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The Equations and Solutions of Unequal Beating Temperament System - Kirnberger III

Introduction

I posted the complete study package of the Equal and Unequal Beating Temperament System (https://my.ptg.org/discussion/equal-and-unequal-beating-temperament-system-complete-study-package) on 16th January 2026. The purpose of this post is to show why 0.966360326 is the universal solution to 9 decimal places for Unequal Beating Temperament System - Kirnberger III (https://my.ptg.org/discussion/unequal-beating-temperament-system-kirnberger-iii-and-kellner), which I have calculated in the Excel file for this temperament in the complete study package of the Equal and Unequal Beating Temperament System by using Goal Seek. I mentioned in my discussion thread for Unequal Beating Temperament System - Kirnberger III that when one is focusing on beat rates, cents lose their meaning. I also mentioned in that thread that the System of Cents has been baked into the Equal and Unequal Beating Temperament System. I gave Cent = 2^{1 / 1200} as the standard definition of the cent. This post will illustrate that cents are eliminated from the equations and the solutions of this temperament, leaving us with frequencies and beat rates in Hz, which is the main objective of the entire Equal and Unequal Beating Temperament System.

Overview of Tools

I have used various tools throughout my mathematical explorations:

Microsoft Excel → Wolfram|Alpha (Online Calculator: https://www.wolframalpha.com) → MathPapa (Online Calculator: https://www.mathpapa.com) → Perplexity AI (Online AI: https://www.perplexity.ai) → MathGPT (Online AI: https://math-gpt.org) → Microsoft Excel

Equation

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

Source: https://my.ptg.org/discussion/unequal-beating-temperament-system-kirnberger-iii-and-kellner

Important notes: The Circle of Equally and Unequally Beating 5ths and 4ths is a Pythagorean-Grad Temperament System, which has been modified to create the Pythagorean-Grad-Syntonic-Schisma Temperament System. The base of the logarithms can be changed to modify this equation, but the same one should be used throughout to remain consistent. There are different ways of expressing the arithmetic operations addition, subtraction, multiplication, and division for programming purposes. For example, either the hyphen-minus symbol (-) or the minus sign (−) can be used in the equation above.

Solution

Δ = 1573660 / (402225 × (log₂(81 / 80) / log₂(531441 / 524288)) + 1259712)

It took several attempts to find this solution by using AI. The one AI that worked in the end was MathGPT, which only just managed to figure out this solution after many tries. Perhaps I am at fault. Maybe AI is at fault. I have checked this solution by pasting it into the Excel file for Unequal Beating Temperament System - Kirnberger III that is inside the complete study package of the Equal and Unequal Beating Temperament System to ensure that it is indeed correct. This is the needle in the haystack for Unequal Beating Temperament System - Kirnberger III.

Commas

Commas are discrepancies in the mathematics of music. In the Equal and Unequal Beating Temperament System, they are being managed via the distribution of beat rates across intervals. I included in my discussion thread for Unequal Beating Temperament System - Kirnberger III the following commas:

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

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

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

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

Cents Ratios

Syntonic Comma / Pythagorean Comma = (1200 × log₂(81 / 80)) / (1200 × log₂(531441 / 524288)) = 21.506289597 cents / 23.460010385 cents = 0.916721231

Schisma / Grad = (1200 × log₂(531441 / 524288) - 1200 × log₂(81 / 80)) / (1200 × log₂(531441 / 524288) / 12) = (23.460010385 cents - 21.506289597 cents) / 23.460010385 cents / 12 = 1.953720788 cents / 1.955000865 cents = 0.999345229

Insights

I have got the Δ and s variables from 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 are an infinite number of ways to express the equations and solutions of this temperament by changing the base of the logarithm in both the beat rate (Δ) and the beat rate modifier (s). The following bases are just a handful of the ones that are available:

• Log Base 2
• Log Base 3
• Log Base 10
• Natural Logarithm (Log Base e or ln)
• Log Base 531441 / 524288 (Log Base Pythagorean Comma)
• Log Base (531441 / 524288)^{1 / 12} (Log Base Grad)
• Log Base 81 / 80 (Log Base Syntonic Comma)
• Log Base 42515280 / 42467328 (Log Base Schisma) [Pythagorean Comma (Ratio) / Syntonic Comma (Ratio) = (531441 / 524288) / (81 / 80) = (531441 / 524288) × (80 / 81) = (531441 × 80) / (524288 × 81) = 42515280 / 42467328]
• Change of Base Formula
• Decimal Form

Logarithms are being divided in both the Δ and s variables. Two things are happening:

1. The change of base formula log_c(a) / log_c(b) = log_b(a) expresses the division of logarithms that share the same base in the form of a logarithm with a different base.
2. Cents are eliminated by dividing two logarithms with the same base (cents / cents).

If the base of the logarithm has changed, then the definition of the cent has changed, which means that multiple definitions of the cent exist in the Equal and Unequal Beating Temperament System. It is possible to eliminate cents when the definition of the cent is the same because the unit is the same. What would happen if we were to divide logarithms with different bases? We would end up with a hybrid cents unit (cents₁/cents₂), but it would still be possible to express it in the form of a cents ratio (cents / cents) by using the change of base formula.

In the next section, I will illustrate why the solution for Unequal Beating Temperament System - Kirnberger III is always the same when the logarithmic base is the same throughout its equation.

Log Base 2

Δ = 1573660 / (402225 × (log₂(81 / 80) / log₂(531441 / 524288)) + 1259712) = 0.966360326

Log Base 3

Δ = 1573660 / (402225 × (log₃(81 / 80) / log₃(531441 / 524288)) + 1259712) = 0.966360326

Log Base 10

Δ = 1573660 / (402225 × (log₁₀(81 / 80) / log₁₀(531441 / 524288)) + 1259712) = 0.966360326

Natural Logarithm (Log Base e or ln)

Δ = 1573660 / (402225 × (log_e(81 / 80) / log_e(531441 / 524288)) + 1259712) = 0.966360326

Log Base 531441 / 524288 (Log Base Pythagorean Comma)

Δ = 1573660 / (402225 × (log_{531441 / 524288}(81 / 80) / log_{531441 / 524288}(531441 / 524288)) + 1259712) = 0.966360326

Log Base (531441 / 524288)^{1 / 12} (Log Base Grad)

Δ = 1573660 / (402225 × (log_{(531441 / 524288)^{1 / 12}}(81 / 80) / log_{(531441 / 524288)^{1 / 12}}(531441 / 524288)) + 1259712) = 0.966360326

Log Base 81 / 80 (Log Base Syntonic Comma)

Δ = 1573660 / (402225 × (log_{81 / 80}(81 / 80) / log_{81 / 80}(531441 / 524288)) + 1259712) = 0.966360326

Log Base 42515280 / 42467328 (Log Base Schisma) [Pythagorean Comma (Ratio) / Syntonic Comma (Ratio) = (531441 / 524288) / (81 / 80) = (531441 / 524288) × (80 / 81) = (531441 × 80) / (524288 × 81) = 42515280 / 42467328]

Δ = 1573660 / (402225 × (log_{42515280 / 42467328}(81 / 80) / log_{42515280 / 42467328}(531441 / 524288)) + 1259712) = 0.966360326

Change of Base Formula

Δ = 1573660 / (402225 × log_{531441 / 524288}(81 / 80) + 1259712) = 0.966360326

Decimal Form

Δ = 1573660 / (402225 × 0.916721231 + 1259712) = 1573660 / (368728.197138975 + 1259712) = 1573660 / 1628440.197138970 = 0.966360326

Conclusion

It is just a bunch of numbers in the end.

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Roshan Kakiya
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Hi David,

Thank you for sharing your insights.

I realised in the end that the mathematics of music boils down to number-crunching.

Kirnberger III was the first temperament that required me to figure out the cents relationship between the Pythagorean Comma and the Syntonic Comma as well as the cents relationship between the Grad and the Schisma. The result was a blueprint in the form of an Excel file that can be used to convert any cents-based temperament that is based on the Pythagorean Comma or the Syntonic Comma into a beats-based one. I calculated the Pythagorean version first, then I converted it into the Syntonic version. I agree with you because Kirnberger III balances the relationship between the 5ths / 4ths and 3rds optimally. I mentioned in my discussion thread for Unequal Beating Temperament System - Kirnberger III that it is all about finding the optimal compromise. I had to separate the Pythagorean Comma into its constituent parts - the Grad, the Syntonic Comma, and the Schisma - to achieve the optimal balance.

Overall, the units Hz and cents both lose their meaning once you delve into the mathematics of music because you end up with a bunch of numbers. That is the key insight.

If it is all about the numbers, then the mathematics of music can be fully automated. Once the humans have done their bit, the machines can take over. There is only so much that we can do as humans.

Goal seek for the win!

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Roshan Kakiya
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Interesting insights. However any tuning system that fails to take into account dark matter and the expanding universe can't be taken seriously.

Ha! This is so simple a 9 year old child could figure it out. Gads! Find me a 9 year old child, can't make heads or tails of this!!

- Groucho Marx