Pianotech

Back to discussions
Expand all | Collapse all

Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

  • 1.  Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-17-2025 19:00

    Circular Harmonic System for Equally Beating Intervals

    The Circular Harmonic System has been produced by Alfredo Capurso. Please read the paper at https://www.scribd.com/document/174787881/Alfredo-Capurso-A-New-Model-of-Interpretation-of-Some-Acoustic-Phenomena-Circular-Harmonic-System-C-HA-S for more details.

    I have posted numerous mathematical explorations from a theoretical tuning perspective. This is my final discussion thread because I do not know where I can go from here theoretically. I have reached the stage where even my own mathematical explorations are beyond my comprehension, and the time has come to finally put theory into practice.

    You will find in the document with the title "Circular Harmonic System Equations for Equally Beating 12th and 15th" the equations that I have discovered for making the 12th and the 15th beat equally in four different configurations by closely examining and implementing the design of Alfredo Capurso's Circular Harmonic System theoretically. The more I research the Circular Harmonic System, the more I realise that something truly special is going on in the equations that it produces. Alfredo Capurso has found a way to make two intervals beat equally in a perfect 1 : 1 ratio in theory. Do not restrict yourself to the 12th and the 15th. This system can be used to make any two intervals beat equally in the configurations that I have specified. You simply have to know how to construct the equations that make this happen. Once those equations have been constructed, the result will be a perfect 1 : 1 theoretical beat ratio. Alfredo Capurso's Circular Harmonic System is the only system I have come across that manages to consistently achieve perfection when it comes to producing 1 : 1 theoretical beat ratios. I am not assuming that this is the case. I have definitively proved that this is occurring by performing beat rate calculations for each of the four configurations that I have specified to test the robustness of this system, which I have included in the posts below.

    I finally understood how to produce the equations for making two intervals beat equally after I explored the mathematics of tuning from a fresh perspective. Until now, I was focusing on equally tempering two intervals in opposite directions. However, that approach only made two intervals beat equally in a specific contiguous arrangement (https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=e8a9c904-de17-4d5e-97d8-1eba569e3dbf&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf). For many years, I converted frequency ratios into cents before I performed my calculations. However, I eventually decided to leave out cents entirely by directly manipulating frequencies and frequency ratios for beat rate calculations. That was when I made a breakthrough because I finally understood what Alfredo Capurso's Circular Harmonic System was doing all along. Beat rate equations are linear, which means that we must strictly use arithmetic operations - addition, subtraction, multiplication, and division - for beat rate calculations. That is the key to achieving equally beating intervals in theory.

    I should explain what is happening in each of the four attached documents.

    1. Circular Harmonic System Equations for Equally Beating 12th and 15th: This document provides us with information about the four different configurations in which any two intervals can be arranged to make them beat equally. Two intervals arranged in the Upper Note Common configuration will share the same upper note. Two intervals arranged in the Lower Note Common configuration will share the same lower note. There are two ways of arranging two intervals in a contiguous fashion. Every equation must be solved for the Δ variable to find the point at which the semitone of each interval is equalised to produce two equally beating intervals. I use Wolfram|Alpha (https://www.wolframalpha.com) to solve every equation.
    2. Equation Conversion Processes for Circular Harmonic System Equations for 12th and 15th: I went ahead and constructed processes for converting all the equations for equally beating intervals into equations whose components fully comply with the rules of logarithms. I do not know what I have achieved by doing this, though. This is where my mathematical explorations have gone beyond the scope of my comprehension and beyond the scope of Alfredo Capurso's original Circular Harmonic System.
    3. Logarithmic Circular Harmonic System Equations for 12th and 15th (Balancing Intervals): These logarithmic equations indirectly equalise the semitone of each interval by expanding / contracting each interval.
    4. Logarithmic Circular Harmonic System Equations for 12th and 15th (Balancing Semitones): These logarithmic equations directly equalise the semitone of each interval by expanding / contracting the semitone of each interval.

    I urge everyone to take a look at Alfredo Capurso's Circular Harmonic System to understand and appreciate what it offers us in the theoretical world of tuning.

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



  • 2.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-18-2025 16:00
      |   view attached

    Beat Rate Analysis of Circular Harmonic System Equations for Equally Beating 12th and 15th

    As promised, please find attached a document containing a beat rate analysis that I have performed to illustrate how Alfredo Capurso's original Circular Harmonic System produces perfect 1 : 1 theoretical beat ratios when two intervals are arranged in the four configurations specified therein.

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

    Original Message


  • 3.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-18-2025 17:00

    Roshan,

    The "proof is in the pudding" (a.k.a. "show me"). Let's hear a piano tuned in this system and then we can evaluate it. Till then its just another theory (among many). 

    Peter Grey Piano Doctor 



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

    Original Message


  • 4.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-19-2025 12:28

    Theory: Alfredo Capurso's Circular Harmonic System.

    Practice: Bill Bremmer's Equal Beating Victorian Temperament.

    Result: A theoretically and practically sound Equal Beating Temperament System. I prefer to refer to it as the Equal Beatment System to distinguish it from the Equal Temperament System.

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

    Original Message


  • 5.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-24-2025 13:03
      |   view attached

    Beat Rate Analysis of Circular Harmonic System Equations for Equally Beating 5th and 4th

    I have performed another beat rate analysis. This one is for the equations that I have discovered for making the 5th and the 4th beat equally in the four configurations that I have specified. Please use the document with the title "Circular Harmonic System Equations for Equally Beating 12th and 15th" as a reference / guide.

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

    Original Message


  • 6.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-24-2025 13:11
      |   view attached

    Analysis of Equal Beating Victorian Temperament

    I have performed my final analysis. This one is for Bill Bremmer's Equal Beating Victorian Temperament. I have constructed this temperament by following the instructions at https://billbremmer.com/ebvt.

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

    Original Message


  • 7.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-26-2025 15:21

    Illustration: Equally Beating Upper Note Common 5th A3-E4 and 4th B3-E4 in Equal Beating Victorian Temperament without Inharmonicity

     

    The Circular Harmonic System equations that I have discovered achieve equally beating Upper Note Common and Lower Note Common intervals by shifting the uncommon notes in opposite directions whilst the common note remains fixed (expanding one interval and contracting the other). The Equal Beating Victorian Temperament achieves equally beating Upper Note Common and Lower Note Common intervals in its sequence by sharpening the common note whilst the uncommon notes remain fixed (expanding both Upper Note Common intervals simultaneously or contracting both Lower Note Common intervals simultaneously). The result of this is that none of the Circular Harmonic System equations that I have posted in this discussion thread for the Upper Note Common and Lower Note Common configurations are applicable to the sequence of the Equal Beating Victorian Temperament. We must explore the mathematics of equally beating intervals from another angle instead.

    In step 9 at https://billbremmer.com/ebvt, the Upper Note Common 5th A3-E4 and 4th B3-E4 are being tuned. We are told to sharpen E4 until the 5th A3-E4 and the 4th B3-E4 beat exactly alike. The illustration below shows us how this process works in theory. According to the instructions, A3 and B3 would be tuned by now, so I can directly extract the figures for their frequencies from my analysis. I will tune E4 from B3 as a pure 4th to recreate the scenario in step 9 and go from there.

     

    Frequency of A3 = 220.000000000 Hz

    Frequency of B3 = 246.937500000 Hz

    Frequency of E4 = 246.937500000 Hz × (4 / 3) = 329.250000000 Hz

     

    5th A3-E4 = 1200 × log2(329.250000000 Hz / 220.000000000 Hz) = 698.015900078 cents = Narrow 5th

    4th B3-E4 = 1200 × log2(329.250000000 Hz / 246.937500000 Hz) = 498.044999135 cents = Pure 4th

     

    The instructions tell us to sharpen E4. The sharpening of E4 will cause both the narrow 5th A3-E4 and the pure 4th B3-E4 to widen. Consequently, the 5th A3-E4 will become a wider narrow 5th, and the 4th B3-E4 will become a wide 4th. Now we need to determine the amount by which E4 must be sharpened to make the 5th A3-E4 and the 4th B3-E4 beat equally.

     

    Frequency of 3rd Partial of A3 = 220.000000000 Hz × 3 = 660.000000000 Hz

    Frequency of 2nd Partial of E4 = 329.250000000 Hz × 2 = 658.500000000 Hz

    Frequency of 4th Partial of B3 = 246.937500000 Hz × 4 = 987.750000000 Hz

    Frequency of 3rd Partial of E4 = 329.250000000 Hz × 3 = 987.750000000 Hz

     

    The uncommon notes A3 and B3 will remain fixed. Therefore, the 3rd partial of A3 will stay where it is, and the 4th partial of B3 will stay where it is. The 2nd and 3rd partials of the common note E4 will become sharper by sharpening E4. Let us check the beat rate of the 5th A3-E4 and the beat rate of the 4th B3-E4.

     

    Beat Rate of 5th A3-E4 = Difference between Frequency of 3rd Partial of A3 and Frequency of 2nd Partial of E4 = 660.000000000 Hz − 658.500000000 Hz = 1.500000000 Hz

    Beat Rate of 4th B3-E4 = Difference between Frequency of 3rd Partial of E4 and Frequency of 4th Partial of B3 = 987.750000000 Hz − 987.750000000 Hz = 0.000000000 Hz

     

    On the one hand, the beat rate of the 5th A3-E4 will decrease as the 2nd partial of E4 moves closer to the 3rd partial of A3. On the other hand, the beat rate of the 4th B3-E4 will increase as the 3rd partial of E4 moves away from the 4th partial of B3. If we want to make the 5th A3-E4 and the 4th B3-E4 beat equally, we must equalise the beat rate of the 5th A3-E4 and the beat rate of the 4th B3-E4. This is not as straightforward as it seems because the 2nd and 3rd partials of E4 will be moving at different rates. The solution is to distribute the difference between the beat rate of the 5th A3-E4 and the beat rate of the 4th B3-E4 in a specific ratio across the 2nd and 3rd partials of E4.

     

    Beat Rate Distribution Ratio for 2nd and 3rd Partials of E4 = 2 : 3

    Beat Rate Distribution Ratio for 2nd Partial of E4 = 2 / (2 + 3) = 2 / 5

    Beat Rate Distribution Ratio for 3rd Partial of E4 = 3 / (2 + 3) = 3 / 5

     

    Difference between Beat Rate of 5th A3-E4 and Beat Rate of 4th B3-E4 =  1.500000000 Hz − 0.000000000 Hz = 1.500000000 Hz

    New Frequency of 2nd Partial of E4 = 658.500000000 Hz + [1.500000000 Hz × (2 / 5)] = 658.500000000 Hz + 0.600000000 Hz = 659.100000000 Hz

    New Frequency of 3rd Partial of E4 = 987.750000000 Hz + [1.500000000 Hz × (3 / 5)] = 987.750000000 Hz + 0.900000000 Hz = 988.650000000 Hz

     

    We now know what the new frequencies of the 2nd and 3rd partials of E4 should be to make the 5th A3-E4 and the 4th B3-E4 beat equally. The time has come to calculate the new beat rates and the beat ratio for verification purposes.

     

    New Beat Rate of 5th A3-E4 = Difference between Frequency of 3rd Partial of A3 and New Frequency of 2nd Partial of E4 = 660.000000000 Hz − 659.100000000 Hz = 0.900000000 Hz

    New Beat Rate of 4th B3-E4 = Difference between New Frequency of 3rd Partial of E4 and Frequency of 4th Partial of B3 = 988.650000000 Hz − 987.750000000 Hz = 0.900000000 Hz

    Beat Ratio of 5th A3-E4 and 4th B3-E4 = 0.900000000 Hz / 0.900000000 Hz = 1 / 1

     

    A perfect 1 : 1 beat ratio lets us know that the 5th A3-E4 and the 4th B3-E4 are beating equally! We cannot stop here because we still need to calculate the new frequency of E4.

     

    New Frequency of E4 (1) = New Frequency of 2nd Partial of E4 / 2 = 659.100000000 Hz / 2 = 329.550000000 Hz

    New Frequency of E4 (2) = New Frequency of 3rd Partial of E4 / 3 = 988.650000000 Hz / 3 = 329.550000000 Hz

    Frequency Ratio of E4 = New Frequency of E4 (1) / New Frequency of E4 (2) = 329.550000000 Hz / 329.550000000 Hz = 1 / 1

     

    It is essential to ensure that the note E4 is still intact by checking its frequency ratio, which must be none other than 1 / 1. The note E4 will split into two if an error has occurred in the calculations, which will be evident when its frequency ratio is not 1 / 1. Please see the final result below.

     

    Frequency of A3 = 220.000000000 Hz

    Frequency of B3 = 246.937500000 Hz

    New Frequency of E4 = 329.550000000 Hz

     

    New 5th A3-E4 = 1200 × log2(329.550000000 Hz / 220.000000000 Hz) = 699.592616079 cents = Narrow 5th

    New 4th B3-E4 = 1200 × log2(329.550000000 Hz / 246.937500000 Hz) = 499.621715135 cents = Wide 4th

     

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

    Original Message


  • 8.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-26-2025 17:27
    And the $64,000 question is what does it do for the music?

    What does the piano do for the music?

    If the piano doesn't do what Chopin or Beethoven expected the instrument to do, giving specific instructions in places which the modern pianist cannot obey, then the piano - through its tuning - is wrong for the music.

    Arguably therefore, such discussions should not be mathematics for mathematics sake, tuning for tuning's sake, the piano for the sanctity of the modern piano's sake, but tuning for the music's sake.

    A result of equal beating, which with my mentor we tried some years ago, was that it felt that the beating was dominating the sound and distracting from the music. Perhaps I was listening too carefully and missing what I should have been listening to.

    If the piano is ruled by mathematics in the superiority of science, it loses the expression of feeling leading to emotional communication. In the UK this has led to a progressive disconnection with music. It is not considered a language of emotion which is essential but instead lumped in with Sport. Sport is optional. Communication of emotion is essential. This is not fiction: the supervisory department of government which has an oversight for music is the Department of Culture, Media and Sport. I do guided tours of a historic house and have done so for four decades. Whereas 20 years ago there was wide appreciation for music, now when I ask "is anyone here a musician?", depressingly the number of positives are now less than one in a hundred.

    The piano has to do better to engage the emotions and this comes around through whether vibrations at the heart of sound coincide together or are dissociated from each other. The piano has to do better than it has done over the period of its industrial production. To the late 19th century buyer of a status symbol in the corner of their living room, the glossy glistening sound of the new German brand cast iron frame instruments was bedazzling and they didn't know the music, especially if the piano was used for the purpose of displaying photographs. The writers of music which forms the bulk of the piano repertoire grew up on straight strung instruments which were tuned to bring the differences between keys to life.

    Roshan - your mathematical analysis is supreme but your Herculean effort is disproportionate to the need for the piano now, rather than as a hangover from the industrial production, to re-engage audiences in the interest in the music. I'm not convinced that Equal Beating does so. 

    Whilst it's said that you're not tuning for Equal Temperament please could you possibly identify intervals in the scale that you propose which differ from key to key and specifically in the size of the major 3rds? What effect does it have on the perception of the 3rds? Are there some which are calmer than others? Can you make, for instance the third C-E or F-A more pure, more relaxed, and the thirds in the remote keys such as B major for instance, B-D# wider and different?

    Best wishes

    David P


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



    Original Message


  • 9.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-26-2025 20:21

    Hi David,

    When all is said and done, it is all about the music.

    When it comes to visualising the features of the EBVT, the graphical history at https://www.rollingball.com/A10z.htm is one of the best resources around.

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

    Original Message


  • 10.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-27-2025 09:30
    Hi Roshan

    Thanks but there I don't see the graph of your temperament. However, in the spirit of Bill Bremmer's, it looks likely to be most useful and worthy of perhaps wider attention than personally I've given to your mathematics

    Best wishes

    David P


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



    Original Message


  • 11.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-27-2025 10:08

    Hi David,

    I have faithfully reproduced the EBVT mathematically by carefully following the detailed temperament sequence instructions at https://billbremmer.com/ebvt. My illustration above provides us with an insight into how I did it.

    The equal beating checks that I have included in my Excel file, which are part of the EBVT's sequence, validate the accuracy of my analysis.

    Compare the offsets in my Excel file with the offsets on page 7 of the instructions. They are exactly the same for every note except for F#. My analysis highlights that only one slight adjustment should be made to the offsets on page 7 of the instructions. The offset for F# should be changed from −0.28 cents to −0.29 cents. 

    The benefit of having an Excel file is that you can create your own graphs.

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

    Original Message


  • 12.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-28-2025 17:09
      |   view attached

    Illustration: Equally Beating Upper Note Common 5th A3-E4 and 4th B3-E4 in Equal Beating Victorian Temperament without Inharmonicity

    I have attached a document that contains the illustration above.

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

    Original Message


  • 13.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-28-2025 20:00
      |   view attached

    Illustration: Equally Beating Upper Note Common 5th A3-E4 and 4th B3-E4 in Equal Beating Victorian Temperament with Inharmonicity

    Please find attached an Excel file for the illustration above. I have included theoretical inharmonicity in the figures.

    The inharmonicity formula that I have used is available at https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=8a9b4611-cf64-41ce-9334-5426aa073960&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf&tab=digestviewer&ReturnUrl=%2fcommunities%2fcommunity-home%2fdigestviewer%3fCommunityKey%3d6265a40b-9fd2-4152-a628-bd7c7d770cbf#bm9c0231ed-b318-4c6c-8f29-48b3f9576f29.

    The inharmonicity constants that I have used are available at https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=6de86f98-a71b-437d-90f9-9f5c99bcb9a4.

    Please use the document with the title "Illustration: Equally Beating Upper Note Common 5th A3-E4 and 4th B3-E4 in Equal Beating Victorian Temperament without Inharmonicity" as a reference / guide.

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

    Original Message


  • 14.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-30-2025 16:00

    Conclusion

    To account for the linear nature of beat rates, add and subtract the frequencies of coincident partials.

    To account for the logarithmic nature of pitch perception, multiply and divide the frequencies of non-coincident partials.

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

    Original Message


  • 15.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-28-2025 23:01
      |   view attached

    Scala File for Equal Beating Victorian Temperament

    Finally, a Scala file.

    Enjoy!

    Notes EBVT Cents 12-ET Cents Offsets
    A 0.00 0.00   0.00
    A# 102.86 100.00 +2.86
    B 199.97 200.00 −0.03
    C 303.80 300.00 +3.80
    C# 398.71 400.00 −1.29
    D 500.86 500.00 +0.86
    D# 601.59 600.00 +1.59
    E 699.59 700.00 −0.41
    F 801.84 800.00 +1.84
    F# 899.71 900.00 −0.29
    G 1003.11 1000.00 +3.11
    G# 1100.67 1100.00 +0.67
    A 1200.00 1200.00   0.00

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

    Original Message


  • 16.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-18-2025 17:38
    Dear Roshan

    Your work is absolutely exhaustive but, with respect, is nothing to do with music. The Equal Temperament kills resonance and has been a tool of obliteration of the local and the old in favour of the presumed assumption of the superiority of the science and the new. The fact that you're unable to take it further is indicative of the blind alley to which research in Equal Temperament is. Of course there are many who will sympathise with you and consider me to be off the wall but there are a few papers which have popped up on Academia.org which are really quite enlightening and put the position into perspective.

    One is 
    which describes, in rather long winded terms imho, the conflict inherent in the acceptance of equal temperament as the de facto "proper" tuning for all and the other is
    and another is https://www.academia.edu/113243327/Early_Modern_Tuning_Temperament_and_the_Natural_Philosophy_of_Empire in the way in which Equal Temperament was a tool of the spread and imposition of Empire. I have found similar evidence in 19th century publications. https://www.academia.edu/99606748/_DRAFT_DO_NOT_CITE_OR_CIRCULATE_WITHOUT_WRITTEN_PERMISSION_OF_AUTHOR_Consonance_and_Dissonance by the same author expands.

    Another is
    by Leon Gunther which is of direct relevance to what you're doing.

    It's a brilliant resource contrasting the Pythagorean, the Just Chromatic and the Equal Tempered scales, a 6 tone scale apparently used by Debussy, and an examination of string harmonics. He notes "The loss is that resonances and consonances are not as strong as they are in Just tuning. " and "" Of course, the use of equal temperament tuning already destroys this resonance except for the set of octaves of a given note."

    Accordingly in the course of tuning a piano we have to ask "Why?"

    Is the purpose of the piano
    1. an instrument of mathematics or of mathematical beauty?
    2. an instrument that can be made very expensively to make manufacturers a lot of money, as well as those who claim to tune pianos in 30 minute? or
    3. an instrument of musical expression and conveyance of essential fundamental emotion?

    If (3) isn't the prime objective then (2) becomes not essential and pianos aren't sold and if (1) doesn't lead to (3) then the mathematics is at a dead end.

    Music isn't in the vibrato of beats between notes but in their arrangement of consonance or dissonance. Equal beating schemes are interesting but not relevant to the music being produced.

    Best wishes

    David P


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



    Original Message


  • 17.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-19-2025 11:17

    Hi David,

    Thank you for your input, but you appear to be staring at the Circular Harmonic System through the lens of equal temperament. It is not an Equal Temperament System. It is an Equal Beatment System. Equal temperament is one method amongst many for creating a full scale for tuning purposes.

    My beat rate analyses above verify the robustness of Alfredo Capurso's original Circular Harmonic System with regard to achieving equally beating intervals theoretically. They also highlight that the scale must be adapted to make any pair of intervals beat equally in different configurations. This discovery automatically rules out equal temperament as a viable option.

    To produce a full scale, we must account for the relationship between every interval, not just two. The Unequal Temperament System provides us with the flexibility that we require for adjusting the scale to achieve equally beating intervals. Bill Bremmer's Equal Beating Victorian Temperament, an unequal temperament, gives us a practical method for creating a full scale based on equally beating intervals.

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

    Original Message


  • 18.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-19-2025 13:17

    Roshan,

    I tune EBVT very often, and particularly my personal piano in EBVT. Are you saying that EBVT is the embodiment of this other system you're talking about?

    Peter Grey Piano Doctor 

    P.S. I often will modify EBVT from Bill's original pattern by setting F3-A3, C4-E4, G3-E4, and G3-B3 at about 5 bps rather than 6 bps. The remainder ofvthe pattern dispersal remains the same. If the piano seems to like it (some do and some don't) I will continue on out. When the piano likes it I feel like I get an almost Kirnberger III effect in the simple keys (repeat: almost) without the intense dissonance in the remote keys. 

    If the piano does not seem to like it I'll quickly change back to 6bps or so. 

    Peter Grey Piano Doctor 



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

    Original Message


  • 19.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-19-2025 22:09

    Hi Peter,

    Yes. Bill Bremmer's EBVT is an equal beatment that coincides with Alfredo Capurso's Circular Harmonic System. Alfredo and Bill, through their own work, have hit the sweet spot where theory and practice combine via the common language of beats to form the Equal Beatment System.

    Alfredo's work shows us how to make intervals beat equally in theory, whereas Bill's work shows us how to make intervals beat equally in practice.

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

    Original Message


  • 20.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-20-2025 17:04

            Roshan: Towards a full scale bearing plan I also favor beats to cents. Regarding the overall compass I especially consider the following notes since they combine to make contiguous M6ths & M10ths with & around A49 (theoretical beat ratios of 5/3 & 5/2).

           A1,A13,C#17,F#22,D#31,F33,C40,A49,F#58,C#65,D#67,C76,F81,A85

          These notes make an interesting sub-set/template to form a broad harmonic framework around A49 whose symmetry is complete in a keyboard up to A97. 


    Original Message


  • 21.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-20-2025 18:09

    Paul,

    I haven't got a clue what you're talking about here. Can you elaborate?

    Peter Grey Piano Doctor 



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

    Original Message


  • 22.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-20-2025 20:14

    Peter: Please note the word contiguous. I'll try to help by listing the M6ths and M10ths separately.

    A1,C#17,F33,A49,C#65,F81 make M10ths.

    A13,F#22,D#31,C40,A49,F#58,D#67,C76 make the M6ths


    Original Message


  • 23.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-20-2025 21:17

    Roshan,

    I think you may be making a distinction without a difference. EBVT beats equally as calculated through IH. Equal temperament (ET), often prioritizing a smooth progression of major thirds, does so based in the specific IH of the piano. It is not something that an aural tuner calculates numerically. Rather, the 5:4 coincident partial of the major third is approx. 14 cents wide, which phases and produces the beating, as I'm sure you know. IH is automatically accounted for in all of these tuning systems since these coincident partials are higher than their mathematically calculated frequency, due to IH.

    Put another way, EBVT deals with IH through equal beating. ET deals with IH through smooth progression of beating thirds. Gravitating to one tuning method or another that is based in beat counting (equal beating, equally progressing, or something else) is not due to the eliminating of IH considerations, but working with IH. Focusing on beats accounts for IH. 

    An ETD can calculate IH because it can give a numeric value to a number of partials from a single note. Aural tuners cannot give a numeric value to the partials, and therefore listen to the beating. ETDs can calculate beat rates by mathematically positioning two (or more) notes through the numeric values of partials and accurately predict their phasing.

    Out of curiosity, is your interest purely mathematical, are you learning piano tuning, or perhaps already tuning but exploring other temperament options?

    Thanks again for interacting with my questions, I enjoy thinking through this.



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

    Original Message


  • 24.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-20-2025 22:32

    Hi Tim,

    My interest is purely mathematical, but my aim is to figure out how to put theory into practice. I can do all sorts of things with the mathematics of tuning in the theoretical environment, where inharmonicity is non-existent.

    I might be accounting for inharmonicity without even realising that I am doing it, e.g., by designing equations whose purpose is to make two intervals beat equally in different configurations / arrangements, then performing beat rate calculations to prove or disprove the production of a perfect 1 : 1 theoretical beat ratio. Tests exist in the practical inharmonicity-laden environment for checking real 1 : 1 beat ratios, so the theory of equally beating intervals does hold up in practice.

    The only thing stopping me from truly making the theory of tuning align with the practice of tuning is my lack of knowledge in terms of how the various coincident partials of an interval are affected by the sharpening / flattening of either one or both of its notes. That knowledge would substantially assist me in figuring out how to account for inharmonicity in practice. My lack of exposure to the practical environment, where I would have no choice but to contend with inharmonicity, is preventing me from connecting the theory of tuning to the practice of tuning. I can only close / bridge that gap by actually tuning a piano.

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

    Original Message


  • 25.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-21-2025 11:07

    Roshan,

    I just now started wading through the paper you referenced at the start of this thread. However, stopped at the end of this section:

    1.5 Temperaments and the State of the Art

    Why? Because the author makes the totally erroneous claim that the current state of the art (i.e., present day ET) was reached at the end of the 17th century. 

    This is so far from accurate as to be laughable, which then calls into question all the other mathematical mumbo jumbo that he is talking about. 

    If the author has something demonstrable that can be evaluated aurally...great! If its nothing more than endless mathematical theorizing, then I would have to consider a waste of time. 

    Not being critical here, but I have been exposed to scientific articles such as in "Nature" magazine that resemble this style of writing, using a  vocabulary style, 80% of which I've never seen before and do not understand the meaning of, but is intended to make the author appear very educated, sophisticated, and intellectually superior, and intended to make me accept what is written simply based on this supposed "smartness". 

    I come back with "show me" what you're talking about. If you can't simplify it and demonstrate it...it's a "nothing burger".

    At least you have acknowledged that Bill Bremmer has something, and I can confirm that from my own experience.

    Peter Grey Piano Doctor 



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

    Original Message


  • 26.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-21-2025 11:20

    Addendum:

    You can essentially eliminate the issue of IH if you do your tempering on a pipe organ (i.e., if the issue here is simply IH). 

    Peter Grey Piano Doctor 



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

    Original Message


  • 27.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-21-2025 12:40

    Hi Peter,

    I read an interesting discussion about tuning pipe organs via equally beating 5ths and 4ths at https://www.reddit.com/r/microtonal/comments/bxvfx5/equal_beating_temperament.

    An unequal temperament from the 1950s that uses this approach is available at https://pubs.aip.org/asa/jasa/article-abstract/29/4/476/720902/Equal-Beating-Chromatic-Scale.

    Bill Bremmer's ET via Marpurg also uses this approach, and its sequence is available at https://forum.pianoworld.com/ubbthreads.php/topics/1227016/re-et-via-marpurg-summary-sequence.html#Post1227016.

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

    Original Message


  • 28.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-24-2025 13:59

    Hi Roshan,

    I typically don't chime in with these discussions because my math chops aren't up to par, but you wrote a couple things that caught my attention. 

    1) You wrote: "The only thing stopping me from truly making the theory of tuning align with the practice of tuning is my lack of knowledge in terms of how the various coincident partials of an interval are affected by the sharpening / flattening of either one or both of its notes. That knowledge would substantially assist me in figuring out how to account for inharmonicity in practice. My lack of exposure to the practical environment, where I would have no choice but to contend with inharmonicity, is preventing me from connecting the theory of tuning to the practice of tuning. I can only close / bridge that gap by actually tuning a piano."

    The way I think about inharmonicity might be useful to you. It is a conceptual approach as opposed to a mathematical one, and it serves my students and me well. You don't need to be able to tune a piano to understand it. If it makes sense to you, you might be able to translate it into a mathematical concept. I'd personally be curious if it had any use to you at all. Email me if you'd like to connect because I might not notice much on here: maggie@timandmaggie.net .  I'm very busy, but I'm sure we can find a time to connect if you are interested. 

    2) You mentioned wishing you could figure in inharmonicity into your formulas. My hubby, Tim, has made several excel sheets for me that give beat rates with various parameters. He was wanting to make one that could account for various levels of regular inharmonicity (still not real world, but would be interesting), but got too busy to get into it. His more mathematical mind and how he thinks about these things might also be of interest to you, and if the two of you could come up with formulas that could be put into a spreadsheet, it would be very interesting. 

    None of this has to do with the current discussion on equal beating temperaments, but I felt like throwing it your way to see what happened. As far as an organ goes, since it has almost zero inharmonicity, it is tuned differently from a piano (as you likely know). For example, because of inharmonicity, all intervals on a piano must be "stretched", which makes fifths and fourths wider. This widening makes fifths closer to pure and fourths further away from pure. Octaves are a different animal, but can only be tuned pure at one coincident partial on a piano. On an organ, octaves can be mostly tuned to all coincident partials at the same time, and fifths and fourths will be tempered the same amount instead of fifths being closer to pure as on a piano. I think this is what Peter was referring to. I have no idea if your math would come out differently. 

    Maggie



    ------------------------------
    Maggie Jusiel, RPT
    Athens, WV
    (304)952-8615
    mags@timandmaggie.net
    ------------------------------

    Original Message


  • 29.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-24-2025 15:20

    Maggie,

    Thanks for chiming in here. I just wanted to add (for Roshan's sake) that the piano has its unique sound BECAUSE of inharmonicity. The organ sounds the way it does because it has no inharmonicity (measurable anyway). Violin, viola, cello  etc sound the way they do because they have no inharmonicity. 

    Interestingly, if you bow a piano string the inharmonicity virtually disappears, which strongly suggests that it's unique sound is due to it's "percussive" nature allowing the strings to do what they "want" and not forcing it into a "driven" vibratory mode. 

    Kind of irrelevant to the specifics of this discussion, but when it comes to tuning, we deal with it with our ears/brain. It's not as hard as it might seem. 

    Peter Grey Piano Doctor 



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

    Original Message


  • 30.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-24-2025 18:02
    Peter - the sound of the piano is driven by the internal structure of the instrument and not by inharmonicity. It is not inharmonicity which makes a Yamaha sound different from a Steinway or a Viennese or a british 19th century Broadwood but the design of the instruments themselves.

    Best wishes

    David P


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



    Original Message


  • 31.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-25-2025 07:50

    David,

    Certainly no argument there.

    What I was referring to was the fact that the "modern" piano has an inherently unique structure (freely vibrating, high tension strings, that exhibit a phenomenon we call inharmonicity), that imparts a sound signature unlike any other instrument. 

    I certainly understand the differences that exist between the various piano "designs" from different manufacturers (and time periods), but since the advent of the full cast iron string frame, with tensions increasing greatly within the design, the piano, as we know it, has the unique soundscape (virtually all of them), primarily due to this phenomenon of IH. 

    Just a little detail that I was unsure that Roshan was aware of. He may in fact be, and it was all wasted "breath". 

    Peter Grey Piano Doctor 



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

    Original Message


  • 32.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-25-2025 08:52
    Dear Peter

    Thanks - but actually there are assumptions in modern piano tuning which have been revered as fact with mythological origins from a century or so ago and over 20 years of exploration I've found not to be entirely justified. 

    If he is willing, Roshan has the mathematical tools at his command to go further

    Best wishes

    David P


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



    Original Message


  • 33.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-25-2025 09:17

    David,

    I don't think you and Peter are really disagreeing, you are both talking about different ways of tuning a piano. Tuning a piano based in IH calculations puts a piano "in tune" according to this mathematical framework of tuning. Therefore, ET, EBVT, etc. are "in tune" based on these differing mathematical constructs which factor IH. Your tuning style works with a different mathematical construct that makes the piano do different things. This is why I like it. 

    It's not a matter of right or wrong methodology, but I think your method does explore (and challenge) what many believe is physically possible on a piano.



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

    Original Message


  • 34.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-25-2025 18:59

    Hi Peter, Tim, and David,

    As always, thank you for your insights.

    It might be good at this point to go right back to the start before I even get into discussing inharmonicity and its implications.

    Why does the mathematics of tuning exist in the first place?

    Theoretically, it exists because of the layout of the standard piano's keyboard and how humans perceive pitch.

    At the heart of the mathematics is the way in which the standard piano's keyboard has been laid out. 12 physical notes repeat themselves in a specific order within the interval that we have defined as the physical Octave. It is of the utmost importance to establish that fact before we move on to understanding how humans have decided to determine the frequency of each of those 12 notes via the creation of musical scales.

    Logarithmic musical scales are based on the theory that humans perceive pitch in a logarithmic manner. That is why we use logarithmic operations to create them and calculate the frequency of each of the 12 notes based on them. However, arithmetic musical scales do exist. There are logarithmic solutions and arithmetic solutions. We must adapt each musical scale to the design of the standard piano's keyboard. This is even before we take inharmonicity into consideration.

    One of Many Logarithmic Solutions: Frequency ratio assigned to Octave = 2. Number of semitones assigned to Octave = 12. Semitone of musical scale based on the equal division of the Octave logarithmically = 21 / 12. Frequency ratio of Octave = (21 / 12)12 = 2. Frequency ratio of 5th = (21 / 12)7 = 27 / 12. Lowest common denominator of semitones = 12 × 7 = 84. Therefore, 7 Octaves should coincide with 12 5ths. 7 Octaves = 27 = 128. 12 5ths = (27 / 12)12 = 128. Problem solved logarithmically! This approach agrees with the theory that humans perceive pitch logarithmically.

    One of Many Arithmetic Solutions: Frequency ratio assigned to Octave = 2. Number of semitones assigned to Octave = 12. Semitone of musical scale based on the equal division of the Octave arithmetically = 2 / 12. Frequency ratio of Octave = (2 / 12) × 12 = 24 / 12 = 2. Frequency ratio of 5th = (2 / 12) × 7 = 14 / 12 = 7 / 6. Lowest common denominator of semitones = 12 × 7 = 84. Therefore, 7 Octaves should coincide with 12 5ths. 7 Octaves = 2 × 7 = 14. 12 5ths = (7 / 6) × 12 = 14. Problem solved arithmetically! This approach does not agree with the theory that humans perceive pitch logarithmically.

    Arithmetic operations and logarithmic operations are somewhat connected. Arithmetic multiplication can be expressed in the form of logarithmic addition. Arithmetic division can be expressed in the form of logarithmic subtraction. Arithmetic exponentiation can be expressed in the form of logarithmic multiplication. Arithmetic root calculation can be expressed in the form of logarithmic division. There are certain arithmetic operations that cannot be expressed logarithmically - arithmetic addition and arithmetic subtraction. There are certain logarithmic operations that cannot be expressed arithmetically - multiplication of logarithms and division of logarithms. These are the limitations / constraints that I am grappling with throughout my mathematical explorations.

    The mathematics of tuning gets even more complex once we factor in beats and inharmonicity. Beat rate equations are linear, which means that we must account for the linear difference between the frequencies of the partials, even though pitches progress logarithmically in line with how humans are thought to perceive pitch. Inharmonicity is the word that is used to describe the phenomenon where the frequencies of the partials depart from whole multiples of the fundamental frequency. Inharmonicity is influenced by several factors, which means that it is an absolute nightmare to accurately account for it mathematically. 

    Everything comes together in theory and in practice once we focus on the production of equally beating intervals, though. It all boils down to knowing either how the partials of an interval are affected by the movement of its notes or how the notes of an interval are affected by the movement of its partials. Therein lies the final piece of the puzzle.

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

    Original Message


  • 35.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-25-2025 18:29

    Peter - Yes, thank you for adding clarity. I'll add a wee bit more. As you already understand, each instrument had its unique sound because of the volume levels and exact pitch placement of partials. Inharmonicity contributes to this effect, as you pointed out. A piano that is more "bright" may have stronger upper partials while a piano that is more "dark" may have a stronger fundamental. Clarinets are virtually missing every other partial. There is a book somewhere in my house that discusses violin strings. If plucked, they have measurable inharmonicity. This inharmonicity seems to be present at the initial attack of the bow (extremely short), but as soon as the bow gets moving, the harmonics "lock in", making them mathematically pure with zero inharmonicity. I assume this has something to do with piano wire behaving the same way. There is a book I got once via interlibrary loan that charted typical inharmonicities of wind instruments. Interesting stuff. It is my current understanding that reed pipes have a negligible amount of inharmonicity, so it's only of interest if one is getting nit-picky. May as well be zero. 

    Hope my random 2-centers aren't too far off topic. 



    ------------------------------
    Maggie Jusiel, RPT
    Athens, WV
    (304)952-8615
    mags@timandmaggie.net
    ------------------------------

    Original Message


  • 36.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-21-2025 12:10

    Hi Peter,

    I appreciate your enthusiasm for pragmatism. I am trying my best to improve my understanding of Alfredo Capurso's Circular Harmonic System for the sole purpose of applying its principles in practice. I read Alfredo's paper a few years ago. It took me quite a while to understand what the equation (3 − Δ)1 / 19 = (4 + Δ)1 / 24 on page 64 is doing in theory. It is making the Lower Note Common 12th and 15th, which is when the 12th and the 15th share the same lower note (Example: 12th A4-E6 and 15th A4-A6), beat equally in theory. It is far too easy to get bogged down in the details. Therefore, I have done my own research based on what little I know, which I have posted in this discussion thread.

    Bill Bremmer's EBVT is the embodiment of Alfredo Capurso's Circular Harmonic System in practice. They are both making intervals beat equally. Bill Bremmer has devised a practical method for doing this. Alfredo Capurso has devised a theoretical method for doing this. 

    Summary:

    Bill Bremmer's EBVT offers us a tried-and-tested practical method for tuning pianos via equally beating intervals. Alfredo Capurso's Circular Harmonic System, which is now a tried-and-tested theoretical method due to my findings, offers us a mathematical approach to making intervals beat equally. They complement each other in both directions (Theory → Practice and Practice → Theory).

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

    Original Message


  • 37.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-21-2025 15:00

    My research into Alfredo Capurso's Circular Harmonic System and Bill Bremmer's Equal Beating Victorian Temperament contributes to the completion of a theoretically and practically sound Equal Beating Temperament System.

    That is my final contribution to the mathematics of tuning.

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

    Original Message


  • 38.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-24-2025 15:24

    Roshan,

    Out of curiosity I emailed Bill and asked him if he had collaborated or been influenced in any way by Capurso's work and I have copied and pasted his response below as follows:

    The EBVT was developed in 1992 when I needed a Victorian style Well Temperament (very mild, nearly equal). Owen Jorgensen's Big Red Book had been published the previous year but it did not have quite what I was looking for. I needed something that I or anyone else could replicate and that had little, if any room for interpretive error.

     

    I had been using (and still do) a Sanderson device. I was used to the Exam Program partial selection which meant that Octave 3 is read on the 4th Partial (octave 5) and Octave 5 on 2nd partial (also Octave 5). I always kept my A3 at 0.0 (read on 4th partial). I simply assigned various numbers to each note until I got what I was looking for. I wanted the least deviation from 0.0 as possible. Therefore, my F3 and F4 got a 1.0. My G3 got a 2.0 and B3 a -2.0. My C4 got a 3.0 and E4 got a -2.0, etc.

     

    Of course, I had many skeptics but I, in fact, still use the same idea with mostly the same figures (only a few very slight alterations) as I did that day more than 30 years ago. So, the idea for the EBVT occurred in 1992 and has never substantially changed since then. It was MY idea, and mine alone.

     

    What did change and needed a number of revisions was the written version of it. The only person who really ever helped me with that was Owen Jorgensen. All other comments, statements, graphs, etc., were not helpful. Having developed some idea of how to spell it out, I had an original, an EBVT II and then an EBVT III but all were merely attempts to spell out accurately how to tune the temperament by ear. The final revision occurred in 2007, shortly before Owen Jorgensen's passing. It has remained ever since and has reverted to the original name: EBVT.

     

    As you can see, I did not reference any other material when developing the idea. It was really just trial and error, entering some small numbers on the ETD and evaluating the results until I got what I was looking for. That being said, Professor Jorgensen did say that it resembled very closely a little known and used temperament idea by 18th Century theorist, Neidhardt. The resemblance is only a coincidence. If the EBVT resembles any other temperament idea, it is also a coincidence unless someone else took and used my information.

     

    It is definitely possible for someone else to come up with a similar idea independently. In my writing about ET and Contiguous Major 3rds, someone accused my of copying the writing of another author but I had never read that material. I just came up with what it takes to tune those intervals by myself. It would be possible for any number of other people to do the same.

     

    I do not know the person that you referenced and have not read any of that material. What makes the EBVT special was not actually by design but turned out to be quite beneficial. I was only looking for small deviations from ET and nothing more but what I quickly realized was the high incidence of Equally Beating intervals. These serve to give the piano a clean and harmonious sound.

     

    I also developed a Quasi Equal Temperament that ended up with many of the same qualities: not by intentional design but simply as a lucky result of what I had done.

     

    I am not concerned about anyone's mathematical analysis and I am certainly not concerned about anyone's claim that the EBVT resembles anyone else's idea. The EBVT is MY idea, plain and simple. So is the ET via Marpurg Quasi Equal Temperament. Both simply have some serendipitous side benefits that make them unique and worth exploration.

     

    Bill Bremmer RPT



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

    Original Message


  • 39.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-25-2025 09:44

    Hi Peter,

    Thank you for sharing this information.

    The Circular Harmonic System is Alfredo Capurso's idea. The Equal Beating Victorian Temperament is Bill Bremmer's idea. It is important to get the facts straight, so I am grateful that you have reached out to Bill.

    I just want to clarify that Alfredo and Bill, through their own work, have made invaluable contributions to the theory and practice of tuning. When you combine their work, you get an Equal Beatment System that works in theory and in practice. Pragmatism is at the heart of my work, so I am trying to combine their work to close the gap between the theory of tuning and the practice of tuning. Inharmonicity is the bridge that connects them rather than the obstacle that separates them. I am building that bridge instead of overcoming that obstacle. I believe that I am closer than ever to figuring out how to do that by homing in on the production of 1 : 1 beat ratios. The key to solving that puzzle lies is the Equal Beatment System. My research into the Circular Harmonic System and the Equal Beating Victorian Temperament offers us a pathway for reaching that final destination.

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

    Original Message


  • 40.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Registered Piano Technician
    Posted 07-25-2025 15:18

    Roshan,

    Is there a place I can go to hear a clear demonstration of Capurso's ideas?  I found a couple of videos on YouTube but they really didn't TELL me anything new...or anything at all. That is, unless I'm actually missing something (I get told that on a regular basis so its quite possible here too).  The math is beyond my grasp, beats and explanations I can handle very well. 

    Peter Grey Piano Doctor 



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

    Original Message


  • 41.  RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals

    Posted 07-26-2025 16:12

    Hi Peter,

    Alfredo's YouTube channel: https://www.youtube.com/@AlfredoCapurso.

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

    Original Message