UDP: Difference between revisions

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The bright generator can also easily be found using modular arithmetic and the modular inverse. If your scale has <math>L</math> large steps, <math>s</math> small steps, and <math>T</math> total steps, the bright generator will always be <math>s^{-1} \mod T</math>, where the result denotes the number of steps ascending from the tonic. As an example, for the diatonic scale we have <math>L=5</math>, <math>s=2</math> and <math>T=7</math>, so <math>2^{-1} \mod 7 = 4</math>, and indeed 4 steps ascending from the tonic is the bright generator: the perfect fifth. (Note that using this convention, the tonic itself maps to <math>0</math> steps rather than to <math>1</math>, so the result is one less than the conventional name: the "fifth" is <math>4</math>, the "fourth" is <math>3</math>, etc.)

The dark generator can be found similarly as <math>s^{-1} \mod T</math>. So for the diatonic scale, we have <math>s=2</math> and <math>T=7</math>, and <math>2^{-1} \mod 7 = 3</math>, where 3 steps ascending from the tonic is the perfect fourth.

The dark generator can be found similarly as <math>L^{-1} \mod T</math>. So for the diatonic scale, we have <math>s=2</math> and <math>T=7</math>, and <math>5^{-1} \mod 7 = 3</math>, where 3 steps ascending from the tonic is the perfect fourth.

The "bright generator" has sometimes also been called the '''chroma-positive generator''', and likewise the "dark generator" has sometimes been called the '''chroma-negative generator''', because of which direction they shift intervals in an MOS by its chroma (the chroma for any MOS is the difference between the large and small step).

The "bright generator" has sometimes also been called the '''chroma-positive generator''', and likewise the "dark generator" has sometimes been called the '''chroma-negative generator''', because of which direction they shift intervals in an MOS by its chroma (the chroma for any MOS is the difference between the large and small step, which is also the difference between the large and small third, fourth, etc).

== The UDP Notation ==

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 The bright generator can also easily be found using modular arithmetic and the modular inverse. If your scale has <math>L</math> large steps, <math>s</math> small steps, and <math>T</math> total steps, the bright generator will always be <math>s^{-1} \mod T</math>, where the result denotes the number of steps ascending from the tonic. As an example, for the diatonic scale we have <math>L=5</math>, <math>s=2</math> and <math>T=7</math>, so <math>2^{-1} \mod 7 = 4</math>, and indeed 4 steps ascending  from the tonic is the bright generator: the perfect fifth. (Note that using this convention, the tonic itself maps to <math>0</math> steps rather than to <math>1</math>, so the result is one less than the conventional name: the "fifth" is <math>4</math>, the "fourth" is <math>3</math>, etc.)
-The dark generator can be found similarly as <math>s^{-1} \mod T</math>. So for the diatonic scale, we have <math>s=2</math> and <math>T=7</math>, and <math>2^{-1} \mod 7 = 3</math>, where 3 steps ascending from the tonic is the perfect fourth.
+The dark generator can be found similarly as <math>L^{-1} \mod T</math>. So for the diatonic scale, we have <math>s=2</math> and <math>T=7</math>, and <math>5^{-1} \mod 7 = 3</math>, where 3 steps ascending from the tonic is the perfect fourth.
-The "bright generator" has sometimes also been called the '''chroma-positive generator''', and likewise the "dark generator" has sometimes been called the '''chroma-negative generator''', because of which direction they shift intervals in an MOS by its chroma (the chroma for any MOS is the difference between the large and small step).
+The "bright generator" has sometimes also been called the '''chroma-positive generator''', and likewise the "dark generator" has sometimes been called the '''chroma-negative generator''', because of which direction they shift intervals in an MOS by its chroma (the chroma for any MOS is the difference between the large and small step, which is also the difference between the large and small third, fourth, etc).
 == The UDP Notation ==