N2D3P9: Difference between revisions

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Before describing how to calculate <math>\text{N2D3P9}</math>, we define three simpler terms that are used in its formula:

# '''2,3-reduced''' ratios, which are also known as "[https://en.wikipedia.org/wiki/Rough_number 5-rough]" ratios. Because factors of <math>2</math> and <math>3</math> in pitch ratios are already notated by changing octaves or moving along the chain of fifths (... B♭♭ F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ Fx ...), <math>\text{N2D3P9}</math> only operates on ratios that have had their factors of <math>2</math> and <math>3</math> removed. For example, there are various numbers of factors of <math>2</math> and <math>3</math> in the following ratios: <math>\frac{16}{15}, \frac{10}{9}, \frac{6}{5}, \frac{5}{4}, \frac{27}{20}, \frac{45}{32}, \frac{64}{45}, \frac{40}{27}, \frac{8}{5}, \frac{5}{3}, \frac{9}{5}, \frac{15}{8}</math>, but when their factors of <math>2</math> and <math>3</math> are removed, they all reduce to <math>\frac{1}{5}</math> or <math>\frac{5}{1}</math>, and so they can all be notated using the same microtonal accidental, pointing either up or down, combined with different letters and sharps or flats. We say that <math>\frac{1}{5}</math> or <math>\frac{5}{1}</math> is the "2,3-reduced" form of these pitch ratios, and because <math>\frac{1}{5}</math> and <math>\frac{5}{1}</math> use the same accidental pointing either up or down, and because <math>\text{N2D3P9}</math> only operates on ratios whose numerator is larger than their denominator (superunison ratios), <math>\frac{5}{1}</math> can represent this entire "2,3-~~equivalent~~" ~~class~~ for the purpose of notation design.

# '''2,3-reduced''' ratios, which are also known as "[https://en.wikipedia.org/wiki/Rough_number 5-rough]" ratios. Because factors of <math>2</math> and <math>3</math> in pitch ratios are already notated by changing octaves or moving along the chain of fifths (... B♭♭ F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ Fx ...), <math>\text{N2D3P9}</math> only operates on ratios that have had their factors of <math>2</math> and <math>3</math> removed. For example, there are various numbers of factors of <math>2</math> and <math>3</math> in the following ratios: <math>\frac{16}{15}, \frac{10}{9}, \frac{6}{5}, \frac{5}{4}, \frac{27}{20}, \frac{45}{32}, \frac{64}{45}, \frac{40}{27}, \frac{8}{5}, \frac{5}{3}, \frac{9}{5}, \frac{15}{8}</math>, but when their factors of <math>2</math> and <math>3</math> are removed, they all reduce to <math>\frac{1}{5}</math> or <math>\frac{5}{1}</math>, and so they can all be notated using the same microtonal accidental, pointing either up or down, combined with different letters and sharps or flats. We say that <math>\frac{1}{5}</math> or <math>\frac{5}{1}</math> is the "2,3-reduced" form of these pitch ratios, and because <math>\frac{1}{5}</math> and <math>\frac{5}{1}</math> use the same accidental pointing either up or down, and because <math>\text{N2D3P9}</math> only operates on ratios whose numerator is larger than their denominator (superunison ratios), <math>\frac{5}{1}</math> can represent this entire "2,3-equivalence-class" for the purpose of notation design.

# The '''copfr''' function, which stands for "Count Of Prime Factors with Repeats". 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>.

# The '''prime-limit''' function, 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 2,3-reduced <math>\frac{1}{1}</math> actually have a prime limit of 3.

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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 2,3-~~equivalent~~ class might be very close in pitch to some member of another ~~such~~ equivalence class, such as <math>\frac{65}{64}</math> being very close to <math>\frac{1}{1}</math>.

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 2,3-equivalence-class might be very close in pitch to some member of another 2,3-equivalence-class, such as <math>\frac{65}{64}</math> being very close to <math>\frac{1}{1}</math>.

== Development/Discovery ==

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Several techniques were used to find and decide on <math>\text{N2D3P9}</math> as the best 2,3-reduced 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 2,3-~~equivalent~~ 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 2,3-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.

@@ Line 7: / Line 7: @@
 Before describing how to calculate <math>\text{N2D3P9}</math>, we define three simpler terms that are used in its formula:
-# '''2,3-reduced''' ratios, which are also known as "[https://en.wikipedia.org/wiki/Rough_number 5-rough]" ratios. Because factors of <math>2</math> and <math>3</math> in pitch ratios are already notated by changing octaves or moving along the chain of fifths (... B♭♭ F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ Fx ...), <math>\text{N2D3P9}</math> only operates on ratios that have had their factors of <math>2</math> and <math>3</math> removed. For example, there are various numbers of factors of <math>2</math> and <math>3</math> in the following ratios: <math>\frac{16}{15}, \frac{10}{9}, \frac{6}{5}, \frac{5}{4}, \frac{27}{20}, \frac{45}{32}, \frac{64}{45}, \frac{40}{27}, \frac{8}{5}, \frac{5}{3}, \frac{9}{5}, \frac{15}{8}</math>, but when their factors of <math>2</math> and <math>3</math> are removed, they all reduce to <math>\frac{1}{5}</math> or <math>\frac{5}{1}</math>, and so they can all be notated using the same microtonal accidental, pointing either up or down, combined with different letters and sharps or flats. We say that <math>\frac{1}{5}</math> or <math>\frac{5}{1}</math> is the "2,3-reduced" form of these pitch ratios, and because <math>\frac{1}{5}</math> and <math>\frac{5}{1}</math> use the same accidental pointing either up or down, and because <math>\text{N2D3P9}</math> only operates on ratios whose numerator is larger than their denominator (superunison ratios), <math>\frac{5}{1}</math> can represent this entire "2,3-equivalent" class for the purpose of notation design.
+# '''2,3-reduced''' ratios, which are also known as "[https://en.wikipedia.org/wiki/Rough_number 5-rough]" ratios. Because factors of <math>2</math> and <math>3</math> in pitch ratios are already notated by changing octaves or moving along the chain of fifths (... B♭♭ F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ Fx ...), <math>\text{N2D3P9}</math> only operates on ratios that have had their factors of <math>2</math> and <math>3</math> removed. For example, there are various numbers of factors of <math>2</math> and <math>3</math> in the following ratios: <math>\frac{16}{15}, \frac{10}{9}, \frac{6}{5}, \frac{5}{4}, \frac{27}{20}, \frac{45}{32}, \frac{64}{45}, \frac{40}{27}, \frac{8}{5}, \frac{5}{3}, \frac{9}{5}, \frac{15}{8}</math>, but when their factors of <math>2</math> and <math>3</math> are removed, they all reduce to <math>\frac{1}{5}</math> or <math>\frac{5}{1}</math>, and so they can all be notated using the same microtonal accidental, pointing either up or down, combined with different letters and sharps or flats. We say that <math>\frac{1}{5}</math> or <math>\frac{5}{1}</math> is the "2,3-reduced" form of these pitch ratios, and because <math>\frac{1}{5}</math> and <math>\frac{5}{1}</math> use the same accidental pointing either up or down, and because <math>\text{N2D3P9}</math> only operates on ratios whose numerator is larger than their denominator (superunison ratios), <math>\frac{5}{1}</math> can represent this entire "2,3-equivalence-class" for the purpose of notation design.
 # The '''copfr''' function, 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>.
 # The '''prime-limit''' function, 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 2,3-reduced <math>\frac{1}{1}</math> actually have a prime limit of 3.
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-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 2,3-equivalent class might be very close in pitch to some member of another such equivalence class, such as <math>\frac{65}{64}</math> being very close to <math>\frac{1}{1}</math>.
+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 2,3-equivalence-class might be very close in pitch to some member of another 2,3-equivalence-class, such as <math>\frac{65}{64}</math> being very close to <math>\frac{1}{1}</math>.
 == Development/Discovery ==
@@ Line 86: / Line 86: @@
 Several techniques were used to find and decide on <math>\text{N2D3P9}</math> as the best 2,3-reduced 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 2,3-equivalent 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 2,3-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.