N2D3P9: Difference between revisions

Line 7:

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 ...), '''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.

# '''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.

@@ 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-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.
+# '''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.