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. ↩︎