N2D3P9: Difference between revisions

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.

== ~~Authors'~~ Justification ==

== Justification ==

Why do we believe that <math>\text{N2D3P9}</math> accurately ranks the popularity of 5-rough pitch classes?

Why should we believe that <math>\text{N2D3P9}</math> accurately ranks the popularity of 5-rough pitch classes?

<math>\text{N2D3P9}</math> was developed or discovered rather late in the development of Sagittal notation. ~~So what did we use~~ previously~~, to decide which ratios should get the simple symbols? We used~~ actual ~~data on~~ ratio usage from [http://www.huygens-fokker.org/microtonality/scales.html the Huygens-Fokker Foundation's scale archive], kindly provided by [[Manuel Op de Coul]].

<math>\text{N2D3P9}</math> was developed (or discovered) rather late in the development of Sagittal notation. The Sagittal designers previously relied on actual ratio usage data from [http://www.huygens-fokker.org/microtonality/scales.html the Huygens-Fokker Foundation's scale archive], kindly provided by [[Manuel Op de Coul]].

All scales in the archive were treated equally, as ~~we didn't have any~~ information about their relative importance. Each occurrence of a pitch ratio in a scale was counted as one vote for that ratio. Then the ratios were grouped into 5-rough pitch classes and a single figure obtained for each 5-rough superunison ratio (representing the class). There were 29,403 votes, allocated to 820 5-rough ratios.

All scales in the archive were treated equally, as there was no information about their relative importance. Each occurrence of a pitch ratio in a scale was counted as one vote for that ratio. Then the ratios were grouped into 5-rough pitch classes and a single figure obtained for each 5-rough superunison ratio (representing the class). There were 29,403 votes, allocated to 820 5-rough ratios.

Like the frequency of use of letters in an alphabet, when sorted in order of decreasing popularity, the ratios obeyed an approximate [https://en.wikipedia.org/wiki/Zipf%27s_law Zipf's law] distribution, with the Nth most popular ratio having votes proportional to approximately <math>\frac{1}{N^{1.37}}</math>. This meant that about half the ratios had only one vote each, and three quarters of them had 3 votes or less. Such low numbers of votes meant that the data on the less popular ratios was vulnerable to "historical noise". In other words, the position of such a ratio in the list might not be a good predictor of its relative frequency of use in the future.

Line 62:

In the early stages of Sagittal design, when allocating symbols for the most popular ratios, we could rely on the Scala archive data, but when we moved on to less popular ratios we needed some "less noisy" way to rank them.

In the early stages of Sagittal design, when allocating symbols for the most popular ratios, the designers could rely on the Scala archive data, but when they moved on to less popular ratios they needed some "less noisy" way to rank them.

We found that <math>\text{N2D3P9}</math> is a psychoacoustically plausible function of a ratio's prime ~~factorizatation~~ that:

Blumeyer and Keenan found that <math>\text{N2D3P9}</math> is a psychoacoustically plausible function of a ratio's prime factorization that:

<ol>

<li value="a">ranks 10 of the 11 most popular ratios in exactly the same way as the archive data, and

Line 75:

~~Our~~ 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 5-rough equivalence class might be very close in pitch to some member of another 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 5-rough equivalence class might be very close in pitch to some member of another equivalence class, such as <math>\frac{65}{64}</math> being very close to <math>\frac{1}{1}</math>.

== Development & Discovery ==

@@ 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.
-== Authors' Justification ==
+== Justification ==
-Why do we believe that <math>\text{N2D3P9}</math> accurately ranks the popularity of 5-rough pitch classes?
+Why should we believe that <math>\text{N2D3P9}</math> accurately ranks the popularity of 5-rough pitch classes?
-<math>\text{N2D3P9}</math> was developed or discovered rather late in the development of Sagittal notation. So what did we use previously, to decide which ratios should get the simple symbols? We used actual data on ratio usage from [http://www.huygens-fokker.org/microtonality/scales.html the Huygens-Fokker Foundation's scale archive], kindly provided by [[Manuel Op de Coul]].
+<math>\text{N2D3P9}</math> was developed (or discovered) rather late in the development of Sagittal notation. The Sagittal designers previously relied on actual ratio usage data from [http://www.huygens-fokker.org/microtonality/scales.html the Huygens-Fokker Foundation's scale archive], kindly provided by [[Manuel Op de Coul]].
-All scales in the archive were treated equally, as we didn't have any information about their relative importance. Each occurrence of a pitch ratio in a scale was counted as one vote for that ratio. Then the ratios were grouped into 5-rough pitch classes and a single figure obtained for each 5-rough superunison ratio (representing the class). There were 29,403 votes, allocated to 820 5-rough ratios.
+All scales in the archive were treated equally, as there was no information about their relative importance. Each occurrence of a pitch ratio in a scale was counted as one vote for that ratio. Then the ratios were grouped into 5-rough pitch classes and a single figure obtained for each 5-rough superunison ratio (representing the class). There were 29,403 votes, allocated to 820 5-rough ratios.
 Like the frequency of use of letters in an alphabet, when sorted in order of decreasing popularity, the ratios obeyed an approximate [https://en.wikipedia.org/wiki/Zipf%27s_law Zipf's law] distribution, with the Nth most popular ratio having votes proportional to approximately <math>\frac{1}{N^{1.37}}</math>. This meant that about half the ratios had only one vote each, and three quarters of them had 3 votes or less. Such low numbers of votes meant that the data on the less popular ratios was vulnerable to "historical noise". In other words, the position of such a ratio in the list might not be a good predictor of its relative frequency of use in the future.
@@ Line 62: / Line 62: @@
-In the early stages of Sagittal design, when allocating symbols for the most popular ratios, we could rely on the Scala archive data, but when we moved on to less popular ratios we needed some "less noisy" way to rank them.
+In the early stages of Sagittal design, when allocating symbols for the most popular ratios, the designers could rely on the Scala archive data, but when they moved on to less popular ratios they needed some "less noisy" way to rank them.
-We found that <math>\text{N2D3P9}</math> is a psychoacoustically plausible function of a ratio's prime factorizatation that:
+Blumeyer and Keenan found that <math>\text{N2D3P9}</math> is a psychoacoustically plausible function of a ratio's prime factorization that:
 <ol>
 <li value="a">ranks 10 of the 11 most popular ratios in exactly the same way as the archive data, and
@@ Line 75: / Line 75: @@
-Our 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 5-rough equivalence class might be very close in pitch to some member of another 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 5-rough equivalence class might be very close in pitch to some member of another equivalence class, such as <math>\frac{65}{64}</math> being very close to <math>\frac{1}{1}</math>.
 == Development & Discovery ==