Original Message:
Sent: 7/25/2025 7:50:00 AM
From: Peter Grey
Subject: RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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
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Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com
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Original Message:
Sent: 07-24-2025 18:01
From: David Pinnegar
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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
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David Pinnegar, B.Sc., A.R.C.S.
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+44 1342 850594
Original Message:
Sent: 7/24/2025 3:20:00 PM
From: Peter Grey
Subject: RE: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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
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Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com
Original Message:
Sent: 07-24-2025 13:59
From: Maggie Jusiel
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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
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Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
Original Message:
Sent: 07-21-2025 12:40
From: Roshan Kakiya
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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.
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Roshan Kakiya
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Original Message:
Sent: 07-21-2025 11:19
From: Peter Grey
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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
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Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com
Original Message:
Sent: 07-21-2025 11:06
From: Peter Grey
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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
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Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com
Original Message:
Sent: 07-20-2025 22:31
From: Roshan Kakiya
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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 a purely theoretical environment, where inharmonicity is non-existent.
I might be accounting for inharmonicity in theory without even realising that I am doing it, e.g., by creating beat rate equations for making two intervals beat equally in different configurations / arrangements, then checking whether or not they produce a perfect 1 : 1 theoretical beat ratio to test and verify their robustness. Tests exist in the practical inharmonicity-laden environment for checking 1 : 1 beat ratios, so the theory 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 what effect the movement of an interval's lower and upper notes has on the movement of its coincident partials. That knowledge would substantially assist me in figuring out how to incorporate inharmonicity into the theoretical side of tuning in a way that works in practice. The 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.
How do I close / bridge that gap?
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Roshan Kakiya
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Original Message:
Sent: 07-20-2025 21:17
From: Tim Foster
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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.
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Tim Foster RPT
New Oxford PA
(470) 231-6074
Original Message:
Sent: 07-20-2025 20:13
From: Paul Klaus
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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:
Sent: 07-20-2025 18:09
From: Peter Grey
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
Paul,
I haven't got a clue what you're talking about here. Can you elaborate?
Peter Grey Piano Doctor
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Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com
Original Message:
Sent: 07-20-2025 17:04
From: Paul Klaus
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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:
Sent: 07-17-2025 18:59
From: Roshan Kakiya
Subject: Final Discussion Thread: Circular Harmonic System for Equally Beating Intervals
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 post 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. However, that approach did not directly produce two equally beating intervals. 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 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.
I should explain what is happening in each of the four attached documents.
- 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.
- Equation Conversion Processes for Circular Harmonic System Equations for 12th and 15th: I went ahead and constructed processes for converting all the equally beating equations 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.
- 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.
- 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 another look at Alfredo Capurso's Circular Harmonic System to understand and appreciate what it offers us in the theoretical world of the mathematics of tuning.
I must confess that I have not figured out how to apply either my own extensive mathematical explorations or those of others practically in the real world of tuning yet.
Thank you.
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Roshan Kakiya
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