N2D3P9: Difference between revisions

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'''<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 N2D3P9 from the parts of three other droids: R2D2, C3P0 and NR-N99. We're only joking, but we hope this helps with remembering and pronouncing the name.

'''<math>\text{N2D3P9}</math>''' is a mathematical function which was developed to help in designing the [https://en.xen.wiki/w/Sagittal_notation Sagittal microtonal notation]. Given a pitch ratio <math>\frac{n}{d}</math>, <math>\text{N2D3P9}</math> estimates its rank in popularity among all rational pitches in musical use. A low value of <math>\text{N2D3P9}</math> indicates that the ratio is used often, and so should have a simple accidental symbol, while a high value indicates that the ratio is used rarely and so can have a more complex symbol if necessary. It may also be useful in designing rational scales or tunings. The name "N2D3P9" is an abbreviation of key components of its formula, described below.

'''<math>\text{N2D3P9}</math>''' is a mathematical function which was developed to help in designing the [https://en.xen.wiki/w/Sagittal_notation Sagittal microtonal notation]. Given a pitch ratio <math>\frac{n}{d}</math>, <math>\text{N2D3P9}</math> estimates its rank in popularity among all rational pitches in musical use. A low value of <math>\text{N2D3P9}</math> indicates that the ratio is used often, and so should have a simple accidental symbol, while a high value indicates that the ratio is used rarely and so can have a more complex symbol if necessary. It may also be useful in designing rational scales or tunings. The name "'''N2D3P9'''" is an abbreviation of key components of its formula, described below.

== Formula ==

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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-free''' 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-removed" or '''2,3-free''' 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.

# '''2,3-free''' 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 ...), '''N2D3P9''' 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-removed''' or '''2,3-free''' 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 '''N2D3P9''' 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 pitch ratio class''' or '''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-removed <math>\frac{1}{1}</math> actually have a prime limit of 3.

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 '''<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 N2D3P9 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.
-'''<math>\text{N2D3P9}</math>''' is a mathematical function which was developed to help in designing the [https://en.xen.wiki/w/Sagittal_notation Sagittal microtonal notation]. Given a pitch ratio <math>\frac{n}{d}</math>, <math>\text{N2D3P9}</math> estimates its rank in popularity among all rational pitches in musical use. A low value of <math>\text{N2D3P9}</math> indicates that the ratio is used often, and so should have a simple accidental symbol, while a high value indicates that the ratio is used rarely and so can have a more complex symbol if necessary. It may also be useful in designing rational scales or tunings. The name "N2D3P9" is an abbreviation of key components of its formula, described below.
+'''<math>\text{N2D3P9}</math>''' is a mathematical function which was developed to help in designing the [https://en.xen.wiki/w/Sagittal_notation Sagittal microtonal notation]. Given a pitch ratio <math>\frac{n}{d}</math>, <math>\text{N2D3P9}</math> estimates its rank in popularity among all rational pitches in musical use. A low value of <math>\text{N2D3P9}</math> indicates that the ratio is used often, and so should have a simple accidental symbol, while a high value indicates that the ratio is used rarely and so can have a more complex symbol if necessary. It may also be useful in designing rational scales or tunings. The name "'''N2D3P9'''" is an abbreviation of key components of its formula, described below.
 == Formula ==
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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-free''' 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-removed" or '''2,3-free''' 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.
+# '''2,3-free''' 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 ...), '''N2D3P9''' 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-removed''' or '''2,3-free''' 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 '''N2D3P9''' 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 pitch ratio class''' or '''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-removed <math>\frac{1}{1}</math> actually have a prime limit of 3.