fine a colour step-size. To us, then, quantum dots can indeed produce any colour.

## How do quantum dots produce so many colours?

Why is that? Aren’t quantum dots semiconductors too?

Yes, they are. However, unlike bulk semiconductor diodes, quantum dots do not have Avogadro’s number level of atoms6. In fact, the smallness of quantum dots means that quantum effects begin to make their presence felt.

### Atoms & semiconductors

#### Atomic energy levels

Electrons is an atom are arranged according to their energies. Those with lower energies inhabit lower energy levels–or shells–and stay closer to the central nucleus, while those with higher energies live in higher shells. Now, shine light on a group of atoms. Some of the photons you fire will interact with the atoms. If the energy is correct, electrons might absorb the photon, gain energy, and jump to a higher shell. Pretty soon, however, that energised electron will want to fall back to the energy level it came from. It will do so by emitting a photon of its own. The energy of the photon emitted will be exactly equal to the energy difference between the two energy levels. If this happens fast, within nanoseconds, then this is fluorescence.

#### Semiconductor energy bands

A similar thing happens with light emitting diodes. However, since LEDs are semiconductors, they don’t have the discrete energy levels that atoms have. Instead, their energy levels form two separate continuum bands. An electron from the lower valence band absorbs energy and jumps up to the higher conduction band. In LEDs, it soon jumps back down, emitting a photon. The energy of the photon emitted is the sum of the energy of the band gap and the excitation energy of the electron :

E = Eband-gap + Eexcitation

In the image below, the brown double arrow shows the band gap. LEDs made of a particular material have a fixed band gap value, and thus can only emit light of a certain colour.

### Quantum dot confinement energy

#### Particle in a box

Contrast this with quantum dots. Since they are so small, electrons are trapped within them. Quantum mechanical effects kick in, and the energy levels of the electron are like that a particle trapped in a box7. This confinement has its own energy, and adds to the total energy of the fluorescent photon.

Efluorescence = Eband-gap + Econfinement + Eexcitation

It is this Econfinement that makes the quantum dot into a technicolour marvel. You see, according to quantum mechanics, the energy of a particle in a box depends on how large the box is. More particularly, the energy goes as the inverse square of the dimension of the box; decrease the box size by 2, and your energy will multiply by 4.

#### Electron in a quantum dot box

For an electron in a quantum dot, the quantum dot itself is its box. The inverse square law implies that smaller dots have higher Econfinement, and vice verse. You can see that in the energy level schematic above. As the size of the quantum dot decreases (red > green > blue), the energy gap increases (blue double arrow > green double arrow > red double arrow).

You can increase or decrease this gap by changing the size of the box, that is, the quantum dot. Theoretically, you can change it in steps of a single atom, that is, by a few tenths of a nanometers. The corresponding change in Efluorescence will be extremely fine, and virtually undetectable by human eyes.

#### Practicality

Of course, technology limits the amount of fine-tuning you can do. For instance, you can’t manually pick up an atom and deposit it within a quantum dot crystal, or pick one out of it. Yet. Everything depends on your crystal deposition process, what materials you use, what conditions you are working in, and how good you are. Still, modern quantum dots are good enough to have the highest true colour resolution among all others.

That is the magic of a quantum dot.

### Is that all?

Er, no. That is just the tip of the iceberg as far as quantum dots are concerned. They have tons of other uses other than being fancy screens. Solar panels made out of quantum dots are more efficient than their non-QD counterparts. Also, biomedical applications of quantum dots have picked up steam recently. Turns out quantum dots are brilliant at imaging inside the body, and are being used as such. Also, they make superb organic dyes, and are a 21st century extension of William Perkin’s dream.

Quantum dots aren’t universally fantastic, though. To manufacture them, you need to use compounds of heavy metals that are incredibly toxic to humans, other animals, and the environment. This severely limits their in vivo use. To counter that, a group of Indian-Welsh scientists have very recently teamed up to fabricate quantum dots from the extract of tea leaves. Not only does this decrease the overall toxicity, these tea-powered British Popeye quantum dots seem to be devastatingly effective at invading and killing lung cancer cells.

More on this in the next edition of The Nerd Druid Investigates!, Quantum Dots from Tea Leaves Kill Lung Cancer Cells! In the meantime, here’s a picture of Tintin and Snowy drinking tea.

## References

### Papers

1. Bawendi, Moungi G. & Steigerwald, Michael L. & Brus, Louis E. : The Quantum Mechanics of Larger Semiconductor Clusters (“Quantum Dots”), Annual Reviews of Physical Chemistry (1990)
2. Reimann, Stephanie M. & Manninen, Matti : Electronic structure of quantum dots, Reviews of Modern Physics (2002)
3. Yoffe, A.D. : Semiconductor quantum dots and related systems: Electronic, optical, luminescence and related properties of low dimensional systems, Advances in Physics (2010)
4. Dey, Samrat et al. : The confinement energy of quantum dots, arXiv (2012)

### Other resources

1. Quantum Dots | Sigma-Aldrich
2. Quantum Dots | Brus group, Columbia University
3. Nanotechnology Timeline, National Nanotechnology Initiative
E 0102 : A mysterious isolated neutron star

## E 0102 : A mysterious isolated neutron star

Neutron stars are exotic remnants of supernova explosions of massive stars. Most rotating neutron stars have extraordinarily high magnetic fields. These pulsars very often have a binary companion star from which they steal stellar matter. Recently, a neutron star in a neighbouring galaxy was found to have neither a binary companion nor a strong magnetic field. Curiously, the position of the supernova remnant with respect to the gases thrown off at the time of supernova does not quite tally. The Nerd Druid investigates E 0102, the lonely offset neutron star.

## Neutron stars

Neutron stars are supernova remnants that are not quite massive enough to form black holes1. They are extremely dense objects and have phenomenally strong gravitational fields. The pressure in the interior of a neutron star one of the highest found naturally in the universe, second only to the pressure inside protons. Temperatures within neutron stars right after formation can be as high as hundreds of billions of degrees, but escaping neutrinos cool the star down.

### Pulsars

Rotating neutrons stars that also have high magnetic field emit synchrotron radiation2. This radiation forms a beam along the magnetic axis of the neutron star; as the star rotates about its rotational axis–which may or may not coincide with the magnetic axis–these beams are observed at precise regular intervals. These are pulsars, celestial timekeepers that are highly accurate. Thanks to the beams, pulsars are easy to detect.

Pulsars are almost always binary stars; their stellar companions are often main-sequence or giant stars that have not yet reached supernova, or, due to lack of mass, never will. Most pulsars are found in the Milky Way galaxy and its neighbourhood.

So, basically, pulsar neutron stars are social beings that have magnetic personalities, but you’re likely to get burnt if you venture too close. Clearly, they have issues.

### Central Compact Objects (CCOs)

However, not all neutron stars are pulsars. There are some that have very low magnetic fields compared to their pulsar cousins. Called central compact objects (CCOs), these supernova remnants lack the active wind nebulae3 that are a telltale signature of pulsars. The pulsars PuppisA and CassiopeiaA (CasA), both located within the Milky Way, are excellent examples of CCOs.

Similarly, not all neutron stars have a binary companion. The Magnificent Seven, named after the spaghetti Western, are a group of seven isolated neutron stars. A decade ago, an eighth such star was found, and was promptly named Calvera, after the villain of the movie. Going by that logic, a ninth star might well be named Gabbar4.

It is quite rare to find neutron stars that both CCOs and isolated. Vogt et al have recently found one that is located outside the Milky Way. However, in doing so, they have uncovered a rather puzzling mystery.

## E 0102

The object of interest is the supernova remnant 1E 0102.2-7219. E 0102 for short, it is one of the few oxygen-rich young supernova remnants (O-rich SNR) in the Magellanic clouds5, and the only low magnetic field isolated neutron star outside the Milky Way. E 0102 is a little more than 200,000 light years away from Earth, and is only about two thousand years old.

### Observing E 0102 with Chandra and Hubble

Unlike pulsars, CCOs such as E 0102 cannot be detected by looking for synchrotron radiation. Instead, one has to rely on blackbody radiation from both the CCO and the supernova debris surrounding the CCO. Images from NASA’s Chandra X-ray Observatory (CXO, named after Indian astrophysicist S. Chandrasekhar) show that E 0102 is dominated by a fast-moving large ring-shaped structure (blue and purple in the X-ray image). This is, quite possibly, the forward envelope of the supernova blast wave. Ejected at supernova at speeds of millions of kilometres per hour, this shell is now, two thousand years later, approximately 12 light years wide. Older optical images from the Hubble Space Telescope show beautiful filaments of superheated highly ionized oxygen filaments (green in the optical image) within the shell.

### Position of E 0102 supernova remnant

Usually, one would expect, given the data available, that the supernova remnant should be right in the middle of the ring that Chandra observed. Unfortunately, as you can see from the X-ray image above, there is no corresponding X-ray source at the centre of the ring. There is, however, an X-ray point source slightly offset from the centre of this large ring, a bit towards the south-west. Given available data, that cannot be the remnant.

Vogt et al disagree. And they evidence, persuasive evidence. And to do that, they have moved beyond what NASA has had to offer.

#### ESO, VLT, & MUSE

The European Southern Observatory is an international collaborative effort at observing the southern skies. Established in 1962 with 5 members, ESO now boasts a roster of 15 member states that sponsor the installation, operation and maintenance of state-of-the-art telescopes, all located in Chile. Of them, the imaginatively named Very Large Telescope (VLT) is probably one of the most advanced optical telescopes available for research now.

The Multi Unit Spectroscopic Explorer (MUSE) is an advanced spectrograph attached to the VLT. Unlike older devices, MUSE provides high-end resolution in both spatial and spectral domains. Which basically means that it produces very clear pictures with sharp colours.

### Observing E 0102 with MUSE

Instead of relying on old Hubble and Chandra data, Vogt et al pointed the VLT towards E 0102. New optical data from MUSE showed that, in addition to the highly ionized fast-moving large gaseous ring, E 0102 also shows a smaller low-ionization slow-moving gas ring (bright red in the optical image) that is about a sixth of the former’s size. The origin of this new ring cannot be explained from the optical picture alone.

The fun begins when you put the optical and X-ray images on top of each other. It is immediately obvious that the smaller red optical ring neatly encircles the small blue X-ray dot. This perfect coincidence convinced Vogt et al that the small blue dot was indeed the hitherto undiscovered supernova remnant they had been looking for.

### Is that really the supernova remnant?

#### Apparent and absolute magnitudes

Well, sort of. Stargazing is a funny job. Since the stars are so far away, the night sky looks more or less like an upturned bowl, with the stars stuck to its underside. Take Venus and Sirius, for instance. Venus is the brightest point object in the night sky, while Sirius is the brightest star in the night sky. Naturally, the Sun and the Moon are disq-ualified.6 Now, going by apparent brightness, Venus outshines Sirius by a factor of 23.337. Actually, of course, Venus is a planet, while Sirius is a really bright star. This is evident in their actual luminosities : Sirius is 22.5 times as luminous as the Sun, which in turn is 550 million times brighter than Venus.

#### Constellations and distances

The same confusion holds for distances. Constellations are groupings of stars that are close together. For instance, Orion (Kalpurush), one of the most distinctive and conspicuous constellations in the night sky, has seven major stars. Of these, the two brightest are Rigel, a blue-white supergiant, and Betelgeuse, a red supergiant. The most distinctive feature of the constellation is the belt; Alnitak, Alnilam and Mintaka are the three stars that make up Orion’s belt. All these stars, being part of the same constellation, seem to lie close to each other on the same plane.

The operative word is, naturally, seem. Here is how far the stars actually are from Earth (lyr = lightyear) :

1. Betelgeuse = 640 lyr
2. Rigel = 863 lyr
3. Alnitak = 800 lyr
4. Alnilam = 1340 lyr
5. Mintaka = 915 lyr

Clearly then, if we move away from Earth, the constellations will not retain their familiar shapes.

### So, is it the CCO or not?

The same issue is relevant when it comes to E 0102’s CCO. Wouldn’t it be possible that the blue X-ray point source simply lies in the background of the red low-ionization ring of gas. Is it impossible to imagine that the extraordinary evidence provided by MUSE might simply be chance alignment?

Well, after four paragraphs and 300 words, you would be entitled to a detailed explanation. Sadly, Vogt et al simply say that the area of the structure seen by MUSE is too small. They calculate that there might be at most 8 × 10-5 sources in the background that are as bright or brighter.

Rather compelling, that number, ain’t it.

What’s more, the MUSE data does not provide any evidence for active pulsar wind nebulae, or the presence of a binary companion to the supernova remnant. Vogt et al can very well claim to have found the first truly isolated CCO outside the Milky Way.

Which is when all hell breaks loose.

## The E 0102 offset mystery

The problem, of course, is that the E 0102 CCO does not quite lie at the centre of the large gas ring.

### A bit of firecrackers

Picture a firecracker going off in the night sky. Imagine one of those expensive ones that, um, look good and are made well. Soon after explosion, the expanding sparkles form a ball, while whatever part did not explode (there’s always some) stays more or less at the centre of this ball. Of course, gravity soon makes all of this come crashing down onto the Earth.

Now imagine you are in charge of firecrackers on a cricket night, and you are a bit short on funds. You buy perhaps one of those fancy firecrackers and scrounge the bottom of the barrel for the rest. Your idea is to impress people with that one good one, the one that looks very symmetric, and then quickly pass the other asymmetric ones under the radar. You succeed, for there are people in the audience that are clearly more interested in the badly made ones, the ones where the explosion remnant doesn’t quite remain in the middle of the ball.

Why does that happen? Well, simply because the explosive mixture in the cracker wasn’t uniform or homogeneous enough. There might have been bumps or hollows, or some parts might have had more or less of one component than necessary for the correct amount of burn. Essentially, the firecracker must have had some asymmetry to begin with.

### How a supernova works

A similar specific set of circumstances might have been applicable with E 0102. Here’s how a supernova works : at the end of its life, massive stars8 throw off most their stellar material in a titanic explosion. What remains behind is the superdense core; if massive enough, it forms a black hole; if not, neutron star. The stellar material is thrown off at the rate of thousands of kilometers a second. This translates into a speed of a few lightyears every thousand years. The gaseous material thrown off forms a spherical shell around the supernova remnant, very much like the fast-moving large ring of high-ionization gas in E 0102 as detected by Chandra’s Advanced CCD Imaging Spectrometer (ACIS).

### Two scenarios

Had E 0102 behaved like that good high-end expensive firecracker, its supernova remnant should have been right at the centre of this big shell. However, given the large offset, clearly this isn’t the case. This leaves us with two scenarios :

1. The supernova was indeed centred at the centre of the large X-ray ring, and that, somehow, the supernova remnant has migrated to where it is now.
2. The supernova was centred at the centre of the smaller optical ring, more or less where the CCO is now, and somehow the larger gas envelope got distorted.

Both scenarios are plausible. Both scenarios are problematic. Which is of course what makes this such a delicious mystery.

### Supernova remnant migrated

#### For

The greenish blob is optical light from the oxygen-rich gases floating around E 0102. The small white + is where the centre of this expanding mass of gas should lie. The white crosshair marker slightly southwest of the centre marks the position of the CCO. As you can see, that portion does not have any oxygen. The scale below converts between seconds of arc and parsecs (pc); the thick black line is equal to 3 pc or 9.78 lyr. If you take a ruler and measure it, you’ll find that the distance between the centre and the CCO is approximately 1.8 pc or about 6 lyr.

6 lightyear! That is a lot of distance to cover. If our first scenario is correct, and the stellar core was indeed at the white + when the progenitor star went supernova, then it must have given a mighty kick to the CCO for it to have travelled 6 lyr in 2000 yr. However, the radius of the green blob is almost 4.5 pc (= 14.5 lyr). Also, 6 lyr in 2000 yr equates to a speed of about 9000 km/s (about 3% lightspeed), which is about what a supernova is capable of. The idea of a massive kick seems legit.

#### Against

However, there is a but. Where did the low-ionization slow-moving gas ring come from? The smaller ring is centred on the current position9 of the CCO, which means that somehow, the CCO must have reached the crosshair position, stopped (which is impossible), ejected a whole lot of gas (also impossible), and then waited as that gas very quickly spread to a diameter of a lightyear.

The timeline doesn’t match. Nor do the evidences tally with one another. And the physics is quite impossible. Stellar cores post-supernova do not have any more gas to expend. Yet we have a new gas ring. Optical analysis proves that this gas is slow-moving. Yet the timeline demands that it move ultrafast. Also, how in the world would a stellar core moving at 0.03c brake?

Hmm, trouble. Shall we try the alternate scenario, then?

### Large gas ring distorted

#### For

In the second scenario, the supernova didn’t kick the remnant away, and the CCO has been drifting more or less at its present position, as indicated by the black crosshair. The pink shell of low-ionization gas almost perfectly, but not exactly, encircles the CCO. Vogt et al show that the CCO has drifted only about 0.36 pc (= 1.17 lyr) from the centre of the smaller ring, thereby setting an upper limit for its drift velocity to be 170 kms-1, a far cry from 9000 km-1 in the kick scenario.

The pink inner shell could well have been ejected at supernova. Alternatively, the progenitor star might have been ejected it a few thousand years before supernova. This indicates that E 0102 might have been unstable, a variable star that would shed stellar matter periodically.

#### Against

Which begs the question : why did the bow shock of the supernova not blow this inner ring away? Also, and this is a bigger problem, the explosion site is now located well away from the centre of the X-ray emissions. In the image below, the white crosshair again marks the position of the CCO, while the bright bands of white, red, blue and some green make up a false-coloured image of the Chandra ACIS X-ray data.

According to Vogt et al, this might have been possible due a very specific set of circumstances. For instance, before exploding, stellar wind from the progenitor might have skewed the direction of the shock wave so that it is now off-kilter–a bit like one of those cheap firecrackers. Alternatively, variations in density in interstellar dust and gas might have altered the flow of the ejecta, so that some parts would move faster than the others. Thinking in terms of electricity10, the interstellar medium, though a vacuum, would have enough dust and gas to be inhomogeneous enough to distort the ejecta.

### A specific set of circumstances

Which begs the question : how is the outer ring so round and regular?

Vogt et al attribute this to a specific set of circumstances. Essentially, it still is a mystery. However, with increased precision analysis of MUSE data, Vogt et al are confident they will have the answer for us. Until that time, here is a composite image of the Crab Nebula, showing infrared (Spitzer), optical (Hubble), and X-ray (Chandra) bands. You can make out the remnant as a bright blue-white dot near the centre.

## References

### Papers

1. Vogt et al. : Identification of the Central Compact Object in the young supernova remnant 1E0102.2-7219, arXiv (2018)

### Articles

1. Press Release : Astronomers Spot a Distant and Lonely Neutron Star, Chandra X-ray Observatory (2018)
2. PTI/Firstpost : NASA discovers rare neutron star outside of Milky Way and releases a stunning image of it (2018)

### Image sources

1. Composite, Optical, and X-ray images of E 0102 : Chandra X-ray Observatory