UDP: Difference between revisions

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A useful method to quickly find the bright generator is that it is always the generator that is the "large" variant of its generic interval class. As an example, in the diatonic scale we know that the generators are the fourth and the fifth. Of the two, the perfect fifth is the "large" type of fifth in the diatonic scale, whereas the "small" variant of fifth is the diminished fifth, so it is the bright generator. Likewise, the fourth is the "small" fourth, whereas the "large" fourth is the augmented fourth, and it is the dark generator.

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> and <math>T=7</math>, ~~and~~ <math>5^{-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 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.

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 A useful method to quickly find the bright generator is that it is always the generator that is the "large" variant of its generic interval class. As an example, in the diatonic scale we know that the generators are the fourth and the fifth. Of the two, the perfect fifth is the "large" type of fifth in the diatonic scale, whereas the "small" variant of fifth is the diminished fifth, so it is the bright generator. Likewise, the fourth is the "small" fourth, whereas the "large" fourth is the augmented fourth, and it is the dark generator.
-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> and <math>T=7</math>, and <math>5^{-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 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.