# Ranking Interval Complexity

I would like to know the order of complexity of musical intervals (in a melody). I learned about this from an exposure to the Kodaly method, although I have not learned from nor taught using the Kodaly method. But in my reading it introduced the concept of how the intervals are naturally learned. Seems like there should be an ordering. I can see that the first two are a minor third down and a perfect fourth up, but I don't know where to find a list of the intervals from easiest to hardest, treating the up direction and the down direction as separate things.

I realize that this ordering of interval difficultly or learning order may be dependent on the culture of the student. It would be extra cool to understand how this ordering of interval difficulty related to the harmonic series, since the first interval there is a perfect fifth up, which does not match my understanding that a minor third down is the easiest interval.

• I have a feeling this may be quite subjective, as I may find some intervals easier or harder than you based on my experiences to date. What is the underlying issue you are trying to solve? Aug 18, 2012 at 9:50
• @Dr Mayhem - I am interested in writing a computer program that would take a set of music and order it according to the rules set. So sure, part of this is likely to be cultural or perhaps tied to individuals experiences, but I think there must be established orders of difficulty. Otherwise Kodaly could not have done what he did. Aug 18, 2012 at 16:13
• I wouldn't take it for totally impossible that someone could have done some cognitive study on this. Aug 18, 2012 at 20:55

Interval complexity is a direct function of the distance between the lowest note of the interval as compared to the highest note of the interval with the closest note in the harmonic series of the lowest note of the interval. (Phew!)

Let me explain:

Poor Man's Harmonic Series: For the sake of this explanation, let's pretend the harmonic series represents these relationships. It does not, but this linear representation is roughly based on the harmonic series:

Octave, fifth, major third, minor 7th, major second, flat fifth, minor sixth, major seventh, minor 9th, minor third.

So if the root note is C, its sympathetic vibrations along the harmonic series are as written above. (Our musical system is not perfectly mapped to the harmonic series, so this is just an approximation.)

As you can see, the harmonic series firmly supports the simplest intervals (octave, fifth and major third), as well as those a whole step up or down from the fundamental (minor 7th, major second). As we climb further into the stratosphere we encounter dissonances not well supported within the fundamental, which would represent the more complex intervals of the flat fifth, minor sixth, major seventh and minor ninth.

Harmonics also imply their own inversions: Note that each interval also implies its own inversion, though, meaning that the perfect fourth, although not even present in this particular example, is well enough supported by the perfect fifth, the two intervals being so closely related.

Higher harmonics are also weaker: Also note that the higher the harmonic is, the more weak it is. So the fundamental is alot less related to the minor 9th than it is to the perfect fifth.

The 12-tone musical system does not relate perfectly to the harmonic series: many notes in the harmonic series are not too closely related to their closest relative in the actual musical scale. For instance, the flat 5th in the harmonic series is 49 cents more flat than the one in the 12-tone scale, a significant difference. This difference has meant that many harmonies which would be perfectly valid in a microtonal system are completely unusable in the 12-tone system we most commonly use.

In my experience as a musician and using Kodaly, it's less about the intervals themselves as it is about the notes of the scales used. For instance, a minor third down is represented as "sol mi" and a perfect fourth up is "sol do". You can see how these are the notes of the tonic triad.

Therefore, in ranking the complexity of an interval, I believe it would be more useful to consider which step of the scale it is. So, if the note lies outside of the key, yet is a perfect fifth up, it would be listed as more complex.

• Most of my experience is modal. Certainly the meaning of an interval will depend on the context, including where the interval occurs in the scale being used and the current tonal center. Maybe my question is outside of Kodaly, strictly speaking, since I am more concerned with comparisons of bodies of music than the education of the musician, but I assumed that this must have been thought through. Nov 18, 2012 at 22:14
• wouldn't that be the sub-dominant triad?
– Ygg
Jul 23, 2013 at 11:44
• @Ygg "do mi sol" are the notes of the tonic triad when using "movable do." Aug 2, 2013 at 2:41

No experience with the method you mentioned, but I learned from the center out.

P5, P4, M3, m3, M6, m6, M2, m2, M7, m7, TT (tritone).

Ascending and descending in all cases.

In this sense, you are starting with more consonant, "stable" intervals, then working your way into the more dissonant, "complex" intervals, if you will.

• Are minor 2nds considered more consonant than a major 7th? Jul 10, 2014 at 8:10
• I would argue that a major 2nd and a minor 7th are roughly "equivalent" in dissonance, same with a minor 2nd and major 7th. M2/m7 less dissonant than m2/M7. My point in laying them out that way was to illustrate the concept of interval inversion pairing in terms of consonance/dissonance. (4ths and 5ths, 3rds and 6ths, 2nds and 7ths, etc..) Aug 6, 2014 at 22:53