Anthony,

There are always theoretical compromises that need to be made. The approach that I have suggested is one of the ways in which the Schisma can be distributed. The Schisma can also be distributed in other ways.

The following is another way of distributing the Schisma:

The Syntonic comma (21.51 cents) can be equally spread across C-G, G-D, D-A and A-E. This would cause C-E to be a Pure Major Third.

1/4 Syntonic comma = 21.506 cents / 4 = 5.377 cents.

Pure Fifth reduced by 1/4 Syntonic comma = 701.955 cents

− 5.377 cents = 696.578 cents = 696.58 cents.

The Schisma can be equally spread across the 8 remaining fifths.

1/8 Schisma = 1.954 cents / 8 = 0.244 cents.

Pure Fifth reduced by 1/8 Schisma = 701.955 cents

− 0.244 cents = 701.711 cents = 701.71 cents.

Verification:

7 Octaves = 1200 cents × 7 = 8400 cents.

12 Tempered Fifths = 696.58 cents × 4 + 701.71 cents × 8 = 8400 cents.

**Modified Kirnberger III (1/4 Syntonic comma and 1/8 Schisma):****https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=72771916-275d-4dc7-bcca-d5d481656d7c&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf**------------------------------

Roshan Kakiya

------------------------------

Original Message:

Sent: 09-12-2019 23:56

From: Anthony Willey

Subject: Fundamental Frequencies vs Coincidental Partials

I'll be honest, I get a little muddled when it comes to the thirds in unequal temperaments. I usually just think of it in terms of spreading the comma between the 5ths.

- Equal temperament gives 12 equal fifths, all 1.96 cents narrow.
- Kirnberger calculated with the Pythagorean comma gives 8 pure fifths and 4 that are 5.87 narrow (yuck)
- Kirnberger calculated with the Syntonic comma gives 7 pure fifths, 4 that are 5.38 cents narrow (slightly less yuck), and one that is 1.95 cents narrow (same as an equal-tempered fifth).

I don't have an opinion on which way is better. I used the Syntonic comma because that's the way I understood that it was done historically, and why use a historical temperament if you're not going to try to be historically accurate? I'm fine with people arguing that one way sounds better than another (though I doubt 99% of people could recognize the difference listening to played music). But I would push back on arguments that one is better than the other because it is easier to express mathematically in terms of interval frequency ratios.

------------------------------

Anthony Willey, RPT

http://willeypianotuning.com

http://pianometer.com

Original Message:

Sent: 09-12-2019 23:19

From: Richard Adkins

Subject: Fundamental Frequencies vs Coincidental Partials

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 × log**_{2}(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#.

------------------------------

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**

**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

------------------------------