N2D3P9: Difference between revisions

'''<math>\text{N2D3P9}</math>''', or Entoo-Deethree-Peenine, is a fictional character in the Star Wars franchise. In an alternative timeline, the young Anakin Skywalker assembles the droid from the parts of three other droids: R<span style="color:#FF0000">2D</span>2, C<span style="color:#FF0000">3P</span>0 and <span style="color:#FF0000">N</span>R-N9<span style="color:#FF0000">9</span>. We're only joking, but we hope this helps with remembering and pronouncing the name.

Before describing how to calculate <math>\text{N2D3P9}</math>, we describe two simpler functions that are used in calculating it.

# '''copfr''', which stands for "<u>C</u>ount <u>O</u>f <u>P</u>rime <u>F</u>actors with <u>R</u>epeats". It applies to any positive integer. For example <math>175</math> has the prime factorization <math>5 × 5 × 7</math>, which has 3 factors including the repeat of <math>5</math>, so <math>\text{copfr}(175) = 3</math>. <math>\text{copfr}(1) = 0</math>. <math>\text{copfr}</math> is also called the "[https://en.wikipedia.org/wiki/Prime_omega_function big omega]" function, <math>Ω</math>.

# '''prime-limit''', which is also known as <math>\text{gpf}</math>, which stands for [https://mathworld.wolfram.com/GreatestPrimeFactor.html greatest prime factor]. <math>\text{prime-limit}(175) = 7</math>. Some authors leave <math>\text{prime-limit}(1)</math> undefined; we avoid the question because we define <math>\text{N2D3P9}(\frac{1}{1})</math> ≡ <math>\text{N2D3P9}(\frac{3}{1}) = 1</math>. This is because the ratios in the equivalence class represented by the 5-rough <math>\frac{1}{1}</math> actually have a prime limit of 3.

So the formula for <math>\text{N2D3P9}</math>(<math>\frac{n}{d}</math><nowiki>) is:

\small{\text{    where }n\text{ and }d\text{ are 5-rough positive integers and }n>d.} \\

The division by 9 does not affect the ranking, but it has the convenient effect that <math>\text{N2D3P9}</math> values are almost the same as the ranks they produce when applied to all 5-rough superunison ratios. Putting it another way, there are approximately <math>N</math> 5-rough pitch ratios with <math>\text{N2D3P9}≤N</math>. For example, <math>\text{N2D3P9}(\frac{77}{5}) = \frac{7}{2} × \frac{11}{2} × \frac{5}{3} × \frac{11}{9} ≈ 39</math>, suggesting there are approximately 38 other 5-rough pitch ratios more popular than <math>\frac{77}{5}</math>. There are actually about 4% fewer than that on average. In this case there are 36.

Why should we believe that <math>\text{N2D3P9}</math> accurately ranks the popularity of 5-rough pitch classes?

All scales in the archive were treated equally, as there was no information about their relative importance. Each occurrence of a pitch ratio in a scale was counted as one vote for that ratio. Then the ratios were grouped into 5-rough pitch classes and a single figure obtained for each 5-rough superunison ratio (representing the class). There were 29,403 votes, allocated to 820 5-rough ratios.

However, their approach was not able to consider ''all'' possible psychoacoustic reasons for a ratio's popularity. For example, <math>\text{N2D3P9}</math> does not evaluate whether some member of a 5-rough equivalence class might be very close in pitch to some member of another equivalence class, such as <math>\frac{65}{64}</math> being very close to <math>\frac{1}{1}</math>.

== Development & Discovery ==

The overall strategy, then, was to minimize this weighted rank correlation, while also minimizing the complexity of the function, to avoid overfitting. An earlier 5-rough-ratio notational popularity ranking function that had been used by the creators of Sagittal was <math>\text{sopfr}</math> ([https://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition]), and as simple as this function is, it does a remarkably good job of estimating the rank of pitch ratios. For comparison, the weighted sum of squares that <math>\text{sopfr}</math> gives for the Scala stats is about 0.026, while the weighted sum of squares <math>\text{N2D3P9}</math> gives is about 0.010. Functions giving sums of squares as low as 0.008 were found, however, these functions were so complex that they probably were fitting to noise in the Scala stats instead of to the true nature of musical pitch. An informal “chunk” metric was devised to compare function complexity in terms of fit to the data, with considered functions ranging from one chunk (<math>\text{sopfr}</math>) to eight chunks; the winning function <math>\text{N2D3P9}</math> has five chunks.

Several techniques were used to find and decide on <math>\text{N2D3P9}</math> as the best 5-rough ratio notational popularity rank estimation function. Initial observations about shortcomings of <math>\text{sopfr}</math>, such as its failure to differentiate balanced ratios from their imbalanced equivalents — such as <math>\frac{11}{5}</math> versus <math>\frac{55}{1}</math> — or those with different prime limits such as <math>\frac{13}{5}</math> and <math>\frac{11}{7}</math>, despite those pairs of ratios exhibiting remarkably different actual ranks in the Scala stats, formed the basis of the investigation. Psychoacoustic plausibility of functions was used as a top-down guide for experimentation. [https://en.wikipedia.org/wiki/Mathematical_optimization Optimization] tools such as [https://www.microsoft.com/en-us/microsoft-365/blog/2009/09/21/new-and-improved-solver/ Excel's Evolutionary Solver] were used to navigate toward ideal values for each parameter. The approach that was finally successful was a brute-force approach implemented by Douglas Blumeyer, whereby nearly 2 billion functions combined out of constituent "submetrics" were checked automatically. In the end, one of the functions on the short-list generated from the brute-force checker was recognized as being re-writable in a much simpler form with parameter values rounded to whole numbers without doing much damage to its sum-of-squares, and thus <math>\text{N2D3P9}</math> was born.

After deciding upon <math>\text{N2D3P9}</math>, the Sagittal forum members checked the ratios for the existing Sagittal symbols against it, to see how well they'd been served by the Scala archive stats and the earlier <math>\text{sopfr}</math> metric. Each symbol in Sagittal's JI notations has a default value, or primary comma, which allows it to exactly notate ratios in a 5-rough ratio equivalence class, and based on <math>\text{N2D3P9}</math>, it was found that only a couple of these commas should be changed (these were among the rarest-used symbols in Sagittal). This was as expected; <math>\text{N2D3P9}</math> was developed primarily in order to add new symbols to Sagittal, to enable it to exactly notate even rarer JI pitches than it already does.

== Table of Top 100 (5-Rough) Pitch Ratio Classes by N2D3P9 ==

!5-rough pitch ratio class