N2D3P9: Difference between revisions
Cmloegcmluin (talk | contribs) correct for Edit mode bug artifacts; shape up titles in table |
Cmloegcmluin (talk | contribs) separating passive and active voice / use of "we" to mean "we the authors" between sections |
||
| Line 48: | Line 48: | ||
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. | 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. | ||
== Justification == | == Authors' Justification == | ||
Why do we believe that <math>\text{N2D3P9}</math> accurately ranks the popularity of 5-rough pitch classes? | Why do we believe that <math>\text{N2D3P9}</math> accurately ranks the popularity of 5-rough pitch classes? | ||
| Line 76: | Line 76: | ||
== Development & Discovery == | == Development & Discovery == | ||
From May to August 2020, a [http://forum.sagittal.org/viewtopic.php?f=4&t=493 collaborative effort] to find such a function, was carried out by members of the [http://forum.sagittal.org/ Sagittal forum], led by Sagittal co-creator [[Dave Keenan]] and [[Douglas Blumeyer]]. Many functions besides <math>\text{N2D3P9}</math> were considered before selecting it as the best function for its purpose. | From May to August 2020, a [http://forum.sagittal.org/viewtopic.php?f=4&t=493 collaborative effort] to find such a function, was carried out by members of the [http://forum.sagittal.org/ Sagittal forum], led by Sagittal co-creator [[Dave Keenan]] and [[Douglas Blumeyer]]. Many functions besides <math>\text{N2D3P9}</math> were considered before selecting it as the best function for its purpose. | ||
| Line 89: | Line 87: | ||
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. | 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) Ratios by N2D3P9 == | == Table of Top 100 (5-Rough) Ratios by N2D3P9 == | ||