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ISBN 978-90-816095-1-7

Applying physics makes auditory sense

A New Paradigm in Hearing

Willem Chr. Heerens

and

J. Alexander de Ru

©2010 Heerens and De Ru

of beat phenomena

You can carry out the following series of experiments:

Entirely based on the premises of the new paradigm,

Pim Heerens has calculated a number of predictable sound phenomena

by using the following frequencies together with prescribed phase relations

in a standard summation procedure to compose a Fourier series:

1:

10000 + 10004 + 10008 + 10012 + 10016 + 10020 + 10024 Hz

Where all the contributions are sine functions.

Our paradigm predicts: an undisputable beat of 4 Hz in a high beep tone.

2:

10000 + 10004 + 10008 + 10012 + 10016 + 10020 + 10024 Hz

Where the contributions are successively alternating sine and cosine functions.

Our paradigm now predicts: an undisputable beat of 8 Hz in the same high beep tone.

3:

10000 + 10004.0625 + 10008 + 10012.0625 + 10016 + 10020.0625 + 10024 Hz

Where all the contributions are sine functions.

Our paradigm now predicts: a beep, in which an undisputable beat exists

that changes every 8 seconds from clearly 4 Hz to 8 Hz and then reverses

again to 4 Hz. So the beat pattern has a period of 8 seconds caused by the

systematic mistuning of 1/16 = 0.0625 Hz.

Additional changes in the mistuning, like for instance from 10004.0625 into

10003.9375 Hz, of either one, two or three of the mistuned frequencies are

predicted to give the same results in the beat pattern as experiment 3.

Please download the software program with which these sound complexes can be properly

calculated in the form of wav files from the following site:

http://www.a3ccm-apmas-eakoh.be/downloads/files/perception-calculations-sept-2010.zip

[ NOTE: The standard setting in the 1/f mode in this software program

takes care that all the individually primary calculated frequencies

contribute equal energy to the resulting sound pressure signal.

This condition is very important for the influences on pitch calculations in

case higher values of the differences between contributing frequencies exist. ]

If you carry out the same series of experiments with a start frequency of

1000 Hz instead of 10000 Hz, you will hear the same series of beat

phenomena, but now with the lower beep of the 1012 Hz instead of the 10012 Hz beep.

Even if you go down with the start frequency to 200 Hz or 400 Hz you will

still hear the same beat phenomena, but now with the low humming tone of

200 Hz respectively with the one octave higher humming tone of 400 Hz.

Hence it is a perception phenomenon that appears all over the entire

auditory frequency range.

Once these beat phenomena are verified as really existing for every

listener with a reasonable normal hearing, for the current paradigm

this is a very serious anomaly.

What for instance is observed from the following combinations of frequencies:

10000 + 10004.0625 + 10008 + 10012.0625 + 10016 + 10020.0625 + 10024 Hz

2000 + 2004.0625 + 2008 + 2012.0625 + 2016 + 2020.0625 + 2024 Hz

400 + 404.0625 + 408 + 412.0625 + 416 + 420.0625 + 424 Hz

having all sine or all cosine contributions, is that they will have an

average frequency – 10012; 2012 respectively 412 Hz – with a beat rhythm

of:

4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 Hz within 32 seconds, so a period of 8 seconds.

While having alternating sine – cosine – sine ... or cosine – sine – cosine ...

contributions, they get a beat with the opposite sequence in the rhythm

of:

8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 Hz within 32 seconds, so with a period of 8 seconds.

In these series you can of course try to attribute the beat phenomena

to ‘special combinations’ of these frequency contributions to eliminate

the problems that arise when the traveling wave model is applied.

Well in order to enervate these suggestions in advance, let us add another

experiment with which this is absolutely impossible.

Let us chose the following frequency contributions:

A series of five tones existing of:

8009 + 8011 + 8013 + 8015 + 8017 Hz

with a minimal difference frequency of 2 Hz.

Where the first, second and fifth contributions are prime numbers, the

8013 Hz contribution is the product of two prime number: 3 x 2671, and the

8015 Hz contribution is the product of three prime numbers: 5 x 7 x 229.

These integer frequency contributions only have 1 Hz as fundamental.

And a series of five tones existing of:

7499 + 7501.0625 + 7503.125 + 7505.1875 + 7507.25 Hz

with a difference frequency of 2.0625 Hz. Again no fundamental.

If all those contributions have sine or cosine functions the beat

phenomenon is given by: a high beep tone with a dominant beat rhythm of:

2 – 4 – 2 – 4 – 2 Hz within 32 seconds, so with a period of 16 seconds

that is mixed with a weaker 6 Hz beat rhythm.

If you modify each of the two frequency series in sine–cosine–sine–cosine–sine

or cosine–sine–cosine–sine–cosine contributions, you will hear a beat

phenomenon in the same high beep tone, but now with a beat rhythm of

4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 Hz within 32 seconds, so with a period of 8 seconds.

Now not only the 2 Hz beat, but also the 6 Hz beat is disappeared.

Now each series apart produce a pure beep tone with a 2 Hz, respectively a

2.0625 Hz beat, in case of all sine or cosine contributions and a 4 Hz

beat respectively a 4.125 Hz beat in case of alternating sine–cosine–sine–cosine–sine

or cosine–sine–cosine–sine–cosine contributions.

If we calculate the sound energy frequency spectrum, we can observe that

only the series of difference frequencies 2.0625 + 4.125 + 6.1875 + 8.25 Hz,

respectively 2 + 4 + 6 + 8 Hz, in the pair by pair combined situation

can generate the beat phenomena we can hear.

For the 10 kHz experiments the 0.0625 Hz detuning means that there exists

an accuracy in the periodicity pattern of 6.25 parts per million.

These salient auditory perception results for each chosen average frequency

over the entire auditory frequency domain are heard.

**Build, write ... play**

**to personally verify the predicted beat phenomena.**

**associated software: "Perception calculations"
together provide you with a tool to personally verify beat phenomena.**

This program is ONLY available for computer systems running under Windows.

**Lay-out of pop-up fill-in screen**

by Willem Christiaan Heerens

Alternative content

You can carry out the following series of experiments: Entirely based on the premises of the new paradigm, Pim Heerens has calculated a number of predictable sound phenomena by using the following frequencies together with prescribed phase relations in a standard summation procedure to compose a Fourier series: 1: 10000 + 10004 + 10008 + 10012 + 10016 + 10020 + 10024 Hz Where all the contributions are sine functions. Our paradigm predicts: an undisputable beat of 4 Hz in a high beep tone. 2: 10000 + 10004 + 10008 + 10012 + 10016 + 10020 + 10024 Hz Where the contributions are successively alternating sine and cosine functions. Our paradigm now predicts: an undisputable beat of 8 Hz in the same high beep tone. 3: 10000 + 10004.0625 + 10008 + 10012.0625 + 10016 + 10020.0625 + 10024 Hz Where all the contributions are sine functions. Our paradigm now predicts: a beep, in which an undisputable beat exists that changes every 8 seconds from clearly 4 Hz to 8 Hz and then reverses again to 4 Hz. So the beat pattern has a period of 8 seconds caused by the systematic mistuning of 1/16 = 0.0625 Hz. Additional changes in the mistuning, like for instance from 10004.0625 into 10003.9375 Hz, of either one, two or three of the mistuned frequencies are predicted to give the same results in the beat pattern as experiment 3. Please download the software program with which these sound complexes can be properly calculated in the form of wav files from the following site: http://www.a3ccm-apmas-eakoh.be/downloads/files/perception-calculations-sept-2010.zip [ NOTE: The standard setting in the 1/f mode in this software program takes care that all the individually primary calculated frequencies contribute equal energy to the resulting sound pressure signal. This condition is very important for the influences on pitch calculations in case higher values of the differences between contributing frequencies exist. ] If you carry out the same series of experiments with a start frequency of 1000 Hz instead of 10000 Hz, you will hear the same series of beat phenomena, but now with the lower beep of the 1012 Hz instead of the 10012 Hz beep. Even if you go down with the start frequency to 200 Hz or 400 Hz you will still hear the same beat phenomena, but now with the low humming tone of 200 Hz respectively with the one octave higher humming tone of 400 Hz. Hence it is a perception phenomenon that appears all over the entire auditory frequency range. Once these beat phenomena are verified as really existing for every listener with a reasonable normal hearing, for the current paradigm this is a very serious anomaly. What for instance is observed from the following combinations of frequencies: 10000+10004.0625+10008+10012.0625+10016+10020.0625+10024 Hz 2000+2004.0625+2008+2012.0625+2016+2020.0625+2024 Hz 400+404.0625+408+412.0625+416+420.0625+424 Hz having all sine or all cosine contributions, is that they will have an average frequency – 10012; 2012 respectively 412 Hz – with a beat rhythm of: 4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 Hz within 32 seconds, so a period of 8 seconds. While having alternating sine – cosine – sine … or cosine – sine – cosine … contributions, they get a beat with the opposite sequence in the rhythm of: 8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 Hz within 32 seconds, so with a period of 8 seconds. In these series you can of course try to attribute the beat phenomena to ‘special combinations’ of these frequency contributions to eliminate the problems that arise when the traveling wave model is applied. Well in order to enervate these suggestions in advance, let us add another experiment with which this is absolutely impossible. Let us chose the following frequency contributions: A series of five tones existing of: 8009+8011+8013+8015+8017 Hz with a minimal difference frequency of 2 Hz. Where the first, second and fifth contributions are prime numbers, the 8013 Hz contribution is the product of two prime number: 3 x 2671, and the 8015 Hz contribution is the product of three prime numbers: 5 x 7 x 229. These integer frequency contributions only have 1 Hz as fundamental. And a series of five tones existing of: 7499+7501.0625+7503.125+7505.1875+7507.25 Hz with a difference frequency of 2.0625 Hz. Again no fundamental. If all those contributions have sine or cosine functions the beat phenomenon is given by: a high beep tone with a dominant beat rhythm of: 2 – 4 – 2 – 4 – 2 Hz within 32 seconds, so with a period of 16 seconds that is mixed with a weaker 6 Hz beat rhythm. If you modify each of the two frequency series in sine–cosine–sine–cosine– sine or cosine–sine–cosine–sine–cosine contributions, you will hear a beat phenomenon in the same high beep tone, but now with a beat rhythm of 4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 Hz within 32 seconds, so with a period of 8 seconds. Now not only the 2 Hz beat, but also the 6 Hz beat is disappeared. Now each series apart produce a pure beep tone with a 2 Hz, respectively a 2.0625 Hz beat, in case of all sine or cosine contributions and a 4 Hz beat respectively a 4.125 Hz beat in case of alternating sine–cosine–sine– cosine–sine or cosine–sine–cosine–sine–cosine contributions. If we calculate the sound energy frequency spectrum, we can observe that only the series of difference frequencies 2.0625 + 4.125 + 6.1875 + 8.25 Hz, respectively 2 + 4 + 6 + 8 Hz, in the pair by pair combined situation can generate the beat phenomena we can hear. For the 10 kHz experiments the 0.0625 Hz detuning means that there exists an accuracy in the periodicity pattern of 6.25 parts per million. These salient auditory perception results for each chosen average frequency over the entire auditory frequency domain are heard.