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Vertical Inharmonicity and Beat Rate Distribution Ratios: Bridging the Gap between the Theory and Practice of Tuning

1. Horizontal inharmonicity is the degree to which inharmonic partials deviate from their harmonic counterparts, both of which are on the same partial level.
2. Vertical inharmonicity is the rate at which partials that are on different partial levels are deviating from each other, which can be derived from the fixed harmonic series in theory and the variable inharmonic series in practice.

Horizontal inharmonicity only exists in practice because there are only harmonic partials in theory. It is possible to artificially incorporate horizontal inharmonicity into the theory of tuning via inharmonicity formulas.

Vertical inharmonicity exists in theory and in practice because partials that are on different partial levels are moving at different rates in theory and in practice.

What is the meaning of all this? Vertical inharmonicity must be accounted for in theory and in practice. In practice, beatless intervals are tuned to overcome the effects of horizontal inharmonicity, but vertical inharmonicity still has to be dealt with.

The presence of vertical inharmonicity was not immediately apparent to me for several years, but it eventually burst forth only a few weeks ago when I tried to "tune" the Equal Beating Victorian Temperament theoretically by following Bill Bremmer's detailed temperament sequence instructions.

I figured out that vertical inharmonicity can be accounted for via beat rate distribution ratios, which distribute beat rates across partials that are on different partial levels. Beat rate distribution ratios are essentially bridges that enable beat rates to move from one partial level to another.

The difference between each beat rate is treated as a comma that must be tempered out. It is split up and spread across partials that are on different partial levels. Once it is tempered out, intervals beat equally.

Beat rate distribution ratios are found in the harmonic series. Replace each harmonic partial in theory with its inharmonic counterpart in practice to account for the rates at which inharmonic partials are deviating from each other in practice.

Beat Rate Distribution Ratio = P₁ : P₂

Beat Rate Distribution Fraction for P₁ = P₁ / (P₁ + P₂)

Beat Rate Distribution Fraction for P₂ = P₂ / (P₁ + P₂)

P₁ is either the harmonic partial or the inharmonic partial that is on one partial level.

P₂ is either the harmonic partial or the inharmonic partial that is on another partial level.

The ratio for the 1st and 2nd partial levels is 1 : 2 (Fractions: 1 / 3 for the 1st partial level and 2 / 3 for the 2nd partial level). The ratio for the 2nd and 3rd partial levels is 2 : 3 (Fractions: 2 / 5 for the 2nd partial level and 3 / 5 for the 3rd partial level). The ratio for the 3rd and 4th partial levels is 3 : 4 (Fractions: 3 / 7 for the 3rd partial level and 4 / 7 for the 4th partial level). The ratio for the 4th and 5th partial levels is 4 : 5 (Fractions: 4 / 9 for the 4th partial level and 5 / 9 for the 5th partial level). The ratio for the 5th and 6th partial levels is 5 : 6 (Fractions: 5 / 11 for the 5th partial level and 6 / 11 for the 6th partial level). The list goes on.

The presence of multiple beat rate distribution ratios suggests to me that there are multiple "optimum points". These "optimum points" may well be where minimum entropy is occurring.

There are two ways of dealing with this scenario. We can either search for the ultimate compromise through guesswork, where every "optimum point" is deoptimised to produce a smooth tuning curve, or choose which "optimum points" are tuned with precision at the expense of all the others, resulting in a jagged tuning curve. If the technology that enables us to tune precisely is already available, it should be utilised to the fullest extent.

Gone are the days of guesswork. Now we are tuning with precision.

Equal Beating Temperament System - Final and Complete Edition:

https://my.ptg.org/discussion/equal-beating-temperament-system-final-and-complete-edition

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Roshan Kakiya
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I should explain what is happening here. What you are seeing is a complete rewrite of the mathematics of tuning so that it actually aligns with what is happening in practice.

I have been exploring the mathematics of tuning for several years, but I finally decided to put theory into practice a month ago. Lo and behold, I shattered my previous understanding of the mathematics of tuning and replaced it with a brand new one within the space of a month. I am absolutely shocked at how out of touch our current understanding of the mathematics of tuning really is with what is happening in reality. My research into the Equal Beating Temperament System finally sets things straight. 

Things got so bad that I ended up questioning my own understanding of mathematics. I went right back to the basics and started from scratch. If you think that you know how the mathematics of tuning works, have a good look at my research into the Equal Beating Temperament System and tell me what you think.

I persevered until the very end to bridge the gap between the theory and practice of tuning. This is the end result. Make of it what you will.

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Roshan Kakiya
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