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Rule of the Octave seems to have been fazed out from music. But I find it really interesting as greats such as Bach, Mozart, etc used it to improvise and make music. Also today's musicians that focus on classical improvisation, such as Mortensen, use it in their playing.

My understanding is that rule of the octave are just "what chords to pick" when you harmonize a bass line. But they usually show it as figured bass. What chords (in roman numerals) does it use in major and minor, ascending and descending?

Edit: a comment was made that the wikipedia page already had the chords. So I may have asked a bit too soon. If anyone can find the minor scale equivalent it will be very appreciated. These are the chords for the major scale:

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Image Source: Hiles, John (1882). A catechism of harmony, thorough-bass, and modulation, with examples, p. 82.

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    I'm unclear on what you mean by "why did diatonic roman numeral chords 'win'?" Often the chords in the Rule of the Octave are diatonic Roman-numeral chords. Or do you mean "why did chord progressions with basslines that aren't ascending/descending scales 'win'?" (Admittedly that doesn't roll off the tongue as easily!) – Richard Nov 16 '18 at 23:48
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    I don’t understand, literally the answer to the question is on the Wikipedia article you linked. Directly underneath the figures bass is a translation to Roman numerals. What more are you looking for? – Pat Muchmore Nov 19 '18 at 0:45
  • @PatMuchmore you're right wow. I never noticed since the roman numerals were such a weird order, maybe that threw me off. Do you think getting the minor form of that would be different or is it just a shift of roman numerals? – foreyez Nov 19 '18 at 2:20
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The key principle behind the Rule of the Octave is that scale degrees ➀ and ➄ receive what we would call a root position triad -- notated in figured bass by a 5 over a 3 (which I will typeset as 5/3) although these are typically left implied and no figures are are written. Meanwhile, most other notes of the scale generally receive what we would call a first position triad -- notated as 6/3 (some 6/3 chords have additional intervals added). The one major exception to this is the 6/4/2 chord on the descending ➃, which is what we would call a third inversion seventh chord.

It's also worth pointing out that there is not a single definitive Rule of the Octave; various authors occasionally made minor tweaks to suit there purposes, often to improve voice leading options. You'll see this as a 6/3 chord, with an additional interval added. Giorgio Sanguinari, in The Art of Partimento, has a "synoptic view" of several different author's versions of the major scale, which I'll simplify and summarize below.

Major Scale, Ascending

  • ➀ 5/3
  • ➁ 6/3 or 6/4/3
  • ➂ 6/3
  • ➃ 5/3 or 6/5/3
  • ➄ 5/3
  • ➅ 6/3
  • ➆ 6/3 or 6/♭5/3
  • ➇ 5/3

Major Scale, Descending

  • ➇ 5/3
  • ➆ 6/3
  • ➅ ♯6/3 or ♯6/4/3
  • ➄ 5/3
  • ➃ 6/4/2
  • ➂ 6/3
  • ➁ 6/4/3
  • ➀ 5/3

He also discusses the minor scale, and while he doesn't give the same convenient table, he does show Fenaroli's version, which is basically the same as the major. Note that the bass uses the melodic minor scale (so ➅ and ➆ are raised when ascending, and lowered when descending), and note that ➄ must always have a major third. In fact, most of the ♯'s in the figures below are to indicate a raised seventh scale degree (i.e. leading tone).

Minor Scale, Ascending

  • ➀ 5/3
  • ➁ ♯6/4/3
  • ➂ 6/3
  • ➃ 6/5/3
  • ➄ 5/♯3
  • ➅ 6/3
  • ➆ 6/♭5/3
  • ➇ 5/3

Minor Scale, Descending

  • ➇ 5/3
  • ♭➆ 6/3
  • ♭➅ ♯6/4/3
  • ➄ 5/♯3
  • ➃ 6/♯4/2
  • ➂ 6/3
  • ➁ ♯6/4/3
  • ➀ 5/3

One thing that may seem a bit unexpected in the above is the ♯6 over the ♭➅ scale degree. In the key of C minor, for example, this would be an A♭ in the left hand, with an F♯ in the right hand's chord, which creates an interval known as an augmented sixth.

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At a basic level the 'Rule of the Octave' TELLS you what chords to play over an ascending or descending scale in the bass. As such, it might help you get through an elementary harmony exercise. Look deeper and it actually deals with the stability (or otherwise) of each degree of the scale. And that remains the basis behind all functional harmony.

Read THIS PAGE (and also click through to THIS ONE

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