UDP: Difference between revisions

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where the term "large-sized" and "small-sized" refer to that each interval class in an MOS has two different sizes: a large and a small one.

In our case, it is easy to see that for the diatonic scale, the fifth is the bright generator. The more fifths you have going up from the tonic, the more sharps you have in your ~~scale~~, or the more "major" or "augmented" intervals you have. The fourth is likewise the dark generator, as increasing the number of fourths going up means you are increasing the number of "flats," or of "minor" or "diminished" intervals, etc.

In our case, it is easy to see that for the diatonic scale, the fifth is the bright generator. The more fifths you have going up from the tonic, the more sharps you have in your mode, or the more "major" or "augmented" intervals you have. The fourth is likewise the dark generator, as increasing the number of fourths going up means you are increasing the number of "flats," or of "minor" or "diminished" intervals, etc.

The standard convention, then, is to choose the *bright generator* for any MOS as our standard generator. This means that the "up" direction doesn't just mean more generators going up, but also means note pitches going up in general, as more "~~sharps~~" or "large" ~~intervals are added to the scale~~.

The standard convention, then, is to choose the *bright generator* for any MOS as our standard generator. This means that the "up" direction doesn't just mean more generators going up, but also means note pitches going up in general, as more of the intervals in your mode or "sharpened" or made into the "large" variant.

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 $L$ large steps, $s$ small steps, and $T$ total steps, the bright generator will always be $s^{-1} \text{mod} T$, where the result denotes the number of steps ascending from the tonic. As an example, for the diatonic scale we have $L=5$ and $T=7$, and $5^{-1} \text{mod} 7 = 4$, 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 $0$ steps rather than to $1$, so the result is one less than the conventional name: the "fifth" is $4$, the "fourth" is $3$, etc.) The dark generator can be found similarly as $s^{-1} \text{mod} T$.

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 UDP Notation ==

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 where the term "large-sized" and "small-sized" refer to that each interval class in an MOS has two different sizes: a large and a small one.
-In our case, it is easy to see that for the diatonic scale, the fifth is the bright generator. The more fifths you have going up from the tonic, the more sharps you have in your scale, or the more "major" or "augmented" intervals you have. The fourth is likewise the dark generator, as increasing the number of fourths going up means you are increasing the number of "flats," or of "minor" or "diminished" intervals, etc.
+In our case, it is easy to see that for the diatonic scale, the fifth is the bright generator. The more fifths you have going up from the tonic, the more sharps you have in your mode, or the more "major" or "augmented" intervals you have. The fourth is likewise the dark generator, as increasing the number of fourths going up means you are increasing the number of "flats," or of "minor" or "diminished" intervals, etc.
-The standard convention, then, is to choose the *bright generator* for any MOS as our standard generator. This means that the "up" direction doesn't just mean more generators going up, but also means note pitches going up in general, as more "sharps" or "large" intervals are added to the scale.
+The standard convention, then, is to choose the *bright generator* for any MOS as our standard generator. This means that the "up" direction doesn't just mean more generators going up, but also means note pitches going up in general, as more of the intervals in your mode or "sharpened" or made into the "large" variant.
 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 $L$ large steps, $s$ small steps, and $T$ total steps, the bright generator will always be $s^{-1} \text{mod} T$, where the result denotes the number of steps ascending from the tonic. As an example, for the diatonic scale we have $L=5$ and $T=7$, and $5^{-1} \text{mod} 7 = 4$, 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 $0$ steps rather than to $1$, so the result is one less than the conventional name: the "fifth" is $4$, the "fourth" is $3$, etc.) The dark generator can be found similarly as $s^{-1} \text{mod} T$.
 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 UDP Notation ==