# "Minor arpeggio secret" and the undertone series

I was recently introduced to a "trick" sometimes known as the "minor arpeggio secret". It can be summarized as following:

If we can maintain the same tension while plucking twice, 3 times, 4 times that length, etc. the undertone series will unfold downwards, containing the minor triad. Similarly, on a wind instrument, if the holes are equally spaced, each successive hole covered will produce the next note in the undertone series. (Source)

In other words, if you keep the distance between your fingers constant and shift your hand, you are able to play a pure minor triad on a string regardless of where you start (since they are found following eachother in the overtone and undertone series). It is demonstrated in this video by Chris West.

While I somehow understand this phenomenon, I have some troubles to intuitively grasp which undertone series I am playing when using this trick, for example starting with a minor third A-C (followed by C-E and E-A).

In other words, how can I formalize the relation between the minor arpeggio secret and the undertone/overtone series?

• Why are you calling this a "secret?" Jun 28 at 13:10
• @MichaelCurtis It is named so by the bassist Chris West who made a video about it (youtube.com/watch?v=umdwB1m70es) Jun 28 at 15:47
• Thanks for posting the video. That adds an enormous amount of info missing from the original post. Working on it.... Jun 28 at 17:21
• @Philip, ah, the "secret" is the equal string division lengths on the fretboard - demonstrated with playing cards - for the tones of a minor arpeggio. That is an interesting factoid. Otherwise, the undertone series is fairly well known. Jun 29 at 13:39

The fundamental of the undertone series producing a minor triad is the 5th of that triad.

As a string multiplies in length, the harmonic series is achieved downward rather than upward. Thus, the starting pitch, an octave lower, a fifth lower, another lower octave, a major third lower. At that point, a minor triad has been achieved within the series.

Ratio "Octave-corrected" ratio Example pitch Scientific pitch
1:1 1:1 E E6
2:1 1:1 E E5
3:1 3:2 A A4
4:1 1:1 E E4
5:1 5:4 C C4
6:1 6:5 A A3

Note that the rule as quoted deals with a single string being plucked at increasing lengths. However, if one then plays the resulting pitches from low to high, an ascending minor triad is the result.

In other words, create a string 6 units in length, then successively pluck at lengths 6, 5, and 4, you'll get, in my example, A-C-E. Or, create a string of length 4 units, and that will be the 5th of a minor chord in which that note, the third and fifth below, and the fourth above will all be equidistant.

• Thanks for the answer, I think it provides some clarification. However, the minor arpeggio is achieved by shortening a string and not increasing its length which makes your first statement a bit counterintuitive. What am I missing here? Jun 28 at 8:37
• The passage you quote indicates that the string is getting longer, not shorter. Shorter strings create an ascending sequence, which results in a major triad. Jun 28 at 8:43
• This is true, but it refers to the undertone series and not the overtone series. The undertone series contains the ascending minor triad as opposed to the ascending major triad. Jun 28 at 8:46
• @PhilipBergström Right. The undertone series is what you get when you progressively make the string longer. The pitches get lower in the same interval order as the overtone series: octave, fifth, octave, major third. That gives a minor triad. Jun 28 at 8:49
• @PhilipBergström The undertone series is constructed from high to low. But if then played low to high, you can an ascending minor triad. In other words, play 6:1 - 5:1 - 4:1 and you get A-C-E in my example. Jun 28 at 8:59