The Science of Honk
220 Hz Square
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
Anatomy of the Inner Ear
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
As that signal is detected by the ciliated nerves of the cochlea, it has pulses that vary from about seven cycles per second to perhaps three hundred cycles per second. These pulses have an amplitude of about thirty millivolts - but they are not electrical. They are caused by sodium and potassium ions changing place with each other. Ions move slowly, so they take perhaps a tenth of a second to move along the eighth cranial nerve.
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.
220 Hz Honk
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:
I propose to call this Fifty Cents of Trichord Honk.
220 Hz Trichord
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.
However, a three-to-one difference in amplitude brings a 4.77
decibel difference in signal strength, and a 9.54 decibel
difference in energy. Accordingly, the sound has a closer
resemblance to a bichord honk of half the Cent value.
220 Hz 50 Cent Honk
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.
The Variety Pianoforte
The author was driven out of Britain by a corrupt government which stole everything from him. Rather than waste his knowledge, he decided to go public. Those who make a commercial profit from this information should reserve a portion for the benefit of the inventor.
(C) 2004 Charles Douglas Wehner.
Use freely but do not plagiarise.
All files other than the square wave wee prepared with the machine code programs at wehner.org/fpoint, to an accuracy of 1 part in 4000 million (32 bit) before being reduced to the 16 bit accuracy of a WAV file.
220 Hz Square Wave
220 Hz 100 Cent Bichord Honk
220 Hz 50 Cent Bichord Honk
220 Hz 25 Cent Bichord Honk
220 Hz 0 Cent
220 Hz 50 Cent Trichord Honk
220 Hz 25 Cent Trichord Honk
220 Hz 12.5 Cent Trichord Honk
220 Hz 100 Cent Bichord Honk with Decay
220 Hz 50 Cent Bichord Honk with Decay
220 Hz 25 Cent Bichord Honk with Decay
220 Hz 0 Cent with Decay
220 Hz 50 Cent Trichord Honk with Decay
220 Hz 25 Cent Trichord Honk with Decay
220 Hz 12.5 Cent Trichord Honk with Decay
440 Hz 0 Cent
The half-life of decay, where appropriate, was set to a quarter of a second.
Full Tricord Honk with Decay
25 Cent Trichord Honk with Decay
Zero Cent Trichord with Decay
Then type SCALE2 17 to create PIANO017.WAV (Bichord 17 Cent scale of C Major) or SCALE3 17 to create FORTE017.WAV (Trichord equivalent).
Up to 255 Cents of honk are allowed, equivalent to 2.55 honks.
Bichord SCALE2 IBM program 1581 bytes
Trichord SCALE3 IBM program 1722 bytes
If they do not download, try http://wehner.org/honk/scale2.x and http://wehner.org/honk/scale3.x , and rename them COM. If a text editor shows you some text, this cannot be saved as a valid program.
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: