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I recently saw a video of someone playing Dorian licks on YouTube. They were going through all 12 keys, but they ascending or descending chromatically, by 4ths, or by 5ths. What is this pattern?

C-Db-Bb-B-Ab-A-Gb-G-E-F-D-Eb

Here's a link to the 1:20 video: Neo Soul Progress 1/5 - Dorian Mode Jazz Piano Licks in all 12 Keys

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If you're asking whether there are other ways to change keys besides going up or down a fourth or a fifth, the answer is "yes, there are many many ways". If you're asking why going up a fouth or down a fifth sounds the way it does, then that is a question with a much larger answer. Which question are you really asking? – Todd Wilcox Feb 26 at 21:01
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The first question. – Ritchel Cousar Jr. Feb 26 at 21:02
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Looks like picking a starting note, then: moving one semitone up, then 3 down, repeat. I think it's just to have a simple pattern that isn't completely boring -- he's basically doing it in descending order, but flipping every other to be ascending and adding in the minor third drop. – Matthew Read Feb 26 at 21:35
    
Fourths and fifths have the nice feature that as well as taking you through all 12 keys in sequence, you only change the pitch of one note in each key-shift, so it's actually quite difficult to make the transitions sound "bad" (whatever your definition of "bad"). You can return to the starting key by 6 steps of a tone, four of a minor 3rd, or three of a major third. The major/minor third sequences have been used in classical music at least as far back as Beethoven. Your key pattern is actually two interleaved whole-tone scales (with 6 keys in each), starting on C and Db. – alephzero Feb 27 at 1:45
up vote 10 down vote accepted

There are multiple levels on which one could answer this question.

For one thing: 1, 5, 7 and 11 are the only* numbers coprime to 12, hence the only way to traverse the entire 12-edo palette through iteration of a single interval is with fourths/fifths or chromatically.
The sequence you ask about is in essence descending in whole steps – this obviously does not cover all 12 keys; the remaining are simply “filled in”, by immediately repeating each key a half-step higher. If you allow for such combinations, there are of course much more possibilities to traverse the palette.

Now, set theory actually has nothing to do with harmony. Really, the reason fourths and fifths are the most common modulation intervals is completely unrelated to the above argument: these intervals have the simplest frequency ratio, namely 3:2 and 4:3. Hence notes from both keys share a lot of overtones when played on most instruments. This is why keys spaced by a fifth “sound related”, allowing modulations to go very smooth and natural.
(It is actually a pretty beautiful fact about the 12-edo tuning system that the most harmonic modulations are the only nontrivial ones that reach every key!)

Clearly, 3:2 and 4:3 are not the only nice frequency ratios. 5:3, 5:4 and 6:5 would be the next best ones. These are the just-intonation major sixth, major and minor thirds. They have representations in 12-edo tuning, albeit unlike fourths and fifths (which are virtually indistinguishable from just intonation) not quite exact. In fact the 12-edo major third is right between a just-intonation (Ptolemaian) 5:4 and the stacking of two Pythagorean whole steps. This creates a sort of ambiguity: a third can be heard both as an “atomic” consonant interval, and as a compound scale degree. I'd speculate that a full modulation§ over one of these intervals is rather perceived as compound and hence doesn't feel nearly as natural as a fourth/fifth modulation. That does of course not mean you can't modulate by such an interval, only, it will have a much starker effect as a fourth/fifth modulation.

I find it very interesting how these facts would translate to different equal-tempered tuning systems. In particular, in Bohlen-Pierce like systems, 5:3 takes the role of the most natural modulation interval. We definitely need more music in such systems to properly explore these effects!


*The only 12-coprime numbers that are themselves smaller than 12. IOW, the only coprimes modulo 12, that is, up to octave shifts, which are usually considered as irrelevant for such matters.

Amongst the notable exceptions are gamelan instruments, which match harmony systems completely different from our Pythagorean/Ptolemaian derived ones.

Because 12-edo is a meantone tuning.

§But also consider modulation to a relative key, which is very natural! Only, it can't really be iterated.

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Math mode needs to point out that 1, 5, and 7, and 11 are the only number both coprime to 12 and less than 12, which matters in the context of music because numbers of half steps greater than 12 are musically equivalent to 0-11. There are an infinite number of numbers greater than 12 that are coprime to 12. – Todd Wilcox Feb 26 at 22:39
    
You're right of course. – leftaroundabout Feb 26 at 22:59
    
+1 for tons of footnotes :-) – Carl Witthoft Feb 27 at 14:19

The number of ways to list 12 keys is 12*11*10...*3*2 = 479001600; you have 12 choices for the first key, leaving 11 for the second, etc.... Or if you always start with C, then it's 11*10*...*3*2 = 39916800.

Or if you require that you start with C and move by the same interval at each step, then there are only 4 choices: up by a half step, down by a half step, up by a fifth, or down by a fifth. Any other choice of interval returns you to C before you've hit every key.

Or you could require that you start with C, and the move by alternating two different intervals. (As in your case, where the intervals are up a half step and down a minor third.) I believe there are 10 of those; written as pairs of intervals counted in terms of half steps, they are: (3, -1), (-3, 1), (-1, 3), (1, -3), (-5, 3), (5, -3), (-3, 5), (3, -5), (6, -1), (6, 1).

In all these case I'm considering patterns the same if they're the same up to octave equivalence: so ascending by fourths is the same as descending by fifths, for example.

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(I can't believe I just did that.) – Bruce Fields Feb 26 at 23:03

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