Pianotech

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 1st partial of the 2nd note is matched with the 2nd partial of the 1st note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

It depends on what you want the tuning to do and to achieve . . . 

Where one's dealing with a variation of size of thirds, using stretching can sound fine on one instrument but harsh on others so there can be no hard and fast rule. No-stretch in the central octaves solves that problem and this is a region where in reality little stretch is required by the Railsback curve, so the departure from the norm here does minimal "damage" to our conventional perceptions. Anthony Willey very kindly wrote to me with regard to implementation with his Easy Piano Tuner app and if he's able to achieve control of the stretch to allow conventional harmonic stretching above and below whilst keeping the central three octaves "pure", this will open up reliable use of unequal temperaments and help people possibly to avoid bad experiences which might have given unequal temperaments a bad name in the past.

This recording is the 1859 Hallé Broadwood tuned to Kirnberger III accompanied by an 1885 Bechstein in Kellner. Both had not been tuned for three months so are not perfect, but I think demonstrates an atmosphere of the Victorian piano - 
https://www.youtube.com/watch?v=I-W8oRAd6Bw

Meanwhile in contrast, the Bechstein being more stable than the indisputably historic instrument
https://www.youtube.com/watch?v=7dDo1GBMspE
https://www.youtube.com/watch?v=vkuPMACOOLk
https://www.youtube.com/watch?v=5xhRPI1LlO0
demonstrates reasonably well what such tunings can achieve.

Best wishes

David P

Best wishes

David P

------------------------------
David Pinnegar BSc ARCS
Curator and House Tuner - Hammerwood Park, East Grinstead, Sussex UK
antespam@gmail.com

Seminar 6th May 2019 - http://hammerwood.mistral.co.uk/tuning-seminar.pdf "The Importance of Tuning for Better Performance"
------------------------------

Original Message:
Sent: 09-08-2019 19:00
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

David, Please share Anthony's instructions, so that I can try your Kirnberger III the way you do it. (Hopefully).

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA
------------------------------

Original Message:
Sent: 09-09-2019 05:22
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

It depends on what you want the tuning to do and to achieve . . .

Where one's dealing with a variation of size of thirds, using stretching can sound fine on one instrument but harsh on others so there can be no hard and fast rule. No-stretch in the central octaves solves that problem and this is a region where in reality little stretch is required by the Railsback curve, so the departure from the norm here does minimal "damage" to our conventional perceptions. Anthony Willey very kindly wrote to me with regard to implementation with his Easy Piano Tuner app and if he's able to achieve control of the stretch to allow conventional harmonic stretching above and below whilst keeping the central three octaves "pure", this will open up reliable use of unequal temperaments and help people possibly to avoid bad experiences which might have given unequal temperaments a bad name in the past.

This recording is the 1859 Hallé Broadwood tuned to Kirnberger III accompanied by an 1885 Bechstein in Kellner. Both had not been tuned for three months so are not perfect, but I think demonstrates an atmosphere of the Victorian piano -
https://www.youtube.com/watch?v=I-W8oRAd6Bw

Meanwhile in contrast, the Bechstein being more stable than the indisputably historic instrument
https://www.youtube.com/watch?v=7dDo1GBMspE
https://www.youtube.com/watch?v=vkuPMACOOLk
https://www.youtube.com/watch?v=5xhRPI1LlO0
demonstrates reasonably well what such tunings can achieve.

Best wishes

David P

Best wishes

David P

------------------------------
David Pinnegar BSc ARCS
Curator and House Tuner - Hammerwood Park, East Grinstead, Sussex UK
antespam@gmail.com

Seminar 6th May 2019 - http://hammerwood.mistral.co.uk/tuning-seminar.pdf "The Importance of Tuning for Better Performance"

Original Message:
Sent: 09-08-2019 19:00
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I’m only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don’t fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it’s impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you “stretch the framework” or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can’t match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no “solution” in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is “stretch” in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

------------------------------
Larry Messerly, RPT
Bringing Harmony to Homes
www.lacrossepianotuning.com
ljmesserly@gmail.com
928-899-7292
------------------------------

Original Message:
Sent: 09-09-2019 10:51
From: Robert Anderson
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I'm only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don't fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it's impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you "stretch the framework" or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can't match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no "solution" in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is "stretch" in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

@Richard, PianoMeter does not currently have a way to do this.  ​The way I calculate the tuning I'm not able to tune anything narrower than 2:1 octaves. I've got some custom tuning style controls in development that I was talking about with David, but still not anything that would specifically support this.

Based on what David has said, I would suggest that the best way to experiment with his "flat" tuning style would be to just download a generic chromatic tuning app. The app called "Pano Tuner" (P-A-N-O...not a typo) is decent, and with a $1.99 upgrade to the "Full" version it will support unequal tempermanets. Kirnberger III is pre-loaded. You'll probably want to tune the top and bottom octaves by ear.

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

Original Message:
Sent: 09-09-2019 11:14
From: Larry Messerly
Subject: Fundamental Frequencies vs Coincidental Partials

If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

------------------------------
Larry Messerly, RPT
Bringing Harmony to Homes
www.lacrossepianotuning.com
ljmesserly@gmail.com
928-899-7292

Original Message:
Sent: 09-09-2019 10:51
From: Robert Anderson
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I'm only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don't fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it's impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you "stretch the framework" or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can't match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no "solution" in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is "stretch" in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

David and Anthony,

Could the following method be useful for tuning the central octaves of a piano's keyboard without stretch?

Each note within the central octaves of a piano's keyboard should be tuned in accordance with its theoretical fundamental frequency.


Example:

If the frequency ratio of A4-E5 is 3/2 and the fundamental frequency of A4 is 440 Hz, the fundamental frequency of E5 can be calculated. 

E5 = 440 Hz × 3/2 = 660 Hz.

If the frequency ratio of [insert interval] is [insert frequency ratio of interval] and the fundamental frequency of [insert 1st note of interval] is [insert fundamental frequency of 1st note of interval], the fundamental frequency of [insert 2nd note of interval] can be calculated.

[Insert 2nd note of interval] = [insert fundamental frequency of 1st note of interval] × [insert frequency ratio of interval] = [insert fundamental frequency of 2nd note of interval].


This method ignores inharmonicity and can be used to directly apply the theoretical model of each temperament without stretch to the central octaves of a piano's keyboard.

Unequal temperaments are more mathematically complicated than equal temperaments due to the inequality of their intervals. It is easier to calculate the scale of a temperament if there are not many different types of fifths. The system of cents can be incorporated into the frequency ratios to calculate the frequencies of the scale of a temperament.

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

Original Message:
Sent: 09-10-2019 23:57
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Richard, PianoMeter does not currently have a way to do this.  ​The way I calculate the tuning I'm not able to tune anything narrower than 2:1 octaves. I've got some custom tuning style controls in development that I was talking about with David, but still not anything that would specifically support this.

Based on what David has said, I would suggest that the best way to experiment with his "flat" tuning style would be to just download a generic chromatic tuning app. The app called "Pano Tuner" (P-A-N-O...not a typo) is decent, and with a $1.99 upgrade to the "Full" version it will support unequal tempermanets. Kirnberger III is pre-loaded. You'll probably want to tune the top and bottom octaves by ear.

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

Original Message:
Sent: 09-09-2019 11:14
From: Larry Messerly
Subject: Fundamental Frequencies vs Coincidental Partials

If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

------------------------------
Larry Messerly, RPT
Bringing Harmony to Homes
www.lacrossepianotuning.com
ljmesserly@gmail.com
928-899-7292

Original Message:
Sent: 09-09-2019 10:51
From: Robert Anderson
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I'm only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don't fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it's impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you "stretch the framework" or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can't match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no "solution" in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is "stretch" in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

Original Message------

David and Anthony,

Could the following method be useful for tuning the central octaves of a piano's keyboard without stretch?

Each note within the central octaves of a piano's keyboard should be tuned in accordance with its theoretical fundamental frequency.


Example:

If the frequency ratio of A4-E5 is 3/2 and the fundamental frequency of A4 is 440 Hz, the fundamental frequency of E5 can be calculated. 

E5 = 440 Hz × 3/2 = 660 Hz.

If the frequency ratio of [insert interval] is [insert frequency ratio of interval] and the fundamental frequency of [insert 1st note of interval] is [insert fundamental frequency of 1st note of interval], the fundamental frequency of [insert 2nd note of interval] can be calculated.

[Insert 2nd note of interval] = [insert fundamental frequency of 1st note of interval] × [insert frequency ratio of interval] = [insert fundamental frequency of 2nd note of interval].


This method ignores inharmonicity and can be used to directly apply the theoretical model of each temperament without stretch to the central octaves of a piano's keyboard.

Unequal temperaments are more mathematically complicated than equal temperaments due to the inequality of their intervals. It is easier to calculate the scale of a temperament if there are not many different types of fifths. The system of cents can be incorporated into the frequency ratios to calculate the frequencies of the scale of a temperament.

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

Original Message:
Sent: 09-10-2019 23:57
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Richard, PianoMeter does not currently have a way to do this.  ​The way I calculate the tuning I'm not able to tune anything narrower than 2:1 octaves. I've got some custom tuning style controls in development that I was talking about with David, but still not anything that would specifically support this.

Based on what David has said, I would suggest that the best way to experiment with his "flat" tuning style would be to just download a generic chromatic tuning app. The app called "Pano Tuner" (P-A-N-O...not a typo) is decent, and with a $1.99 upgrade to the "Full" version it will support unequal tempermanets. Kirnberger III is pre-loaded. You'll probably want to tune the top and bottom octaves by ear.

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

Original Message:
Sent: 09-09-2019 11:14
From: Larry Messerly
Subject: Fundamental Frequencies vs Coincidental Partials

If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

------------------------------
Larry Messerly, RPT
Bringing Harmony to Homes
www.lacrossepianotuning.com
ljmesserly@gmail.com
928-899-7292

Original Message:
Sent: 09-09-2019 10:51
From: Robert Anderson
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I'm only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don't fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it's impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you "stretch the framework" or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can't match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no "solution" in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is "stretch" in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

David,

"We're not talking about calculating the frequency - we're talking about tuning it."

If a piano is tuned with the aid of a device/machine, it would be beneficial to make use of its capabilities by calculating and using fundamental frequencies. This is because your method directly involves dealing with fundamental frequency ratios such as 2/1 for the perfect octave and 3/2 for the perfect fifth.


"Therefore tuning to fundamentals gives a more solid sound than listening to beats from harmonics, which are musically irrelevant compared to the fundamentals."

The method that I have described on my previous post facilitates this approach because it only involves tuning each note within the central octaves of a piano's keyboard in accordance with its theoretical fundamental frequency. It ignores inharmonicity by not considering any of the partials beyond the 1st partial/fundamental frequency of each note within the central octaves of a piano's keyboard.

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

Original Message:
Sent: 09-11-2019 06:05
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

We're not talking about calculating the frequency - we're talking about tuning it. 

For years I have tuned both aurally and using Tunelab with standard stretch. Most of the time the results were superb.

But since adopting no stretch in the central three octaves the results have been reliable. A reality is that in this section inharmoncity isn't generally very heavy so tuning to fundamentals doesn't give a lot of difference to tuning harmonics. There's a further thought here in so far as when tuning fundamentals it's as tuning organ pipes or sine waves from two laboratory oscillators. The harmonics of piano strings are at a lower level than the fundamentals. Therefore tuning to fundamentals gives a more solid sound than listening to beats from harmonics, which are musically irrelevant compared to the fundamentals. Tuning getting harmonics beatless may well have been an adjunct to Equal Temperament practise so as to give the piano an edge with which to get the sound to glisten and shimmer. Essentially I'm tuning a compromise between the two systems, and arguably or potentially accessing the best of both. 

In the treble top two octaves, bearing in mind that the instruments with duplex stringing sound out a range of frequencies for every note as the Duplex is rarely nicely tuned, we're looking there more at the concept of tuned percussion so tuning bright, to the harmonics below is fine. Below the central three octaves for the bass two and a bit octaves were the harmonics not to align with the centre then it would jangle more which is why people take objection rightly to pure machine tuned fundamentals down there. Those notes musically set the harmony for above and so getting their harmonics to coincide with harmonic chords above is the priority above any relevance of their fundamental pitch.

Incidentally I'm aware of a tuner in the UK who has record of a tuner in the 1920s bemoaning the loss of discussions when he was young in the 1870s or so as to where to put "The Wolf". This sets fire to Fred Sturm's insistence that after the 1830s when Montal laid out the textbook for tuning Equal Temperament the new tuning became universal. Far from it. In due course I'll be meeting up with him and will be able to find out more about the source.

Best wishes

David P

Original Message------

David and Anthony,

Could the following method be useful for tuning the central octaves of a piano's keyboard without stretch?

Each note within the central octaves of a piano's keyboard should be tuned in accordance with its theoretical fundamental frequency.


Example:

If the frequency ratio of A4-E5 is 3/2 and the fundamental frequency of A4 is 440 Hz, the fundamental frequency of E5 can be calculated.

E5 = 440 Hz × 3/2 = 660 Hz.

If the frequency ratio of [insert interval] is [insert frequency ratio of interval] and the fundamental frequency of [insert 1st note of interval] is [insert fundamental frequency of 1st note of interval], the fundamental frequency of [insert 2nd note of interval] can be calculated.

[Insert 2nd note of interval] = [insert fundamental frequency of 1st note of interval] × [insert frequency ratio of interval] = [insert fundamental frequency of 2nd note of interval].


This method ignores inharmonicity and can be used to directly apply the theoretical model of each temperament without stretch to the central octaves of a piano's keyboard.

Unequal temperaments are more mathematically complicated than equal temperaments due to the inequality of their intervals. It is easier to calculate the scale of a temperament if there are not many different types of fifths. The system of cents can be incorporated into the frequency ratios to calculate the frequencies of the scale of a temperament.

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

Original Message:
Sent: 09-10-2019 23:57
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Richard, PianoMeter does not currently have a way to do this.  ​The way I calculate the tuning I'm not able to tune anything narrower than 2:1 octaves. I've got some custom tuning style controls in development that I was talking about with David, but still not anything that would specifically support this.

Based on what David has said, I would suggest that the best way to experiment with his "flat" tuning style would be to just download a generic chromatic tuning app. The app called "Pano Tuner" (P-A-N-O...not a typo) is decent, and with a $1.99 upgrade to the "Full" version it will support unequal tempermanets. Kirnberger III is pre-loaded. You'll probably want to tune the top and bottom octaves by ear.

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

Original Message:
Sent: 09-09-2019 11:14
From: Larry Messerly
Subject: Fundamental Frequencies vs Coincidental Partials

If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

------------------------------
Larry Messerly, RPT
Bringing Harmony to Homes
www.lacrossepianotuning.com
ljmesserly@gmail.com
928-899-7292

Original Message:
Sent: 09-09-2019 10:51
From: Robert Anderson
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I'm only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don't fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it's impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you "stretch the framework" or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can't match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no "solution" in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is "stretch" in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

To put it simply, theory = practice, in practice, if the central octaves of a piano's keyboard and the intervals within them are being tuned without stretch.

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

Original Message:
Sent: 09-11-2019 07:29
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

"We're not talking about calculating the frequency - we're talking about tuning it."

If a piano is tuned with the aid of a device/machine, it would be beneficial to make use of its capabilities by calculating and using fundamental frequencies. This is because your method directly involves dealing with fundamental frequency ratios such as 2/1 for the perfect octave and 3/2 for the perfect fifth.


"Therefore tuning to fundamentals gives a more solid sound than listening to beats from harmonics, which are musically irrelevant compared to the fundamentals."

The method that I have described on my previous post facilitates this approach because it only involves tuning each note within the central octaves of a piano's keyboard in accordance with its theoretical fundamental frequency. It ignores inharmonicity by not considering any of the partials beyond the 1st partial/fundamental frequency of each note within the central octaves of a piano's keyboard.

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

Original Message:
Sent: 09-11-2019 06:05
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

We're not talking about calculating the frequency - we're talking about tuning it. 

For years I have tuned both aurally and using Tunelab with standard stretch. Most of the time the results were superb.

But since adopting no stretch in the central three octaves the results have been reliable. A reality is that in this section inharmoncity isn't generally very heavy so tuning to fundamentals doesn't give a lot of difference to tuning harmonics. There's a further thought here in so far as when tuning fundamentals it's as tuning organ pipes or sine waves from two laboratory oscillators. The harmonics of piano strings are at a lower level than the fundamentals. Therefore tuning to fundamentals gives a more solid sound than listening to beats from harmonics, which are musically irrelevant compared to the fundamentals. Tuning getting harmonics beatless may well have been an adjunct to Equal Temperament practise so as to give the piano an edge with which to get the sound to glisten and shimmer. Essentially I'm tuning a compromise between the two systems, and arguably or potentially accessing the best of both. 

In the treble top two octaves, bearing in mind that the instruments with duplex stringing sound out a range of frequencies for every note as the Duplex is rarely nicely tuned, we're looking there more at the concept of tuned percussion so tuning bright, to the harmonics below is fine. Below the central three octaves for the bass two and a bit octaves were the harmonics not to align with the centre then it would jangle more which is why people take objection rightly to pure machine tuned fundamentals down there. Those notes musically set the harmony for above and so getting their harmonics to coincide with harmonic chords above is the priority above any relevance of their fundamental pitch.

Incidentally I'm aware of a tuner in the UK who has record of a tuner in the 1920s bemoaning the loss of discussions when he was young in the 1870s or so as to where to put "The Wolf". This sets fire to Fred Sturm's insistence that after the 1830s when Montal laid out the textbook for tuning Equal Temperament the new tuning became universal. Far from it. In due course I'll be meeting up with him and will be able to find out more about the source.

Best wishes

David P

Original Message------

David and Anthony,

Could the following method be useful for tuning the central octaves of a piano's keyboard without stretch?

Each note within the central octaves of a piano's keyboard should be tuned in accordance with its theoretical fundamental frequency.


Example:

If the frequency ratio of A4-E5 is 3/2 and the fundamental frequency of A4 is 440 Hz, the fundamental frequency of E5 can be calculated.

E5 = 440 Hz × 3/2 = 660 Hz.

If the frequency ratio of [insert interval] is [insert frequency ratio of interval] and the fundamental frequency of [insert 1st note of interval] is [insert fundamental frequency of 1st note of interval], the fundamental frequency of [insert 2nd note of interval] can be calculated.

[Insert 2nd note of interval] = [insert fundamental frequency of 1st note of interval] × [insert frequency ratio of interval] = [insert fundamental frequency of 2nd note of interval].


This method ignores inharmonicity and can be used to directly apply the theoretical model of each temperament without stretch to the central octaves of a piano's keyboard.

Unequal temperaments are more mathematically complicated than equal temperaments due to the inequality of their intervals. It is easier to calculate the scale of a temperament if there are not many different types of fifths. The system of cents can be incorporated into the frequency ratios to calculate the frequencies of the scale of a temperament.

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

Original Message:
Sent: 09-10-2019 23:57
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Richard, PianoMeter does not currently have a way to do this.  ​The way I calculate the tuning I'm not able to tune anything narrower than 2:1 octaves. I've got some custom tuning style controls in development that I was talking about with David, but still not anything that would specifically support this.

Based on what David has said, I would suggest that the best way to experiment with his "flat" tuning style would be to just download a generic chromatic tuning app. The app called "Pano Tuner" (P-A-N-O...not a typo) is decent, and with a $1.99 upgrade to the "Full" version it will support unequal tempermanets. Kirnberger III is pre-loaded. You'll probably want to tune the top and bottom octaves by ear.

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

Original Message:
Sent: 09-09-2019 11:14
From: Larry Messerly
Subject: Fundamental Frequencies vs Coincidental Partials

If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

------------------------------
Larry Messerly, RPT
Bringing Harmony to Homes
www.lacrossepianotuning.com
ljmesserly@gmail.com
928-899-7292

Original Message:
Sent: 09-09-2019 10:51
From: Robert Anderson
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I'm only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don't fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it's impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you "stretch the framework" or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can't match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no "solution" in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is "stretch" in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

@Roshan,  ​​What you're describing sounds more like a Pythagorean/Just tuning system. David is talking about applying the calculated (cent) offsets of unequal temperaments on top of regular old unstretched 12-tone equal temperament. So E5 would be 659.2551 Hz, not 660 Hz as you calculated above. The fundamental frequency doubles exactly each octave, and the frequency ratio of each minor second in the underlying temperament (before the unequal temperament is applied) is exactly 2^(1/12). I think I've seen you calculate the frequencies yourself in an earlier post, but they're also listed at https://en.wikipedia.org/wiki/Piano_key_frequencies#List

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

Original Message:
Sent: 09-11-2019 04:55
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David and Anthony,

Could the following method be useful for tuning the central octaves of a piano's keyboard without stretch?

Each note within the central octaves of a piano's keyboard should be tuned in accordance with its theoretical fundamental frequency.


Example:

If the frequency ratio of A4-E5 is 3/2 and the fundamental frequency of A4 is 440 Hz, the fundamental frequency of E5 can be calculated.

E5 = 440 Hz × 3/2 = 660 Hz.

If the frequency ratio of [insert interval] is [insert frequency ratio of interval] and the fundamental frequency of [insert 1st note of interval] is [insert fundamental frequency of 1st note of interval], the fundamental frequency of [insert 2nd note of interval] can be calculated.

[Insert 2nd note of interval] = [insert fundamental frequency of 1st note of interval] × [insert frequency ratio of interval] = [insert fundamental frequency of 2nd note of interval].


This method ignores inharmonicity and can be used to directly apply the theoretical model of each temperament without stretch to the central octaves of a piano's keyboard.

Unequal temperaments are more mathematically complicated than equal temperaments due to the inequality of their intervals. It is easier to calculate the scale of a temperament if there are not many different types of fifths. The system of cents can be incorporated into the frequency ratios to calculate the frequencies of the scale of a temperament.

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

Original Message:
Sent: 09-10-2019 23:57
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Richard, PianoMeter does not currently have a way to do this.  ​The way I calculate the tuning I'm not able to tune anything narrower than 2:1 octaves. I've got some custom tuning style controls in development that I was talking about with David, but still not anything that would specifically support this.

Based on what David has said, I would suggest that the best way to experiment with his "flat" tuning style would be to just download a generic chromatic tuning app. The app called "Pano Tuner" (P-A-N-O...not a typo) is decent, and with a $1.99 upgrade to the "Full" version it will support unequal tempermanets. Kirnberger III is pre-loaded. You'll probably want to tune the top and bottom octaves by ear.

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

Original Message:
Sent: 09-09-2019 11:14
From: Larry Messerly
Subject: Fundamental Frequencies vs Coincidental Partials

If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

------------------------------
Larry Messerly, RPT
Bringing Harmony to Homes
www.lacrossepianotuning.com
ljmesserly@gmail.com
928-899-7292

Original Message:
Sent: 09-09-2019 10:51
From: Robert Anderson
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I'm only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don't fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it's impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you "stretch the framework" or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can't match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no "solution" in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is "stretch" in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III for C3-C4-C5-C6.

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

Original Message:
Sent: 09-11-2019 23:52
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Roshan,  ​​What you're describing sounds more like a Pythagorean/Just tuning system. David is talking about applying the calculated (cent) offsets of unequal temperaments on top of regular old unstretched 12-tone equal temperament. So E5 would be 659.2551 Hz, not 660 Hz as you calculated above. The fundamental frequency doubles exactly each octave, and the frequency ratio of each minor second in the underlying temperament (before the unequal temperament is applied) is exactly 2^(1/12). I think I've seen you calculate the frequencies yourself in an earlier post, but they're also listed at https://en.wikipedia.org/wiki/Piano_key_frequencies#List

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

Original Message:
Sent: 09-11-2019 04:55
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David and Anthony,

Could the following method be useful for tuning the central octaves of a piano's keyboard without stretch?

Each note within the central octaves of a piano's keyboard should be tuned in accordance with its theoretical fundamental frequency.


Example:

If the frequency ratio of A4-E5 is 3/2 and the fundamental frequency of A4 is 440 Hz, the fundamental frequency of E5 can be calculated.

E5 = 440 Hz × 3/2 = 660 Hz.

If the frequency ratio of [insert interval] is [insert frequency ratio of interval] and the fundamental frequency of [insert 1st note of interval] is [insert fundamental frequency of 1st note of interval], the fundamental frequency of [insert 2nd note of interval] can be calculated.

[Insert 2nd note of interval] = [insert fundamental frequency of 1st note of interval] × [insert frequency ratio of interval] = [insert fundamental frequency of 2nd note of interval].


This method ignores inharmonicity and can be used to directly apply the theoretical model of each temperament without stretch to the central octaves of a piano's keyboard.

Unequal temperaments are more mathematically complicated than equal temperaments due to the inequality of their intervals. It is easier to calculate the scale of a temperament if there are not many different types of fifths. The system of cents can be incorporated into the frequency ratios to calculate the frequencies of the scale of a temperament.

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

Original Message:
Sent: 09-10-2019 23:57
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Richard, PianoMeter does not currently have a way to do this.  ​The way I calculate the tuning I'm not able to tune anything narrower than 2:1 octaves. I've got some custom tuning style controls in development that I was talking about with David, but still not anything that would specifically support this.

Based on what David has said, I would suggest that the best way to experiment with his "flat" tuning style would be to just download a generic chromatic tuning app. The app called "Pano Tuner" (P-A-N-O...not a typo) is decent, and with a $1.99 upgrade to the "Full" version it will support unequal tempermanets. Kirnberger III is pre-loaded. You'll probably want to tune the top and bottom octaves by ear.

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

Original Message:
Sent: 09-09-2019 11:14
From: Larry Messerly
Subject: Fundamental Frequencies vs Coincidental Partials

If one were to try and tune without stretch, what key signature should the use? Remember, depending upon the key signature F# is not the same as G flat.

------------------------------
Larry Messerly, RPT
Bringing Harmony to Homes
www.lacrossepianotuning.com
ljmesserly@gmail.com
928-899-7292

Original Message:
Sent: 09-09-2019 10:51
From: Robert Anderson
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
Since all pianos have inharmonicity, some more than others, the benefits of a stretched framework, if I understand that term correctly, are intervals that sound better. Your question about stretched vs. unstretched is a little like this: if I start my car engine and prepare to drive off without buckling my seat belt, the car makes an annoying beeping sound. If I'm only going to drive 500 ft., what are the benefits of fastening my seatbelt or not fastening it? It save time if I don't fasten it and there is much less chance of an accident than if I were driving, say, one mile. The advantages of fastening it: no annoying beeping, reenforcing a good habit, i.e., fastening the seat belt every time I prepare to drive, and being ready for the small chance of an accident in 500 feet.

All intervals on a piano are out of tune insofar as it's impossible to match all coincidental partials at the same time. Some intervals, e.g., 3rds, are quite badly out of tune, irrespective of whether you "stretch the framework" or not. Octaves can be pretty well in tune usually, even if all their coincidental partials can't match at the same time. Higher coincidental partials are much more audible lower in the scale. There is no "solution" in the sense of finding the solution to a mathematical problem (like the area of a rectangle of 2 x 3 units). This is where piano tuners tend to get a little impatient or glassy eyed with theoretical speculation. We just need to get the piano tuned to sound as good as possible. There is "stretch" in the piano whether we choose to ignore it or not. If we ignore it, it is like painting by the numbers. The art/craft of piano tuning comes into play when we have to deal with the inconsistencies in a piano scale, like when inharmonicity suddenly changes between adjoining plain strings and wrapped strings.

I hope this helps.

Bob Anderson, RPT
Tucson, AZ


Original Message------

Should inharmonicity be ignored within the central octaves of a piano's keyboard?

Should intervals within the central octaves of a piano's keyboard be tuned without stretch?

What are the benefits of a stretched framework?

What are the benefits of an unstretched framework?


An example of a stretched octave:

If the 2nd partial of the 1st note is matched with the 1st partial of the 2nd note, the result will be a stretched octave due to the sharpening of partials caused by inharmonicity. In this case, coincidental partials will have been matched.


An example of an unstretched octave:

If the fundamental frequency of the 1st note is 440 Hz and the fundamental frequency of the 2nd note is 880 Hz, the result will be an unstretched octave which has a frequency ratio of 2/1 (880 Hz / 440 Hz = 2/1). In this case, fundamental frequencies will have been arranged in a way that causes them to have a specific frequency ratio.

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

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

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

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

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

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

Original Message------

Kirnberger III Circle of Fifths and Fourths

Source: https://www.hpschd.nu/g/tech/tmp/kirnberger.gif

Kirnberger III Frequencies (C3 to C6)

C3	131.70	Hz
C#3	138.75	Hz
D3	147.16	Hz
D#3	156.09	Hz
E3	164.44	Hz
F3	175.60	Hz
F#3	185.00	Hz
G3	196.89	Hz
G#3	208.12	Hz
A3	220.00	Hz
A#3	234.14	Hz
B3	246.66	Hz
C4	263.40	Hz
C#4	277.50	Hz
D4	294.33	Hz
D#4	312.18	Hz
E4	328.88	Hz
F4	351.21	Hz
F#4	369.99	Hz
G4	393.77	Hz
G#4	416.24	Hz
A4	440.00	Hz
A#4	468.27	Hz
B4	493.33	Hz
C5	526.81	Hz
C#5	554.99	Hz
D5	588.66	Hz
D#5	624.37	Hz
E5	657.77	Hz
F5	702.41	Hz
F#5	739.99	Hz
G5	787.54	Hz
G#5	832.49	Hz
A5	880.00	Hz
A#5	936.55	Hz
B5	986.65	Hz
C6	1053.62	Hz

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

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

@Roshan, Those are the right frequencies if you calculate the Kirnberger III using the Pythagorean Comma. I calculate it using the Syntonic Comma, placing the Schisma between C# and F#.


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

Original Message:
Sent: 09-12-2019 11:18
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Kirnberger III Circle of Fifths and Fourths

Source: https://www.hpschd.nu/g/tech/tmp/kirnberger.gif

Kirnberger III Frequencies (C3 to C6)

C3	131.70	Hz
C#3	138.75	Hz
D3	147.16	Hz
D#3	156.09	Hz
E3	164.44	Hz
F3	175.60	Hz
F#3	185.00	Hz
G3	196.89	Hz
G#3	208.12	Hz
A3	220.00	Hz
A#3	234.14	Hz
B3	246.66	Hz
C4	263.40	Hz
C#4	277.50	Hz
D4	294.33	Hz
D#4	312.18	Hz
E4	328.88	Hz
F4	351.21	Hz
F#4	369.99	Hz
G4	393.77	Hz
G#4	416.24	Hz
A4	440.00	Hz
A#4	468.27	Hz
B4	493.33	Hz
C5	526.81	Hz
C#5	554.99	Hz
D5	588.66	Hz
D#5	624.37	Hz
E5	657.77	Hz
F5	702.41	Hz
F#5	739.99	Hz
G5	787.54	Hz
G#5	832.49	Hz
A5	880.00	Hz
A#5	936.55	Hz
B5	986.65	Hz
C6	1053.62	Hz

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

------------------------------
Roshan Kakiya
------------------------------​

Anthony,

I prefer to divide the Pythagorean comma because It is easier to deal with one comma (Pythagorean comma) rather than two commas (Syntonic comma and Schisma).

Syntonic comma (21.51 cents) + Schisma (1.95 cents) = Pythagorean comma (23.46 cents).

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

Original Message:
Sent: 09-12-2019 11:59
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Roshan, Those are the right frequencies if you calculate the Kirnberger III using the Pythagorean Comma. I calculate it using the Syntonic Comma, placing the Schisma between C# and F#.

For that case, the cent offset values would be:

A3	0
A#3	6.35
B3	-1.47
C4	10.26
C#4	0.49
D4	3.42
D#4	4.4
E4	-3.42
F4	8.31
F#4	0.49
G4	6.84
G#4	2.44
A4	0

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

Original Message:
Sent: 09-12-2019 11:18
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Kirnberger III Circle of Fifths and Fourths

Source: https://www.hpschd.nu/g/tech/tmp/kirnberger.gif

Kirnberger III Frequencies (C3 to C6)

C3	131.70	Hz
C#3	138.75	Hz
D3	147.16	Hz
D#3	156.09	Hz
E3	164.44	Hz
F3	175.60	Hz
F#3	185.00	Hz
G3	196.89	Hz
G#3	208.12	Hz
A3	220.00	Hz
A#3	234.14	Hz
B3	246.66	Hz
C4	263.40	Hz
C#4	277.50	Hz
D4	294.33	Hz
D#4	312.18	Hz
E4	328.88	Hz
F4	351.21	Hz
F#4	369.99	Hz
G4	393.77	Hz
G#4	416.24	Hz
A4	440.00	Hz
A#4	468.27	Hz
B4	493.33	Hz
C5	526.81	Hz
C#5	554.99	Hz
D5	588.66	Hz
D#5	624.37	Hz
E5	657.77	Hz
F5	702.41	Hz
F#5	739.99	Hz
G5	787.54	Hz
G#5	832.49	Hz
A5	880.00	Hz
A#5	936.55	Hz
B5	986.65	Hz
C6	1053.62	Hz

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

------------------------------
Roshan Kakiya
------------------------------​

Roshan, 
 Thanks for clarifying .... so where do you place your "schizma"? And what are the cents there...? The third question is, should
one be setting Kirnberger III from C3 or from A3 (or even A2)?...that's pretty low. Doesn't that change the perfect thirds and schizma?
I guess I don't understand the fine points of using historical temperaments. I usually set them using the Cs, rather than the As;
I think as long as the perfect intervals are the same, it shouldn't matter where you begin. It is confusing though is someone
is talking about starting from As and someone else talks about starting from Cs, at least it is to me.

Richard

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA
------------------------------

Original Message:
Sent: 09-12-2019 14:37
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Anthony,

I prefer to divide the Pythagorean comma because It is easier to deal with one comma (Pythagorean comma) rather than two commas (Syntonic comma and Schisma).

Syntonic comma (21.51 cents) + Schisma (1.95 cents) = Pythagorean comma (23.46 cents).

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

Original Message:
Sent: 09-12-2019 11:59
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Roshan, Those are the right frequencies if you calculate the Kirnberger III using the Pythagorean Comma. I calculate it using the Syntonic Comma, placing the Schisma between C# and F#.

For that case, the cent offset values would be:

A3	0
A#3	6.35
B3	-1.47
C4	10.26
C#4	0.49
D4	3.42
D#4	4.4
E4	-3.42
F4	8.31
F#4	0.49
G4	6.84
G#4	2.44
A4	0

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

Original Message:
Sent: 09-12-2019 11:18
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Kirnberger III Circle of Fifths and Fourths

Source: https://www.hpschd.nu/g/tech/tmp/kirnberger.gif

Kirnberger III Frequencies (C3 to C6)

C3	131.70	Hz
C#3	138.75	Hz
D3	147.16	Hz
D#3	156.09	Hz
E3	164.44	Hz
F3	175.60	Hz
F#3	185.00	Hz
G3	196.89	Hz
G#3	208.12	Hz
A3	220.00	Hz
A#3	234.14	Hz
B3	246.66	Hz
C4	263.40	Hz
C#4	277.50	Hz
D4	294.33	Hz
D#4	312.18	Hz
E4	328.88	Hz
F4	351.21	Hz
F#4	369.99	Hz
G4	393.77	Hz
G#4	416.24	Hz
A4	440.00	Hz
A#4	468.27	Hz
B4	493.33	Hz
C5	526.81	Hz
C#5	554.99	Hz
D5	588.66	Hz
D#5	624.37	Hz
E5	657.77	Hz
F5	702.41	Hz
F#5	739.99	Hz
G5	787.54	Hz
G#5	832.49	Hz
A5	880.00	Hz
A#5	936.55	Hz
B5	986.65	Hz
C6	1053.62	Hz

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

------------------------------
Roshan Kakiya
------------------------------​

Original Message------

Roshan, 
 Thanks for clarifying .... so where do you place your "schizma"? And what are the cents there...? The third question is, should
one be setting Kirnberger III from C3 or from A3 (or even A2)?...that's pretty low. Doesn't that change the perfect thirds and schizma?
I guess I don't understand the fine points of using historical temperaments. I usually set them using the Cs, rather than the As;
I think as long as the perfect intervals are the same, it shouldn't matter where you begin. It is confusing though is someone
is talking about starting from As and someone else talks about starting from Cs, at least it is to me.

Richard

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA
------------------------------

Richard,

"Thanks for clarifying .... so where do you place your 'schizma'?"


Pythagorean comma (23.46 cents) = Syntonic comma (21.51 cents) + Schisma (1.95 cents).

The Syntonic comma and the Schisma are part of the Pythagorean comma. The Syntonic comma and the Schisma can be eliminated at the same time by eliminating the Pythagorean comma. Therefore, one can kill two birds with one stone by eliminating the Pythagorean comma. However, this is only one side of the story.


The other side of the story must also be considered. Anthony's diagram provides a clear insight into the theoretical compromises that need to be made:

If the Syntonic comma (21.51 cents) is equally spread across C-G, G-D, D-A and A-E, C-E becomes a Pure Major Third. However, this is achieved by reducing F#-C# by a Schisma (1.95 cents). Therefore, this method produces 7 Pure Fifths.

On the other hand, if the Pythagorean comma is equally spread across C-G, G-D, D-A and A-E, F#-C# becomes a Pure Fifth. Therefore, this method produces 8 Pure Fifths. However, this is achieved by reducing C-E by a Schisma (1.95 cents). I can illustrate this by comparing the value of the Tempered C-E in cents with the value of the Pure C-E in cents.


Kirnberger III Cents

C	0.00	cents
C#	90.22	cents
D	192.18	cents
D#	294.13	cents
E	384.36	cents
F	498.04	cents
F#	588.27	cents
G	696.09	cents
G#	792.18	cents
A	888.27	cents
A#	996.09	cents
B	1086.31	cents
C	1200.00	cents

Offsets in Cents from Pure Octave Equal Temperament

C	0.00	cents
C#	−9.78	cents
D	−7.82	cents
D#	−5.87	cents
E	−15.64	cents
F	−1.96	cents
F#	−11.73	cents
G	−3.91	cents
G#	−7.82	cents
A	−11.73	cents
A#	−3.91	cents
B	−13.69	cents
C	0.00	cents

Difference between Tempered C-E and Pure C-E in cents = 384.36 cents − 1200 × log₂(5/4) cents = 384.36 cents − 386.31 cents = −1.95 cents = Schisma.

The Tempered C-E will be a Pure Major Third reduced by a Schisma if Kirnberger III has been constructed by equally spreading the Pythagorean comma across C-G, G-D, D-A and A-E.

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

Original Message:
Sent: 09-12-2019 15:49
From: Richard Adkins
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
 Thanks for clarifying .... so where do you place your "schizma"? And what are the cents there...? The third question is, should
one be setting Kirnberger III from C3 or from A3 (or even A2)?...that's pretty low. Doesn't that change the perfect thirds and schizma?
I guess I don't understand the fine points of using historical temperaments. I usually set them using the Cs, rather than the As;
I think as long as the perfect intervals are the same, it shouldn't matter where you begin. It is confusing though is someone
is talking about starting from As and someone else talks about starting from Cs, at least it is to me.

Richard

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA

Original Message:
Sent: 09-12-2019 14:37
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Anthony,

I prefer to divide the Pythagorean comma because It is easier to deal with one comma (Pythagorean comma) rather than two commas (Syntonic comma and Schisma).

Syntonic comma (21.51 cents) + Schisma (1.95 cents) = Pythagorean comma (23.46 cents).

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

Original Message:
Sent: 09-12-2019 11:59
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Roshan, Those are the right frequencies if you calculate the Kirnberger III using the Pythagorean Comma. I calculate it using the Syntonic Comma, placing the Schisma between C# and F#.

For that case, the cent offset values would be:

A3	0
A#3	6.35
B3	-1.47
C4	10.26
C#4	0.49
D4	3.42
D#4	4.4
E4	-3.42
F4	8.31
F#4	0.49
G4	6.84
G#4	2.44
A4	0

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

Original Message:
Sent: 09-12-2019 11:18
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Kirnberger III Circle of Fifths and Fourths

Source: https://www.hpschd.nu/g/tech/tmp/kirnberger.gif

Kirnberger III Frequencies (C3 to C6)

C3	131.70	Hz
C#3	138.75	Hz
D3	147.16	Hz
D#3	156.09	Hz
E3	164.44	Hz
F3	175.60	Hz
F#3	185.00	Hz
G3	196.89	Hz
G#3	208.12	Hz
A3	220.00	Hz
A#3	234.14	Hz
B3	246.66	Hz
C4	263.40	Hz
C#4	277.50	Hz
D4	294.33	Hz
D#4	312.18	Hz
E4	328.88	Hz
F4	351.21	Hz
F#4	369.99	Hz
G4	393.77	Hz
G#4	416.24	Hz
A4	440.00	Hz
A#4	468.27	Hz
B4	493.33	Hz
C5	526.81	Hz
C#5	554.99	Hz
D5	588.66	Hz
D#5	624.37	Hz
E5	657.77	Hz
F5	702.41	Hz
F#5	739.99	Hz
G5	787.54	Hz
G#5	832.49	Hz
A5	880.00	Hz
A#5	936.55	Hz
B5	986.65	Hz
C6	1053.62	Hz

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

------------------------------
Roshan Kakiya
------------------------------​

The resultant schisma for CE seems to go against what David Pinnegar advicates a "pure" CE interval
to get his "better" effect than ET, no?

R

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA
------------------------------

Original Message:
Sent: 09-12-2019 18:51
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Richard,

"Thanks for clarifying .... so where do you place your 'schizma'?"


Pythagorean comma (23.46 cents) = Syntonic comma (21.51 cents) + Schisma (1.95 cents).

The Syntonic comma and the Schisma are part of the Pythagorean comma. The Syntonic comma and the Schisma can be eliminated at the same time by eliminating the Pythagorean comma. Therefore, one can kill two birds with one stone by eliminating the Pythagorean comma. However, this is only one side of the story.


The other side of the story must also be considered. Anthony's diagram provides a clear insight into the theoretical compromises that need to be made:

If the Syntonic comma (21.51 cents) is equally spread across C-G, G-D, D-A and A-E, C-E becomes a Pure Major Third. However, this is achieved by reducing F#-C# by a Schisma (1.95 cents). Therefore, this method produces 7 Pure Fifths.

On the other hand, if the Pythagorean comma is equally spread across C-G, G-D, D-A and A-E, F#-C# becomes a Pure Fifth. Therefore, this method produces 8 Pure Fifths. However, this is achieved by reducing C-E by a Schisma (1.95 cents). I can illustrate this by comparing the value of the Tempered C-E in cents with the value of the Pure C-E in cents.


Kirnberger III Cents

C	0.00	cents
C#	90.22	cents
D	192.18	cents
D#	294.13	cents
E	384.36	cents
F	498.04	cents
F#	588.27	cents
G	696.09	cents
G#	792.18	cents
A	888.27	cents
A#	996.09	cents
B	1086.31	cents
C	1200.00	cents

Offsets in Cents from Pure Octave Equal Temperament

C	0.00	cents
C#	−9.78	cents
D	−7.82	cents
D#	−5.87	cents
E	−15.64	cents
F	−1.96	cents
F#	−11.73	cents
G	−3.91	cents
G#	−7.82	cents
A	−11.73	cents
A#	−3.91	cents
B	−13.69	cents
C	0.00	cents

Difference between Tempered C-E and Pure C-E in cents = 384.36 cents − 1200 × log₂(5/4) cents = 384.36 cents − 386.31 cents = −1.95 cents = Schisma.

The Tempered C-E will be a Pure Major Third reduced by a Schisma if Kirnberger III has been constructed by equally spreading the Pythagorean comma across C-G, G-D, D-A and A-E.

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

Original Message:
Sent: 09-12-2019 15:49
From: Richard Adkins
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
 Thanks for clarifying .... so where do you place your "schizma"? And what are the cents there...? The third question is, should
one be setting Kirnberger III from C3 or from A3 (or even A2)?...that's pretty low. Doesn't that change the perfect thirds and schizma?
I guess I don't understand the fine points of using historical temperaments. I usually set them using the Cs, rather than the As;
I think as long as the perfect intervals are the same, it shouldn't matter where you begin. It is confusing though is someone
is talking about starting from As and someone else talks about starting from Cs, at least it is to me.

Richard

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA

Original Message:
Sent: 09-12-2019 14:37
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Anthony,

I prefer to divide the Pythagorean comma because It is easier to deal with one comma (Pythagorean comma) rather than two commas (Syntonic comma and Schisma).

Syntonic comma (21.51 cents) + Schisma (1.95 cents) = Pythagorean comma (23.46 cents).

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

Original Message:
Sent: 09-12-2019 11:59
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Roshan, Those are the right frequencies if you calculate the Kirnberger III using the Pythagorean Comma. I calculate it using the Syntonic Comma, placing the Schisma between C# and F#.

For that case, the cent offset values would be:

A3	0
A#3	6.35
B3	-1.47
C4	10.26
C#4	0.49
D4	3.42
D#4	4.4
E4	-3.42
F4	8.31
F#4	0.49
G4	6.84
G#4	2.44
A4	0

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

Original Message:
Sent: 09-12-2019 11:18
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Kirnberger III Circle of Fifths and Fourths

Source: https://www.hpschd.nu/g/tech/tmp/kirnberger.gif

Kirnberger III Frequencies (C3 to C6)

C3	131.70	Hz
C#3	138.75	Hz
D3	147.16	Hz
D#3	156.09	Hz
E3	164.44	Hz
F3	175.60	Hz
F#3	185.00	Hz
G3	196.89	Hz
G#3	208.12	Hz
A3	220.00	Hz
A#3	234.14	Hz
B3	246.66	Hz
C4	263.40	Hz
C#4	277.50	Hz
D4	294.33	Hz
D#4	312.18	Hz
E4	328.88	Hz
F4	351.21	Hz
F#4	369.99	Hz
G4	393.77	Hz
G#4	416.24	Hz
A4	440.00	Hz
A#4	468.27	Hz
B4	493.33	Hz
C5	526.81	Hz
C#5	554.99	Hz
D5	588.66	Hz
D#5	624.37	Hz
E5	657.77	Hz
F5	702.41	Hz
F#5	739.99	Hz
G5	787.54	Hz
G#5	832.49	Hz
A5	880.00	Hz
A#5	936.55	Hz
B5	986.65	Hz
C6	1053.62	Hz

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

------------------------------
Roshan Kakiya
------------------------------​

A tempered C-E in Kirnberger III (1/4 Pythagorean comma) that has a value of 384.36 cents is significantly "better" than a tempered C-E in Pure Octave Equal Temperament that has a value of 400.00 cents.


Pure C-E = 386.31 cents.

Tempered C-E in Kirnberger III (1/4 Pythagorean comma) = 384.36 cents.

Deviation = 384.36 cents − 386.31 cents = −1.95 cents.


Pure C-E = 386.31 cents.

Tempered C-E in Pure Octave Equal Temperament = 400.00 cents.

Deviation = 400.00 cents − 386.31 cents = +13.69 cents.

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

Original Message:
Sent: 09-12-2019 19:30
From: Richard Adkins
Subject: Fundamental Frequencies vs Coincidental Partials

The resultant schisma for CE seems to go against what David Pinnegar advicates a "pure" CE interval
to get his "better" effect than ET, no?

R

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA

Original Message:
Sent: 09-12-2019 18:51
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Richard,

"Thanks for clarifying .... so where do you place your 'schizma'?"


Pythagorean comma (23.46 cents) = Syntonic comma (21.51 cents) + Schisma (1.95 cents).

The Syntonic comma and the Schisma are part of the Pythagorean comma. The Syntonic comma and the Schisma can be eliminated at the same time by eliminating the Pythagorean comma. Therefore, one can kill two birds with one stone by eliminating the Pythagorean comma. However, this is only one side of the story.


The other side of the story must also be considered. Anthony's diagram provides a clear insight into the theoretical compromises that need to be made:

If the Syntonic comma (21.51 cents) is equally spread across C-G, G-D, D-A and A-E, C-E becomes a Pure Major Third. However, this is achieved by reducing F#-C# by a Schisma (1.95 cents). Therefore, this method produces 7 Pure Fifths.

On the other hand, if the Pythagorean comma is equally spread across C-G, G-D, D-A and A-E, F#-C# becomes a Pure Fifth. Therefore, this method produces 8 Pure Fifths. However, this is achieved by reducing C-E by a Schisma (1.95 cents). I can illustrate this by comparing the value of the Tempered C-E in cents with the value of the Pure C-E in cents.


Kirnberger III Cents

C	0.00	cents
C#	90.22	cents
D	192.18	cents
D#	294.13	cents
E	384.36	cents
F	498.04	cents
F#	588.27	cents
G	696.09	cents
G#	792.18	cents
A	888.27	cents
A#	996.09	cents
B	1086.31	cents
C	1200.00	cents

Offsets in Cents from Pure Octave Equal Temperament

C	0.00	cents
C#	−9.78	cents
D	−7.82	cents
D#	−5.87	cents
E	−15.64	cents
F	−1.96	cents
F#	−11.73	cents
G	−3.91	cents
G#	−7.82	cents
A	−11.73	cents
A#	−3.91	cents
B	−13.69	cents
C	0.00	cents

Difference between Tempered C-E and Pure C-E in cents = 384.36 cents − 1200 × log₂(5/4) cents = 384.36 cents − 386.31 cents = −1.95 cents = Schisma.

The Tempered C-E will be a Pure Major Third reduced by a Schisma if Kirnberger III has been constructed by equally spreading the Pythagorean comma across C-G, G-D, D-A and A-E.

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

Original Message:
Sent: 09-12-2019 15:49
From: Richard Adkins
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
 Thanks for clarifying .... so where do you place your "schizma"? And what are the cents there...? The third question is, should
one be setting Kirnberger III from C3 or from A3 (or even A2)?...that's pretty low. Doesn't that change the perfect thirds and schizma?
I guess I don't understand the fine points of using historical temperaments. I usually set them using the Cs, rather than the As;
I think as long as the perfect intervals are the same, it shouldn't matter where you begin. It is confusing though is someone
is talking about starting from As and someone else talks about starting from Cs, at least it is to me.

Richard

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA

Original Message:
Sent: 09-12-2019 14:37
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Anthony,

I prefer to divide the Pythagorean comma because It is easier to deal with one comma (Pythagorean comma) rather than two commas (Syntonic comma and Schisma).

Syntonic comma (21.51 cents) + Schisma (1.95 cents) = Pythagorean comma (23.46 cents).

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

Original Message:
Sent: 09-12-2019 11:59
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Roshan, Those are the right frequencies if you calculate the Kirnberger III using the Pythagorean Comma. I calculate it using the Syntonic Comma, placing the Schisma between C# and F#.

For that case, the cent offset values would be:

A3	0
A#3	6.35
B3	-1.47
C4	10.26
C#4	0.49
D4	3.42
D#4	4.4
E4	-3.42
F4	8.31
F#4	0.49
G4	6.84
G#4	2.44
A4	0

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

Original Message:
Sent: 09-12-2019 11:18
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Kirnberger III Circle of Fifths and Fourths

Source: https://www.hpschd.nu/g/tech/tmp/kirnberger.gif

Kirnberger III Frequencies (C3 to C6)

C3	131.70	Hz
C#3	138.75	Hz
D3	147.16	Hz
D#3	156.09	Hz
E3	164.44	Hz
F3	175.60	Hz
F#3	185.00	Hz
G3	196.89	Hz
G#3	208.12	Hz
A3	220.00	Hz
A#3	234.14	Hz
B3	246.66	Hz
C4	263.40	Hz
C#4	277.50	Hz
D4	294.33	Hz
D#4	312.18	Hz
E4	328.88	Hz
F4	351.21	Hz
F#4	369.99	Hz
G4	393.77	Hz
G#4	416.24	Hz
A4	440.00	Hz
A#4	468.27	Hz
B4	493.33	Hz
C5	526.81	Hz
C#5	554.99	Hz
D5	588.66	Hz
D#5	624.37	Hz
E5	657.77	Hz
F5	702.41	Hz
F#5	739.99	Hz
G5	787.54	Hz
G#5	832.49	Hz
A5	880.00	Hz
A#5	936.55	Hz
B5	986.65	Hz
C6	1053.62	Hz

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

Best wishes

David P

--

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

Original Message------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

------------------------------
Roshan Kakiya
------------------------------​

To be clear I'm meaning that the two ways of getting a Kirnberger III ....Anthony Willey's way has a perfectd CE third... and Roshan's way....has
a schisma CE third...which is 1.95cents....so there'll be a beat of almost 2 per second...very audible and slightly "wangy" sounding....

That's what I was getting at.... yes, either one would be better than the tempered third, but only if you want to do a Kirnberger temperament
in order to get David Pinnegar's effect.

But Anthony's comes out pure and Roshan's doesn't. Roshan give you one more perfect 5th...but that to me is not as
critical as still thirds.... 

Maybe I just don't understand the point to do a Kirnberber III if you're going to compromise a third like that for the sake of 5th.

I'll have to try both ways, I guess.

Thanks Roshan!



------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA
------------------------------

Original Message:
Sent: 09-12-2019 19:30
From: Richard Adkins
Subject: Fundamental Frequencies vs Coincidental Partials

The resultant schisma for CE seems to go against what David Pinnegar advicates a "pure" CE interval
to get his "better" effect than ET, no?

R

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA

Original Message:
Sent: 09-12-2019 18:51
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Richard,

"Thanks for clarifying .... so where do you place your 'schizma'?"


Pythagorean comma (23.46 cents) = Syntonic comma (21.51 cents) + Schisma (1.95 cents).

The Syntonic comma and the Schisma are part of the Pythagorean comma. The Syntonic comma and the Schisma can be eliminated at the same time by eliminating the Pythagorean comma. Therefore, one can kill two birds with one stone by eliminating the Pythagorean comma. However, this is only one side of the story.


The other side of the story must also be considered. Anthony's diagram provides a clear insight into the theoretical compromises that need to be made:

If the Syntonic comma (21.51 cents) is equally spread across C-G, G-D, D-A and A-E, C-E becomes a Pure Major Third. However, this is achieved by reducing F#-C# by a Schisma (1.95 cents). Therefore, this method produces 7 Pure Fifths.

On the other hand, if the Pythagorean comma is equally spread across C-G, G-D, D-A and A-E, F#-C# becomes a Pure Fifth. Therefore, this method produces 8 Pure Fifths. However, this is achieved by reducing C-E by a Schisma (1.95 cents). I can illustrate this by comparing the value of the Tempered C-E in cents with the value of the Pure C-E in cents.


Kirnberger III Cents

C	0.00	cents
C#	90.22	cents
D	192.18	cents
D#	294.13	cents
E	384.36	cents
F	498.04	cents
F#	588.27	cents
G	696.09	cents
G#	792.18	cents
A	888.27	cents
A#	996.09	cents
B	1086.31	cents
C	1200.00	cents

Offsets in Cents from Pure Octave Equal Temperament

C	0.00	cents
C#	−9.78	cents
D	−7.82	cents
D#	−5.87	cents
E	−15.64	cents
F	−1.96	cents
F#	−11.73	cents
G	−3.91	cents
G#	−7.82	cents
A	−11.73	cents
A#	−3.91	cents
B	−13.69	cents
C	0.00	cents

Difference between Tempered C-E and Pure C-E in cents = 384.36 cents − 1200 × log₂(5/4) cents = 384.36 cents − 386.31 cents = −1.95 cents = Schisma.

The Tempered C-E will be a Pure Major Third reduced by a Schisma if Kirnberger III has been constructed by equally spreading the Pythagorean comma across C-G, G-D, D-A and A-E.

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

Original Message:
Sent: 09-12-2019 15:49
From: Richard Adkins
Subject: Fundamental Frequencies vs Coincidental Partials

Roshan,
 Thanks for clarifying .... so where do you place your "schizma"? And what are the cents there...? The third question is, should
one be setting Kirnberger III from C3 or from A3 (or even A2)?...that's pretty low. Doesn't that change the perfect thirds and schizma?
I guess I don't understand the fine points of using historical temperaments. I usually set them using the Cs, rather than the As;
I think as long as the perfect intervals are the same, it shouldn't matter where you begin. It is confusing though is someone
is talking about starting from As and someone else talks about starting from Cs, at least it is to me.

Richard

------------------------------
Richard Adkins
Piano Technician
Coe College
Cedar Rapids, IA

Original Message:
Sent: 09-12-2019 14:37
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Anthony,

I prefer to divide the Pythagorean comma because It is easier to deal with one comma (Pythagorean comma) rather than two commas (Syntonic comma and Schisma).

Syntonic comma (21.51 cents) + Schisma (1.95 cents) = Pythagorean comma (23.46 cents).

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

Original Message:
Sent: 09-12-2019 11:59
From: Anthony Willey
Subject: Fundamental Frequencies vs Coincidental Partials

@Roshan, Those are the right frequencies if you calculate the Kirnberger III using the Pythagorean Comma. I calculate it using the Syntonic Comma, placing the Schisma between C# and F#.

For that case, the cent offset values would be:

A3	0
A#3	6.35
B3	-1.47
C4	10.26
C#4	0.49
D4	3.42
D#4	4.4
E4	-3.42
F4	8.31
F#4	0.49
G4	6.84
G#4	2.44
A4	0

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

Original Message:
Sent: 09-12-2019 11:18
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

Kirnberger III Circle of Fifths and Fourths

Source: https://www.hpschd.nu/g/tech/tmp/kirnberger.gif

Kirnberger III Frequencies (C3 to C6)

C3	131.70	Hz
C#3	138.75	Hz
D3	147.16	Hz
D#3	156.09	Hz
E3	164.44	Hz
F3	175.60	Hz
F#3	185.00	Hz
G3	196.89	Hz
G#3	208.12	Hz
A3	220.00	Hz
A#3	234.14	Hz
B3	246.66	Hz
C4	263.40	Hz
C#4	277.50	Hz
D4	294.33	Hz
D#4	312.18	Hz
E4	328.88	Hz
F4	351.21	Hz
F#4	369.99	Hz
G4	393.77	Hz
G#4	416.24	Hz
A4	440.00	Hz
A#4	468.27	Hz
B4	493.33	Hz
C5	526.81	Hz
C#5	554.99	Hz
D5	588.66	Hz
D#5	624.37	Hz
E5	657.77	Hz
F5	702.41	Hz
F#5	739.99	Hz
G5	787.54	Hz
G#5	832.49	Hz
A5	880.00	Hz
A#5	936.55	Hz
B5	986.65	Hz
C6	1053.62	Hz

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

Original Message:
Sent: 09-12-2019 06:30
From: Roshan Kakiya
Subject: Fundamental Frequencies vs Coincidental Partials

David,

Have you tried using a device/machine to tune each note within the central octaves in accordance with its theoretical frequency?

For example, according to your data, the theoretical frequency of C4 is 262.8 Hz. You could set 262.8 Hz as the tuning frequency on a device/machine and tune C4 until its frequency is 262.8 Hz.

This procedure can be repeated to manually tune each note within the central octaves.

A device/machine that does not account for inharmonicity would need to be used so that the intervals within the central octaves are not stretched.

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

Original Message:
Sent: 09-12-2019 06:15
From: David Pinnegar
Subject: Fundamental Frequencies vs Coincidental Partials

The frequencies for Kellner are from Tenor C up three octaves:

131.4
138.5
147.0
155.8
164.6
175.2
184.6
196.6
207.7
220.0
233.6
246.8
262.8
276.9
294.1
311.6
329.2
350.4
369.2
393.2
415.4
440.0
467.2
493.6
525.7
553.9
588.2
623.2
658.3
700.9
738.5
786.5
830.8
880.0
934.5
987.2
1051.3
1107.8
1176.4
1246.3
1316.6
1401.8
1477.0
1573.0
1661.7
1760.0
1869.0
1974.4
2102.7

Calculating the frequencies is not the difficult part.

It's predicting how they will line up with harmonics of the lower strings when in practice inharmonicity differs between instruments. This is a matter that varies in the field and lends itself to analog rather than digital solution. The analogue solution to the problem is the task of the piano tuner. Why we are obsessed by digital solutions I don't know. But digital solutions such as Anthony's based on the inharmonicities measured in the field are particularly helpful.

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------

Anthony,

The method that I have described can be used for any tuning system such as equal temperament, unequal temperament and just/rational intonation. I know that unequal temperaments can have a mixture of intervals such as perfect fifths and tempered fifths.

I can use the Circle of Fifths and Fourths to construct the scale of an unequal temperament. I can also incorporate the system of cents into the frequency ratios so that cents can be converted into frequencies.

I can calculate the frequencies of Kirnberger III and Kellner if you can provide the notes of the central octaves.

------------------------------
Roshan Kakiya
------------------------------​