Category: Acoustics

The Colour of Music : Notes of Equal Temperament

The Colour of Music : Notes of Equal Temperament

Reading Time: 11 minutes

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.

Diagram showing hearing ranges of differnt animals. Humans are in orange, between 20Hz-20kHz. Elephants and moles (red) have a far lower range, while cats and dogs (green) have a wider hearing range (upto 40 kHz) than humans. Bats and dolphins are champions sound detectors, able to hear up to 160 kHz. Image Credit : Unknown. Image Source : Cochlea-dot-org.
Diagram showing hearing ranges of different animals. Humans are in orange, between 20Hz-20kHz. Elephants and moles (red) have a far lower range, while cats and dogs (green) have a wider hearing range (upto 40 kHz) than humans. Bats and dolphins are champions sound detectors, able to hear up to 160 kHz. Image Credit : Unknown. Image Source : Cochlea-dot-org.

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.

Image of keys in a piano. Remember that usually, pianos are tuned in equal temperament, not just intonation. Image Source : playpiano-dot-com.
Keys in a piano. Remember that usually, pianos are tuned in equal temperament, not just intonation. Image Source : playpiano-dot-com.

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?

Keys in a piano. The five black keys are in between the seven white keys. Image Source : playpiano-dot-com.
Keys in a piano. The five black keys are in between the seven white keys. Image Source : playpiano-dot-com.

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.

Image of keys in a piano, including the sharps and the flats. In order to go from A to D, you would need to take the following simple path : A - G♯ - G - F♯ - F - E - D♯ - D, a total of 7 steps. Image Source : johnpratt-dot-com.
Keys in a piano, including the sharps and the flats. In order to go from A to D, you would need to take the following simple path : A – G♯ – G – F♯ – F – E – D♯ – D, a total of 7 steps. Image Source : johnpratt-dot-com.

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 $\Delta$.

Plot of the frequencies of the twelve notes in an octave. The blue dots represent 12 tone equal temperament (12-TET) frequencies, while the red dots represent the justly tuned ones; they’re connected by straight line segments of the same colour for photogenic purposes. The yellow dots represent , the percentage difference between them. As is obvious, there is at most a 1% difference between the two frequencies, with the highest difference at F♯, Unison and octave are, of course, identical, while the fourth and the fifth are almost identical. Image Credit : <a href="http://nerdruid.com/">The Nerd Druid</a>.
Plot of the frequencies of the twelve notes in an octave. The blue dots represent 12 tone equal temperament (12-TET) frequencies, while the red dots represent the justly tuned ones; they’re connected by straight line segments of the same colour for photogenic purposes. The yellow dots represent , the percentage difference between them. As is obvious, there is at most a 1% difference between the two frequencies, with the highest difference at F♯, Unison and octave are, of course, identical, while the fourth and the fifth are almost identical. Image Credit : The Nerd Druid.

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”.

Images of Captain (Retd.) Archibald Francis Haddock, "playing the piano". From Destination Moon (Objectif Lune). Image Credit : Hergé.
Captain (Retd.) Archibald Francis Haddock, “playing the piano”. From Destination Moon (Objectif Lune). Image Credit : Hergé.

 


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

Reading Time: 11 minutes

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.

Photograph of a vintage bakelite radio. From the Bakelite Museum, Somerset, UK. Image Credit : Robneild.
A vintage bakelite radio. From the Bakelite Museum, Somerset, UK. Image Credit : Robneild.

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, $v = \lambda \nu$.

Diagram of sound as pressure waves in air. The regions of compression (labelled "condensation" in the image) are the "crests", those of rarefaction are the "troughs". The distance between two crests or two troughs is a wavelength.
Sound as pressure waves in air. The regions of compression (labelled “condensation” in the image) are the “crests”, those of rarefaction are the “troughs”. The distance between two crests or two troughs is a wavelength. Image Credit : Unknown
Diagram of sound frequencies. The shorter flute produces sound of a higher pitch, while the longer flute produces a low frequency sound. The same principle holds in a guitar; you hold a string to a fret to change the pitch. The closer to the guitar bridge the fret, shorter the effective length, higher the pitch. Image Credit : Unknown.
The shorter flute produces sound of a higher pitch, while the longer flute produces a low-frequency sound. The same principle holds in a guitar; you hold a string to a fret to change the pitch. The closer to the guitar bridge the fret, shorter the effective length, higher the pitch. Image Credit : Unknown.

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.

Photograph of Julie Andrews in the 1964 Hollywood movie The Sound of Music. The most famous song in this musical, Doe-a-deer, teaches the Do-Re-Mi scale. Image Credit : sound-of-music-dot-com.
Julie Andrews in the 1964 Hollywood movie The Sound of Music. The most famous song in this musical, Doe-a-deer, teaches the Do-Re-Mi scale. Image Credit : sound-of-music-dot-com.

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.

Photograph of Einstein's Blackboard. A blackboard used by Albert Einstein in a 1931 lecture in Oxford. The last three lines give numerical values for the density (ρ), radius (P), and age of the universe. The blackboard is on permanent display in the Museum of the History of Science, Oxford. Photo Credit : decltype.
Einstein’s Blackboard. A blackboard used by Albert Einstein in a 1931 lecture in Oxford. The last three lines give numerical values for the density (ρ), radius (P), and age of the universe. The blackboard is on permanent display in the Museum of the History of Science, Oxford. Photo Credit : decltype.

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.

Illustration from 1913 showing Pythagoras teaching a class of women. Many prominent members of his school were women and some modern scholars think that he may have believed that women should be taught philosophy as well as men. Credit : Unknown.
Illustration from 1913 showing Pythagoras teaching a class of women. Many prominent members of his school were women and some modern scholars think that he may have believed that women should be taught philosophy as well as men. Credit : Unknown.

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.

Keys in a piano. Image Source : playpiano-dot-com.
Keys in a piano. Image Source : playpiano-dot-com.
Image of an 88-key piano, with the octaves numbered and Middle C (cyan) and A4 (yellow) highlighted. Image Credit : AlwaysAngry
An 88-key piano, with the octaves numbered and Middle C (cyan) and A4 (yellow) highlighted. Image Credit : AlwaysAngry

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

$$f_{C^{\star}} = 2 f_{C} \tag{1}$$
Animation of three blinking lights flashing at various frequencies. In musical parlance, the middle one is an octave higher than the top, while the bottom light is an octave higher than the middle, and two octaves higher than the top. Here f is the frequency in Hertz ("Hz"), meaning the number of flashes per second. T is the period in seconds ("s"), meaning the number of seconds per flash. T and f are always inverses. Image Credit : Sbyrnes321
Three blinking lights flashing at various frequencies. In musical parlance, the middle one is an octave higher than the top, while the bottom light is an octave higher than the middle, and two octaves higher than the top. Here f is the frequency in Hertz (“Hz”), meaning the number of flashes per second. T is the period in seconds (“s”), meaning the number of seconds per flash. T and f are always inverses. Image Credit : Sbyrnes321

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.

Image of Heinrich Rudolf Hertz, German physicist. Image Credit : Robert Krewaldt.
Heinrich Rudolf Hertz, German physicist. Image Credit : Robert Krewaldt.
James Clerk Maxwell, Scottish physicist. Image Credit : Engraving of James Clerk Maxwell by G. J. Stodart from a photograph by Fergus of Greenock, sourced from the frontpiece in James Maxwell, The Scientific Papers of James Clerk Maxwell. Ed: W. D. Niven. New York: Dover, 1890.
James Clerk Maxwell, Scottish physicist. Image Credit : Engraving of James Clerk Maxwell by G. J. Stodart from a photograph by Fergus of Greenock, sourced from the frontispiece in James Maxwell, The Scientific Papers of James Clerk Maxwell. Ed: W. D. Niven. New York: Dover, 1890.

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 $C_4$.

Diagram of an 88-key piano, with the octaves numbered and Middle C (cyan) and A4 (yellow) highlighted. Note that the piano is usually tuned in equal temperament, not just intonation. Image Credit : AlwaysAngry
An 88-key piano, with the octaves numbered and Middle C (cyan) and A4 (yellow) highlighted. Image Credit : AlwaysAngry

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.

Image of Signora Bianca Castafiore and Captain (Retd.) Archibald Francis Haddock. Image Credit : Hergé.
Signora Bianca Castafiore and Captain (Retd.) Archibald Francis Haddock. Image Credit : Hergé.

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. ↩︎