Harmonic series role in a just intonation interval ranking?

Question

In my studies, I was reflecting on this theory and just wanted your guys clarification..

A pitch in an octave is comprised of a fundamental frequency followed by its overtones. When we focus on an interval, its harmonic stability or instability is evidently stemming from the interaction of the two pitches harmonic series combined. As we add additional pitches to that interval, the relationships become furthermore complex due to additional overtones combined.

If I wanted to build a chord through selecting the intervals myself that would produce the desired level of consonance/dissonance, I would have to seek each pitches overtones in that chord and compare them to each other pitches overtones in that chord which would be rather tedious.

Is it because of this that the just intonation interval ranking was developed? Was it developed with the intention of making the stated above process easier by giving us a list of intervals and their ratios dictating the two pitches overtone relationships?

Is it also true that the manner in which the interval ranking functions in its process, is through providing a ratio for an interval with the ratio stating that in the case of the ratio 3:2 (Unison:Perfect 5th) the unison pitch has to perform 3 cycles and the perfect pitch 2 cycles for their wave crests to coincide, therefore determining their level of consonance/dissonance and this process in fact displays an accurate representation/translation of two pitches overtone interaction?

This would explain a lot for me. Thank you guys.

!!EDIT!!

Here is the math formula provided by WillRoss1, simplified. Here you go Albrecht Hügli.

"Formula for producing a chord ratio by any infinte amount of interval ratios."

In this example we will utilize the Major 7th Chord, with the intervals Major 3rd (4:5), Perfect 5th (2:3) and Major 7th (8:15). This formula below applies to infinite amount of intervals you want to combine by following the exact process below.

FORMULA

Step 1. Mutipy the first interval ratio (4:5) by the result of multiplying all the other interval ratios first numbers by each other (2 and 8). 2 x 8 = 16.

16 x 4 = 64 / 16 x 5 = 80.

Step 2. Multiply the second interval ratio (2:3) by the result of multiplying all the other ratios first numbers by each other (4 and 8). 4 x 8 = 32.

32 x 2 = 64 / 32 x 3 = 96.

Step 3. Multiply the third interval ratio (8:15) by the result of multiplying all the other ratios first numbers by each other (4 and 2). 4 x 2 = 8.

8 x 8 = 64 / 15 x 8 = 120.

Your Major 7th Chord ratio is 64:80:64:96:64:120.

Step 4. Since there are duplicate numbers (64) in the chord ratio, we can turn the ratio instead into 64:80:96:120.

Step 5. To further summarize this chord ratio, we find the common factor (a single number that can produce each of the four numbers) for 64,80,96 and 120 which would be 8.

Step 6. Divide 64, 80, 96 and 120 each by 8.

64 ÷ 8 = 8

80 ÷ 8 = 10

96 ÷ 8 = 12

120 ÷ 8 = 15.

Your Major 7th Chord ratio is now 8:10:12:15.

Answer 1

I may be misunderstanding your question or your intent but what I read form it makes me think you are putting the cart before the horse, at least historically.

I think the ranking of just intervals in terms of perceived dissonance or consonance came before our understanding of harmonics in physics. People observed sympathetic resonance in vibrating systems but I do not think the ancient Greeks knew that a plucked string had a specific harmonic content or that there was interference between the harmonics leading to "dissonance". People judged these qualities based on what seemed like subjective criterion, just as one might judge warm days to be more pleasant than cold days (another subjective judgement that whole cultures or people may agree on). Helmholtz attempted to provide a physics/math based explanation for the apparent "universal agreement" on what is consonant vs dissonant using harmonic analysis. Basically intervals with many harmonics that line up are more likely to fall into the consonant group while those with many conflicting harmonics that produce interference are likely to by judged as dissonant. Keep in mind that consonant is taken to mean "pleasant" or "harmonious" while dissonance is taken to mean "tense" or "unpleasant", "unharmonious". There is a mapping between what is now thought of as a technical term with subjective descriptors and people do not always agree.

To your question about whether or not harmonics were the cause of reason that intervals were placed into categories (which is my interpretation of your question) since people were not consciously aware of this when they made the judgement I would say that that was a "reason" for labeling intervals this way. It may be that harmonic analysis is used now to try and objectively place intervals in a ranking system but the way you stated the question makes me think you are asking about the historical assessment of intervals. And again, even if we use what seems like an objective criterion to rank intervals it does not mean that people will agree with the subjective judgement of pleasant or unpleasant.

Answer 2

I am answering here knowing this is rather a comment (and I'm afraid they will delete it again but I don't mind).

I am - and have always been - very interested in the relationship of math and music and the fundamentals of acoustic and physics. I hope I'll understand one day the complexe matrix that you got the other day but I'm afraid I won't understand your analysis of the 5th symphony of Beethoven in this system.

You're questions are very inspiring but I don't know whether this is the ideal approach to music, harmony and composition: Mind that the first experiments with intervals by Pythagoras et all (Aristoteles, Ptolemeus, Boethius, Hucbald, Glarean, Tinctoris, were built on the monochord and they were not researching chords but scales and describing the modes. I have also collected many books since of Zarlino, Fux, Schönberg and Hindemith and articles about their theories since I've been active in this Stack Exchange. (They all say that the music of the antic Chinese, Egypt, Greek, Roman and Byzanthinian was not polyphonic but the longer the more I'm doubting about this assumption.

I think the confusion in your approach is:

a) you can't ignore the function of the chord in the progression with other chords or with its own inversions.

b) there is an objectif consonance/dissonance concept that isn't congruent with what we are hearing and interpreting - depending of our earlier experiencies and the actual context of a chord in a piece of music it's connection with other the voice leading.

The art of arranging is not the arranging of art.

Somehow you remind me of this guy (but this is a joke!) who is trying to make an new order in artworks by ordering the elements in a picture the correct way.

(min.5)

"We Swiss are famous for chocolate and cheese. Our trains run on time. We are only happy when things are in order. But to go on, here is a very good example to see. This is a picture by Joan Miro. And yeah, we can see the artist has drawn a few lines and shapes and dropped them any old way onto a yellow background. And yeah, it's the sort of thing you produce when you're doodling on the phone. (Laughter) And this is my -- (Laughter) -- you can see now the whole thing takes up far less space. It's more economical and also more efficient. With this method Mr. Miro could have saved canvas for another picture. But I can see in your faces that you're still a little bit skeptical. So that you can just appreciate how serious I am about all this, I brought along the patents, the specifications for some of these works, because I've had my working methods patented at the Eidgenössische Amt für Geistiges Eigentum in Bern, Switzerland. (Laughter) I'll just quote from the specification. "Laut den Kunstprüfer Dr. Albrecht --" It's not finished yet. "Laut den Kunstprüfer Dr. Albrecht Götz von Ohlenhusen wird die Verfahrensweise rechtlich geschützt welche die Kunst durch spezifisch aufgeräumte Regelmässigkeiten des allgemeinen Formenschatzes neue Wirkungen zu erzielen möglich wird.") Ja, well I could have translated that, but you would have been none the wiser. I'm not sure myself what it means but it sounds good anyway. I just realized it's important how one introduces new ideas to people, that's why these patents are sometimes necessary. I would like to do a short test with you. Everyone is sitting in quite an orderly fashion here this morning. So I would like to ask you all to raise your right hand. Yeah. The right hand is the one we write with, apart from the left-handers. And now, I'll count to three. I mean, it still looks very orderly to me. Now, I'll count to three, and on the count of three I'd like you all to shake hands with the person behind you. OK? One, two, three. (Laughter) You can see now, that's a good example: even behaving in an orderly, systematic way can sometimes lead to complete chaos!*"

*) translation:

"According to the art examiner Dr. Albrecht Götz von Ohlenhusen, the procedure is protected by law which art by specifically tidy regularities of the general vocabulary of new effects is possible to achieve."