N2D3P9: Difference between revisions
Cmloegcmluin (talk | contribs) this and the previous change are to correct for a bug in the wiki which yanked MathJax out of the previous paragraph |
Cmloegcmluin (talk | contribs) add info about why N2D3P9 was developed and how it affected Sagittal |
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== Development | == Development & Discovery == | ||
How did we come up with that particular 5-rough notational-popularity ranking function? | How did we come up with that particular 5-rough notational-popularity ranking function? | ||
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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. A brute-force technique was also utilized whereby nearly 2 billion functions combined out of constituent "submetrics" were checked automatically. In the end, one of the functions 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. | 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. A brute-force technique was also utilized whereby nearly 2 billion functions combined out of constituent "submetrics" were checked automatically. In the end, one of the functions 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 Sagittal against it, to see how well they'd been served by <math>\text{sopfr}</math>. 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) Ratios == | == Table of Top 100 (5-Rough) Ratios == | ||