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  • 1.  Unequal Beating Temperament System - Kirnberger III and Kellner

    Posted 01-02-2026 20:00
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    Unequal Beating Temperament System - Kirnberger III and Kellner

     

     

    Kirnberger III

     

    Equation

    ((((((((((((((((((((((((440 × 2 + (Δ × 3 × ((1200 × log2(81 / 80)) / (1200 × log2(531441 / 524288))))) / 3) × 4 + (Δ × 3 × ((1200 × log2(81 / 80)) / (1200 × log2(531441 / 524288))))) / 3) × 2 + (Δ × 3 × ((1200 × log2(81 / 80)) / (1200 × log2(531441 / 524288))))) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 1 × (((1200 × log2(531441 / 524288)) - (1200 × log2(81 / 80))) / ((1200 × log2(531441 / 524288)) / 12)))) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 3 × ((1200 × log2(81 / 80)) / (1200 × log2(531441 / 524288))))) / 3) = 220

    or

    ((((((((((((((((((((((((440 × 2 + (Δ × 3 × (21.506289597 / 23.460010385))) / 3) × 4 + (Δ × 3 × (21.506289597 / 23.460010385))) / 3) × 2 + (Δ × 3 × (21.506289597 / 23.460010385))) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 1 × ((23.460010385 − 21.506289597) / (23.460010385 / 12)))) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 3 × (21.506289597 / 23.460010385))) / 3) = 220

    Δ = 0.966360326

     

    Pythagorean Comma = 1200 × log2(531441 / 524288) = 23.460010385 cents

    Syntonic Comma = 1200 × log2(81 / 80) = 21.506289597 cents

    Grad = Pythagorean Comma / 12 = 1200 × log2(531441 / 524288) / 12 = 23.460010385 cents / 12 = 1.955000865 cents

    Schisma = Pythagorean Comma − Syntonic Comma = 1200 × log2(531441 / 524288) − 1200 × log2(81 / 80) = 23.460010385 cents − 21.506289597 cents = 1.953720788 cents

     

    Temperament Octave

    Notes Frequencies (Hz) Kirnberger III Cents 12-ET Cents Offsets (Cents)
    A3 220.00 0.00 0.00   0.00
    A#3 233.78 105.19 100.00 +5.19
    B3 246.50 196.92 200.00 −3.08
    C4 263.00 309.10 300.00 +9.10
    C#4 277.07 399.33 400.00 −0.67
    D4 294.22 503.27 500.00 +3.27
    D#4 311.71 603.24 600.00 +3.24
    E4 328.67 694.97 700.00 −5.03
    F4 350.67 807.15 800.00 +7.15
    F#4 369.76 898.88 900.00 −1.12
    G4 393.18 1005.22 1000.00 +5.22
    G#4 415.61 1101.28 1100.00 +1.28
    A4 440.00 1200.00 1200.00   0.00

    Intervals Notes Sizes (Cents) Beat Rates (Hz)
    Narrow 5th D4-A4 696.73 2.66
    Wide 4th D4-G4 501.95 2.66
    Narrow 5th C4-G4 696.11 2.66
    Pure 4th C4-F4 498.04 0.00
    Pure 5th A#3-F4 701.96 0.00
    Pure 4th A#3-D#4 498.04 0.00
    Pure 4th D#4-G#4 498.04 0.00
    Pure 5th C#4-G#4 701.96 0.00
    Wide 4th C#4-F#4 499.55 0.97
    Pure 5th B3-F#4 701.96 0.00
    Pure 4th B3-E4 498.04 0.00
    Narrow 5th A3-E4 694.97 2.66

     

     

    Kellner

     

    Equation

    ((((((((((((((((((((((((440 × 2 + (Δ × (2 + (2 / 5)))) / 3) × 4 + (Δ × (2 + (2 / 5)))) / 3) × 2 + (Δ × (2 + (2 / 5)))) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × 0)) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × (2 + (2 / 5)))) / 3) × 4 + (Δ × 0)) / 3) × 2 + (Δ × (2 + (2 / 5)))) / 3) = 220

    Δ = 1967075 / 2134329

     

    Temperament Octave

    Notes Frequencies (Hz) Kellner Cents 12-ET Cents Offsets (Cents)
    A3 220.00 0.00 0.00   0.00
    A#3 233.44 102.69 100.00 +2.69
    B3 246.67 198.10 200.00 −1.90
    C4 262.62 306.60 300.00 +6.60
    C#4 276.67 396.83 400.00 −3.17
    D4 294.07 502.39 500.00 +2.39
    D#4 311.26 600.74 600.00 +0.74
    E4 328.89 696.14 700.00 −3.86
    F4 350.17 804.65 800.00 +4.65
    F#4 368.90 894.87 900.00 −5.13
    G4 392.83 1003.69 1000.00 +3.69
    G#4 415.01 1098.78 1100.00 −1.22
    A4 440.00 1200.00 1200.00   0.00

    Intervals Notes Sizes (Cents) Beat Rates (Hz)
    Narrow 5th D4-A4 697.61 2.21
    Wide 4th D4-G4 501.30 2.21
    Narrow 5th C4-G4 697.09 2.21
    Pure 4th C4-F4 498.04 0.00
    Pure 5th A#3-F4 701.96 0.00
    Pure 4th A#3-D#4 498.04 0.00
    Pure 4th D#4-G#4 498.04 0.00
    Pure 5th C#4-G#4 701.96 0.00
    Pure 4th C#4-F#4 498.04 0.00
    Narrow 5th B3-F#4 696.77 2.21
    Pure 4th B3-E4 498.04 0.00
    Narrow 5th A3-E4 696.14 2.21


     

     

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



  • 2.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Registered Piano Technician
    Posted 01-03-2026 11:39

    Hi Roshan,

    Interesting stuff! Am I correct to understand that what makes these "equal beating" (EB) is that the tempered fifths beat equally? I am not seeing any of the thirds beat equally, though 7 pure fifths allow major/minor thirds to be synchronously (which is the case using regular offsets for these temperaments). Do you feel there is any practical reason to have fifths equally beating when functionally speaking, the fifths would rarely be played together in real music? The tempered fifths in your alternative temperaments don't seem to have any benefit to the major/minor third beat synchrony, which I believe is one of the practical benefits of unequal systems.

    Your EB Kellner seems usable, although the A-E fifth at 5.8 cents narrow pushes my own limit (I have found a 1/4 syntonic comma at 5.4 cents narrow is about the top of my usable limit, at least when used for modern music). For this reason, your EB Kirnberger III I would not use for most circumstances, with a 7 cent narrow A-E fifth, unless for earlier music. (I tune Kirnberger III regularly.)

    Question: Is there a practical reason you placed the schisma in your EB Kirnberger III between F#-C# instead of B-F# where it is traditionally placed? F#-C# schisma placement is Kirnberger II (1771 version), not III to my knowledge. If memory serves, the 1776 Kirnberger II combines the schisma into the two tempered fifths (D-A, A-E), making these 1/2 Pythagorean commas and slightly inverting the C-E and G-B "pure" major thirds. These are some gnarly fifths for sure!



    ------------------------------
    Tim Foster RPT
    New Oxford PA
    (470) 231-6074
    ------------------------------



  • 3.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Posted 01-03-2026 14:33

    Hi Tim,

     

    It's great to hear from you! You have made some very interesting points, and I shall go through them paragraph by paragraph.

      

    Paragraph 1: Interesting stuff! Am I correct to understand that what makes these "equal beating" (EB) is that the tempered fifths beat equally? I am not seeing any of the thirds beat equally, though 7 pure fifths allow major/minor thirds to be synchronously (which is the case using regular offsets for these temperaments). Do you feel there is any practical reason to have fifths equally beating when functionally speaking, the fifths would rarely be played together in real music? The tempered fifths in your alternative temperaments don't seem to have any benefit to the major/minor third beat synchrony, which I believe is one of the practical benefits of unequal systems. 

    Response 1: Yes. I am using a system that I have developed called the Equal and Unequal Beating Temperament System, which is based on what I have learned by studying Alfredo Capurso's Circular Harmonic System and Bill Bremmer's Equal Beating Victorian Temperament. As you can see, the Circle of Equally and Unequally Beating 5ths and 4ths acts as the blueprint for each temperament. It standardises the Equal and Unequal Beating Temperament System. The Equal Beating Victorian Temperament clearly showcases how the Equal and Unequal Beating Temperament System can be used flexibly to make specific intervals beat equally in a manner that creates a coherent whole in the form of a fully usable temperament without any wolf intervals. You could say that I have found the middle ground between piano tuning and music making, where the emphasis is on standardising beats-based temperaments to achieve musical goals such as narrower 3rds in the "home" keys and wider 3rds in the "remote" keys.

     

    Paragraph 2: Your EB Kellner seems usable, although the A-E fifth at 5.8 cents narrow pushes my own limit (I have found a 1/4 syntonic comma at 5.4 cents narrow is about the top of my usable limit, at least when used for modern music). For this reason, your EB Kirnberger III I would not use for most circumstances, with a 7 cent narrow A-E fifth, unless for earlier music. (I tune Kirnberger III regularly.)

    Response 2: When one is focusing on beat rates, cents lose their meaning. The System of Cents (Cent = 21 / 1200)  has been baked into the Equal and Unequal Beating Temperament System. The emphasis is on the beat rates and beat ratios, not the cents values. The primary unit of measure is Hz. The secondary unit of measure is cents. I have decided to construct Kellner by distributing the Pythagorean Comma across D4-A4 (5th), D4-G4 (4th), C4-G4 (5th), B3-F#4 (5th), and A3-E4 (5th) within the Temperament Octave A3-A4 to ensure that the Circle of 5ths and 4ths closes. Every tempered 5th and 4th beats equally in Kellner within the Octave A3-A4 for tuning purposes. In Kirnberger III, I have distributed the Syntonic Comma across D4-A4 (5th), D4-G4 (4th), C4-G4 (5th), and A3-E4 (5th) in order to make them beat equally within the Octave A3-A4 and get as close as possible to making the C4-E4 Major 3rd beatless. I have tried to remain faithful to the original intent of each temperament rather than its design alone.

     

    Paragraph 3: Is there a practical reason you placed the schisma in your EB Kirnberger III between F#-C# instead of B-F# where it is traditionally placed? F#-C# schisma placement is Kirnberger II (1771 version), not III to my knowledge. If memory serves, the 1776 Kirnberger II combines the schisma into the two tempered fifths (D-A, A-E), making these 1/2 Pythagorean commas and slightly inverting the C-E and G-B "pure" major thirds. These are some gnarly fifths for sure!

    Response 3: As I understand it, the Schisma was originally placed on the 5th F#-C# in Kirnberger III. I have placed it on the 4th C#4-F#4 in my standardised sequence of 5ths and 4ths to be faithful to its original design. It seems to me that this is a strategic move because it simultaneously minimises the harshness of multiple Major 3rds as you go round the Circle of 5ths and 4ths. From my perspective, the Major 3rds get harsher if there are more consecutive beatless 5ths and 4ths in that Circle. The aim is to disrupt that chain on purpose by introducing a comma in the specific place on that Circle where many Major 3rds are narrowed in order to reduce their harshness. This strategic move is evident in various temperaments, e.g., Kellner, Kirnberger III, and Werckmeister III.

     

    I hope that I have answered your questions sufficiently! I am always open to challenging my own understanding and getting to know other people's perspectives.

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

    Original Message


  • 4.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Posted 01-03-2026 14:55
    Roshan - thanks for putting unequal temperament mathematics into the arena. 

    However, as a practical tuner and a mathematical Philistine, I'm not at all sure that the musical result of any mathematical tweaking here is to be actually experienced in practice.


    		 Roshan Standard Difference
    A3 220 220 0
    A#3 233.78 233.94 0.2
    B3 246.5 246.73 0.2
    C4 263 263.18 0.2
    C#4 277.07 277.26 0.2
    D4 294.22 294.25 0.0
    D#4 311.71 311.92 0.2
    E4 328.67 328.98 0.3
    F4 350.67 350.91 0.2
    F#4 369.76 370.10 0.3
    G4 393.18 393.55 0.4
    G#4 415.61 415.89 0.3
    A4 440 440.00 0.0

    Most notes differ by no more than one beat in five seconds, one beat in 3 seconds with only intervals involving G differing by one beat in more than 2 seconds.

    Because the intensity of piano sound dies away, and because of phase changes and different mode behaviours between vertical and horizontal vibration of strings, and interactions when more than one note is played, I don't think that variations of one beat in such timescales will make any difference to the experience of the music at all. Even less so when most notes in music last for much shorter times than two seconds.

    One can be in danger of being too scientific. There are many factors in the piano which really rather make accuracy a false god. Whilst as technicians we focus on absolute unisons, for instance, there are many who consider such not to be lifelike and who prefer slight detuning. This phenomenon is also inherent in the comparison between digital and analogue sound in which, from my magnetic tape recording experience I hear what is technically known as "flutter" as a recording defect but which is heard by the lay listener as a slight vibrato, even if subliminal or imperceptible. A very good tape recorder will have speed stability of perhaps 0.05% variation whilst lesser machines might be 0.1% and even higher can be considered to be acceptable by engineers and inaudible to most. A 0.4hz difference, for instance identified by Roshan on G4 at 393hz, represents only a 0.1% difference, being demonstrated by recording standards to being inaudible.

    Some years ago I had the good fortune to meet up and chat with a piano maker of long tradition. He talked about his Eureka! Temperament based upon notes left by Handel and published "How Mr Handel tuned his harpsichord" which described tuning using chords rather than intervals and a recording app on my phone recorded the gems of his experience. For anyone interested I put it on https://www.youtube.com/watch?v=3REzLHO56VE. An organ builder I know talks about tuning as a "process" and favours the method given by Young. For anyone in England travelling through London Bridge Station, the Public Organ placed there by https://www.pipe-up.org.uk/ will have been tuned to such a scheme.

    If anyone would like to follow up in experiencing what Kirnberger III and in some cases Kellner can do here's the Hammerwood curated playlist - and it's potentially quite interesting in experimenting with an 1802 Stodart Grand and then seeing how the music transfers onto a "modern era" cross-strung instrument - and I guarantee that neither is tuned beyond 1 beat in 2 second accuracy and nor are the analogue recordings made on a machine with lower than 0.05% W&F:

    Hammerwood playlist

    Elsewhere


    Experiments 


    Best wishes

    David P


    --
    - - - - - - - - - - - - - - - - - - - - - - - -
    David Pinnegar, B.Sc., A.R.C.S.
    - - - - - - - - - - - - - - - - - - - - - - - -
    +44 7868385643





  • 5.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Registered Piano Technician
    Posted 01-05-2026 13:45

    Roshan,

    Thanks for the response! While an equally beating system can make aural tuning much easier, the ear is still going to process more tempered 5ths/4ths as less consonant regardless of their beat rate, since the ear hears pitch relationships in frequency ratios. Because of this, I believe a logarithmic function (cents) can better describe how the sounds will be heard by human ears. For example, F3-A3 beating approx. 7 bps is not heard by the listener as more consonant than F4-A4 at 14 bps since the ratio is the same, though it's beating twice as fast. For this reason, I don't think it's possible for cents to lose their meaning in any practical way, though I see the benefits of equal beating for aural tuning purposes.

    As for where to place the schisma, either location works, I believe Kirnberger's letter to Forkel places it between B-F#, but we're splitting hairs at this point. 

    David,

    While the difference in Hz is fairly insignificant from a beating standpoint, the difference in cents is a little more significant. While the difference in Hz in recordings is insignificant since (to my knowledge) the frequency of every pitch will still have its same ratio, a 7 cent narrow A-E is much worse than Werkmeister, for example. I find Kellner tempered fifths (4.8 cents narrow) considerably more tolerable/usable in isolation than Kirnberger III fifths (5.4 cents). 7 cents narrow for me would be a deal breaker. This becomes very important when tuning strong temperaments for modern music.



    ------------------------------
    Tim Foster RPT
    New Oxford PA
    (470) 231-6074
    ------------------------------

    Original Message


  • 6.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Posted 01-05-2026 15:44
    Tim - yes the Schisma is between F# and B

    Whilst the Kellner temperament is a "let's make the instrument sound nice" tuning Kirnberger III has a clarity which blows Kellner away and without the extreme in B major that one gets, unacceptably in my opinion, with Werkmeister.

    Here's Kellner and Kirnberger III compared on a Steinway B vs Bechstein. Kirnberger is so good that on the last tuning the instruments were reversed with Kirnberger III on the Steinway.

    When we go to babies here's Kellner

    and Steinway Model O

    Best wishes

    David P


    --
    - - - - - - - - - - - - - - - - - - - - - - - -
    David Pinnegar, B.Sc., A.R.C.S.
    - - - - - - - - - - - - - - - - - - - - - - - -
    +44 7868385643



    Original Message


  • 7.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Posted 01-05-2026 15:47

    Hi Tim,

    You have made a critical point. The main reason for developing the Equal and Unequal Beating Temperament System was to balance the mathematical / scientific side of piano tuning with the musical / artistic side of music composition. This system combines the linear nature of beat rates (adding and subtracting the frequencies of coincident partials) and the geometric progression of pitches (multiplying and dividing the frequencies of non-coincident partials). This is the closest that I have come to achieving that balance.

    As for where to place the Schisma, I decided to replicate the design of Werckmeister III for Kirnberger III. In Werckmeister III, there are 3 consecutive contracted 5ths (C-G, G-D, and D-A), then there are 2 consecutive beatless 5ths (A-E and E-B) and 1 contracted 5th (B-F#). Similarly, in my version of Kirnberger III, there are 4 consecutive contracted 5ths (C-G, G-D, D-A, and A-E), then there are 2 consecutive beatless 5ths (E-B and B-F#) and 1 contracted 5th (F#-C#). The goal is to make the 3rds progress smoothly from one key to another and reduce their harshness. I decided to keep Kellner as it is because it sits between both ends of the spectrum. In Kellner, there are 4 consecutive contracted 5ths (C-G, G-D, D-A, and A-E), then there is 1 beatless 5th (E-B) and 1 contracted 5th (B-F#). All in all, Kellner has neither got 5 consecutive contracted 5ths nor 2 consecutive beatless 5ths between the contracted 5ths. It is all about balance in the end. It is all about finding the middle ground. It is all about finding the optimal compromise.

    We are indeed splitting hairs at this point, but these incremental adjustments contribute to the overall improvement of each temperament. Temperaments are sensitive to minute changes. For example, the Schismatic 4th C#4-F#4 in my version of Kirnberger III very nearly beats at 1 Hz! Therefore, the differences that we are dealing with might be miniscule on the surface, but they can have a material impact on a deeper level. This is mainly why I am extending my figures to at least 9 decimal places to mitigate the impact that rounding errors and small differences may have on the final outcomes.

    Thank you for getting back to me. You are making me think carefully about why I am trying my best to distribute the various commas in the most meaningful of ways!
    ------------------------------
    Roshan Kakiya
    ------------------------------

    Original Message


  • 8.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Registered Piano Technician
    Posted 01-06-2026 07:01

    Hi Roshan,

    Thank you, as always, for your interaction- what an interesting subject! I'm struggling a little with your statement below:

    "The main reason for developing the Equal and Unequal Beating Temperamsnt System was to balance the mathematical / scientific side of piano tuning with the musical / artistic side of music composition. This system combines the linear nature of beat rates (adding and subtracting the frequencies of coincident partials) and the geometric progression of pitches (multiplying and dividing the frequencies of non-coincident partials). This is the closest that I have come to achieving that balance."

    While I see the benefit of equally beating thirds from a musical standpoint (e.g. Bremmer's EBVT), I'm still struggling to see what benefit equally beating fifths has from a musical standpoint. For example, Kirnberger III's beating fifths are noticeable in isolation, but are hidden when played with the thirds. I can't really think of any musical example where beating fifths have an artistic significance, since these fifths are rarely used in isolation, and the beat rates are masked within chords. Does this make sense?



    ------------------------------
    Tim Foster RPT
    New Oxford PA
    (470) 231-6074
    ------------------------------

    Original Message


  • 9.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Posted 01-06-2026 10:11
    Above I mentioned a piano manufacturer describing how he tuned Handel's temperament which was documented in 1780

    In the organ world Colin Pykett is a distinguished scholar and authority and he has written about it
    mentioning that beats were not used, the quality of the chords being important. His documentation is worth the read.

    It appears to me that old tuners were much more fluid and arbitrary than with electronic standards we expect to see today. In my observation, as long as CC# FF# were narrow and DC# GF# were wide, and arguably a few others, expected key characteristics emerged quite predictably. The fun of calculations today, however, is finding that with some characteristics the sound of the modern piano can be better.




    Best wishes

    David P


    --
    - - - - - - - - - - - - - - - - - - - - - - - -
    David Pinnegar, B.Sc., A.R.C.S.
    - - - - - - - - - - - - - - - - - - - - - - - -
    +44 7868385643



    Original Message


  • 10.  RE: Unequal Beating Temperament System - Kirnberger III and Kellner

    Posted 01-06-2026 12:35

    Hi Tim,

    Thanks again for your input!

    I am trying to balance several factors at the same time. One factor is the relationship between the 5ths / 4ths and Major 3rds. Four consecutive 5ths and 4ths make one Major 3rd. It is possible to control the size of each Major 3rd by manipulating the size of each of the four consecutive 5ths and 4ths that it is made up of around the Circle of 5ths and 4ths. From a tuning standpoint, the 5ths and 4ths are tuned. From a musical perspective, the Major 3rds are supposed to be narrower in the home keys around the key of C and wider in the remote keys around the key of F#. Your observation is spot on when it comes to playing intervals in isolation vs playing chords. This is where it gets technical. Intervals on their own have beat rates because we are dealing with the difference between their coincident partials in Hz. Chords have beat ratios, which are the beat rate relationships between / among the intervals that form them. I am sure that Bill Bremmer has mentioned in one of his posts on Piano World that we are listening to the quality of the intervals rather than the beat rates when we are playing chords. We are now getting to the heart of the design of the Equal and Unequal Beating Temperament System. Those beat rates and beat ratios are being manipulated to achieve certain targets, e.g., equally beating intervals and / or unequally beating intervals with specific beat ratios. 

    It might be fair to say that the line between mathematics / science and music / art is getting blurry at this point. That is a good thing in my opinion because they have to mesh together eventually. I am dealing with objectivity and subjectivity at the same time. They both influence each other, creating a feedback loop. Initially, my main aim was to bridge the gap between the theory and practice of tuning. I have honestly lost count of the number of gaps that I have ended up bridging now!
    ------------------------------
    Roshan Kakiya
    ------------------------------

    Original Message