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Help: Implementing the Equal Beating Temperament System in Practice

  • 1.  Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-10-2025 16:00

    The Equal Beating Temperament System is based on the production of equally beating intervals. A different theoretical approach is required to align the theory of tuning with the practice of tuning. Temperaments are constructed by distributing beat rates across intervals rather than cents. This means that temperaments can be designed by using and expressed in the form of the Circle of Beating 5ths and 4ths rather than the Circle of Tempered 5ths and 4ths. Many of the theoretical aspects of tuning have changed, and I am finding it difficult to give them meaning and purpose practically. I have tried my best to find information about how other people have implemented the Equal Beating Temperament System in practice, but I am having a hard time as barely anything is available. This system challenges my previous understanding of piano tuning. The theory makes sense, but I have no clue as to how to implement it in practice yet. The whole point of this system is to take guessing out of the equation by setting the creation of 1 : 1 beat ratios as the sole target.

    Each temperament is set by counting the beat rate of one interval. That is the only beat counting that is required. All the other intervals are either beating exactly alike or beatless. Mathematically, the calculations are totally based on Hz values. The system of cents is redundant. The inharmonicity across different partial levels is in play in addition to the inharmonicity across the same partial level. The mismatch between the linear nature of beat rates and the logarithmic nature of pitch perception must be taken into consideration when each temperament is extended beyond its Temperament Octave across the entirety of the piano's keyboard.

    I cannot go back to the old theory of tuning at this point. The new theory of tuning is here to stay. Every cents-based temperament in existence can be reworked into a beats-based temperament. If there is an Equal Beating Temperament System, there is also an Unequal Beating Temperament System.

    How do I implement all of this in practice?

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



  • 2.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-10-2025 18:46

    Roshan, 

    Owen Jorgensen wrote in his big red book that absolute true EQUAL temperament is next to impossible to achieve due to the fluctuating situation in the piano (it's a moving target). However we can come close...essentially close enough. 

    Personally I am no longer a fan of ET. However, the Pure 12th version is as about as close as it gets IMO to an "ideal" ET.  I prefer several versions of UT, depending on the circumstances. 

    That said, Feaster promoted the "Dynamic Scale" Back in the 50's and in fact is the precise aural version of the equal beating data you published here earlier. Essentially it is tuned by adjusting all 4ths and 5ths to beat equally at "about 1bps". This does not amount to much deviation from the theoretical where the 5ths would beat slightly slower than the 4ths, but there is some logic behind this. His emphasis is on the SBI's and letting the RBI's vary slightly accordingly. The stretch factor is limited in this manner (basically locked in) and not as wide as P12th, but again there is some logic here. 

    The above would have the most equal beating intervals in it (I think). If you use the data you wrote you should be able to do it digitally. Scale anomalies will of course have some effect which would need to be accounted for aurally. 

    Peter Grey Piano Doctor 



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

    Original Message


  • 3.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-10-2025 21:25

    Hi Peter,

    It seems to work both ways. Creating equally beating intervals via cents calculations is like finding a needle in a haystack. Creating equally tempered intervals via beat rate calculations is like finding a needle in a haystack. The solution is to align the mathematics of tuning with aural piano tuning by letting go of cents-based mathematical models and replacing them with beats-based mathematical models. I am very much interested in reading everything that Owen Jorgensen has published on equal-beating temperaments.

    What makes Pure 12th Equal Temperament the "ideal" Equal Temperament in your view? 

    It was interesting to find out that Pure Octave Equal Beating Temperament coincides with A. L. Leigh Silver's Equal Beating Chromatic Scale and C. Raymond Feaster's Dynamic Scale. I am not aware of how they are being tuned. My research shows that the 5ths and 4ths of the tuning chain beat at approximately 1 Hz within a Pure Octave. How did they extend that temperament beyond the Temperament Octave? My solution is to halve the beat rate of each 5th and 4th of the tuning chain in each successive Octave downwards and double it in each successive Octave upwards to temper out a comma that keeps emerging. Would that approach hold up in practice? What effect does inharmonicity have on beat rates and beat ratios? What methods are available for extending the temperament? Are the 1 : 1 beat ratios of the 5ths and 4ths of the tuning chain within the Temperament Octave being maintained in each successive Octave upwards and downwards? If so, how?

    If I were to tune this temperament with the aid of an electronic tuning device, I would use beats-based solutions rather than cents-based solutions. More research is required to determine how to implement the Equal Beating Temperament System digitally.

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

    Original Message


  • 4.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-10-2025 23:22

    Roshan, first of all, I appreciate your diligence and contributions over the years with your theoretical work. It seems to me that in the past as now you come up against the same wall of theory versus practice. Peter addresses this in the first and last sentences of his reply. Simply said, due to numerous variables, there are decisions that can only be made in real time specific to the instrument you are tuning. The so-called "fluctuating situation in the piano". Given a specific instrument at a specific point in space and time there are ephemeral influences that will defy mathematical theory. 

    Add to this the ultimate goal being how the piano sounds to the ear, meaning the final judgement is not physical, it is psychoacoustical. The brain does not make this determination via mathematical analysis any more than our tongue does when it tastes food.  It may or may not conform to acoustical theory, interestingly, it may alternately conform and not conform to theory varying from moment to moment (fluctuate). This is an inherent characteristic of musicality.
    The piano is somewhat unique in that it is fixed-tuned and unusually high in inharmonicity, but in general playing with correct intonation in all acoustic musical instruments, as well as the voice, require minute adjustments in real time regardless of whether they conform to acoustic theory. Ultimately it is about musicality, not physics.
    It's worth noting that while electronic musical instruments are capable of adhering to mathematical theory invariably, in practice they almost always seek to synthesize these ephemeral variables through the introduction of "reverb" or other "noise". 

    Perhaps finding the needle in a haystack is the crux of good intonation. To mix metaphors, tuners are indeed 'hair splitters'.



    ------------------------------
    Steven Rosenthal RPT
    Honolulu HI
    (808) 521-7129
    ------------------------------

    Original Message


  • 5.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-13-2025 21:16

    Hi Steven and Peter,

    Thank you for sharing your insights. Piano tuning does fall short in many respects when it is solely treated as an objective scientific endeavour. It should also be treated as a subjective artistic endeavour to account for tastes and preferences. We are humans rather than machines after all.

    That last sentence is ever so important. I think I understand why I have been trying to bridge the gap between the theory and practice of tuning now. I have seen that people have been struggling to make the mathematics of tuning work in practice. Too much guesswork is involved. It is because the mathematics itself is at fault. Cents-based mathematical models are at odds with the practice of tuning. Beats-based mathematical models should be used instead so that the theory of tuning actually makes sense in practice. Both theoreticians and practitioners can work with each other rather than against each other at last.

    The "fluctuating situation in the piano". The "scale anomalies". The "needle in the haystack is the crux of good intonation". I could not have said it better myself. All of those factors came up when I was researching the Equal Beating Temperament System. The points at which equally beating intervals are formed are needles. They are highly specific points, which is why I refer to each one as a "needle in a haystack". As inharmonicity fluctuates, the needles change positions. There are "scale anomalies" in theory, which is exactly why I am utilising the flexibility of the Unequal Temperament System to account for them. More research needs to be performed to confirm this, but minimum entropy is potentially occurring at each needle. Perhaps that is the "crux of good intonation" that we seek.

    Before we can account for tastes and preferences, we have to get somewhere first. The mathematics of tuning provides us with models that operate as guides in practice. The final judgement, of course, is made by ear.

    Have you ever asked yourself about how many problems in practice are theoretical? My research has without a doubt highlighted the numerous fluctuations that are already happening in theory before the fluctuations in practice come into play. We might finally have a mathematical model, in the form of the Equal Beating Temperament System, that seamlessly combines the theory and practice of tuning.

    However, this is only the beginning. Much more research is needed to confirm whether or not these "postulations" are correct. If only it were easier to have access to the research of others. If anybody has access to any research into equal-beating temperaments, please do share it. I am always on the lookout for Owen Jorgensen's work.

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

    Original Message


  • 6.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-13-2025 21:43
    As an organist, an electronic organ always sounds wrong to me because it is absolutely in tune. No real pipe pecan be that perfect. 

    I tuned one of the really short string pianos that had to be amplified to be heard. The owner said that protools had some plugin to sound like that instrument and they purposely made it out of tune to make it more real. 

    The sound is more important than the mathematics. 

    Tom
    Tom Brantigan
    Piano Technician 
    All strange wordings come to you from Siri!





    Original Message


  • 7.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-13-2025 22:37

    Hi Thomas,

    I am going to play the devil's advocate.

    What is in tune? What is out of tune?

    If possible, I would like to know more about what temperament was being used in both instances. If one's ears are accustomed to Equal Temperament, maybe that is preferred above all else.

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

    Original Message


  • 8.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-14-2025 06:37

    Roshan,

    "I am going to play the devil's advocate.

    What is in tune? What is out of tune?"

    Until the advent of "fixed tone" instruments, good musicians/vocalists were able to satisfy their inclination toward "just" intervals by adjusting their pitch on the fly. "In tune" = just intervals. 

    The development of the twelve note keyboard (as we know it today) did not occur overnight, but rather was a gradual development over a long period, but essentially began with the diatonic scale (tuned in just intonation) of what we now refer to as the key of C (this was considered to be "pure and chaste" and became the basis of the "vulgar" or "common" tuning). Addition of other keys (i.e., "sharps/flats") came little by little as musicians wanted more access to other keys and harmonies. 

    Problems started escalating though with the addition of more keys to the keyboard. Today we know all the math behind it, but since we're talking 12th-14th century, they essentially went by the "feelings" they experienced with the various tunings of intervals. According to Owen Jorgensen, they did not hold intervals and listen to beats (as we do today), they "rolled" the interval and tuned it musically. Their musical prowess was the deciding factor in what was considered "in tune" (obviously this could vary somewhat), but the thing to remember is that everyone felt that the key of C (as we know it) was to be in absolute pure harmony (or as close as possible to it) and whatever negative collateral effects elsewhere simply were avoided (largely). Jorgensen even stated (in a class...not in print) that the "vulgar" tuning was mandated by the church for the organs used at that time, and that alteration of that form of tuning carried the threat of severe consequences. Therefore "in tune" = pythagorean tuning at that time. 

    The above (though generalized) laid the basis for believing that each key had a separate and distinct "character" or key coloration, and that certain keys or  combinations were to be avoided at all costs because they were musically unacceptable...period. If you (as a musician) did not follow the rules there would be consequences for doing so. 

    All fixed tone instruments require "tempering" (or alteration from pure) in order to be musically acceptable whereas non fixed tone instruments do not, since they can be "tuned" on the fly with a preference toward just intonation. Tempering is therefore a compromise, and this compromise is subject to interpretation and taste. Little by little overbthe centuries there has been "evolution" toward ET as we know it. Hundreds of UT versions of tuning have been utilized, but they all still center around the concept that the key of C should be as pure as possible (in various degrees) while allowing ALL the other keys to be musically acceptable. According to Jorgensen, ET is an outgrowth of the desire to not be shackled in any way by musical rules. It is atonal. There is no key coloration as in UT. It carries with it a degree of dissonance distributed "equally" though all keys and ignores the "requirement" of keeping C clean. 

    Therefore in ET, "intune" = mathematically equal distribution of the 12 tones to the octave with unisons tuned pure (or very close to). 

    There's a heck of a lot more to this story, but...

    Peter Grey Piano Doctor 



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

    Original Message


  • 9.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-14-2025 08:18

    Hi Peter,

    You have explained why Equal Temperament has become the standard better than I ever could. However, Owen Jorgensen, Bill Bremmer, Alfredo Capurso, and I have gone a step further by exploring equally beating intervals. I consider that step to be the final refinement of the theory of tuning.

    "It [Equal Temperament] carries with it a degree of dissonance distributed 'equally' though all keys". The key word is "dissonance". I recall reading a post by Bill Bremmer in which he alludes to the "dissonance cancelling effect" of equally beating intervals. My calculations suggest that "something" is lining up to make this happen. Perhaps it has got something to do with spectral lines and minimum entropy. It is possible to achieve "purity" by tuning beatless intervals. Another type of "purity" is achieved when intervals beat equally. Knowledge of how or why this is happening would lend credibility to the Equal Beating Temperament System.

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

    Original Message


  • 10.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-15-2025 16:46
    A further complication is that a "cent" is not a real value.  
    "440 Hertz., for example" , in contrast, is the same here, there and in the deep blue sea. But "cent" is 100th of a semitone -- or in equal temperament, 1/1200 of an octave. But that begs the question... whose octave tuned to what? A-63 tuned to 880 is not an in-tune octave to A-49 tuned at 440 on the piano at any set of coincident partials. And the width of the "octave" will be different whether it is a 2:1; 4:2; or 6:3 octave -- or some gestalt that is in between any set of coincident partials that just happens to "sound better" to real people with real ears. So, a "cent" in equal temperament will vary in actual finite width in terms of a fraction of a Hertz but all cents "should" be equal on any given semitone. 

    If you come up with unequal temperaments, any significant meaning of "cent" goes completely out the window as each semitone will vary slightly in width from the next and thus the "cent", as 1/100 of that variable semitone will vary for each note. 

    All the above may be irrelevant to the practicalities of tuning a Rudolph Wurlitzer spinet. But it is a practical obstacle to devising by mathematical means alone what the "correct" pitch in Hz should be in any given tuning scenario -- including mathematical means that comprise the internal hidden algorithm used in tuning devices.  This is much to the annoyance of many who have what I call (for lack of a better term) an "engineering mentality". Personally, I find it better just to relax and make the thing sound good rather than stressing over terminology that's inherently "fuzzy". 




    Keith Akins, RPT
    Piano Technologist
    715/775-0022 Mon-Sat 9a-9p
    Find me on LinkedIn



    Original Message


  • 11.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-14-2025 11:13

    Currently, the PiaTune beta for iOS is able to project beat rates of various intervals using the measured inharmonicty from the piano; useful for checking if the current settings and temperament strength is likely to accomplish the desired results. Send an email to support@piatune.com if you are interested in working with the beta version; he is very responsive to tech input.

    PianoScope has just released an update that includes graphs similar to the ones on the rollingball website - and I believe the graphs will work with custom temperaments. The developer is working on another update that may include a "strength slider" to customize temperaments to achieve beat rates/equal beating as desired by temperament designers. 

    I stumbled on to this issue with Tim Foster's recent mild temperaments that were designed to have 3 pure, or nearly pure 5ths. (That brings up another topic - how pure/equal beating is close enough to achieve desired results?) When I checked the beat projections with inharmonictity and midrange stretch loaded, I found that a 50% temperament strength most often achieved those pure 5ths without making any others beat too fast for my liking.

    PiaTune does go an extra step to apply corrections to partials of the inharmonictity constants which I believe allows for a more accurate representation of the piano scale which leads to a slightly different tuning calculation. The clue leading to this extra step was mis-matches observed in the display when stepping through different partials; if the inharmonicty data is accurate, it shouldn't matter which partial is selected for tuning since they all should agree. Instead of favoring the strongest partial, then the most stable partials can be favored for more stability in the visual display.

    So yes, I believe the technology is there or nearly there to help accomplish your goal.

    Ron Koval



    ------------------------------
    Ron Koval
    CHICAGO IL
    ------------------------------

    Original Message


  • 12.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-14-2025 12:27

    Hi Ron,

    That is excellent news, but I still have a few questions. There are two types of inharmonicity.

    There is the inharmonicity of coincident partials. For example, the 2 : 1 Octave. A beatless 2 : 1 Octave can be tuned by aligning the 1st partial of the upper note with the 2nd partial of the lower note.

    There is also the inharmonicity of non-coincident partials. For example, the 3 : 2 5th A3-E4 and the 4 : 3 4th B3-E4. If I am trying to make them beat equally by sharpening the common note E4, I am shifting the 2nd and 3rd partials of E4 at the same time, which are non-coincident partials. The key is to eliminate the difference between the beat rate of the 5th and the beat rate of the 4th. I am sure that Bill Bremmer has explained how this process works by stating that the aim is to listen to the quality of the intervals rather than the beat rates. In other words, the target is the production of 1 : 1 beat ratios.

    How is the inharmonicity of coincident partials being handled? How is the inharmonicity of non-coincident partials being handled?

    How is the temperament being extended beyond the Temperament Octave? Are Octaves, Double Octaves, Triple Octaves, and Quadruple Octaves being tuned?

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

    Original Message


  • 13.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-14-2025 12:50
    I'm currently using PiaTune, so in addition to choosing the midrange settings, there are multiple settings which transition to bass and treble. There is a 'magic wand' assist function that suggests, based on internal testing, which options may produce the better results for overall stretch. The chart that displays 3rd, 4th and 5th midrange beat rates for the chosen stretch has been very helpful, but limited as it applies to bass and treble choices.

    Might not be helpful for your needs, but this type of app approach is fairly new, so I'm watching with interest for updates to come.

    Ron Koval



    Original Message


  • 14.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-16-2025 07:36

    RE: non-coincident partials

    Someone correct me if I'm wrong here, but since non-coincident partials do not exhibit "audible" beats they are irrelevant to us tuners. The only "beats" we can manipulate toward our desired outcome are those we can detect. "Beats" only occur between frequencies that are relatively close together. 

    Again, if I'm in left field here let me know.

    Peter Grey Piano Doctor 



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

    Original Message


  • 15.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-16-2025 10:17

    Hi Peter,

    Horizontal inharmonicity is the degree to which inharmonic partials deviate from their harmonic counterparts in practice, both of which are on the same partial level. There is no horizontal inharmonicity in theory because partials are whole multiples of the fundamental frequency. This has an impact on the position of the "needle in the haystack" that I have described below.

    When one interval is tuned, multiple beat rates are present when its coincident partials are not in alignment. For example, to tune a beatless 3 : 2 5th, we have to align the 2nd partial of its upper note with the 3rd partial of its lower note to eliminate the difference between them in Hz. Only that coincident partial pair will be beatless when horizontal inharmonicity is present.

    When two intervals are tuned, we are no longer listening to beat rates. Instead, we are listening to the quality of the intervals. I will use the 3 : 2 5th A3-E4 and 4 : 3 4th B3-E4 in EBVT to illustrate what is happening. The 3 : 2 5th A3-E4 and 4 : 3 4th B3-E4 share a common note, which is E4. When the 1st partial of E4 moves, its 2nd and 3rd partials move at the same time. Its 2nd and 3rd partials are non-coincident partials, which are on different partial levels. Those partials are moving at different rates, which is vertical inharmonicity in theory and in practice.

    There are two coincident partial pairs:

    1. 3rd partial of A3 and 2nd partial of E4 for the 3 : 2 5th A3-E4.
    2. 4th partial of B3 and 3rd partial of E4 for the 4 : 3 4th B3-E4.

    The beat rate of the 3 : 2 5th A3-E4 and the beat rate of the 4 : 3 4th B3-E4 are moving at the same time when the non-coincident 2nd and 3rd partials of E4 are shifted at the same time. There will come a point where both beat rates are in alignment, i.e., the difference between them is 0 Hz. I am going a step further to ensure that the 1st partial of E4 has not split into two by making use of beat rate distribution ratios that are found in the harmonic series, i.e., the frequency ratio of E4 is 1 / 1. That is the specific "needle in the haystack" for the equally beating 3 : 2 5th A3-E4 and 4 : 3 4th B3-E4. The position of that needle is dependent upon the coincident partial pair of the 5th A3-E4 and the coincident partial pair of the 4th B3-E4 when horizontal inharmonicity is present.

    I have only managed to find out that this is happening by expressing Bill Bremmer's aural piano tuning instructions for the EBVT mathematically. Something truly special is going on here, and I am still trying to make sense of it all.

    Both of these examples show us how "purity" is achieved. Beatless intervals and equally beating intervals are connected because the overall objective is to have a 0 Hz difference between two Hz values.

    What is your interpretation? I would love to know what others have to say about all of this, particularly on the application of theory in practice. If I have got any holes in my knowledge, please feel free to point them out and correct them. I have still got a lot to learn about the Equal Beating Temperament System.

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

    Original Message


  • 16.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-16-2025 18:47

    My memory is fuzzy about Bill's progression while working on the EBVT series, but I believe the equal beating was first a bonus, then turned into a feature as he realized he could work with that to solidify the temperament. 

    I believe Tim Foster's "symmetric ET" also has the bonus of equal beating. Here is the type of projection that PiaTune is able to give of a Kawai UST-8 I just tuned this morning. I set the temperament strength at 1/2 of how Tim designed it because I find that most often achieves his close to pure designated 5ths. Then I set the temperament choice that most closely matches those pure fifths. Below is how the data is displayed in the app for this piano. Beats of M3, P4, P5 up from the listed note: 

    C#3 (5.7/0.3/-0.4)

    D3 (5.7/0.2/-0.7)

    D#3 (4.9/0.4/-0.5)

    E3 (5.9/0.6/-0.1)

    F3 (5.9/0.3/-0.5)

    F#3 (6.2/0.9/-0.5)

    G3 (7.7/0.9/0.0)

    G#3 (7.5/0.4/-0.6)

    A3 (7.6/1.1/-0.5)

    A#3 (9.0/1.2/0.1)

    B3 (9.0/0.4/-0.6)

    C4 (9.4/1.4/-0.6)

    C#4 (11.2/1.5/0.1)

    D4 (10.1/0.6/-0.7)

    The app also has a beats measuring just for the M3 so it is possible to verify the numbers actually can be achieved.

    The offsets for Tim's temperament are simple: +1 cent for B,D,F, G# and -1 cent for A#,C#, E, G. Since I used 1/2 strength, all offsets were changed to 0.5 or -0.5 to match. Awesome results for such minor adjustments, but of course the extra time to really get the pitch of each string close to the calculation is important.

    Ron Koval



    ------------------------------
    Ron Koval
    CHICAGO IL
    ------------------------------

    Original Message


  • 17.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-16-2025 20:30

    Ron, I agree, less offset is the best way to get the pure fifths in my temperaments. In well scaled pianos, instead of 1 cent offsets, I find 0.8 cent offsets works well in the 4:2 octave stretch. And agreed, P12  needs only 0.5 cent offsets. 



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

    Original Message


  • 18.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-18-2025 12:57

    Hi Ron and Tim,

    I understand that you have made your own temperaments.

    How do you tune them aurally?

    I am trying to simplify the aural piano tuning process as much as possible through the exclusive use of beatless and equally beating intervals within and beyond the Temperament Octave. I am open to learning about the various possibilities that exist.

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

    Original Message


  • 19.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-18-2025 16:05

    I designed these based on the qualities I believe help the piano resonate the best with minimal offsets. I tune them exclusively with an ETD, so I have no current procedure to tune them aurally, although I suppose you could tune a careful aural ET and then slightly make the changes necessary for symmetrically placed pure fifths.



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

    Original Message


  • 20.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-16-2025 22:54

    This according to Gemini AI bot:

    "Horizontal inharmonicity" is a specific type of inharmonicity that occurs in a piano due to the two-dimensional vibration of the strings. While a struck piano string vibrates primarily up and down (vertically), it also vibrates a small amount from side to side (horizontally). This is because the hammer strike is not perfectly centered, and the bridge that transfers the string's vibration to the soundboard is much stiffer in the horizontal direction than the vertical.

    Because the horizontal and vertical vibrations have slightly different frequencies and decay rates, they are not perfectly in sync. This phenomenon causes a subtle "beating" or wavering effect in the sound. While vertical inharmonicity-the more commonly discussed type, caused by the overall stiffness of the string-makes overtones slightly sharper than they should be, horizontal inharmonicity adds a unique timbral complexity and contributes to the characteristic "warmth" and rich, complex sound of a piano.

    Roshan,

    Is the above a reasonable facsimile of what you were saying?

    Peter Grey Piano Doctor 



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

    Original Message


  • 21.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-17-2025 08:54

    Hi Peter,

    I fed the terms "horizontal inharmonicity" and "vertical inharmonicity" into Perplexity AI.

    "Horizontal inharmonicity and vertical inharmonicity are related to the concept of inharmonicity in music, which describes how the frequencies of overtones or partials deviate from exact whole number multiples of the fundamental frequency.

    Vertical inharmonicity refers to the deviation of frequencies of the overtones (partials) within a single musical sound (or note) from the ideal harmonic series. This means the partials are not perfectly aligned as integer multiples of the fundamental frequency, causing the timbre to differ from an ideal harmonic sound. In string instruments, this inharmonicity happens due to the physical characteristics like thickness, tension, and elasticity of the string, which make the overtones slightly sharp or flat compared to exact harmonics.

    Horizontal inharmonicity, on the other hand, occurs across different notes or tones in a musical instrument or musical context. This can mean the tuning or spectral relationship between notes is not perfectly harmonic or consistent. For example, in piano tuning, horizontal inharmonicity can describe how the tuning of different notes does not correspond exactly to ideal harmonic intervals due to physical inharmonicity of the strings and the resulting stretched tuning scale.

    In summary:

    Vertical inharmonicity: Inharmonicity among the partials within a single note.

    Horizontal inharmonicity: Inharmonicity or deviation from harmonic relations among different notes across a musical scale or instrument tuning."

    It has pretty much read my mind. Marvellous.

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    Roshan Kakiya
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    Original Message


  • 22.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-20-2025 00:00

    The Importance of the 1st Partial

    I wish to add to the current body of research by emphasising the importance of the 1st partial.

    The Equal Beating Temperament System is based on the creation of 1 : 1 beat ratios. This is achieved by nullifying the difference between two beat rates via the production of equally beating intervals, i.e., the difference is 0 Hz. The nullification of the difference between any two Hz values naturally gives rise to beatless intervals and intact Unisons in this system. I have accounted for intact Unisons throughout my calculations, though it may not be obvious upon first inspection. This connects the Circular Harmonic System, the Equal Beating Victorian Temperament, the Equal Beating 5ths and 4ths Temperament System, and the Pure Octave Equal Beating Temperament. Intact Unisons are integral to the overall design of the Equal Beating Temperament System. It would be good to know what impact this has on the tuning of Unisons in practice.

    There is also the inverse relationship between the Pythagorean Comma and inharmonicity. I will use the 2 : 1 Octave A3-A4 in Pure Octave Equal Beating Temperament to illustrate this relationship as it is directly connected to the 1st partial level. 

    When the Pythagorean Comma is present, the 2nd partial of A3 is flatter than the 1st partial of A4. If inharmonicity sharpens the 2nd partial of the lower note, the Pythagorean Comma becomes smaller. The reason for pointing this out is that a decrease in the size of the Pythagorean Comma decreases the beat rate of each 5th and 4th of the tuning chain theoretically. Now, the inharmonicity of the coincident partials of all the 5ths and 4ths of the tuning chain must also be taken into consideration to determine the uniform beat rate of that chain in practice. In this scenario, the 2 : 1 Octave A3-A4 is wider than pure by flattening A3 by an amount that is less than the Pythagorean Comma (Beatless: 2nd partial of A3 = 1st partial of A4), the 5ths are narrower than pure, and the 4ths are wider than pure on the 1st partial level.

    If inharmonicity sharpens the 2nd partial of A3 by the same amount as the Pythagorean Comma, we find ourselves in a unique scenario where A3 is flattened by the same amount as the Pythagorean Comma. Theoretically, each 5th and 4th of the tuning chain is beatless. However, the inharmonicity of the coincident partials of all the 5ths and 4ths of the tuning chain must also be taken into consideration to determine the uniform beat rate of that chain in practice. In this scenario, the 2 : 1 Octave A3-A4 is wider than pure by flattening A3 by the same amount as the Pythagorean Comma (Beatless: 2nd partial of A3 = 1st partial of A4), the 5ths are pure, and the 4ths are pure on the 1st partial level.

    If inharmonicity sharpens the 2nd partial of A3 by an amount that is more than the Pythagorean Comma, the reverse Pythagorean Comma emerges. Theoretically, the beat rate of each 5th and 4th of the tuning chain increases to compensate for the emergence of the reverse Pythagorean Comma. However, the inharmonicity of the coincident partials of all the 5ths and 4ths of the tuning chain must also be taken into consideration to determine the uniform beat rate of that chain in practice. In this scenario, the 2 : 1 Octave A3-A4 is wider than pure by flattening A3 by an amount that is more than the Pythagorean Comma (Beatless: 2nd partial of A3 = 1st partial of A4), 5ths are wider than pure, and 4ths are narrower than pure on the 1st partial level.

    How is all of this relevant to the practice of tuning? The interplay between the inharmonicity of the coincident partials of the Octave A3-A4 and the inharmonicity of the coincident partials of each 5th and 4th of the tuning chain determines the uniform beat rate of that chain in practice. Inharmonicity must be accounted for when extending Pure Octave Equal Beating Temperament beyond the Temperament Octave A3-A4. Various methods exist for extending the temperament. More research is needed to figure out how to implement them practically.

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    Roshan Kakiya
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    Original Message


  • 23.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-20-2025 00:20

    Roshan, I highly recommend that you get your hands on a piano, a temperament strip, and a tuning lever, and experience these intervalic relationships for yourself. Particularly with aural tuning, it is something of a juggling act, manipulating the different notes until they coincide with your intent. Then you will find the relationships between the physical and the theoretical. With the aid of an ETD to get you close, it won't be that hard.



    ------------------------------
    Steven Rosenthal RPT
    Honolulu HI
    (808) 521-7129
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    Original Message


  • 24.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-20-2025 07:46

    Fundamental is the first partial.



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

    Original Message


  • 25.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-20-2025 08:29

    The only place you can generate a beat at the fundamental (1st partial) is in a unison. Otherwise ALL intervals are producing beats at the 2nd partial and/or  above. 

    Peter Grey Piano Doctor 



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

    Original Message


  • 26.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-20-2025 09:10
    "The only place you can generate a beat at the fundamental (1st partial) is in a unison. Otherwise ALL intervals are producing beats at the 2nd partial and/or  above."

    We know what you probably meant, but that didn't come out right, did it?

    2:1, 3:1, 4:1, 5:1, etc. are intervals generating beats with the first partial of the upper note.



    Original Message


  • 27.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-20-2025 11:02

    Hi Kent,

    What is the best way to extend a temperament beyond its Temperament Octave when inharmonicity is present? Theoretically, it is easy to create any temperament within the Temperament Octave. The real challenge ensues when it comes to extending it. Multiple methods are available. In the interest of pragmatism, what are the most practical methods? For example, the Beatless Multi-Octave (Octave, Double Octave, Triple Octave, and Quadruple Octave) Extension Method with Inharmonicity Correction seems rather appealing to me, but I am wondering about whether other methods are more practical and / or more effective at dealing with inharmonicity. I am searching for methods that are ideally applicable to both aural and electronic piano tuning. The Equal Beating Temperament System locks in beatless intervals, equally beating intervals, and intact Unisons, but a more nuanced approach might yield better results, e.g., blended compromises. This is where the art of tuning comes into play. I have studied your every-which-way temperament sequence with great interest.

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    Roshan Kakiya
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    Original Message


  • 28.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Registered Piano Technician
    Posted 09-20-2025 13:44

    Kent,

    Quite right...you interpreted correctly. I guess I was having a brain fart (a.k.a.,  "senior moment"?). Thanks for the correction. I'll try to do better next time. 

    Peter Grey Piano Doctor 



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

    Original Message


  • 29.  RE: Help: Implementing the Equal Beating Temperament System in Practice

    Posted 09-23-2025 19:00
    Alfredo Capurso's Circular Harmonic System: The Theoretical Framework Underpinning the Equal Beating Temperament System
     
    I want to make you aware of something that I have found through my research into the Equal Beating Temperament System, as it brings everything together.
     
    Two intervals coincide at their common note regardless of how they have been arranged. I have found that all the mathematical operations should be arithmetic to solve linear beat rate equations when this happens.
     
    The Δ variable represents the beat rate of each interval. The s variable appears to have two uses. In Bill Bremmer's Equal Beating Victorian Temperament, it splits into two, giving us s1 and s2, which are beat rate distribution ratios for dealing with vertical inharmonicity via equally beating intervals. In the Equal Beating 5ths and 4ths Temperament System, I am using the s variable to modify the beat rate of each 5th and 4th of the tuning chain to stretch or swing the scale uniformly. The scale can swing between Pure Octave Equal Beating Temperament, when s = 1, and Pythagorean Tuning, when s = 0. It is the Circle of Equally Beating 5ths and 4ths.
     
    The Circular Harmonic System is present throughout my research into the Equal Beating Temperament System. The theory and practice of tuning are combining seamlessly. The Δ and s variables play a pivotal role in every equation, although I must clarify that, based on my research, the s variable is not something that can be deliberately included in those equations. It is neatly hidden away, but it eventually emerges naturally / organically once the right conditions are met. In the Equal Beating Victorian Temperament, the conditions are met by the need to shift the common note to tune equally beating intervals. In the Equal Beating 5ths and 4ths Temperament System, the conditions are met by the need to stretch or swing the Circle of Equally Beating 5ths and 4ths uniformly.
     
    I should also state that Alfredo is shifting two uncommon notes in the Circular Harmonic System, Bill is shifting one common note in the Equal Beating Victorian Temperament, and I am shifting one uncommon note in the Equal Beating 5ths and 4ths Temperament System and Pure Octave Equal Beating Temperament. The idea here is to maximise flexibility and efficiency by enabling one to tune a chain of intervals.
     
    These insights facilitate the practical application of the Equal Beating Temperament System. More research is needed to clarify the exact link between horizontal inharmonicity and vertical inharmonicity, as well as the various practical uses of the Δ and s variables, in the equations of all the aforementioned systems / temperaments. From my standpoint, it seems as though the Δ variable deals with horizontal inharmonicity and the s variable deals with vertical inharmonicity, which indicates to me that both of these types of inharmonicity can be expressed mathematically in a unified manner, effectively bridging the gap between the theory and practice of tuning.
     
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    Roshan Kakiya
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