WHY DO KARAOKE SINGERS SOUND SO BAD?
How waves and fractions make good and bad harmony
The inventor of the karaoke machine has a lot to answer for. Before it existed, most amateur singers restricted their performances to the shower. Now, with microphone in hand and musical accompaniment for moral support, they can inflict pain on a far larger audience.
The reason why so much karaoke singing sounds so bad, of course, is that many people are unable to sing ‘in tune’. Put another way, the sounds being emitted from the karaoke singer’s voice box clash with the notes on the musical accompaniment – or with the notes that the audience’s brains expect to hear.
There are several factors that determine whether singers sound out of tune. Some are to do with the expectations set within our culture, others are to do with the way the brain interprets sounds. But some of the reasons can be explained mathematically, and these are the main concern of this chapter.
It all begins with the curvaceous sine wave…
Ask anyone to think of a wave and they are most likely to picture a steady undulation, such as is found on the sea. The simplest such undulation is known as a sine wave, which looks like this:
The word ‘sine’, by the way, comes from the Latin word sinus meaning a bay, and sine curves look a little like bays found on a coast. Sine waves are in fact the most basic and fundamental waveform, and crop up in various real-world situations. For example, if you hang a weight from a spring, pull it downwards and release it, the weight will now bob up and down.
The distance of the weight from its central position plotted against time looks like this:
In this case, the time between the peaks of the wave, a cycle, is one second. The number of cycles per second is known as the frequency of the wave, so in this case the frequency is 1 cycle per second, or 1 Hertz (often written as ‘Hz’).
Sine waves can also be created by motion in a circle. If you took a ride on the Millennium Wheel and plotted your height above the ground against time, the graph would look like this:
The position of a capsule on the wheel over time plots out a sine wave, and, since a complete ride on the wheel takes about half an hour, the frequency of the wave is 1 cycle of the wheel per 30 minutes, which is 1 cycle per 1,800 seconds or about 0.0006 Hz.
Anything that vibrates or cycles sends pulses through the air, causing the air molecules to move to and fro like the weight on a spring. The pulses are sound waves, and the human ear can detect these sounds as long as they are between about 20 Hz (a very low note) and 20,000 Hz (a high whistle). The frequencies of the Millennium Wheel and the weight bobbing on the spring are too low to be picked up by our ears, but if the spring was more powerful or the Wheel was whizzing round at a sickening speed, both would create audible notes. The sound would resemble that made by a tuning fork, or the penetrating noise of a wet finger running around the rim of a fine wineglass.
Other vibrating objects such as bees’ wings, struck saucepans and electric razors all make notes, too. The greater the frequency of the vibration, the higher the note, and, since notes put together make tunes, with the right collection of bees, saucepans and electric razors, you could put on a recognisable, if rather eccentric, performance of Beethoven’s Fifth Symphony. Or any other tune, of course.
The waveforms of these unconventional musical instruments are complicated, but it turns out that the sine wave is fundamental to them all. A Frenchman called Fourier made the remarkable discovery that absolutely any wave, however irregular it looks, can be constructed simply by adding together a combination of different sine waves. It might, for instance, look like this:
The difficult part is working out exactly which sine waves you need to add together. The squiggle above may be a combi nation of ten or more waves, each with a different frequency and amplitude, and the analysis required to find them is well beyond this book – though not necessarily beyond our ears.
How the ear reacts to combinations of notes
Without knowing anything about Fourier the human ear is able to listen to a sound wave and to some extent break it down into its constituent sine waves. If you listen to notes being played on three recorders simultaneously, you will probably be able to pick out three distinct notes, even though an oscilloscope monitoring the sound would show that a complicated-looking combined wave is reaching your ears.
However, although ears are good at picking out note combi nations, they are not perfect at it. If there are two pure notes of identical frequency, the human ear will detect a single sound. The ear can only detect different notes if the frequencies are sufficiently far apart. Here is a rough guide to what happens, though the exact frequency ranges involved depend both on the individual (some people have more acute hearing than others) and on the frequency level being listened to.
If the difference in frequency is tiny, less than 1 Hz, say, only one note is heard, and the ear is quite happy. Two violins in a professional orchestra cannot ever be played at exactly the same note, but they are so close that few humans can tell the difference.
If the notes differ by between 1 and 10 Hz, the joint sound that the ear detects will be a single note that pulses loud and soft, a phenomenon known as beats.
If the notes differ by between 10 and 20 Hz, there is a rough sound, caused partly by high-frequency beats. The ear doesn’t like this at all. In fact it is believed that this rough sound caused by frequencies that are within a certain critical range of each other, is the basis of what sounds universally ‘bad’, or dissonant, in music of all cultures.
When the frequencies differ by more than a critical range – say 20 Hz – then the ear can clearly identify that they are different, and regards their combination as being quite acceptable, though not necessarily beautiful.
This simple theory suggests that, as long as their frequencies are far enough apart, two pure notes played together should always sound OK.
Does this mean that a bad karaoke singer is somehow managing to be out of tune just enough – but not too much – to be producing frequencies that perfectly clash in the critical band of the backing sound? Partly, yes. However, the notes from a singer’s larynx are not pure sine waves. They are a combination of many different frequencies, and even if they are far apart, these impure notes can produce nasty clashes…
Why do two different notes sometimes clash?
When any instrument is plucked or struck or blown, it will produce a note at its natural frequency. However, instruments will produce notes at other frequencies at the same time. These other frequencies are known as harmonics, and on well-crafted musical instruments such as flutes and guitars (but not razors and saucepans) these harmonics are simple multiples of the basic frequency. So, if a plucked string has a base frequency of 100 Hz, it will also produce quieter frequencies at multiples of 100 Hz:
The base note and the harmonics are all pure sine waves, but their combination is a much more complex-looking waveform, found by adding the harmonics together. These harmonics make the note played on a piano, for example, sound much ‘richer’ in quality than the sound produced by the finger around the wineglass. If you strike a single piano key very hard, then in addition to the main note you may well be able to detect these higher notes in the background. In flutes, the base note is much more dominant, with very little in the way of audible harmonics. Whatever the instrument, you are unlikely to be able to hear anything beyond the fourth harmonic.
So what about listening to two notes that are played at the same time? What you hear is a combination of their lowest frequencies and all of the harmonics. This combination can sound good, especially if the relative string or tube lengths of the notes are ratios of small whole numbers, like 2/1 or 3/2. To see why, here are the harmonics of three strings, one of which is full-length, one half-length and the other two-thirds-length.
The longest string creates the following harmonics:
When two sound waves are produced at the same time, the waves can be literally added together. Two identical pure notes added together make a note of identical frequency – but louder!
The timing of the peaks and troughs of the above notes are identical, or ‘in phase’. Even if they are not in phase, two identical sine waves added together always combine to create
another sine wave of the same frequency.
If the peaks of one coincide with the troughs of the other, they
completely cancel each other out when added together:
These two notes combined would actually appear silent to the listener. This principle is used by engineers to create antinoise – noise generators that produce identical but ‘upside-down’ sound waves in order to reduce the noise level in the environment.
The frequencies of the half-length string are all double those of the longer string:
And the frequencies of the two-thirds-length string are in between those of the other two strings:
When plucked together, any pair of these notes has coinciding harmonics. Indeed, the 600 Hz harmonic is shared by all three strings. The ear, which likes coinciding frequencies, will appreciate this. Furthermore, none of the frequencies are close to any of the others, and this means there won’t be any of the harsh beating noises that the ear dislikes.
These three ratios of string create the most comfortable harmonies of all, which helps to explain why notes in the ratio of 3/2 and 2/1 have appeared in music of almost all cultures, past and present. Even an ancient Chinese flute discovered by archaeologists was found to have holes positioned to produce 3/2 notes. This ratio is known in musical parlance as a perfect fifth.
The general principle is that notes that sound good together have some harmonics that coincide with each other, and have no harmonics that enter each other’s critical ‘bad noise’ frequency band. By far the best such harmonies are produced by the ratios of small numbers, such as 3/2, 4/3 and 5/3.
Pythagoras and the clanging hammers
The story goes that Pythagoras was one day walking past a forge when he heard two hammers banging together. The notes that the two hammers produced sounded the same – and yet different.
When Pythagoras checked, he discovered that one piece of metal being struck was exactly half the length of the other, and the shorter piece was producing the higher note. What he had heard would later become known as an octave. Pythagoras was able to reproduce this effect by plucking strings of different lengths. He went on to experiment with other simple ratios of string lengths. Notes whose lengths had simple ratios such as 3/2 or 4/3 seemed to sound good together or to ‘agree’, the Greek word for agreement being ‘harmonia’, hence harmony. All of this reinforced Pythagoras’s notion that numbers were behind everything in the natural world.
Why there are twelve different notes
Music has far more notes than just the simple octave and the 3/2 note. In fact there are twelve notes in a Western octave, and most people give no thought to where these notes come from. They just seem to be ‘there’ in much the same way as snowflakes always have six points. However, this system of twelve notes evolved through a combination of mathematics and chance. Since this music scale is partly what we use to judge a karaoke singer, it’s worth knowing its origins.
Pythagoras was the first person in Western culture to produce a musical scale. He decided that the scale should contain exactly seven different notes, partly because of the mystical importance of the number seven. He also thought that all notes should be constructed using ratios of 3/2.
The first music scale?
This is what the original Pythagorean music scale might have looked like. Each string length has been created by multiplying 2/3 or 3/2 together. To bring all the notes between 2 and 1, a single octave, their lengths have then been doubled or halved (doubling or halving a string length changes a note by an octave but retains its essential sound.) So, for example, 2/3 x 2/3 = 4/9 or 0.444. To bring this between 2 and 1, it needs to be doubled twice, making it 16/9 or 1.778. In the scale that follows, the longest string makes the lowest note:
As it turns out, the scale attributed to Pythagoras produced reasonably good harmonies, but these were not necessarily any easier on the ear than those being created independently in other cultures. Some cultures settled on a 5-note scale, for example, while others produced scales with as many as 22 notes.
If you were to listen to this Pythagorean scale being played, the notes would sound pretty close to those that we are familiar with in modern music. However, medieval musicians realised there was something missing. They wanted the freedom to start a tune on any note and still be able to sing along to a familiar melody. This was not possible using the seven notes of Pythagoras’s scale.
Try playing ‘Three Blind Mice’ on Pythagoras’s scale. If the first three notes played are ‘E, D, C’ it sounds all right, but if you attempt to play ‘Three Blind Mice’ starting ‘F, E, D’ it sounds quite wrong. Why? Because the notes in Pythagoras’s scale were not spaced evenly apart.
Medieval musicians had to insert notes to give the scale a roughly even spacing, represented today by the black notes on a piano keyboard. One way to fill in the gaps was to extend Pythagoras’s idea of creating all of the notes using the 3/2 interval. As it happens, if you start with a string that produces a note and reduce its length by the ratio two-thirds twelve times, you end up producing a note that is almost exactly the same as your starting note, but seven octaves higher. This is because (3/2)12, which works out as 129.7, is approximately 27, or 128. The coincidence of 129.7 being so close to 128 is the main reason why there are twelve notes in the modern scale. Do the numbers seven and twelve ring a bell? For quite different reasons, they are also the basis of the western system of measuring time, as discussed in Chapter 1.
The twelve notes created by 3/2 intervals make a reasonable scale, though some of the intervals between the notes sound pretty awful. To improve the harmonies between different pairs of notes, the Medievalists began to experiment with different ratios of string lengths. Pythagoras’s idea that every note had to come from the ratio 3/2 was sensibly dropped – why not include the ratios 5/4 and 5/3 too? Later inventors of the music scale struggled to find string lengths that produced simple ratios (and therefore pleasant harmonies) for all twelve notes when plucked together. But perfect combinations for every pair of notes proved elusive.
Wolf intervals
The seventh note of the twelve in the
Renaissance scale – now called F sharp (F#) – was a particular nuisance. It sounded lousy when combined with almost every other note, creating toecurling intervals that reminded listeners of howling wolves, and they became known as ‘wolf intervals’. Since these intervals could surely not be what God had intended, the church named F# the devil’s note, and for a while banned it from all music.
There is one way to remove the occasional cringe-making harmonies from a musical scale, and that is to make the intervals between every one of the twelve notes identical. Eventually, somebody spotted that the way to do this was to make the lengths increase in a logarithmic scale. What this means in practice is that each note’s frequency has to be 1.059 times higher than its predecessor. As a result, on a modern keyboard you can play ‘Auld Lang Syne’ or ‘Happy Birthday’ starting at C, E, F# or anywhere else and it will always sound like the regular tune.
Back to the karaoke singer
All of which brings us back to the failure of the karaoke singer to hit the right notes. What he is failing to hit, of course, are the notes to which we have become accustomed in our Western scale. The sound waves he produces include frequencies that grate with us or make us want to howl like wolves.
It isn’t all down to maths, however. We shouldn’t forget the cultural factor. Some of the notes and harmonies were established because they sound good to most people, but others were created because the mathematics of the different ratios dictated that they had to be there to fit into the musical scale. Some of these latter notes arguably only sound good because we are so used to hearing them.
So, although the karaoke singer may sound bad, it isn’t entirely his fault. Although some intervals seem to be popular in just about all cultures – the fifth, for example – others that we are used to are very specific to the West. Other cultures have their own completely different scales, created from different, less mathematical starting points. Perhaps there is an island somewhere in the South Pacific where even the most terrible of karaoke singers sounds as mellifluous as the birds, and as perfect as Pavarotti does to us.
4. ln other words, if you were to play the white notes on a piano in this order, the scale would sound very similar to the Pythagoras scale. Incidentally, the seven notes didn’t acquire letters until medieval times. When playing a complete scale, the first note is repeated at the end to make the octave.