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Just to make things clear, let's set some limits:

  1. The scale is F major and the tonic is F
  2. One of the scales has to start and end on F
  3. The other note can start from any note in the F major scale up to an octave away from the other scale's starting point
  4. They have to move in contrary motion

Now, my gut feeling is that there can be a maximum of 4 scale notes in a row before dissonance crops up. Why? Because other than the octave and unison, there are only 4 consonant intervals from the tonic to any other note in the scale. Similar things apply for the other notes of the scale, except for the fourth and the seventh, where only 3 other diatonic intervals are consonant besides the unison and octave. And since the scales are moving in contrary motion, they are bound to hit every interval, including the dissonant ones, with the tritone being the only exception(and also the most noticeable dissonance when the notes are spread out more than an octave apart).

Well, let's see, at the unison we have this:

F G A Bb C D E F

F E D C Bb A G F

Which leads to this interval pattern(condensing the intervals to within an octave to more clearly see scale relationships):

Unison Third Fifth Seventh Second Fourth Sixth Octave

A string of 3 consonant intervals, followed by dissonance, and then again 3 consonant intervals. What if it starts at the second? Well, then we get this pattern:

F G A Bb C D E F

E D C Bb A G F E

Which leads to this interval pattern:

Second Fourth Sixth Octave Third Fifth Seventh Second

Huh, 5 scale notes before dissonance hits.

Scales at the Third:

F G A Bb C D E F

D C Bb A G F E D

Third Fifth Seventh Second Fourth Sixth Octave Third

2 and then 4

Scales at the Fourth:

F G A Bb C D E F

C Bb A G F E D C

Fourth Sixth Octave Third Fifth Seventh Second Fourth

5 scale notes in a row

Okay, so I didn't take the octave into account when I made my prediction, but I got pretty close. Patterns at the Fifth, Sixth, Seventh, and Octave are going to be the same as what I have here, so I didn't bother checking those.

So why is it that a maximum of 5 scale notes in a row can produce consonant intervals in major scales in contrary motion? And does this number hold for minor scales, including harmonic and melodic minor? Or do rarer intervals that show up in harmonic and melodic minor such as the augmented fifth lower this maximum consonance number?

2

So why is it that a maximum of 5 scale notes in a row can produce consonant intervals in major scales in contrary motion?

Because they must inevitably pass each other, which involves 2 consecutive intervals of 2nds/7ths. That leaves up to 5 to be consonant. If you start the descending scale on A, you get 2 instances of tritones as well as the 2nds/7ths, so this one only has 3 consonant intervals.

You should notice that the intervals always appear in the same order. Changing which note you start on simply rotates them, but their order is fixed: unison, 3rd, 5th, 7th, 9th(2nd), 11th(4th), 13th(6th). So they all have 5 consecutive intervals, except for the one rotation where the 4th and 5th are augmented and diminished, respectively.

And does this number hold for minor scales, including harmonic and melodic minor?

The natural minor is literally identical to the major because it is the same collection of intervals.

The melodic minor works out to be roughly equivalent because with it too you inevitably need the voices to pass, so you get 2nd/7ths. This is the case if you raise the 6th and 7th scale degrees ascending only (which is true melodic minor) and if you raise them in both directions.

The harmonic minor, because it has an augmented 2nd (which is enharmonically equivalent to a minor 3rd, which is consonant) it can actually escape this and have all 7 intervals be consonant. But only 1 rotation achieves this, when the descending scale starts on the 5th scale degree. All the other rotations have the same problem.

In general, it simply depends on the particular series of intervals and how it interacts with its inverted self as you rotate them. In all cases, the intervals will always appear in the same order: unison, 3rd, 5th, 7th, 2nd, 4th, 6th. The exceptions will occur when one or more of those intervals is augmented or diminished. I expect all heptatonic scales to have this feature.

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