It was whilst passing a blacksmith`s shop that Pythagoras made a big discovery.

He went into the shop and measured the lengths of strips of metal that were being beaten. Then, he subjected the measurements to mathematical analysis. He discovered which pairs of strips, when beaten together, made a harmonious or chordant sound, and which a discord.

Frequencies of notes in the ratio of 1:2 became what we today call an octave. Frequenciy ratios of 2:3 create a major fifth. Frequency ratios of 3:4 create a major fourth.

According to Diogenes Lærtius, Aristoxenus the musician stated that Pythagoras was the first person to introduce weights and measures amongst the Greeks.

Why should a musician be interested in weights and measures?

If a standard weight stretches a standard string between two frets separated by standard measure, that string will be capable of emitting a standard pitch of note.

When standard frequencies (pitches) are defined, music can be written down and be performed identically at a later date.

When all musicians adhere to the same standards, they can perform in unison.

For these reasons and more, the Pythagoreans were setting themselves up as the standards institute of Greece - the forerunner of DIN in Germany, ASA in the States and the International Standards Organisation ISO.

The Pythagoreans were so taken with the 3:4 ratio of the major fourth, and of the 3:4:5 triangle, that they began to see whole-number ratios as the key to all of life. Their task was to seek simple rules, based upon whole numbers and their ratios, by which music could be harmonious, buildings could have right-angled corners, and order would arrive in society.

However, if an octave is a ratio of 1:2, two octaves is 1:4 and three octaves is 1:8. This is an exponential series. They are whole-number ratios themselves (in mathematical parlance they are RATIONAL) but their subdivisions are not.

Here is an exponential curve:

What, for example, is the ratio for HALF an octave? In modern music, we have twelve semitones per octave, so we mean six semitones for half an octave. This is known as a tritone, and has been called the “Devil’s Interval”.

The answer is ROOT TWO.

This number, 1.414213562, goes on forever. That is to say, it has infinite terms - of which I have shown only ten. It cannot be expressed by the ratio of whole numbers, and is therefore irrational.

Whilst the Pythagoreans were at sea, Hippasus found some proof of this irrationality. The Pythagoreans threw him overboard to drown, because such mathematical pursuits did not deliver any simple rules, and by wasting time threatened their livelihood.

The harmony of simple numbers was seen as something extremely deep indeed. The Pythagoreans even conjectured that the very soul of Man was a harmony.

In the dialogues of Plato, Socrates was asked whether the soul of Man is a harmony. He did not answer directly - but as another question.

“If the harp is broken, where does the harmony go”?

Those of a religious disposition may answer “To heaven”, and have nothing more to do with the question. However, there are much deeper considerations.

Socrates was speaking of a harp - an earthlike thing, in that it is solid. It delivers into the atmosphere - an airlike, or gaseous thing (air itself) - a vibration. That vibration is energy - a firelike thing similar to AC electricity or the oscillating electromagnetism that is light. The firelike things are plasmic. Finally - although Socrates did not know it - the vibration is delivered to the lymph of the inner ear - a waterlike or liquid thing.

Into which analogy - with earth, air, fire or water - does a harmony fall? Is it solid, gaseous, plasmic or liquid? No - it is none of these. These things are the physical attributes. Harmony is an abstract mathematical ratio. A ratio exists only as an adjunct to objects in the real world, and when they have gone so has the ratio.

Harmony is far more easily perceived than absolute pitch. In an instant, we can tell whether a sound is harmonious or disharmonious - yet only trained musicians who deal with musical notation every day can say whether the note is B or C.
It was Aristotle, the favourite pupil of Socrates, who stated that:

“Perception is a RATIO”.

We will see, by a brief study of the inner ear, how the detection of frequency ratios is so important a survival technique that it is built into the fabric of our anatomy.

It is established today that the nervous system creates a train of pulses, with a frequency proportional to the logarithm of the stimulus. That is to say, if you look at the exponential curve above, and choose a height, the logarithm is the distance towards the right of that place on the curve.

Another way of describing a logarithm is in "Octaves". So, if frequency 220 is the fundamental A or TONIC, 440 is one octave, 880 is two, and 1760 three octaves above.

Adding the logarithm of A to the logarithm of B leads to the logarithm of A times B. So if we add two octaves to three, we are really multiplying a fourfold pitch increase by an eightfold.

Our nervous system can multiply easily, by streaming pulses together - but has trouble adding. This is quite unlike the modern computer, which adds more easily than it multiplies.

If one train of pulses suppresses another, this leads to the difference of logarithms. This is the logarithm of a ratio. This is therefore at the very heart of our nervous system - but Aristotle did not know this when he said that perception is a ratio.

When a string, or one of the metal strips of Pythagoras, is set in vibration, it can resonate in many ways. A single wave - the fundamental - may appear, or two, or three, or any whole number of waves. If the fundamental is A 220, all frequencies will be multiples of 220.

A string can resonate in several ways at the same time. Here are the odd harmonics, 220, 660, 1100 and 1540. Whereas the fundamental at 220 is full strength, I have chosen the third overtone (660) to be at one third strength, the fifth (1100) one fifth and the seventh (1540) one seventh.

If we add the fundamental to the third harmonic, we get the following:

Now we add the fifth overtone:

And then the seventh:

It can be seen that as we add more overtones - one ninth of the ninth, one eleventh of the eleventh and so on to infinity - we will ultimately get a square wave. This sounds exactly like the ringing tones of portable phones.

So although the square wave is just one wave, it can be dismantled into a rich array of overtones (also called harmonics).

If there is a sound having 220, 440, 660, 880, 1100 and 1320 cycles per second within it, how do we know whether it is a single raucous crow or a choir of nightingales?

This may not matter to you, but it is of great interest to the crow and to the nightingales.

It is possible to create a mathematical analysis of the combined sound, such as a Fourier analysis, but that is a slow process. The crow and the nightingales need a RAPID analysis of the sound - so that they can instantly identitfy it. That is the task of the inner ear.

Anatomy of the Inner Ear

It was Hermann von Helmholtz who in 1863 wrote a book on the physiology of hearing, in which he defined such things as the impedance matching and the frequency analysis.

An extended edition of that book, “Die Lehre von den Tonempfindungen als physiologische Grundlage fuer die Theorie der Musik”, was published in 1870.

The story begins with sound in the air. It takes little pressure to move a large amount of air. The ratio of pressure to quantity of movement is defined as IMPEDANCE, so air is a low-impedance system.

The sound will have to be transfered into the lymph, or fluid, of the inner ear. It takes much more pressure to cause a liquid to flow. For example, 22.4 litres of air weigh about 14.4 grammes. However, 22.4 litres of water weigh 22.4 KILOgrammes. Water is therefore over a thousand times denser than air. It is a high-impedance environment.

The sound is gathered by the ear and directed down a tunnel to the eardrum. This has air on both sides of it.

Nature uses LEVERS in the form of the bones Malleus and Incus to carry out the impedance transformation from air to lymph. The third bone of the inner ear - the smallest bone of the human body - is the Stapes. This is the "stirrup" that connects to the oval window of the cochlea.

The cochlea is an exponential horn that is wound up like a snail. Cochlea is Latin for snail. It contains two kinds of fluid known as lymph.

Also associated with the two cochleas are the semicircular canals, which are also filled with lymph. They give an awareness of head movement.

When the author was a member of the Stereoscopic Society in London, a fellow member known as Martin Wilsher said that he had found the sixth sense, and it was nothing superstitious.

The author had also found an additional sense - so he listened with respect. Martin Wilsher then explained that it is the sense of BALANCE.

Aristotle when he defined the five senses had conjectured whether there was a sense of balance, but decided that the eyes can tell what movement is taking place - so a sense of balance would be redundant. However, that would signify that a blind man could never be dizzy. So Aristotle was wrong.

Here, in the semicircular canals we have the proof - an organ devoted to the detection of the Wilsher sixth sense. And it is associated not with the eyesight, as Aristotle had conjectured, but with the hearing. Wilsher`s observation had brought the doctrine of Aristotle up to date.

The semicircular canals do not sense absolute balance, but the rate of movement. Lymphatic fluid swirls about within the canals, and brushes past ciliated nerves - nerves with hair-like endings. These nerves sense the rate of change of movement.

Perhaps it is this intimate association between the organs of hearing and of balance that explains the human penchant to combine song with dance.

And the seventh sense? TEMPERATURE. We must separate the sense of temperature from the Aristotelian sense of touch, and define it as a separate sense. Hardness/softness to the touch is quite distinct from hot/cold.

The exponential horn that is the cochlea is also filled with ciliated nerves. It would be capable of resonating at 220 cycles per second (220 Herz) and at its overtones if it was very much bigger. But with sound travelling through liquid at about a thousand miles per hour, the structure is too small.

Perhaps it evolved this way, from earlier beings whose interest was mainly in the higher frequencies. To reach lower frequencies, Nature incorporates a septum (separator) into the cochlea, known as the Basilar Membrane.

Helmholtz discovered that the stiffness of this membrane increases as you head down the cochlea towards the small end. The vibration is in transverse mode, like the S-waves (shear-waves) in seismology. The speed is slower in this mode, so the wavelength is shorter. This allows the cochlea to be small and yet detect low pitches.

The lymph therefore provides damping to the resonances of the basilar membrane, and by being itself resonant at a higher frequency produces high frequency roll-off of the spectral response. Such a loss of high frequencies ensures that each zone of the membrane resonates in fundamental mode only.

Nerves are arranged in pairs, of which there are twelve. The eighth nerve pair are to do with hearing and balance. An extension of the eighth nerve runs through the basilar membrane, and picks up the sounds that have already been separated into their component frequencies.

If the sound was that of a flute, or a cuckoo, or something with very few overtones, only a single bundle of neurons in the nerve will have been actuated. So already we are beginning to define the timbre as well as the pitch of the sound.

If there were several overtones, several bundles will have been actuated. The presence of a large cluster of actuated bundles suggests a shrill tone. However, full confirmation that it is a single shrill notes rather than several disparate sounds requires the nervous data to be matched up.

What is going on in the eighth cranial nerve? The signal is approximately like this:

The frequency of these pulses has nothing to do with the pitch of the note. The pitch has already been evaluated by the cochlea, and been encoded into the bundle that carries the signal. The frequency of the pulses defines the loudness of that tone.

The dynamic range is perhaps a million-to-one. In other words, 60 decibels. Zero decibels is defined at the “Quiet Room”. Ten decibels is ten times as loud as the quiet room. Twenty is a hundred times the quiet room. Thirty is a thousand, forty is ten thousand, fifty is a hundred thousand and sixty is a million.

So there are very roughly five pulses per decibel. That explains why sound engineers use the decibel system. Just as the decibel system is a logarithmic progression, so also is the encoding system of the entire nervous system.

There are millions of neurons in the eighth cranial nerve. Here we see a nerve receiving similar pulse frequencies at A and B - suggesting that sound A and sound B are of similar strength. The logarithm of A minus the logarithm of B emerges at Q. This is the logarithm of the ratio of strength A to strength B, because perception is a ratio.

As A and B are the same, the output is NIL. This is the logarithm of 1. Similarly, at R we have the logarithm of the ratio of strength B to strength A. This also is zero.

It is, however, not quite so simple. This is because there are about ten thousand million nerves in the human brain, which are extensively interconnected. To maintain system integrity, Nature needs to supply engineering test signals, which show that the nervous connexions are alive and active. These are generally the alpha waves, at about seven cycles per second. There are other signals though, Delta at one tenth to three Herz (cycles per second), Theta at 4 to 8, Alpha (as described) from 7.5 to 13 and Beta above 12.

One typical arrangement of frequencies in a naturally occurring sound has 3 dB loss of amplitude per octave. That would be perhaps 15 pulses per second of the nervous system per octave. So A 220 might deliver 100 Hz at the cochlea, A 440 might deliver 85. A 880 might deliver 70, and A 1760 might deliver 55.

The nervous system might take the 85 away from the 100 - giving 15. Also, in real time (whilst the data is still passing along the nerve), it might subtract the 55 from the 70. This also gives 15 - but at the output of a separate neuron.

As these two 15 Hz signals travel along the nerve, one is subtracted from the other. The output is nil (plus the alpha wave). So from the ratio of ratios the eighth nerve has shown that they are all related to each other.

We have seen how a square wave consists of the odd harmonics. The notes 220, 440, 660, 880 and so on would represent all the harmonics. So comparisons are made in every conceivable way until the special shrillness of a crow`s cry can be distinguished from the special shrillness of a square wave.

This continues, with ratios of ratios of ratios and so on, as the data passes along the nerve. It has been said that there is so much data-processing that one cannot say where it is that the eighth nerve ends and the brain itself begins.

By the time the signal reaches the thalamus, it is already identified not as 220, 440, 660, 880 Herz, but as 220 shrill. That shrillness is itself parameterised.

From the thalamus, the signal goes to the auditory cortex on each side of the brain. Here, the 220 shrill on the left is compared with the 220 shrill on the right. Spacial awareness arrives - the brain detects that it is "220 shrill" at 45 degrees azimuth on the left, because the left is louder than the right.

The image shows the outline of the brain, and the approximate positions of the auditory and visual cortices. Note that both the auditory and the visual information is “upside down”. That is to say, treble is detected near the base of the brain, and the picture on the visual cortex is inverted.

Perhaps there is the hiss of escaping steam at 60 degrees azimuth on the right. This white noise would mask the sound. However, the auditory cortex also detects this sound - and by separating the two sounds spacially actually delivers noise reduction in the perceived "220 shrill" on the left.

The crow has no interest whatsoever in the musical pitch or timbre of the note, nor in the azimuth angle - as far as we know. However, it is interested in crows. The sounds of all crows it has met have been parameterised, and are stored in the brain.

The LIMBIC SYSTEM of the brain, shown here as the white areas in the middle, is the very soul of the mind. It compares the A 220 shrill sound with all the other sounds the crow has experienced. Perhaps it finds a match.

The limbic system is the inner rind of the brain. It connects all parts of the brain together. By gathering all that it known on any particular subject - the sight, the sound, the emotional involvement and so on - collected by all the seven senses, it gives an overall impression of that subject to the conscious mind.

Humans can identify a single voice out of several hundred thousand. For example, our limbic systems enable us to recognise the voice of Elvis Presley, or of Pavarotti. All the subtle clues of timbre and pitch enable our brains to link one piece of data with another.

So the crow feels that it can hear a crow "over there" (at 45 degrees).

Perhaps the crow turns its head through 45 degrees. The semicircular canals confirm that the head is turning, and define the speed - but as Aristotle observed, the eyes can see movement. It seems to be the visual clues that make the crow stop turning its head when it reaches 45 degrees.

If you turn your head to the left, the world will seem to pass you by to the right. However, if the image on the visual cortex is upside-down, that image will also turn to the left. So the crow needs only to turn its head until the perceived angle of the inverted image matches the perceived direction of the sound.

If the crow now hears the sound again, and actually sees a crow, the feeling that the crow receives from its limbic system is "There is a crow".

Similarly, the nightingales experience in their limbic systems a feeling of disappointment.

Harmony Discovered

Harmony is a TEASE. The mechanics of the ear detect not just one frequency, but many frequencies. Often these frequencies are the overtones of a fundamental. The eighth cranial nerve processes the data, and puts all overtones together. Thus many components resolve to a single sound.

The mechanism that combines frequencies 1:2:3 into a single tone also combines frequencies 2:3 into a single tone. So the major fifth is SEEMINGLY a natural sound.

This data now goes to the limbic system for comparison with previously experienced sounds. Now comes the surprise. This major fifth was NOT FOUND IN NATURE.

Perhaps there was the rare occasion when two nightingales happened to be singing together, and their tones blended. Or perhaps, on passing a blacksmith`s shop, one has heard the chance resonance of different strips of metal being beaten. But these things are rare. They are the exceptions that prove the rule.

A harmony is a natural sound not found in nature. It is the set of harmonics 1:2:3 with the 1 missing, or the set 1:2:3:4 with the 1 and 2 missing (the major fourth). The undertones, 1 or 1 and 2, are never missing from a naturally-occurring sound.

Also it is specifically the LOW overtones with their undertones missing that provide harmony. As the overtones become higher, the ratios tend towards unity. Thus, the ratio of the eighteenth to the seventeenth is about 1.0588. We will see shortly that a semitone is about 1.059. Two notes a semitone apart will create a discord when played together.

Disharmony is a property of the middle overtones. When we reach the hundredth overtone of 2.2 Hz., the difference between A 220 and the next (the 101th) is 2.2 Herz. 220 cycles per second and 222.2 are almost the same pitch, so harmony reappears. However, they will beat with each other at 1.1 Herz, giving a "breathing" effect that makes the sound seem less thin. For this reason, there are two or three strings per note on the pianoforte.

Is the soul a harmony? Are you one-and-a-half or one-and-a-third? It is trivial to compare the entirety of a mind with a simple ratio such as a major fifth or a major fourth.

Is the harmony detected in the soul? This is also fairly trivial. All experiences are detected in what Joseph Towne and others called the SENSORIUM. This is the real soul. It is the limbic system, where a signal from one or more of the seven senses triggers the memory of all other experiences that were associated with that signal.

To a crow, an awareness of another crow is given by a single cry. To a human, a flood of memories of concert-going experiences, or of the replay of recorded music, is triggered by a harmony.

Sometimes, but not often, a particular harmony may trigger recollections of the sounds of the countryside, and the song of birds. But harmonies in natural sound are rare, so these recollections are vague.

The harmony is detected in the eighth cranial nerve, whilst the data is being processed in real time, on its journey to the soul.

Pythagoras wrote a book called Natural Philosophy, which went missing but was widely reported. There can be no higher philosophy than to study Mankind within his environment. Pythagoras, with his harmonies, had discovered how the process of evolution had reflected the arithmetic of overtones into the structure of the ear and brain - although he did not know this.

It is a mark of the correctness of the observations of Pythagoras that they still resonate with the discoveries of modern medicine and neurology after two and a half millennia.

The Equitempered Scale

The difference between the major fourth and major fifth - such as between F and G in the key of C - is a single tone. This would allow only six notes per octave.

However, before we leave this subject let us examine it. The difference between rational numbers is always rational. So one-and-a-half minus one-and-a-third is ONE SIXTH.

This would give A 220, B 256 and two-thirds, C# 293 and one-third, D# 330, F 366 and two-thirds, G 403 and one-third and A 440.

Consider the set of harmonics 1:2:3:4:5 with the 1, 2 and 3 missing. This gives us a MAJOR THIRD which differs from a major fourth by ONE TWELFTH.

The twelfth is defined as a semitone. It slots notes such as A# between existing notes. So from a linear average of A and B, we get A# 238 and-a-third. Other semitones slot between the six tones in the same way.

The problem with this harmonic scale is that it can only obey the strictures of Pythagorean arithmetic for the span of a single octave. Then, because all notes an octave above are double the frequency, the next octave has twice as much slope. This puts a kink into the curve.

There were many schemes to get round the problem of this discontinuity. So-called MODES of tuning included the Ionian, the Aeolian, the Lydian, the Mixolydian, the Dorian, the Phrygian and the Locrian.

Johan Sebastian Bach was so dedicated to music that he walked over two hundred miles to learn new techniques from Dietrich Buxtehude. The music he created was so avante-garde that it “Confused the unprepared congregation”.

He was sacked as a choir-master at the Neue Kirche in Arnstadt after he had played jingles to mask the attrocious singing. This was defined as impiety, and he suffered greatly from criticism from the pietist J. A. Frohne of Mühlhausen.

His problem was that if the congregation drifted down in pitch, he would have liked to drift down also - so that his accompaniment would be in step with the singing. This is called TRANSPOSITION. However, the harmonies only worked in a single key. If he transposed, he lost the precision of tuning because the kink slid into the scale.

Bach began to campaign for a system that dropped the simple arithmetic of Pythagoras where each semitone was a twelfth of the tonic - such as one twelfth of A 220. In its place, every semitone would be some MAGIC NUMBER times the frequency of the semitone before. This would stretch the exponential system down into the scale itself.

All semitones would be alike. That is, they would be some multiplier rather than something added to the frequency. The semitone multiplied by itself twelve times would give a doubling or octave.

This defined the semitone as the twelfth root of two, and created the new EQUITEMPERED SCALE.

The idea was revolutionary. It also challenged what had become almost part of religion itself - that the work of God is the ratio of whole numbers. These approximations were anathema.

Look, for example at E 659. It should be E 660 according to Pythagoras.

The answer Bach gave was to play quickly. Another blasphemy. It is not pious to play quickly.

However, if two instruments are played together - one tuned to the equitempered E 659 and the other to the harmonic tempered E 660, it will take a full second for them to slip out of phase with each other and then return to being in step. If tuning is more precise, at 659.255 cycles, it takes even longer. A note of a fifth or a tenth of a second will go undetected as being an approximation.

To prove his point, he published in 1722, when he was 37, a set of compositions under the title Das wohltemperirte Klavier. A second set followed in 1744. The name alone speaks volumes - the WELLtempered keyboard. He had decided that equitempering is good.

Later composers were quick to approve of the innovation. Mozart first came across the music of Bach when he visited the library of the royal court in Vienna. The librarian showed him the work. Mozart exclaimed “This is wonderful - but I can do the same”, and rushed off to compose some music in the style of Bach.

Beethoven spoke of the “Divine harmonies” of Bach.

The equitempered scale has since become the standard tuning arrangement in the western world.

Bach was an organist. He was fortunate to have the instrument-maker Gottfried Silbermann as a friend. Three years after his first volume of compositions was published - that is, in 1725 - he was shown a new instrument by Silbermann, and was at first not in favour.

This instrument was a copy of the 1709 Gravicembalo con Piano e Forte by Bartolomeo Cristofori. It had two strings per note, struck by hammers. Silbermann made two such copies. They seemed to have either four or four-and-a-half octaves.

The influence of Bach was such that even in 1842 - two years before the publication of his second volume - the organ at faraway St. Nicholas’ Church in Newcastle, England, was being tuned in the equitempered manner.

After the death of Bach, the “Twelve apostles” of Silbermann went to England, and Johann Christoph Zumpe became particularly famous for his instruments. It was a Zumpe that was used at a J.C. Bach piano recital at the Thatched House, London.

In the Seventeen-Seventies, Johann Andreas Stein in Vienna designed a lightweight mechanism. Mozart - who had travelled widely - disliked the heaviness of the English actions, and seems to have composed for the Stein.

We can see that when Bach wrote his pieces for the “KLAVIER”, he had something else in mind than the pianoforte. The pianoforte only became popular at the end of the century, after his death. He meant ANY keyboard instrument which could be precision tuned for test purposes, and from thence all instruments of all kinds. The modern German word Klavier is used for the piano because it has not yet been superseded.

Bach`s twelve pieces in all the major keys, and twelve in all the minors, demonstrated that there was no restriction upon the composer. The deviation from Pythagorean harmony would not be detected by any audience, and a new standard was born.

Inversions

In the harmonic scale of Pythagoras, a major fifth is three divided by two. Let us invert the mathematics. Two divided by three is the ratio of the inversion to the tonic. Double it. Four divided by three is the ratio to the octave below the tonic. That four over three is a major fourth.

The inversion works as follows: The major fifth is seven semitones up. Its inversion is seven semitones down. There are twelve semitones in an octave, so the inversion is five semitones above the lower tonic. That is a fourth.

In the equitempered scale of Bach, a major fifth is 1.498307077. Find the reciprocal (0.667419927). Double it to compare it with the octave below (1.334839854). This is the major fourth in the equitempered scale - so the inversion still works.

Take the equitempered major third (1.25992105). Find the reciprocal (0.793700526). Double it (1.587401052). This is exactly the same as an augmented equitempered fifth - an equitempered semitone more than an equitempered fifth.

The equitempered third finds an equivalent after inversion because it is itself a diminished equitempered fourth.

Take the harmonic major third (five divided by four). Invert it (four divided by five). There cannot be an equivalent in the harmonic scale - which is based on twelfths - because twelve and five are coprime.

The Phase of Sound-Waves

Take another look at the anatomy of the inner ear. Suppose a sound puts pressure on one cochlea and suction on the other. What happens?

Sound runs through the semi-circular canals. The lymph will always flow from high pressure to low. First it flows one way, and at the next half-cycle it will flow the other way.

The semi-circular canals do not have a basilar membrane designed to detect the pitch of notes. Nevertheless, they will feel the buzz of the sound.

The eighth cranial nerve on each side takes data from the basilar membrane on each side and from the semi-circular canals. So if a sound is always accompanied by the "buzz" of the semi-circular canals, it must be an out-of-phase sound.

Suppose another sound puts pressure on both cochleas, and half a cycle later puts suction on both. No sound flows through the semicircular canals, and the eighth cranial nerve detects that they are in phase with each other, and permits the otic cortex to find the spacial position of that sound.

Unplug one loudspeaker of a high-fidelity system, and reconnect it reversed. The sound seems to come from two speakers. Restore the loudspeaker to the correct phasing. The sound takes up its correct position in the environment between the speakers.

But what are the restrictions on the phase-awareness of the human hearing system? Sound travels at about 600 miles per hour. That is 10560 inches per second. So the note A 220 has a wavelength of 48 inches (1 metre 22).

To be exactly out of phase, the path taken to each ear by the sound must differ by two feet (61 cm). This is much bigger than the width of the human head, so A 220 can only be partially out of phase due to naturally occurring path differences.

A 440, however, has a half wavelength of one foot. A 880 has six inches. A 1760 has three inches. So as the pitch rises, the incidence of naturally-occurring phase errors rises. Nature has to learn to ignore such cases.

It is therefore in the eighth cranial nerve, where 220, 440, 660, 880 and so on are all distilled into 220-shrill that all the harmonics are "tarred with the same brush" (condemned as being out of phase), on the basis of phase error detected in the undertones.

Absolute phase cannot be detected. The voltages in an electronic sound system, and as a result the pressure in the air, may be inverted - and it will sound exactly the same. On the basis that we cannot detect it, we have no memory of phase.

In passing, one might note that the heads of birds and small animals are smaller than those of humans, so they may have spacial awareness of sound into much higher frequencies.

It is not only in the human and animal otic apparatus that sum-and-difference effects occur. Even in the air, when the pressure of one sound meets the suction of another, they will cancel.

Here we see two sinusoids that differ in frequency by a semitone. They begin in step with each other, and then after about eight cycles become exactly out of step.

We can even estimate how many cycles it takes. The semitone is 1.059463094. We take away the 1. We find the reciprocal (16.81715374). So after 8.408576872 cycles they are exactly in antiphase, and after the same amount again they are back in step.

An absurdly precise definition of the semitone is given at the end of the article. It should be borne in mind that when comparing one note with the NEXT semitone, SIX PERCENT MORE is a good mnemonic. When raised to the power twelve, however, it delivers 2.012 which is six percent high - one semitone too much.

So 5.94 percent more is a better guess.

In 1870 A.J. Ellis published in London “On the Sensations of Tone as a Physiological Basis for the Theory of Music” - a translation of the book by Helmholtz. It was republished by Dover Publications, New York, in 1954.

In the appendix, he suggested subdividing the equitempered semitone into even smaller divisions, which he called CENTS. Such a Cent, if multiplied by itself a hundred times will deliver a semitone.

Here is the equation given by Ellis:

where n is the number of Cents in the interval from b to a.

If a is an octave above b, Log₂ is unity, and the octave becomes 1200 Cents.

The Cent is 1.0005777895, and another absurdly exact definition is given below.

In references to the subdivision of the semitone, I propose to adhere to the term CENTS when referring to the Ellis definition (an exponential subdivision), and to use the term HUNDREDTHS when taking a fragment to be a multiple of “0.0594 MORE”.

So a Cent is 577 parts per million more than the previous, and a hundredth is 594 parts per million of the tonic more.

Such fine distinctions are meaningless when we are concerned with the evaluation of pitch. However, differences may be detected when studying the BEAT FREQUENCIES, so the distinction is made.

The consequence of phase-creep between the various frequency components is that at times they add, and increase the sound, and sometimes subtract and weaken it. This is sometimes described as BEATING and sometimes as VIBRATO (Italian for vibration).

A. J. Ellis appears to be the Dr. Ellis who was president of the Philological Society, and the father of scientific linguistics. He also used musical knowledge in the study of speech.

In the University of Wales at Aberystwyth they have a Philological Society booklet on “Neo-Latin Names for the Artichoke”, inscribed to A. J. Ellis by “His Imperial Highness Prince L. L. Bonaparte” in 1885. Whatever turns you on.......

Closely Tuned Strings

The equation for the combination of two cosines is

Here, Cos (A+B)/2 is the linear, or Pythagorean mean.

Cos (A-B)/2 is defined as the envelope. Such a cosine envelope alternates between positive and negative, so the resulting sound alternates between the “correct” way up (in phase) and “upside-down” (antiphase).

At the point where the envelope crosses zero, there must be a phase inversion - so the composite waveform will show a point of inflexion where it tries to make one wave, and then “changes its mind” before making the inverted wave. Look at the composite of two waves an equitempered tone apart:

The tone is the sixth root of two, or 1.122462. One wave will be 12.2462 percent faster than the other. The beat frequency will be half of that (6.1231 percent), but it will not sound that way because a burst of sound appears on each half-cycle.

The cycles of one burst of sound will progressively intensify the vibration of the basilar membrane. However, the next cycle, being in antiphase, will first bring that oscillation to rest. Then the basilar membrane will vibrate in the opposite phase. The mind perceives two quite distinct bursts of sound, but no perception of phase.

We have seen that the centre frequency is Pythagorean - it is halfway between the two. That is, it is 6.1231 percent higher than the lower frequency.

If we add that centre frequency, the phase inversions become clear. We have assumed equal quantities of all sinusoids we used. So we had a double-strength standard honk wave. Now a single amount of centre frequency either adds to it, giving three units, or subtracts from it to leave just one:

The effect of alternating between a threefold and single-fold burst of sound is like having dried peas or stones in a metal can. As you shake the can, you get a loud and a soft clang.

The distinction is made to help match up the timbre of treble, which has two strings, and bass which has three.

In the basilar membrane, there may be sufficient energy left over from the large bursts to absorb all the energy of the small ones. In such a case, one will be unaware of there being a large and a small burst. One may be aware only of the loud sounds, and the rattling may seem to be at half speed.

Accordingly, we can already define a rule for the specification of the “HONK” of a note.

Honk Definitions

If there are two strings, the standard honk has the upper and lower strings separated by a tone. This is the SIXTH ROOT OF TWO, or 1.122462048 of the centre frequency. Therefore, we have a span of 0.122462048 of the centre frequency.

We take Half of that - 0.061231024. We multiply that by the desired centre frequency, such as 220 (giving 13.47082531). We subtract that from the desired centre frequency (220), giving the lower pitch (206.5291747). We add it onto the desired centre frequency (220), giving the upper pitch (233.47082531).

If we desire to have a 50 Cent bichord honk, we have a semitone span. A semitone is 1.059263094, so we have strings 0.029731547 times the centre frequency above and below.

If we want 17 Cents of honk, we multiply 1.000577789506554 (a Cent) by itself THIRTY-FOUR times. This gives 1.019833287. We subtract 1. Now we divide by 2, giving 0.009916643. We multiply this by the centre frequency (220), giving 2.181661578 cycles per second to be added and subtracted from the centre frequency.

Note that the 2.181661578 is the exact rate of beating of the two strings. Twice this (4.363323155) will be the frequency of the bursts of sound.

TRICHORD

If there are three strings, the standard honk has the upper and lower frequencies separated by TWO tones. This is the cube root of two (1.25992105), making the strings 12.9960524 percent higher and lower than the centre frequency.

A 50 Cent trichord honk (3 strings) has the upper and lower strings set up as for a full bichord honk (plus and minus 6.1231024 percent of centre) as described above. However, the centre string - at the nominal frequency - is present.

A 17 Cent trichord honk need to have the Cent raised to four times that power (1.00057779 to the power sixty-eight, or 1.040059933). That gives 2.002996671 percent of the centre frequency above and below it.

The LOW Standard

A simpler definition of the honk uses 6.12 percent above and below the centre frequency for the standard bichord honk.

13 percent would be added and subtracted for the outer strings of a trichord system.

For 17 HUNDREDTHS of honk, we decide whether it is a bichord or a trichord. Then we use 17% of 6.12 percent or 17% of 13 percent.

The hundredths system is easier, but is a less exact standard. It is based on half of a tone (12.24%) and half of two tones (26%).

Real-World Standard
(24 Sept 2004)

The author's computer broke down before further analysis and test samples could be created. However, the target specification was intended to be one Cent of deviation from the centre frequency for each Cent of honk.

Taken to the extreme, a 1200 Cent bichord honk would have the upper string one octave higher in frequency, and the lower string one octave lower. However, in the real world (non-electronically generated), the amplitude declines at 3 dB per octave. So we would have HALF a unit of sound at twice the frequency combining with TWO units at half frequency.

It can be seen that the centre frequency will remain steady under such conditions.

The trichord honk would have its percentage values doubled, of course.

Normal honk values will be perhaps from fifteen to thirty Cents. where the two sinusoids that are beating with each others match in amplitude to about two percent. Similarly, this C- standard will agree to within about 3 Cents with the two previous standards, for values up to 100 Cents.

Technical Summary

The precision "Coefficient of Honk" is expressed in Cents.

For a bichord system, the deviation (D_f) of each string from the centre frequency will be given by:

where C is the Cent (1.0005777895) given below, and h is the coefficient. Thus, for 17 Cents of honk, we raise 1.0005777895 to the power 34, subtract 1 and halve the remaining number.

TRICHORD

For a trichord system, the deviation (D_f) of each string from the centre frequency will be given by:

Thus, for 17 Cents of honk, we raise 1.0005777895 to the power 68, subtract 1 and halve the remaining number.

Technical Summary
(Approximation)

The approximate "Coefficient of Honk" is expressed in hundredths.

For bichord system, the deviation (D_f) of each string from the centre frequency will be given by:

Accuracy

By the use of 0.000612 per Hundredth, the full bichord honk will match. However, the 50 Hundredths honk will be 0.0306 either side of the centre frequency when the simple system is used.

Raising the Cent to the power of 103 delivers 1.06130064. Halving 0.0613 gives 0.0306. So when we divide 103 by two, we get 51.5. This is 51.5 Cents - the equivalent of the simple calculation of 50 hundredths.

We expect about one-and-a-half Cents of deviation (worst case) from the ideal tuning, when we use simple tuning of the bichord.

By the use of 0.0013 per Hundredth, the full trichord honk will match. However, the 50 Hundredths honk will be 0.065 either side of the centre frequency when the simple system is used.

Raising the Cent to the power of 212 delivers 1.1302. Halving 0.1302 gives 0.0651. So when we divide 212 by four, we get 53. This is 53 Cents - the equivalent of the simple calculation of 50 hundredths.

We expect about three Cents of deviation (worst case) from the ideal tuning, when we use simple tuning of the trichord.

Inaccuracy

Perception is not always a ratio. The laws of acoustics subtract one frequency from another to create the beat. So the subtraction has already taken place in the air before the beat is perceived. It is only within our logarithmic minds that perception is a ratio.

Electronic systems can produce waves of such precision that they beat together with the exactitude of a metronome. However, real instruments invariably drift due to physical constraints such as barometric pressure, temperature and moisture.

There is a song “I wish that I had a million violins, then I could play for you my one-fingered symphony.....”. In these days of virtual instruments constructed in software, this is no longer so improbable.

However, a million simple virtual violins playing in unison would each deliver exactly the same wave. Even the delay caused by electrical wiring would be trivial because the phase-velocity of electrical signals approaches the speed of light. Those million waves would merge into a single giant wave.

The effect is not that of a million violins. It is that of a single virtual violin a million times (60dB) louder.

To create the effect of a massed string section to the orchestra, a small but controlled amount of randomness must be given to each instrument. Random number generators are known and used. These would simulate a slight deviation of pitch, of timing and of overall delay of each instrument. Only this way can one ensure that the waves beat with each other in the air as they would at a genuine concert.

The variation of pitch simulates how one instrument has perhaps warped due to warmth and damp relative to the others. The variation in timing would simulate the different playing styles of the players. The overall delay would vary to simulate the varying distances of the players from the audience.

Only by precisely controlled inaccuracy can the real world be recreated in software.

Attack and Decay

It is not only military bands that describe the onset of a note as the ATTACK. It makes sense to present a sound as a COSINE rather than a SINE because a cosine begins at the top of the wave, as if the string were held at maximum deflection and then released. The mathematics is also easier.

However, the sudden jump from nil to the peak of the wave produces a click, much like a hammer striking a string. You can hear this in the samples provided.

After the string is released, energy is taken from it by the sounding board, or by the air or dampers. That energy is a percentage of the energy available, so the decay is exponential. First, over a given time, half is released, then a quarter, then an eighth and so on. That time is defined as the half life.

Dampers are used to restrict the half-life to perhaps a quarter of a second. The action of the loud pedal is to lift the dampers, thereby reducing the losses and extending the half-life to perhaps two seconds. The action of the soft pedal is to provide added damping.

We have seen how two equal waves, when added together, produce a modulated wave at the centre frequency. Let M be the modulation frequency (the beat frequency we studied in the research into honk). Let C be the centre frequency.

It is only if the modulation is of the form of a sinusoid that the outcome is simple:

The single spectral line on the left for Cos C has turned into two spectral lines of half size after modulation by Cos M. The separation between the lines is 2M.

An exponentially decaying curve starts steeply at the top, and then become more shallow. An exponetially decaying note after the attack will have two spectral lines far apart that then drift closer and closer as the exponential becomes flatter and flatter, and so approximates to lower and lower frequencies. Meanwhile, the lines are themselves shrinking as the sound gets weaker.

Modulation therefore has a spectrum-widenening effect. The result in the eighth cranial nerve is that the sound is not localised, but spread over a large bunch of neurons, allowing only fuzzy awareness of the pitch. Bach was therefore right when he insisted that the briskness of the melodies would mask the perception of error.

For those who are in desperate need of such things, here are the equitempered Semitone and equitempered Cent to two thousand places of decimal precision: