# Why the difference between interval ratios and the harmonic series?

I've observed that the harmonic series which are the most consonant pitches of the fundamental such as the note A for this example, with A's first harmonic series being (A) E and C#.. Do not correlate with the consonance to dissonance ratios. In the ratios it would be A,E and D..

• Harmonic Series - A (root) E (fifth) C# (Third)
• Interval Ratios - A (root) E (fifth) D (Fourth)

It is also worth mentioning that the following harmonic after the third (C#) is a 7th (G).. While in interval ratios the order would be Unison, Octave, Fifth, Fourth and a Major Sixth.

What is going on here?

Also, could you folks confirm the validity of this interval ranking shown in the image below? Image Source: https://www.pnas.org/content/112/36/11155

Thank you.

• Look into David Cope's concept of "interval strength", which is similar to what you're trying to do here. (My answer would be that harmony is only loosely related to harmonics, as exemplified by the eleventh harmonic, which is right in-between two notes.) Jul 14, 2019 at 3:50
• Where are you getting your information about what are "the most consonant pitches of the fundamental"? Are you saying that because they're the earliest harmonics? Jul 14, 2019 at 8:19
• BTW I think it would really be worth reading and fully understanding that paper I linked to - at least I think it would be good for me, and probably good for you too! Jul 14, 2019 at 19:43
• What do you mean "confirm the validity of this interval ranking"? They asked a bunch of random people, and took the average ranking. That means it's valid for the average person from that group. There's a chance that the average person in another group will agree, unless the group is non-random, like only trained musicians or fans of a particular genre. But mr. Blue Dot likes minor sevenths as much as major thirds, while mr. Cross put major thirds first and minor sevenths in 10th place; those two will never agree, and I don't know how you're going to make music that appeals to both of them. Jul 16, 2019 at 1:42
• I don't think you can really have universally accepted rules when human perception and preference comes into play, anymore than you can determine the world's favourite colour. Jul 16, 2019 at 21:19

## 2 Answers

It's because the fourth is the inversion of the fifth. E is the third harmonic of A, and C# is the fifth harmonic of A, but A is the third harmonic of D.

So the ratio of the ascending fifth (the interval, that is, in this example A to E) is 3:2. The ascending major third (for example A to C#) is a ratio of 5:4, and the ratio of the ascending fourth (A to D) is 4:3. Notice that the odd factor is in the denominator here, because A is an overtone of D rather than D being an overtone of A.

• Yep. Another way of looking at it is that the series of most consonant intervals (in the classical music sense) is obtained (with the usual temperament factors) from the harmonic series, but not from 1/1, 1/2, 1/3, 1/4, 1/5, etc, but rather 1/2, 2/3, 3/4, 4/5, and so forth. Jul 15, 2019 at 10:17
• @ScottWallace it's more common to express the ratios for ascending intervals rather than descending, but, regardless, the powers of two in the numerator or denominator can also be seen simply as adjusting by octaves. Whether the major third is the distance between the fourth and fifth harmonics or the distance between the fundamental and the fifth harmonic, adjusted by two octaves, is not particularly significant, is it? I rather think the whole idea that the scale is "derived from" the harmonic series is seriously overrated. We don't use intervals based on the 7th or 11th harmonic. Jul 15, 2019 at 13:33
• @Seery they are each other's reciprocals. For example, an ascending major third is 5:4 while a descending major third is 4:5. Inversions of chords may be more dissonant to the extent that the fourth is sometimes regarded as dissonant, that is, for reasons of melody or voice leading, but otherwise, that is acoustically, inversions are more or less equally consonant and dissonant. That is, the second and seventh are dissonant, while the third and the sixth are consonant. Jul 16, 2019 at 0:01
• @Seery It depends on the context and on the listener. If a piece in G major or G minor ends with an open fifth (G-D-G), that doesn't sound more dissonant to me than a major or minor triad. Does it sound more dissonant to you? Or are you talking about a chord like G-C-D? That is more dissonant because you have to look at all the intervals, not just relative to the root. G-C-D has a perfect fourth, a major second, and a perfect fifth. G-B-D has a major third, a minor third, and a perfect fifth. The major second in G-C-D is more dissonant than any of the intervals in G-B-D. Jul 16, 2019 at 1:11
• @Seery if the two tones are sounding at the same time there is no ascending or descending. Your question is analogous to comparing which relationship is closer, that between a parent and child or that between a child and parent. Jul 16, 2019 at 20:52

The harmonic series is not in strict order of pitch consonance with the fundamental. This can be seen from the fact that there are harmonics that are two, three, four.... octaves above the fundamental, all of which would themselves be very consonant with the fundamental, and yet have harmonics either side of them representing pitches that are less consonant.

Consonance and dissonance are subjective sensations which aren't necessarily even particularly well-defined. Plomp and Levelt's 1965 paper discusses some of the difficulties in pinning down what consonance really is, and discusses how their curve of interval consonance was calculated. I thought it was obtained directly from experiments on people, but on reading the paper, that isn't the case - it's derived from experiments using simple tones (sine waves).

• i read a portion of the paper and found it to confirm concepts i've come across thus far. I've added an image in my post of a cons/disso curve i have found.. Could you confirm its validity? I apologise if i seem to be going in circles here. Also i came across compound intervals regarding 9ths,10ths,11ths,12ths,13th,14th and 15ths being extensions of 2nd,3rd,4ths,5ths and so on.. If for example a 2nd is dissonant and a 5th is consonant, how does that measurement of conso/disso correlate with the 2nd and 5ths compound intervals 9th and 12th? Thank you sir! Jul 15, 2019 at 23:01
• @Seery this is a topic that interests me too! I think I'd need to do my own revision on the derivation of that dissonance curve to really be able to make a stronger statement on the extent of its validity... I will aim to do so at some point! Jul 15, 2019 at 23:08
• @Seery I'm a bit confused as to what you mean - I can't see anything in image A in your post that says anything about consonance degrees. I think image A just tells you what's on the X axis on image B. Jul 15, 2019 at 23:22
• @Seery as to your point about imperfection - a lot of that is because it involves human beings with all their vagaries! (and I don't think music is alone in this) - The graph you've posted, which represents actual listener assessments, is what I thought the Plomp curve represented - in fact his curve for musical tones was calculated from his curve for simple tones. Jul 15, 2019 at 23:36
• @Seery it's like we said in another question - you can't just look at the interval from the root to every other note - you have to look at the intervals between every note and every other note. The minor third between M3 and P5 is more consistent than the second between P4 and P5 Jul 16, 2019 at 0:53