An Appreciation Thread for Tuning Practitioners from the Perspective of a Tuning Theoretician
In my experience as a tuning theoretician, I simply could not figure out why tuning practitioners were telling me that the mathematics of tuning does not hold up in practice. I kept meeting dead end after dead end throughout my mathematical explorations. I tossed my theories aside and started again every time, only to be eventually told that I was wrong. Was I wrong? No, I was not wrong. My calculations were perfect, and I even had my own set of checks and balances to ensure that my calculations were indeed correct. Were tuning practitioners wrong? No, they were not wrong. They were successfully tuning pianos day in and day out, so they knew exactly what they were doing. If the tuning theoreticians and the tuning practitioners were not wrong, why were they still at odds with each other? It did not make sense, and the worst part is that it did not make sense to me why it did not make sense. Therefore, I kept pushing the mathematics of tuning further and further until I ended up in a fantasyland of mathematical perfection, which was completely out of touch with reality.
However, there was one ray of hope left. I kept seeing a high prevalence of equal-beating checks in aural piano tuning instructions, so I explored the mathematics of equally beating intervals many years ago. I discovered that intervals beat equally in a specific contiguous arrangement when they are tempered in opposite directions by the same amount in cents within an overarching pure interval (https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=e8a9c904-de17-4d5e-97d8-1eba569e3dbf&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf). They are known as equally beating equally tempered intervals. I was performing my calculations in cents, and trust me when I say that finding equally beating intervals via cents calculations is like finding a needle in a haystack, so this was a surprising turn of events. I tried to see whether equally beating intervals are produced when the intervals in the aforementioned contiguous arrangement are flipped. The answer is a resounding no (https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=f2134099-71a1-43e6-92a9-3723ac44e6e5&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf&tab=digestviewer). However, something else was going on here. The frequency ratio of the overarching pure interval matched the beat ratio of the two contiguous tempered intervals that were inside it. I had discovered something special, but I could not explain what was really going on in the calculations. In fact, this is still a mystery to me.
I was having a fruitful discussion with Bill Bremmer many years ago about my Equal Temperament Semitone Combination System (ETSCS) (https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=30e63600-7b49-4484-a02e-270fae5c0ce2&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf&tab=digestviewer), and he told me that he makes 5ths and 4ths beat equally when the 4th is inside the 5th. I used my ETSCS to find a solution for this scenario, and I succeeded! The answer is Pure Major 2nd Equal Temperament, where the wide 5th D4-A4 and wide 4th E4-A4 beat equally at 18 Hz apiece. I realised how ridiculous my system truly was at that point. There was no consistency whatsoever. ETSCS was designed to temper intervals in opposite directions, but this example broke its design because the 5th and 4th were tempered in the same direction. On top of that, 18 Hz is a fantastically absurd value for the beat rate of a 5th and a 4th. To put the final nail in the coffin, ETSCS contradicts itself by creating multiple widths of Equal Temperament. When there are multiple widths of Equal Temperament, there is no Equal Temperament. Bill destroyed my ETSCS with one example, and I am grateful for that because he taught me that there are different ways of configuring equally beating intervals.
Many tuning practitioners told me to incorporate inharmonicity into my calculations so that they align with what is happening in practice. I asked for average inharmonicity constants, and Anthony Willey gave me the perfect answer by providing me with one for each note on the piano's keyboard along with an example of how to use what appears to be Harvey Fletcher's formula for inharmonicity (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=/caut/ourdiscussiongroup#bm9c0231ed-b318-4c6c-8f29-48b3f9576f29). I went a step further by creating a new system called the Expanding / Contracting Temperament System, where semitones expand and contract in a uniform manner across the piano's keyboard (https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=b1aa16f5-ee20-4569-bddb-c6192b96b283). I even went beyond that by incorporating the aforementioned inharmonicity formula into another system that I had created (https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=0f556934-0a14-4c9e-9051-3eacf2b5e8f1&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf). These systems take absurdity to the next level. Tuning practitioners had already told me countless times about the inconsistency of inharmonicity. Therefore, I had to find another way to account for inharmonicity instead.
After going through the motions again and again, I still could not figure out why I was unable to make intervals beat equally in a consistent manner. Therefore, I searched for answers. I found Alfredo Capurso's Circular Harmonic System (https://www.scribd.com/document/174787881/Alfredo-Capurso-A-New-Model-of-Interpretation-of-Some-Acoustic-Phenomena-Circular-Harmonic-System-C-HA-S). There was the answer that I was searching for all along. Tuning practitioners use beat rates, and here was a system for constructing musical scales that are based on equally beating intervals. I performed my initial beat ratio analyses of this system to check that it really does what it says (https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=d16c29b0-be3f-4736-a537-51c0e61be182&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf&tab=digestviewer). My first beat ratio analysis proves that Alfredo's original equation makes the Lower Note Common 12th and 15th beat equally by shifting the uncommon notes in opposite directions. My second beat ratio analysis proves that it makes the Upper Note Common 12th and 15th beat unequally by shifting the uncommon notes in opposite directions. I revised that equation by changing the plus and minus symbols into multiplication and division symbols to make it fully comply with the rules of logarithms (https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=956b5a0d-c3a7-4faa-8cd1-6f2cd894299f&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf&tab=digestviewer), but Alfredo resoundingly informed me that this was nonsense, so I went back to the drawing board.
At this point, I was convinced that Alfredo and Bill had all the answers. I was feeling hopeless, though, because the Circular Harmonic System had put everything that I had learned about the mathematics of tuning into question by using the arithmetic operations addition and subtraction in its original equation. Those mathematical operations are nowhere to be seen in cents calculations because there is no logarithmic equivalent for them. I eventually realised that my mathematical explorations were distorted by cents the entire time. Therefore, I decided to leave out cents altogether by focusing on directly manipulating frequencies and frequency ratios for beat rate calculations. There was the ultimate answer to every question that I ever had about piano tuning. I took another look at Alfredo's paper on the Circular Harmonic System, and I discovered the equations for making the 12th and 15th beat equally in four different configurations by shifting the uncommon notes (Upper Note Common, Lower Note Common, Contiguous 12th and 15th, and Contiguous 15th and 12th). I decided to create my "final discussion thread" on this forum because I thought that I had reached the end of the road in terms of mathematically exploring the theory of tuning (https://my.ptg.org/discussion/final-discussion-thread-circular-harmonic-system-for-equally-beating-intervals). My research into the Circular Harmonic System and the Equal Beating Victorian Temperament (https://billbremmer.com/ebvt) paved the way for the creation of the two principles that underpin the Equal Beating Temperament System (https://my.ptg.org/discussion/equal-beating-temperament-system-final-and-complete-edition). I finally completed my research into the mathematics of tuning by presenting a brand new mathematical model that is governed by vertical inharmonicity and beat rate distribution ratios (https://my.ptg.org/discussion/vertical-inharmonicity-and-beat-rate-distribution-ratios-bridging-the-gap-between-the-theory-and-practice-of-tuning).
The Future of the Mathematics of Tuning
In hindsight, my mathematical explorations were gravitating towards aural piano tuning the whole time. Aural piano tuning set the standard that the mathematics of tuning should have been based on right from the very beginning. The Pythagorean Comma is a beat rate, so the solution for tempering it out should have been based on the distribution of beat rates rather than the distribution of cents. This gives us a brand new Circle of 5ths and 4ths that is based on equally beating 5ths and 4ths as opposed to equally tempered 5ths and 4ths. Frequencies progress geometrically across the piano's keyboard, which means that all the beat rates must also progress geometrically across the piano's keyboard to temper out the mismatch that occurs as a result of the linear nature of beat rates and the logarithmic nature of pitch perception. When you think about it, cents do not make sense. The greatest injustice in the history of the mathematics of tuning occurred when the Pythagorean Comma was tempered out via the distribution of cents. Everything about piano tuning naturally fell into place in the mathematics of tuning once I tempered out the Pythagorean Comma via the distribution of beat rates.
The Equal Beating Temperament System lays the foundation for the future of the mathematics of tuning. I am certain that every temperament that is based on the distribution of cents has an equivalent that is based on the distribution of beat rates. Pure Octave Equal Beating Temperament (12-EBT) and Pure Octave Equal Temperament (12-ET) are two sides of the same coin. Beat rates are on one side; cents on the other. Cents create a discrepancy between the theory and practice of tuning. Beat rates eliminate that discrepancy. It is about time we tempered out the comma that lies between the theory and practice of tuning once and for all!
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Roshan Kakiya
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