Pianotech

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant of a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant of each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament. There are 88 average inharmonicity constants because there are 88 notes from A0 to C8.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, of a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant of each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant of each note been explored in the past? If the answer is yes, where can the average inharmonicity constant of each note be found?

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 7/6/2020 10:03:00 AM
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant for a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant for each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, for a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant for each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant for each note been explored in the past? If the answer is yes, where can the average inharmonicity constant for each note be found?

------------------------------
Roshan Kakiya
------------------------------

My aim is precisely to overcome the specificity of inharmonicity by generalising inharmonicity.

Each temperament can have its own general inharmonicity model.

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 07-06-2020 11:25
From: David Pinnegar
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Different tuners focus on different things. Inharmonicity may not be so relevant to some as to others.

Different models of pianos express different inharmonicities due to scaling. It's much more of an individual thing than can be averaged across the spectrum

Best wishes

David P

--

- - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594

Original Message:
Sent: 7/6/2020 10:03:00 AM
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant for a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant for each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, for a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant for each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant for each note been explored in the past? If the answer is yes, where can the average inharmonicity constant for each note be found?

------------------------------
Roshan Kakiya
------------------------------

Roshan, 
I don't know anybody who has ever used that metric (average inharmonicity constant) and I don't know that it would be meaningful or useful. The inharmonicity varies exponentially in a single piano. When I graph it I do so on a log plot.  Off the top of my head, it can range from 0.00001 in the tenor to 0.01 in the treble. I don't think you're going to get anything representative of the temperament region by averaging the whole piano. 

I don't know of a published database anywhere listing inharmonicity of many pianos. But I could get you some example numbers to work with if you want to play around with the math. Also FYI the iH depends on the size & scale of the piano, not the manufacturer. A long Baldwin will have lower inharmonicity than a short Bosendorfer. At least in the tenor and bass. I couldn't predict for the treble, which can be all over the map, and which would have the biggest effect on the "average" while at the same time have the least relevance to the tuning and temperament.

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
------------------------------

Original Message:
Sent: 07-06-2020 11:37
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

My aim is precisely to overcome the specificity of inharmonicity by generalising inharmonicity.

Each temperament can have its own general inharmonicity model.

------------------------------
Roshan Kakiya

Original Message:
Sent: 07-06-2020 11:25
From: David Pinnegar
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Different tuners focus on different things. Inharmonicity may not be so relevant to some as to others.

Different models of pianos express different inharmonicities due to scaling. It's much more of an individual thing than can be averaged across the spectrum

Best wishes

David P

--

- - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594

Original Message:
Sent: 7/6/2020 10:03:00 AM
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant for a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant for each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, for a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant for each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant for each note been explored in the past? If the answer is yes, where can the average inharmonicity constant for each note be found?

------------------------------
Roshan Kakiya
------------------------------

Each note has its own average inharmonicity constant. There are 88 average inharmonicity constants because there are 88 notes from A0 to C8.

For example, the inharmonicity constant of A4 in relation to Piano 1 might not be the same as the inharmonicity constant of A4 in relation to Piano 2.

Average inharmonicity constant of A4 = (Inharmonicity constant of A4 in relation to Piano 1 + Inharmonicity constant of A4 in relation to Piano 2) / 2.

More examples:

All of the various inharmonicity constants of the note F3 can be used to calculate the average inharmonicity constant of the note F3.

All of the various inharmonicity constants of the note C4 can be used to calculate the average inharmonicity constant of the note C4.


I would be grateful if you could provide the average inharmonicity constant of each individual note.

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 07-06-2020 12:15
From: Anthony Willey
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Roshan,
I don't know anybody who has ever used that metric (average inharmonicity constant) and I don't know that it would be meaningful or useful. The inharmonicity varies exponentially in a single piano. When I graph it I do so on a log plot.  Off the top of my head, it can range from 0.00001 in the tenor to 0.01 in the treble. I don't think you're going to get anything representative of the temperament region by averaging the whole piano.

I don't know of a published database anywhere listing inharmonicity of many pianos. But I could get you some example numbers to work with if you want to play around with the math. Also FYI the iH depends on the size & scale of the piano, not the manufacturer. A long Baldwin will have lower inharmonicity than a short Bosendorfer. At least in the tenor and bass. I couldn't predict for the treble, which can be all over the map, and which would have the biggest effect on the "average" while at the same time have the least relevance to the tuning and temperament.

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 07-06-2020 11:37
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

My aim is precisely to overcome the specificity of inharmonicity by generalising inharmonicity.

Each temperament can have its own general inharmonicity model.

------------------------------
Roshan Kakiya

Original Message:
Sent: 07-06-2020 11:25
From: David Pinnegar
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Different tuners focus on different things. Inharmonicity may not be so relevant to some as to others.

Different models of pianos express different inharmonicities due to scaling. It's much more of an individual thing than can be averaged across the spectrum

Best wishes

David P

--

- - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594

Original Message:
Sent: 7/6/2020 10:03:00 AM
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant for a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant for each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, for a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant for each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant for each note been explored in the past? If the answer is yes, where can the average inharmonicity constant for each note be found?

------------------------------
Roshan Kakiya
------------------------------

Oh, I misunderstood before. Taking the note-by-note average makes more sense. There will still be variation from piano to piano, but the average numbers will still hold some meaning. Here's a list of inharmonicity numbers from A0 to C8 that can be used as a jumping off point. These aren't representative of any particular piano, but are an idealized (smooth) set based on some average parameters from different pianos I measured.

0.000639
0.000577
0.000521
0.000471
0.000426
0.000386
0.00035
0.000318
0.000289
0.000264
0.000242
0.000222
0.000204
0.000189
0.000176
0.000164
0.000154
0.000146
0.00014
0.000135
0.000131
0.000128
0.000127
0.000127
0.000128
0.00013
0.000133
0.000138
0.000144
0.000151
0.000159
0.000169
0.00018
0.000193
0.000207
0.000223
0.00024
0.00026
0.000282
0.000306
0.000333
0.000362
0.000394
0.000429
0.000468
0.000511
0.000557
0.000608
0.000664
0.000725
0.000792
0.000865
0.000946
0.001033
0.001129
0.001234
0.001349
0.001475
0.001612
0.001762
0.001926
0.002106
0.002302
0.002517
0.002751
0.003008
0.003288
0.003595
0.00393
0.004297
0.004698
0.005136
0.005615
0.006139
0.006712
0.007338
0.008022
0.008771
0.009589
0.010484
0.011462
0.012531
0.0137
0.014978
0.016376
0.017903
0.019574
0.0214

Using these you can calculate the frequency of any partial/harmonic with the following formula:
Fn = F0 * n * sqrt(1 + B * n^2)
where F0 is an ideal fundamental frequency, n is the harmonic number, and B is the inharmonicity constant.
Or if you prefer to use F1 as the fundamental frequency, which probably makes sense in this case, then
F1 = F0 * 1 * sqrt(1 + B)
so by substitution
Fn = F1 * n * sqrt(1 + B * n^2) / sqrt(1 + B)
This allows you to find the frequency of any harmonic from the frequency of the first harmonic. Note that if there is no inharmonicity (B=0) this reduces to
Fn = F1 * n
which just says that partials are whole number multiples of the fundamental frequency.


------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
------------------------------

Original Message:
Sent: 07-06-2020 12:35
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Each note has its own average inharmonicity constant.

For example, the inharmonicity constant of A4 in relation to Piano 1 might not be the same as the inharmonicity constant of A4 in relation to Piano 2.

Average inharmonicity constant of A4 = (Inharmonicity constant of A4 in relation to Piano 1 + Inharmonicity constant of A4 in relation to Piano 2) / 2.

More examples:

All of the various inharmonicity constants of the note F3 can be used to calculate the average inharmoncity constant of the note F3.

All of the various inharmonicity constants of the note C4 can be used to calculate the average inharmonicity constant of the note C4.


I would be grateful if you could provide the various inharmonicity constants of each individual note.

------------------------------
Roshan Kakiya

Original Message:
Sent: 07-06-2020 12:15
From: Anthony Willey
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Roshan,
I don't know anybody who has ever used that metric (average inharmonicity constant) and I don't know that it would be meaningful or useful. The inharmonicity varies exponentially in a single piano. When I graph it I do so on a log plot.  Off the top of my head, it can range from 0.00001 in the tenor to 0.01 in the treble. I don't think you're going to get anything representative of the temperament region by averaging the whole piano.

I don't know of a published database anywhere listing inharmonicity of many pianos. But I could get you some example numbers to work with if you want to play around with the math. Also FYI the iH depends on the size & scale of the piano, not the manufacturer. A long Baldwin will have lower inharmonicity than a short Bosendorfer. At least in the tenor and bass. I couldn't predict for the treble, which can be all over the map, and which would have the biggest effect on the "average" while at the same time have the least relevance to the tuning and temperament.

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 07-06-2020 11:37
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

My aim is precisely to overcome the specificity of inharmonicity by generalising inharmonicity.

Each temperament can have its own general inharmonicity model.

------------------------------
Roshan Kakiya

Original Message:
Sent: 07-06-2020 11:25
From: David Pinnegar
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Different tuners focus on different things. Inharmonicity may not be so relevant to some as to others.

Different models of pianos express different inharmonicities due to scaling. It's much more of an individual thing than can be averaged across the spectrum

Best wishes

David P

--

- - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594

Original Message:
Sent: 7/6/2020 10:03:00 AM
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant for a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant for each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, for a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant for each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant for each note been explored in the past? If the answer is yes, where can the average inharmonicity constant for each note be found?

------------------------------
Roshan Kakiya
------------------------------

This is exactly what I have been looking for!

Thank you very much for providing this, Anthony.

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 07-06-2020 13:05
From: Anthony Willey
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Oh, I misunderstood before. Taking the note-by-note average makes more sense. There will still be variation from piano to piano, but the average numbers will still hold some meaning. Here's a list of inharmonicity numbers from A0 to C8 that can be used as a jumping off point. These aren't representative of any particular piano, but are an idealized (smooth) set based on some average parameters from different pianos I measured.

0.000639
0.000577
0.000521
0.000471
0.000426
0.000386
0.00035
0.000318
0.000289
0.000264
0.000242
0.000222
0.000204
0.000189
0.000176
0.000164
0.000154
0.000146
0.00014
0.000135
0.000131
0.000128
0.000127
0.000127
0.000128
0.00013
0.000133
0.000138
0.000144
0.000151
0.000159
0.000169
0.00018
0.000193
0.000207
0.000223
0.00024
0.00026
0.000282
0.000306
0.000333
0.000362
0.000394
0.000429
0.000468
0.000511
0.000557
0.000608
0.000664
0.000725
0.000792
0.000865
0.000946
0.001033
0.001129
0.001234
0.001349
0.001475
0.001612
0.001762
0.001926
0.002106
0.002302
0.002517
0.002751
0.003008
0.003288
0.003595
0.00393
0.004297
0.004698
0.005136
0.005615
0.006139
0.006712
0.007338
0.008022
0.008771
0.009589
0.010484
0.011462
0.012531
0.0137
0.014978
0.016376
0.017903
0.019574
0.0214

Using these you can calculate the frequency of any partial/harmonic with the following formula:
Fn = F0 * n * sqrt(1 + B * n^2)
where F0 is an ideal fundamental frequency, n is the harmonic number, and B is the inharmonicity constant.
Or if you prefer to use F1 as the fundamental frequency, which probably makes sense in this case, then
F1 = F0 * 1 * sqrt(1 + B)
so by substitution
Fn = F1 * n * sqrt(1 + B * n^2) / sqrt(1 + B)
This allows you to find the frequency of any harmonic from the frequency of the first harmonic. Note that if there is no inharmonicity (B=0) this reduces to
Fn = F1 * n
which just says that partials are whole number multiples of the fundamental frequency.


------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 07-06-2020 12:35
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Each note has its own average inharmonicity constant.

For example, the inharmonicity constant of A4 in relation to Piano 1 might not be the same as the inharmonicity constant of A4 in relation to Piano 2.

Average inharmonicity constant of A4 = (Inharmonicity constant of A4 in relation to Piano 1 + Inharmonicity constant of A4 in relation to Piano 2) / 2.

More examples:

All of the various inharmonicity constants of the note F3 can be used to calculate the average inharmoncity constant of the note F3.

All of the various inharmonicity constants of the note C4 can be used to calculate the average inharmonicity constant of the note C4.


I would be grateful if you could provide the various inharmonicity constants of each individual note.

------------------------------
Roshan Kakiya

Original Message:
Sent: 07-06-2020 12:15
From: Anthony Willey
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Roshan,
I don't know anybody who has ever used that metric (average inharmonicity constant) and I don't know that it would be meaningful or useful. The inharmonicity varies exponentially in a single piano. When I graph it I do so on a log plot.  Off the top of my head, it can range from 0.00001 in the tenor to 0.01 in the treble. I don't think you're going to get anything representative of the temperament region by averaging the whole piano.

I don't know of a published database anywhere listing inharmonicity of many pianos. But I could get you some example numbers to work with if you want to play around with the math. Also FYI the iH depends on the size & scale of the piano, not the manufacturer. A long Baldwin will have lower inharmonicity than a short Bosendorfer. At least in the tenor and bass. I couldn't predict for the treble, which can be all over the map, and which would have the biggest effect on the "average" while at the same time have the least relevance to the tuning and temperament.

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 07-06-2020 11:37
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

My aim is precisely to overcome the specificity of inharmonicity by generalising inharmonicity.

Each temperament can have its own general inharmonicity model.

------------------------------
Roshan Kakiya

Original Message:
Sent: 07-06-2020 11:25
From: David Pinnegar
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

Different tuners focus on different things. Inharmonicity may not be so relevant to some as to others.

Different models of pianos express different inharmonicities due to scaling. It's much more of an individual thing than can be averaged across the spectrum

Best wishes

David P

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David Pinnegar, B.Sc., A.R.C.S.
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+44 1342 850594

Original Message:
Sent: 7/6/2020 10:03:00 AM
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant for a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant for each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, for a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant for each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant for each note been explored in the past? If the answer is yes, where can the average inharmonicity constant for each note be found?

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Roshan Kakiya
------------------------------

By listening!

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Tom Servinsky
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Original Message:
Sent: 07-06-2020 10:02
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant for a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant for each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, for a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant for each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant for each note been explored in the past? If the answer is yes, where can the average inharmonicity constant for each note be found?

------------------------------
Roshan Kakiya
------------------------------

Here's inharmonicity data for a bunch of pianos: http://www.goptools.com/gallery.htm. This is from Tremaine Parsons' Pscale site.

Bob Runyan, RPT
runyanpiano.com
Chico, CA

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Bob Runyan
Chico CA
530-635-7852
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Original Message:
Sent: 07-07-2020 06:11
From: Thomas Servinsky
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

By listening!

------------------------------
Tom Servinsky

Original Message:
Sent: 07-06-2020 10:02
From: Roshan Kakiya
Subject: Incorporating the average inharmonicity constant of each note into the theoretical mathematical model of each temperament

It has been stated on many occasions on this forum that inharmonicity varies from piano to piano. Therefore, each piano must have its own specific solution for inharmonicity.

Each piano can have its own inharmonicity constant for each note. The average inharmonicity constant for a note can be determined by calculating the average of all its various inharmonicity constants across different pianos. The average inharmonicity constant for each note can be used to create the general inharmonicity model of each temperament. The general inharmonicity model of each temperament can provide a general idea of the effect of inharmonicity on the partials of each of its notes. Therefore, this is one of the ways in which the gap between theory and practice can be bridged. Average inharmonicity constants can be used to create the general inharmonicity model of any temperament.

The frequency of each partial of each note can be calculated by using the following equation for inharmonicity:

f_n = n × f₁ × [(1 + B_average × n²) / (1 + B_average)]^1/2

n is the number of the partial.

f₁ is the theoretical frequency of the 1st partial.

B_average is the average inharmonicity constant.

The theoretical frequency of the 1st partial of each note of a temperament can be calculated by using a system of cents. The average inharmonicity constant, Baverage, for a note can be inserted into the equation for inharmonicity to calculate the general inharmonic frequencies of its partials. This process can be repeated for all remaining notes to create a general inharmonicity model of a temperament. The general inharmonicity model of a temperament can give an indication of the general effect of inharmonicity on the partials of each of its notes.

Where can I find the inharmonicity constant for each note in relation to various pianos that have been manufactured by brands such as Steinway, Bechstein, Yamaha, Bösendorfer and so on?

Has the average inharmonicity constant for each note been explored in the past? If the answer is yes, where can the average inharmonicity constant for each note be found?

------------------------------
Roshan Kakiya
------------------------------