Pianotech

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------

John, thank you for the hint! It clears up a lot. Maybe you also could help me understanding how the energy distributes among sympathetic (tonal) harmonics and so called noise harmonics. Which factors form it? 

Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. 

I thought that the sympathetic (tonal) harmonics must be strong and continuous, but some of them are not. Moreover some of them are barely seen at the spectrum graph. On the other hand the noise harmonics are expected to decay fast as they can't "reside" long on C#4 string. However I see some of them lasting on A0 string quite prominently. Like they resonate as well. 

So is there a simple theory why such energy/amplitude distribution is happening? Is that the intrinsic frequencies of piano body/decca come into action? Dampening the strong tonal harmonics and sustaining the weak noise harmonics, for example? That's the only guess coming to my mind so far. 

Original Message:
Sent: 03-03-2024 21:03
From: John Rhodes
Subject: Theory of sympathetic resonances doesn't match what I hear

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------

I'm confused:

"Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. "

Am I missing something? 😕 

Peter Grey Piano Doctor 

John, thank you for the hint! It clears up a lot. Maybe you also could help me understanding how the energy distributes among sympathetic (tonal) harmonics and so called noise harmonics. Which factors form it? 

Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. 

I thought that the sympathetic (tonal) harmonics must be strong and continuous, but some of them are not. Moreover some of them are barely seen at the spectrum graph. On the other hand the noise harmonics are expected to decay fast as they can't "reside" long on C#4 string. However I see some of them lasting on A0 string quite prominently. Like they resonate as well. 

So is there a simple theory why such energy/amplitude distribution is happening? Is that the intrinsic frequencies of piano body/decca come into action? Dampening the strong tonal harmonics and sustaining the weak noise harmonics, for example? That's the only guess coming to my mind so far. 

Original Message:
Sent: 03-03-2024 21:03
From: John Rhodes
Subject: Theory of sympathetic resonances doesn't match what I hear

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------

I'm confused:

"Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. "

Am I missing something? 😕 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 03-05-2024 19:21
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

John, thank you for the hint! It clears up a lot. Maybe you also could help me understanding how the energy distributes among sympathetic (tonal) harmonics and so called noise harmonics. Which factors form it? 

Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. 

I thought that the sympathetic (tonal) harmonics must be strong and continuous, but some of them are not. Moreover some of them are barely seen at the spectrum graph. On the other hand the noise harmonics are expected to decay fast as they can't "reside" long on C#4 string. However I see some of them lasting on A0 string quite prominently. Like they resonate as well. 

So is there a simple theory why such energy/amplitude distribution is happening? Is that the intrinsic frequencies of piano body/decca come into action? Dampening the strong tonal harmonics and sustaining the weak noise harmonics, for example? That's the only guess coming to my mind so far. 

Original Message:
Sent: 03-03-2024 21:03
From: John Rhodes
Subject: Theory of sympathetic resonances doesn't match what I hear

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------

I marked in bold the corresponding tonal harmonics of A0 and C#4. Those not marked in bold for A0 are excited too when I hit C#4, but I call them noise harmonics. Let me know what else I could clarify. Thanks! 

Original Message:
Sent: 03-06-2024 07:36
From: Peter Grey
Subject: Theory of sympathetic resonances doesn't match what I hear

I'm confused:

"Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. "

Am I missing something? 😕 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 03-05-2024 19:21
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

John, thank you for the hint! It clears up a lot. Maybe you also could help me understanding how the energy distributes among sympathetic (tonal) harmonics and so called noise harmonics. Which factors form it? 

Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. 

I thought that the sympathetic (tonal) harmonics must be strong and continuous, but some of them are not. Moreover some of them are barely seen at the spectrum graph. On the other hand the noise harmonics are expected to decay fast as they can't "reside" long on C#4 string. However I see some of them lasting on A0 string quite prominently. Like they resonate as well. 

So is there a simple theory why such energy/amplitude distribution is happening? Is that the intrinsic frequencies of piano body/decca come into action? Dampening the strong tonal harmonics and sustaining the weak noise harmonics, for example? That's the only guess coming to my mind so far. 

Original Message:
Sent: 03-03-2024 21:03
From: John Rhodes
Subject: Theory of sympathetic resonances doesn't match what I hear

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------

Original Message:
Sent: 3/6/2024 8:50:00 AM
From: Vlad Oplev
Subject: RE: Theory of sympathetic resonances doesn't match what I hear

I marked in bold the corresponding tonal harmonics of A0 and C#4. Those not marked in bold for A0 are excited too when I hit C#4, but I call them noise harmonics. Let me know what else I could clarify. Thanks! 

Original Message:
Sent: 03-06-2024 07:36
From: Peter Grey
Subject: Theory of sympathetic resonances doesn't match what I hear

I'm confused:

"Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. "

Am I missing something? 😕 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 03-05-2024 19:21
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

John, thank you for the hint! It clears up a lot. Maybe you also could help me understanding how the energy distributes among sympathetic (tonal) harmonics and so called noise harmonics. Which factors form it? 

Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. 

I thought that the sympathetic (tonal) harmonics must be strong and continuous, but some of them are not. Moreover some of them are barely seen at the spectrum graph. On the other hand the noise harmonics are expected to decay fast as they can't "reside" long on C#4 string. However I see some of them lasting on A0 string quite prominently. Like they resonate as well. 

So is there a simple theory why such energy/amplitude distribution is happening? Is that the intrinsic frequencies of piano body/decca come into action? Dampening the strong tonal harmonics and sustaining the weak noise harmonics, for example? That's the only guess coming to my mind so far. 

Original Message:
Sent: 03-03-2024 21:03
From: John Rhodes
Subject: Theory of sympathetic resonances doesn't match what I hear

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------

Yes, it's exactly what I'm doing. These are frequencies of 25 overtones existing at A0 string:

(27.50), 55.00, 82.50, 110.00, 137.50, 165.00, 192.50, 220.00, 247.50, 275.00, 302.50, 330.00, 357.50, 385.00, 412.50, 440.00, 467.50, 495.00, 522.50, 550.00, 577.50, 605.00, 632.50, 660.00, 687.50

These are frequencies of 12 overtones existing at C#4 string:

(138.59), 277.18, 415.78, 554.37, 692.96, 831.55, 970.14, 1108.73, 1247.32, 1385.92, 1524.51, 1663.10

I marked in bold those frequencies which are close to each other.

And supposedly they all must look very prominent (bright) at the spectrum graph (captured when I silently hold A0, then hit and release C#4). However I see that some harmonics are weaker and not so shining. For example, on the pic below 412.5Hz is weaker and seems equal to the non-sympathetic (intrinsic) harmonics 385 & 440Hz of A0 string. 

And the same time non-sympathetic (intrinsic) harmonics of A0, which must be weaker than sympathetic ones, are present at the graph as continuous pitches (as if they resonate too). Check out below the frequencies 385, 440, 522 and a clashing pair of 330 & 357.5Hz. Meanwhile other non-sympathetic (intrinsic) harmonics decay much faster and don't drone at all.

So I'm curious what defines the amplitude (energy) distribution of: 1) coinciding frequencies (sympathetic harmonics); 2) non-sympathetic harmonics. 

Original Message:
Sent: 03-06-2024 10:23
From: David Pinnegar
Subject: Theory of sympathetic resonances doesn't match what I hear

Original Message:
Sent: 3/6/2024 8:50:00 AM
From: Vlad Oplev
Subject: RE: Theory of sympathetic resonances doesn't match what I hear

I marked in bold the corresponding tonal harmonics of A0 and C#4. Those not marked in bold for A0 are excited too when I hit C#4, but I call them noise harmonics. Let me know what else I could clarify. Thanks! 

Original Message:
Sent: 03-06-2024 07:36
From: Peter Grey
Subject: Theory of sympathetic resonances doesn't match what I hear

I'm confused:

"Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. "

Am I missing something? 😕 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 03-05-2024 19:21
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

John, thank you for the hint! It clears up a lot. Maybe you also could help me understanding how the energy distributes among sympathetic (tonal) harmonics and so called noise harmonics. Which factors form it? 

Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. 

I thought that the sympathetic (tonal) harmonics must be strong and continuous, but some of them are not. Moreover some of them are barely seen at the spectrum graph. On the other hand the noise harmonics are expected to decay fast as they can't "reside" long on C#4 string. However I see some of them lasting on A0 string quite prominently. Like they resonate as well. 

So is there a simple theory why such energy/amplitude distribution is happening? Is that the intrinsic frequencies of piano body/decca come into action? Dampening the strong tonal harmonics and sustaining the weak noise harmonics, for example? That's the only guess coming to my mind so far. 

Original Message:
Sent: 03-03-2024 21:03
From: John Rhodes
Subject: Theory of sympathetic resonances doesn't match what I hear

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------

Vlad, you might try damping all of the strings except the A0 and C#4 with masking tape. Make sure you don't have anything else going on in your system such as EQ etc.

Try some other note samples like C1 and E4 to see if you get similar results. Consider mic placement... Your analyzer probably offers different graphic formats, you can try those for comparisons.

As piano tuners, we have to deal with what we are hearing whether it comports with theory or not. 

Make sure there are not other invasive frequencies coming from fluorescent lights, dishwashers, refrigerators, fans or parakeets.

Yes, it's exactly what I'm doing. These are frequencies of 25 overtones existing at A0 string:

(27.50), 55.00, 82.50, 110.00, 137.50, 165.00, 192.50, 220.00, 247.50, 275.00, 302.50, 330.00, 357.50, 385.00, 412.50, 440.00, 467.50, 495.00, 522.50, 550.00, 577.50, 605.00, 632.50, 660.00, 687.50

These are frequencies of 12 overtones existing at C#4 string:

(138.59), 277.18, 415.78, 554.37, 692.96, 831.55, 970.14, 1108.73, 1247.32, 1385.92, 1524.51, 1663.10

I marked in bold those frequencies which are close to each other.

And supposedly they all must look very prominent (bright) at the spectrum graph (captured when I silently hold A0, then hit and release C#4). However I see that some harmonics are weaker and not so shining. For example, on the pic below 412.5Hz is weaker and seems equal to the non-sympathetic (intrinsic) harmonics 385 & 440Hz of A0 string. 

And the same time non-sympathetic (intrinsic) harmonics of A0, which must be weaker than sympathetic ones, are present at the graph as continuous pitches (as if they resonate too). Check out below the frequencies 385, 440, 522 and a clashing pair of 330 & 357.5Hz. Meanwhile other non-sympathetic (intrinsic) harmonics decay much faster and don't drone at all.

So I'm curious what defines the amplitude (energy) distribution of: 1) coinciding frequencies (sympathetic harmonics); 2) non-sympathetic harmonics. 

Original Message:
Sent: 03-06-2024 10:23
From: David Pinnegar
Subject: Theory of sympathetic resonances doesn't match what I hear

Original Message:
Sent: 3/6/2024 8:50:00 AM
From: Vlad Oplev
Subject: RE: Theory of sympathetic resonances doesn't match what I hear

I marked in bold the corresponding tonal harmonics of A0 and C#4. Those not marked in bold for A0 are excited too when I hit C#4, but I call them noise harmonics. Let me know what else I could clarify. Thanks! 

Original Message:
Sent: 03-06-2024 07:36
From: Peter Grey
Subject: Theory of sympathetic resonances doesn't match what I hear

I'm confused:

"Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. "

Am I missing something? 😕 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 03-05-2024 19:21
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

John, thank you for the hint! It clears up a lot. Maybe you also could help me understanding how the energy distributes among sympathetic (tonal) harmonics and so called noise harmonics. Which factors form it? 

Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. 

I thought that the sympathetic (tonal) harmonics must be strong and continuous, but some of them are not. Moreover some of them are barely seen at the spectrum graph. On the other hand the noise harmonics are expected to decay fast as they can't "reside" long on C#4 string. However I see some of them lasting on A0 string quite prominently. Like they resonate as well. 

So is there a simple theory why such energy/amplitude distribution is happening? Is that the intrinsic frequencies of piano body/decca come into action? Dampening the strong tonal harmonics and sustaining the weak noise harmonics, for example? That's the only guess coming to my mind so far. 

Original Message:
Sent: 03-03-2024 21:03
From: John Rhodes
Subject: Theory of sympathetic resonances doesn't match what I hear

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------

Yes, it's exactly what I'm doing. These are frequencies of 25 overtones existing at A0 string:

(27.50), 55.00, 82.50, 110.00, 137.50, 165.00, 192.50, 220.00, 247.50, 275.00, 302.50, 330.00, 357.50, 385.00, 412.50, 440.00, 467.50, 495.00, 522.50, 550.00, 577.50, 605.00, 632.50, 660.00, 687.50

These are frequencies of 12 overtones existing at C#4 string:

(138.59), 277.18, 415.78, 554.37, 692.96, 831.55, 970.14, 1108.73, 1247.32, 1385.92, 1524.51, 1663.10

I marked in bold those frequencies which are close to each other.

And supposedly they all must look very prominent (bright) at the spectrum graph (captured when I silently hold A0, then hit and release C#4). However I see that some harmonics are weaker and not so shining. For example, on the pic below 412.5Hz is weaker and seems equal to the non-sympathetic (intrinsic) harmonics 385 & 440Hz of A0 string. 

And the same time non-sympathetic (intrinsic) harmonics of A0, which must be weaker than sympathetic ones, are present at the graph as continuous pitches (as if they resonate too). Check out below the frequencies 385, 440, 522 and a clashing pair of 330 & 357.5Hz. Meanwhile other non-sympathetic (intrinsic) harmonics decay much faster and don't drone at all.

So I'm curious what defines the amplitude (energy) distribution of: 1) coinciding frequencies (sympathetic harmonics); 2) non-sympathetic harmonics. 

Original Message:
Sent: 03-06-2024 10:23
From: David Pinnegar
Subject: Theory of sympathetic resonances doesn't match what I hear

Original Message:
Sent: 3/6/2024 8:50:00 AM
From: Vlad Oplev
Subject: RE: Theory of sympathetic resonances doesn't match what I hear

I marked in bold the corresponding tonal harmonics of A0 and C#4. Those not marked in bold for A0 are excited too when I hit C#4, but I call them noise harmonics. Let me know what else I could clarify. Thanks! 

Original Message:
Sent: 03-06-2024 07:36
From: Peter Grey
Subject: Theory of sympathetic resonances doesn't match what I hear

I'm confused:

"Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. "

Am I missing something? 😕 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 03-05-2024 19:21
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

John, thank you for the hint! It clears up a lot. Maybe you also could help me understanding how the energy distributes among sympathetic (tonal) harmonics and so called noise harmonics. Which factors form it? 

Let's assume A0 (note 21) is open and I hit C#4 (note 49). C#4 will excite on A0 two types of harmonics: 1) corresponding tonal harmonics sharing close overtone frequencies like 49, 61, 68, 73, 77, etc; 2) noise harmonics excited on A0 by a band-width impulse of C#4 hammer: 45, 49, 52, 55, 57, 59, 61, etc. 

I thought that the sympathetic (tonal) harmonics must be strong and continuous, but some of them are not. Moreover some of them are barely seen at the spectrum graph. On the other hand the noise harmonics are expected to decay fast as they can't "reside" long on C#4 string. However I see some of them lasting on A0 string quite prominently. Like they resonate as well. 

So is there a simple theory why such energy/amplitude distribution is happening? Is that the intrinsic frequencies of piano body/decca come into action? Dampening the strong tonal harmonics and sustaining the weak noise harmonics, for example? That's the only guess coming to my mind so far. 

Original Message:
Sent: 03-03-2024 21:03
From: John Rhodes
Subject: Theory of sympathetic resonances doesn't match what I hear

Vlad,
When you strike C#3, the hammer blow is an impulse which could be considered a bandwidth-limited step function.  This impulse contains all frequencies -- rolling off in amplitude thru the top of the keyboard.  Only after the impulse traverses the length of the C#3 string many times does the string and its terminations "sort out" the resonant pitch and partials associated with the note.  It is the broadband impulse (noise, if you will) of the C#3 hammer strike which is exciting the many partials you observe of A0.

Try this experiment:  With A0 damped, strike C#3, then quickly depress A0 after perhaps one-quarter of a second, and *immediately* release C#3.  You should hear A0 sounding only the partials you predicted in your analysis.

------------------------------
John Rhodes
Vancouver WA
(360) 721-0728
jrhodes@pacifier.com

Original Message:
Sent: 03-03-2024 16:11
From: Vlad Oplev
Subject: Theory of sympathetic resonances doesn't match what I hear

Though the theory of string overtones and piano sympathetic resonances seems quite understandable, I've found that a real grand-piano gives more complicated sound. Here is a simple example:

I silently press a piano key A0 (note 21 in MIDI). Its harmonics in musical notes are 33, 40, 45, 49, 52, 55, 57, 59, 61, 62/63, 64, etc (I'm not taking in the consideration a detune of some overtones).

Then I hit and release right away the key C#3 (note 49). Its harmonics are: 61, 68, 73, 77, 80, 83, 85, 87, 89, 90/91, 92. Plus, of course, the main tone – 49.

Here is a table for a better visualization:

Comparing numbers in the table I see that theoretically the open A0 must resonate with (marked in bold): 5th harmonic (not 49), 10th harmonic (note 61), 15th harmonic (note 68) and 20th harmonic (note 73), the most upper. The energy of even higher harmonic is weak, and they can be discarded, I suppose.

However on a real grand-piano I hear and see (in iZotope RX) the following:

1) Obviously note 45 (A2, 110Hz) does sound also, despite that according to the theory it can't be excited by the higher note 49. Then why there's note 45!? And if the one was supposedly excited by hammer of key 49 just mechanically via piano frame and body, then why there are no harmonics with notes 33 and 40? And no main tone 21?

2) Checking the spectrum I see, as expected, the strong and bright harmonics with notes 49 and 61. But the same time there is the whole bunch of harmonics like 52, 55, 57, 59, 61, etc. Almost all belonging to key (string) A0 all the way up to 25th harmonic. Check the 700Hz boldly shining in the spectrum (note 77, F5). What excited these harmonics like 52, 55, 57, 59? Because the vibration of the string for key 49 had no frequencies corresponding to them?

Also, why the harmonics higher than 10th, ok, higher than 15th are so prominent in the graph? As I understand the amplitude of harmonic is proportional to 1/N, where N is harmonic number in sequence. So all those upper harmonics like 20-25th must be really weak to be heard or tracked in the spectrum. But they are obviously present in my experiment.

These couple points are bugging me? Any suggestions what the theory of sympathetic resonances is missing when applied to just two piano strings?

P.S. When listening and recording I damped the treble (open) strings (keys above note 90) of grand-piano I had access to. Plus this model doesn't have duplex scale, which could resonate.

------------------------------
Vlad Oplev
Yucca Valley CA
(760) 365-0769
------------------------------